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Logan Meeting (October 10-11)- Page 775 Charlotte Meeting (October 17-18)- Page 783 Denton Meeting (October 31-November 1)-Page 791

Notices of the American Mathem.atical Society

October 1986, Issue 250 Volume 33, Number 5, Pages 705-888 Providence, Rhode Island USA ISSN 0002-9920 Calendar of AMS Meetings

THIS CALENDAR lists all meetings which have been approved by the Council prior to the date this issue of Noti~es was sent to the press. The summer and annual meetings are joint meetings of the Mathematical Association of Amer•~a and the American Mathematical Society. The meeting dates which fall rather far in the future are subject to ~hange; th~s is particularly true of meetings to which no numbers have yet been assigned. Programs of the meetings Will _appear In the issues indicated below. First and supplementary announcements of the meetings will have appeared 1n earher 1ssues. ABSTRACTS OF PAPERS presented at a meeting of the Society are published in the journal Abstracts of papers presented to the American Mathematical Society in the issue corresponding to that of the Notices which contains the program of the meeting. Abstracts should be submitted on special forms which are available in many departments of mathematics and from the headquarter's office of the Society. Abstracts of papers to be presented at the meeting must be received at the headquarters of the Society in Providence, Rhode Island. on or before the deadline given below for the meeting. Note that the deadline for abstracts for consideration for presentation at special sessions is usually three weeks earlier than that specified below. For additional information. consult the meeting announcements and the list of organizers of special sessions. ABSTRACT MEETING# DATE PLACE DEADLINE ISSUE 828 October 1G-11, 1986 Logan, Utah EXPIRED October 829 October 17-18, 1986 Charlotte, North Carolina EXPIRED October 830 October 31-November 1, Denton, Texas EXPIRED October 1986 831 January 21-24, 1987 San Antonio, Texas October 15 January (93rd Annual Meeting)* 832 March 26-28, 1987 Honolulu, Hawaii February 2 February 833 April 3-4, 1987 Kent, Ohio February 4 February 834 April 25-26, 1987 Newark, New Jersey February 6 February 835 June 18-20, 1987 Tacoma, Washington April17 June January 6-9, 1988 Atlanta, Georgia (94th Annual Meeting) August 8-12, 1988 Providence, Rhode Island (AMS Centennial Celebration) January 11-14, 1989 Phoenix, Arizona (95th Annual Meeting) * Preregistration/Housing deadline is November 15.

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Other Events Sponsored by the Society January 2G-22, 1987, AMS Short Course: The Moment Problem, San Antonio, Texas. February 18, 1987, Symposium on Some Mathematical Questions in Biology: Models in Population Biology, Chicago, Illinois. May 12-16, 1987, Symposium on the Mathematical Heritage of Hermann Weyl, location to be announced. June 14-July 25, 1987, Joint Summer Research Conference in the Mathematical Sciences, University of Colorado, Boulder, Colorado. July 12-August 15, 1987, Joint Summer Research Conference in the Mathematical Sciences, Cornell University, Ithaca, New York. July 16-24, 1987, Summer Research Institute on Theta Functions, location to be announced.

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Volume 33, Number 5, October 1986 EDITORIAL COMMITTEE Paul F. Baum. Ralph P. Boas Raymond L. Johnson. Mary Ellen Rudin Steven H. Weintraub. Daniel Zelinsky Everett Pitcher (Chairman) 707 Physical and Mathematical Applications of MANAGING EDITOR Gauge Theories James A. Voytuk Clifford H. Taubes ASSOCIATE EDITORS 716 National Science Week Stuart Antman. Queries 735 International Congress of Mathematicians Hans Samelson. Queries 1986 Ronald L. Graham. Special Articles 738 The University Research Initiative SUBSCRIPTION INFORMATION 741 Mathematical Text Processing Subscription prices for Volume 33 (1986) 752 Washington Outlook are $85 list; $68 institutional member: 754 News and Announcements $51 individual member. (The subscription price for members is included in the 759 NSF News&. Reports annual dues.) A late charge of 10% of 763 For Your Information the subscription price will be imposed 768 Letters to the Editor upon orders received from nonmembers 772 1986 AMS Elections after January 1 of the subscription year. Add for postage: Surface delivery 773 Queries outside the United States and lndia-$8: 775 Future Meetings of the Society to lndia-$18; expedited delivery to Logan. October 1D-11. 775; Charlotte. destinations in North America-$12: October 17-18. 783; Denton. October elsewhere--$15. Subscriptions and 31-November 1. 791; San Antonio. orders for AMS publications should be January 21-24. 801; Mathematical Sciences addressed to the American Mathematical Employment Register. 826; Synopses. Society. P.O. Box 1571. Annex Station. Short Course on Moments in Mathematics. Providence. Rl 02901. All orders must be prepaid. 830; Invited Speakers and Special Sessions. 833; Mathematical Biology Symposium. 835 ADVERTISING &. INQUIRIES 836 Joint Summer Research Conference Series Notices publishes situations wanted and classified advertising. and display 840 Special Meetings advertising for publishers and academic 845 New AMS Publications or scientific organizations. Requests for 849 Miscellaneous information: Personal Items. 849; Deaths. 849; Reciprocity Advertising: Wahlene Siconio Agreement. 850; Visiting Mathematicians Change of address or subscriptions: (Supplementary List). 851; Application Membership and Sales Department Deadlines. 853 Bool< order number 800-556-7774. 855 AMS Reports and Communications Recent Appointments. 855 CORRESPONDENCE. including 856 Advertisements changes of address should be sent to American Mathematical Society. P.O. 877 Preregistration Forms Box 6248. Providence. Rl 02940. San Antonio Preregistration and Housing. Second class postage paid at Providence. 877; San Antonio MAA Minicourses. Rl. and additional mailing offices. 879; Employment Register. 881 Copyright © 1986 by the American 883 ElMS Subscription Form Mathematical Society. All rights reserved. 885 New AMS Publications Order Form and Printed in the United States of America. Mailer Notices: highlights

Clifford Taubes, in an expository article, explores the various facets of gauge theory as it pertains to pure mathematics with new insight into 4-manifold struc­ ture, and as it pertains to mathematical physics with the success of explaining Weak interactions and the hope of applying the theory to the Strong force. Page 707.

National Science Week Symposium, "Mathematics: The Unifying Thread in Science," centers on the role played by mathematics in the Nobel Prize work of three scientists: Allan M. Cormack, Herbert A. Hauptman, and Steven Weinberg. Page 716.

International Congress of Mathematicians 1986 was a major scientific success. The meeting was highlighted by the presentation of the Fields Medals to Simon Donaldson, Gerd Faltings, and Michael Freedman, and by the presentation of the Nevanlinna Prize to Leslie Valiant. Page 735.

The University Research Initiative awards were made this summer by the Department of Defense with mathematics receiving nine grants, a significant share of the eighty-six grants funded under this program. The program is designed to im­ prove university research capability and to strengthen graduate and undergraduate education. Page 738.

Richard S. Palais, in his regular column on text processing, reviews the Mac­ intosh world of technical word processing software. He responds to readers by summarizing the comments he has received on the best way to do mathematical word processing and by explaining some of the techni<;al details behind font design. Page 741.

Kenneth Hoffman, in his Washington Outlook column, reviews the past year in terms of public information activities and gives reasons why we should celebrate. Page 752.

NSF reports that Ph.D. production for 1985 in science and engineering is approx­ imately the same as for 1984, but that the number of non-U.S. citizens receiving degrees has increased. Page 759.

San Antonio Meetings, January 21-24, 1987. The preliminary announcement contains lists of invited speakers, a timetable for major sessions and addresses, and registration information. Page 801. Physical and Mathematical Applications of Gauge Theories Clifford Henry Taubes *

1. Physics. Experiment and Theory are two halves green glow that you will see in the dark is powered of physics. Knowledge is born from the womb they in part by the weak force's work. create. By probing and peering with detectors, The details and facets of each of four forces, collectors, colliders, and scopes, experimenters their forms and their textures, their sinks and their uncover the facts. The facts are the clues to the sources, must all be explained by any acceptable natural laws that structure the way the universe theory. For three of four forces (not gravitation), works. A theory is a guess at these natural laws, gauge theories provide the basic foundation for a a model that, hopefully, fits all the facts. theory of force that could well be correct. (See Accuracy, elegance, simplicity, and scope are ['tH].) the factors by which one will choose, of models In mathematics, gauge theory is the study and theories, the best one. But accuracy, here, of principal bundles with Lie groups as fibres. is the hanging judge, whose vote bears no veto. The simplest such bundle, the archetype of all, is The path to our past sees theories on theories the product, the manifold x the Lie group. In of beauty and grace to rival Troy's Helen-all general, let G be the Lie group in question and dropped and abandoned; they failed to describe let M be a manifold. A principal bundle whose one fact or other in the natural world. base is this M, and whose fibre is G, is a manifold A theory, by nature, presages new facts, or P which looks, at least locally, like M x G. That their absence. Such visions, like the Sirens of is, there is a map 1r, called the projection, which Odysseus's journey, lead to new experiments, new is a surjection from Ponto M with 1r-1 (a point) discoveries, and new understanding. Together, being the Lie group in question: theory and experiment probe the unknown. Of the natural world, one fact has been G-+P~M. established: there are only four forces that do all nature's work. These four are the cause of all P must admit a global G-action, which restricts known interactions, collisions, commotions, and to each fibre as the standard right action of G miscellaneous reactions anytime and anywhere. on itself. And finally P must be locally trivial so On a good day you'll witness all four of the that small open sets looks like {open} x G. forces, or at least their effects. Gravity's force is In physics the base of the bundle, the man­ the force that holds you to earth. The second force ifold M, is space-time. It is the wheres and the is the electromagnetic force. Electric/magnetic whens of all possible actions, events, all things effects are its feature. It holds us together, and that take place. Each group that is chosen gives anything else which is made up of atoms held near a different world model and the task is to choose each other. The force comes in waves, and some one that fits all the facts. of its forms are light rays and x-rays and radio. The "facts" that will fit in a given gauge The third and the fourth of the forces we theory are carried by structure on the principal know are nuclear forces with ranges much smaller bundle. In particular, the connections on the than atoms. But often they're found in the principal bundle give the set of all forces that the macroscopic world through effects that reveal model allows. them to unaided eyes. The weak force and the A connection defined on the principal bun­ strong force are the names of these two other dle is a smooth subbundle of tangents to P. forces. But G-equivariance and horizontal are added re­ Work by the strong force gives nuclear power. quirements that must be met. A connection is Fusion and fission are both its endeavors; it gives a G-equivariant split of the sequence of vector the power that lights up the sun and the rest of bundles that is written below: the stars. The weak force is subtle and harder to find. 0-+ Vert-+ TP -+ 1r*TM-+ 0. Look at a watch with a radium dial; that eerie 1 p

*Supported in part by an NSF Postdoctoral Fellow­ Here Vert in T P is the vertical bundle of vectors ship in Mathematics. tangent to the fibres of P.

707 Let U be a domain where P is a product; them closely. If they are all dipoles, the theory then, here the connection is simply a 1-form which has been tested in the following way: If the is taking on values in Lie Alg(G). When each sample is random, then amongst all the magnets, point of G is an n X n matrix, the connection on monopoles occur in percent less than 1. (For a P, over U, is one too: Ann x n matrix of 1-forms, test that is more striking, take all of the magnets and cut each in half; each half should also be a dimM dipole, as Maxwell's gauge theory predicts it to A= I: A,_,dx"', be.) v=l Quantum mechanics plus Maxwell's gauge theory are coupled together in the theory of where each A,_, is a matrix of functions on U. Quantum Electro-Dynamics (QED). QED is more From connection comes curvature; this is the accurate than any other theory in physics. obstruction in the sense of Frobenius to integra­ The grail in physics is Unification: to view tion of the connection's horizontal subbundle (of the four forces as aspects of one unified force field. tangents to P). Let U be as before and let A Since Maxwell, the quest's great achievement denote the connection; let FA be the curvature is the unification of the weak force and QED. of the connection A. On the open set U, FA is Invented by Salam, Glashow, and Weinberg, the a 2-form with values in Lie Alg(G). The form is electro-weak theory is working quite well (cf. computed by the rule [We]). This electro-weak theory is a gauge theory, but with structure group U(2) instead of U(1). (Nonabelian theories were introduced to physics The Maxwellian theory of electromagnetics by Yang and Mills in the 50s, two decades before makes the forces and fields, electric/magnetic, the electro-weak theory.) At subatomic scales, the derived from the choice of a given connection on a electro-weak theory makes all electric, magnetic, principal bundle with base, time and space. The and weak forces equal. At scales that are large group for the bundle is the group of the circle, compared to the atom, only electric/magnetic the Lie group U(1). effects remain, and the macroscopic predictions of The vector potential of Maxwellian physics Maxwell's theory are retained. is i times the 1-form that defines the connection; The electro-weak force needs a U(2) connec­ and the fields, the electric and also magnetic, are tion-a matrix of 1-forms that is anti-hermitian the parts of the 2-form, i times F: Let t be the of size 2 x 2. Each choice of connection determines time as distinguished from three space parameters the force fields; the electro-weak theory tells which x, y, and z. Let E and B be the electric and ones to choose. The U(1) of Maxwell fits into the magnetic vector fields. They are obtained from A theory as U(1) fits into U(2) as a subgroup. With by the formula below: a Cartan basis for Lie Alg(U(2)), write the U(2) connection in the following way: iFA =Ex dt 1\ dx + Ey dt 1\ dy + Ez dt 1\ dz + Bx dy 1\ dz +By dz 1\ dx + Bz dx 1\ dy. Through choice of connection, the theory of (AEM and Az generate the Cartan subalgebra.) Maxwell describes all electric/magnetic effects. AEM is Maxwell's connection; the remaining com­ But the format alone, through the use of gauge ponents describe the weak force. theory, makes predictions concerning the natural Light is the wave in Maxwellian theory, and world which are insensitive to the precise choice electro-weak theory predicts waves as well. One of connection. wave should exist per component of the U(2) The predictions here follow from Bianchi's connection, so three waves plus light must exist equation: Let P be given as a principal bundle, in this world. At CERN's big collider, in the the group and the base can be any at all. For A a Alps near Geneva, the other three waves were just connection, the Bianchi equation says A and FA recently seen. The electro-weak theory is now must always obey official dogma: Its inventors all wear the Nobel Prize. The success of gauge theory for the weak interactions suggested gauge theory for the strong For a U(1) connection, Bianchi's equation force as well. It is believed that the strong force translates, in physical terms, to the assertion that is described by gauge theory with group SU(3) magnetic field lines can never have ends. For (cf. [GI]). Its connection is a matrix of 1-forms the world, this predicts that all magnets have with eight components. There are eight waves in two poles, one north and one south, no exceptions the theory, each called a "gluon." The theory allowed. U(1) gauge theory says dipoles (or higher says matter (except for the leptons) is made up of poles) are all that are seen. quarks, just six different kinds. Then quarks bind The skeptical reader can test this prediction together with binding, the gluons, to make up the by buying up magnets, say one hundred. Look at matter that exists in the world.

708 In gauge theory parlance, a vector space appear as this force field; its internal structure bundle exists over space-time, and its sections would be completely disguised. describe the dynamics of quarks: The norm of Though a Nobel Prize sits stored for the a section, once squared, gives the chances of monopole's finder, today not a one has ever finding a quark at a point in space-time. Quark been seen. But monopole searching is not too interaction will come from the section, through expensive. A detector of poles can be built by parallel transport by the "gluon" connection that your hands. Here is a Nobel Prize race that you is giving the force. could easily enter. A battery, light bulb, and Unification is still the main goal. To unify small coil of wire is all you require to construct the strong force with the electro-weak force, use the detector. The schematic in Figure 1 tells how a big group containing SU(3) x U(2) to make a to connect them. Once connected, patience is gauge theory that's bigger than both. Georgi all that you need as you sit back to watch for and Glashow invented this game; such a theory is monopoles. called a Grand Unified Theory, GUT (cf. [Ge]). The simplest GUT, the "standard model," is a gauge theory with group SU(5). The connection is a 5 x 5 matrix of 1-forms which is anti-hermitian Loop of wire and has zero trace. At distances larger than found in the atom, just one component is all that remains, that is Maxwell's connection, the AEM· At scales subatomic, twelve components are seen, eight for the strong force and four, the electro­ weak force's connection. In the 5 X 5 matrix of 1-forms these fit to commute with each other: Battery ( A(su0{3)) 0 ) FIGURE 1 Au{2)

And at very short range, all forces get mixed; the connection is symmetric with 24 parts of strengths When a monopole crosses the plane, through that are equal; one Asu{5)· the loop, your light bulb will brighten, or darken The break-down of groups into subgroups a bit. From a dipolar passage the light bulb will that occurs, the big SU(5) to SU(3) x U(2) and flicker; first brighten, then darken, or vice versa: then down to Maxwell's remaining U(1)-this corresponds, in geometry's terms, to bundle re­ ductions from big groups to small. The SU(5)-theory, so far as is known, fits Monopole signals all the facts. But it does make new predictions of a surprising nature, of things that today have yet to be seen. It predicts, for example, that matter decays: Quark by quark less is here than was here yesterday. 1031 quarks can be found Dipole signals in ten thousand pounds of wax. In a Morton Salt mine, deep underground, there sits a wax There are the plans, good luck in the chase, vat, surrounded by counters, detectors and such the Nobel is the prize at the end of the race. where the physicists wait for a quark to decay. The theory claims one in that whole vat will go 2. Mathematics and physics. Physics is the in each year. Please, don't blink, as you might study of the dynamical world-it is the interac­ miss the prize of seeing the flash of this year's tions which interest physicists. The dynamical lone quark's predicted demise. After five years of world is quantum mechanical; this is a fact which patience, statistics are small; no announcement's cannot be denied. Quantum mechanics assigns been made of matter's downfall. probability for events to occur in the natural These GUTs make a second prediction that world: where classical mechanics would say, "The is equally wild. They predict there are magnets sun will rise at 7:00 tomorrow morning," quantum with only one pole. mechanics would say, "The probability is almost, In U(1) gauge theory, Bianchi's equations but not quite, unity that the sun will rise at 7:00 say magnetic force lines do not have an end. tomorrow morning." In the macroscopic world, Nonabelian theories do not make this claim, as the difference between quantum mechanics and Bianchi's equation yields no such constraint. The classical mechanics is quite irrelevant. But, in U(1) force lines in Grand Unified Theories can end the submicroscopic world, classical mechanics is a in a knot of SU(5) forces, the knot being smaller dismal failure, while quantum mechanics stands than nuclear size. A monopole magnet would uncontradicted.

709 The assignment of quantum mechanical prob­ is self-dual (or anti-self-dual) if abilities for subnuclear forces (electromagnetic, weak, or strong) is made via a probability mea­ Wt2 = +(-)w34, sure on the space of all field configurations, which Wt3 = -(+)w24, is to say the space of all connections on the Wt4 +(-)w23· apropriate principal bundle. (The appropriate = principal bundle is determined by which theory Since the curvature of a connection is a Lie­ is of interest: quantum electrodynamics, electro­ algebra-valued 2-form, it makes sense to assert weak, quantum chromodynamics or grand unified that the curvature of a connection is self-dual, theory.) or anti-self-dual. One writes FA = +(-) * FA The Feynman-Kac formalism [FH] requires if the curvature of the connection A is self-dual that the "correct" gauge theory probability mea­ (anti-self-dual). sure is computable by "integrating" a certain For example, the Hopf fibration "volume form" over the space of connections on the appropriate principal bundle over R4 . This 83-+ 81-+ 84 integration uses the ''volume form" below: defines a principal SU(2)-bundle over 8 4 (since group SU(2) is diffeomorphic to 8 3). The 2 4 the Lie d~-tlA = exp (- L.IFAI d x) · doo A. standard metric on 81 defines a connection, A, on the principal bundle over 8 4 • The horizon­ Here FA is the curvature tensor of the connection tal spaces of this connection are the orthogonal A, and d00 A is "Lebesgue" measure on the space complements in 8 7 to the tangent spaces to the of connections on the principal bundle. (Lebesgue 8 3 fibres. Let s be the south pole of 8 4 , and measure on an infinite-dimensional space? Indeed, let : R4 -+ 8 4 \s be the inverse of stereographic a great goal of mathematical physics is to make projection from s. Then, pull-back by of the some kind of mathematical sense of the formula connection A defines a connection on the principal above. And, there is much progress: If R4 is SU(2)-bundle R4 x SU(2) which is anti-self-dual replaced by R3 , then much progress has been made with a finite nonzero value for YM. towards making the measure above rigorously It was the steepest descent approximation to defined [B].) the quantum mechanical measure which motivated For the physicist, the formula above makes the physics community to study the minima of the sense if and only if a prescription can be found Yang-Mills functional, the self-dual and anti-self­ which arrives at predictions (which is to say, dual connections. And knowledge of the minima numbers) which are experimentally verified. One of the Yang-Mills functional was once thought to prescription for making sense of d~-t is given by be the key to understanding subnuclear physics. the method of steepest descent. This method Today, it is now widely believed that the replaces the integral above by an appropriate steepest descent approximation to the "correct" sum of Gaussian (quadratic) integrals. A finite­ functional integral (whatever that may be) is not dimensional analog is to approximate a measure a very good approximation; and certainly not dult = exp(-f(t)) ·dt on R by the steepest descent good enough to accurately describe the known measure, world. Even various improvements to the steepest descent method which take into account approxi­ exp(-f(x)) mate, nonminimal critical points of the Yang-Mills {x: xis a minima off} functional are found wanting. · exp( -! · 1:: ·t 2 ) · dt. The failure of steepest descent to approximate reality is due to the fact that the variational Here f"lx is the second derivative off at x. equations of YM are elliptic in an appropriate For gauge theories, the analog of the function sense. As a consequence of the ellipticity of these f in the preceding finite-dimensional example is equations, the critical points of YM obey a priori the Yang-Mills functional. This functional assigns estimates which imply that these critical points to a connection A, the number are connections which are too well behaved to account for real world phenomena. YM = IFAI 2 d4 x. The aforementioned estimates exist for the jR•r following reason: The variational equations of YM are second order in the partial derivatives of The set of minima of the Yang-Mills func­ the connection, and cubic in the nonlinearities. is the set of connections whose curvature tional Schematically, a critical connection A for YM tensor is either self-dual or anti-self-dual. Self­ obeys the equation dual means the following: With respect to a 2 3 4 standard basis of T*R4 , {dx 1 ,dx ,dx ,dx }, a 82 A+ A- aA + A3 = 0. 2-form 0 13 suppose that one w = L W 0 f3 dx 1\ dx For the sake of argument, a

710 equation equates some linear combination of the The intersection form is symmetric, and uni­ second derivatives of A with a function of the modular. Two such bilinear forms are isomorphic first and zeroth derivatives of A. Thus, if A if one is obtained from the other by conjugation is continuous, then some linear combination of by GL(·,Z). the second derivatives of A is continuous. The Mike Freedman [Fl] asserts M 1 and M 2 are assertion that the variational equations of YM homeomorphic if and only if ( 1) their intersec­ are elliptic means that this linear combination of tion forms are isomorphic and (2) they have the second derivatives contains enough of the second same Kirby-Siebenmann invariant. (The Kirby­ derivatives to allow one to conclude that all Siebenmann invariant lies in H4 ( M; Z2 ). It equals of the second derivatives of A are continuous. (Index w )mods when the intersection form is even, Thus, A is C2 • But now this would mean that i.e., when w(a, a) is even for all a E H 2 (M, Z).) the appropriate linear combination of the second Freedman proved that every symmetric, unimod­ derivatives of A was C2 . But, then, the ellipticity ular matrix, w, is the intersection form of a means that all of the second derivatives of A are compact, simply connected 4-manifold, and up to C2 . Thus, A is C4 . One can obviously iterate this homeomorphism, precisely two such if w is not logic indefinitely to conclude from the fact that even, and precisely one such if w is even. A is C0 that A is coo. This iteration is called After Freedman, the open questions in simply "elliptic bootstrapping." connected 4-manifold topology were: Karen Uhlenbeck has shown [U] that it is (1) Identify which topological4-manifolds are enough to know that the critical point of YM realizable as smooth ones. has a finite value of YM in order to start the (2) Classify the smooth manifolds up to bootstrapping iteration which leads to estimates diffeomorphism. and bounds for the derivatives of A. Gauge theory has proved to be immensely As more and more mathematical results con­ useful in the investigations of these two questions: cerning YM were obtained, less and less did Fix a 4-manifold, M, oriented, simply connected physicists find the functional interesting, for the and compact. For a given group G, two principal mathematicians told the physicists what they G-bundles defined over M are called isomorphic if did not hope to hear; that their steepest de­ there exists a G-equivariant map 4> which makes scent approximation to the quantum theory of commutative the subnuclear forces is not an accurate predic­ tor of reality. But paradoxically, the features of YM which ruin the physical applicability of \. ,/ the steepest descent approximation are precisely M the features which make YM so interesting to mathematicians. The isomorphism classes of principal G­ 3. Mathematics. In the field of pure math, bundles over M are computable from homotopy the surprise of gauge theory is the insight it gives theoretic data; precisely, they are in 1-1 corre­ to 4-manifold structure. Simon K. Donaldson, via spondence with the set of homotopy classes of gauge theory, is giving a remarkable perspective maps from M into the classifying space, BG. on 4-dimensional differential topology. For example, BU(1) = CP00 (the direct limit In comparing two manifolds, resolution in­ CP1 ---+ CP2 ---+ •. · ), and creases from homotopy equivalent through ho­ 00 2 meomorphic to diffeomorphic. Let M 1 and M2 be [M; CP ] ~ H (M; Z) simply connected, compact, oriented 4-manifolds. (since CP00 is a K(Z, 2)). The isomorphism above M 1 and M2 are homotopy equivalent if and only if (1) H2(M1; Z) = H2(M2; Z) and (2) their is realized by the first Chern class of the associated respective intersection forms are isomorphic [Wi]. line bundle L = P x 8 , C. Poincare duality The intersection form on H2(M; Z) is a bilin­ identifies H 2 and H 2 , and so the isomorphism ear pairing classes of principal U(1) bundles over the 4- manifold determine its homology. w: H2(M; Z) x H2(M; Z) ---+ Z. The intersection form on H2 (M; Z) is re­ gained from the set of isomorphism classes of A pair ([a], [.8]) E H2(M; Z) can be represented by U(2) bundles on M. The isomorphism class of smoothly embedded, oriented submanifolds a, ,B. a U(2) bundle, P, over M is determined by the Then first and second Chern classes of the associated w([a],[,B]) = (-1)n(p)_ L:: C2 vector bundle pEan/3 Here a n .B is the set of points in both a and ,8; E = p Xu(2) C2 . transversality can make this set finite. For a point p E an ,8, n(p) is zero or one if the orientations From U(1) bundles P1 ,P2, a U(2) bundle P agree or not on both sides of the equation is constructed by taking

TMIP ~ Talp EB T.BIP"

711 The recipe requires the rule of S' x S' action group, respectively. (See, e.g., [FU] for a basic course in the analytic properties of A and g.) The space A/ g is, as a point set, the isomorphism classes of pairs (principal bundle P', isomorphic to P, connection on P'). The group g does not The associated line bundles £ 1 , £ 2 construct E = act freely on A. For most A E A, the stabilizer L 1 EB £2 and the first and second Chern classes of gA = {g E g: g · A = A} is the center of g. There E follow: are exceptional A where 9A is larger. These A are the reducible connections whose holonomy lies c1(E) = c1(LI) + c1(£2), in a subgroup of G. The set R c A of reducible c2(E) = c1(LI) U c1(£2). connections has infinite codimenison in A and the set B = A/ g is a manifold away from reducible orbits (see Figure 2 and [FU]). Poincare duality (Pd: H 2 c::= H 2 ) makes the equality The space A/ g exists for any principal bun­ dle, on manifolds of any dimension. In dimension 4, there is extra structure. On an n-dimensional Riemannian manifold, the metric gives Hodge's And so, from the isomorphism classes of U(2)­ *, an algebraic isomorphism from p-forms to bundles, the whole intersection form on H2 (M; Z) (n- p)-forms which is defined according to the is recoverable. Such is K-theory on M. Via formula Whitehead and Freedman, the isomorphism classes w 1\ *TJ = (w, TJ) dvol. of U(l) and U(2) bundles over M contain the Here, (·, ·) is the metric inner product on 1\pT* M homeomorphism data that classifies M. and dvol is the volume n-form. A smooth structure on the base, M, is a priori For n = 4, takes 2-forms to 2-forms and required to even discuss connections on a bundle * 2 = 1. The choice of metric via Hodge's picks P over M. Recall that a connection was defined * * out of a fiducial, finite-dimensional subset as G-equivariant splitting of the vector-bundle A/ g of orbits of connections called the anti-self-dual exact sequence moduli space, M(P). This is the set of orbits of connections on P whose curvature 2-forms are 0 --+ Vert --+ T P --+ n*TM --+ 0, anti-self-dual a sequence in which T M explicitly appears. The appearance ofT M, above, means that differential M(P) ={[A] E B(P): FA = -*FA}· topology must be coded in the set of isomorphism classes of pairs (principal P --+ M, connection on The equation FA + *FA = 0, as an equation P). for [A] E B, is an elliptic nonlinear differen­ Fix a principal G-bundle P --+ M. (At tial equation [AHS], [Tl], [T2]. It resembles in this time, all investigations involve G = SU(2) respects both profound and superficial the inte­ or S0(3), essentially U(2)/center U(2).) The set grability condition for a complex structure [Dl]. g = Aut P is the set of isomorphisms of P with General Sard-Smale theory for Fredholm itself. Let A = {connections on P}. Then g acts maps can be used, after hard work ([FU], see also on A. It is now straightforward to topologize A [D2]) to show that for generic metrics, M(P) is a and g as infinite-dimensional manifold and Lie smooth manifold in B* = (A\R)/ g with canonical

9orbit~~ ) ) ) ))1•1(( ( (

Reducible --~------~ connection

FIGURE 2

712 cone-type singularities where it intersects R/ 9. dimension, (See Figure 3.) 2p1(P xc LieAlg(G)) S. K. Donaldson's vision saw that "character­ dimG .. istic numbers" of .M are computable invariants of ~ - 2- (Euler charactenstlc (M) the differential structure of M. Donaldson's cal­ culations provide a remarkable perspective from ~ signature(M)), which to view 4-manifolds. (Pl = first Pontrjagin number) should now be There are suitably defined characteristic viewed as the simplest of the set of invariants of numbers of .M which are invariants of cobordisms cobordisms of .M of B. This simplest cobordism of .M in B. They come by taking the intersection invariant is obviously a differential invariant of number of .M with the finite codimension varieties M since it is a topological one. The other in B* which represent the cohomology of B* (see cobordism invariants of M are more subtle; they Figure 4). This cohomology is determined by are differential invariants of M, but they do not the homotopy type of the manifold B•, and this, distinguish M up to homeomorphism. ultimately, is determined by the cohomology of Why cobordism invariants? Recall that a M [D3]. The intersection number of .M with such choice of metric on T M was necessary. To a class is a differential invariant of M [D4], [Ti]. change the metric changes .M, so only the metric­ independent properties of .M are differential in­ The simplest cobordism invariant of a man­ variants of M. As .M is the zero set of a Fredholm ifold is its dimension. The dimension of .M is map, Smale-Sard theory [SS] says that under a computed by the Atiyah-Singer index theorem for continuous change of the metric on T M, .M can the linearized anti-self-dual equations [AHS]. This change, at worst, by a cobordism; see Figure 5

A/9

R/9

FIGURE 3

~ class in H*(B*; Z)

········ A/9

R/9

FIGURE 4

713 and [FU], [Ti]. This Smale-Sard theory is, via uncountable set of distinct smoothings of R4 are the implicit function theorem, only the nonlinear known to exist [T3]. version of the well-known fact that the index of a Do gauge theory and moduli spaces give Fredholm operator is constant under continuous all obstructions and classification for smoothing deformations. 4-manifolds? This question is still open. By defining and studying the "ambient" cobordism invariants for moduli spaces for princi­ REFERENCES pal bundles over 4-manifolds, Donaldson has, in [AHS] M. F. Atiyah, N. Hitchin and I. particular, proved that only certain topological M. Singer, Self-duality in four-dimensional Rie­ 4-manifolds are smoothable: mannian geometry, Proc. Roy. Soc. London (1) If a smooth, simply connected 4-manifold 362A (1978), 425-461. has negative definite intersection form, then it is [B] T. Balaban, Ultraviolet stability of three­ homeomorphic to S4 or to an n-fold connected dimensional lattice pure gauge field theories, -2 sum of CP 's (see [D2]). Comm. Math. Phys. 102 (1985), 255-275 (2) If the intersection form is even, with and references therein. one or two negative eigenvalues (over R), then [D1] S. K. Donaldson, Anti-self-dual Yang­ the smooth simply connected 4-manifold is ho­ Mills equations over complex algebraic surfaces, Proc. London Math. Soc. 50 (1985), 1-26. meomorphic to S2 x S2 or (S 2 x S2 )#(S2 x S 2 ) [D3]. [D2] S. K. Donaldson, An application of Donaldson also proved that certain compact gauge theory to the topology of 4-manifolds, J. 4-manifolds have more than one smoothing. There Diff. Geom. 18 (1983), 279-315. are at least two smooth manifolds which are home- [D3] S. K. Donaldson, Cohomology and con­ 2 -2 nections, J. Diff. Geom., to appear. omorphic to the connect sum CP #g CP [D4]. [D4] S. K. Donaldson, La topologie differenti­ (CP 2 is complex projective space with reversed elle des surfaces complexes, C. R. Acad. Sci. orientation.) This last result was carried further Paris, t. 301, Serie 1, no. 6 (1985), 317. See also: by R. Friedman and J. Morgan [FM] who, in cal­ S. K. Donaldson, Irrationality and the h-cobordism culating Donaldson's cobordism invariant, showed conjecture, preprint. that there are at least a countable number of dis­ [F1] M. H. Freedman, The topology of 4- tinct diffeomorphism types of smooth manifolds dimensional manifolds, J. Diff. Geom. 17 (1983), 2 which are all homeomorphic to CP 2 # 9 CP . 357-454. In another direction, R. Fintushel and R. [FH] R. P. Feynman and A. P. Hibbs, Quan­ Stern have modified Donaldson's original ideas tum mechanics and path integrals, McGraw-Hill, to investigate pseudofree orbits and homology 3- 1965. spheres [FS1], [FS2]. Two very readable articles [FM] R. Friedman and J. Morgan, On the by R. Stern on gauge theory and topology are diffeomorphism type of certain complex surfaces, [81] and [82]. preprint. See also: C. Okonek and A. Van de The results above all imply, one way or Ven, Stable bundles and differentiable structures another, the existence of exotic smoothings of on certain elliptic surfaces, preprint. topological R4 . The existence of the first such [FS1] R. Fintushel and R. Stern, Pseudo-free was deduced by M. Freedman. Subsequently, R. orbifolds, preprint. Gompf [G] proved the existence of a countable [FS2] R. Fintushel and R. Stern, S0(3) set of distinct smoothings of R4 (based on work connections and the topology of 4-manifolds, J. by M. Freedman and L. Taylor [FT].) Now, an Diff. Geom. 20 (1984), 523-539.

; +- class in H*(B*; Z)

.········· .. ·· .... ······ Ajg

R./9

FIGURE 5. The moduli spaces for metrics g1 and g2 on TM

714 [FT) M. Freedman and L. Taylor, A universal [T1) C. H. Taubes, Self-dual connections on smoothing of 4-space, to appear. non-self-dual 4-manifolds, J. Diff. Geom. 17 [FU) D. Freed and K. K. Uhlenbeck, Instan­ (1982), 139--170. tons and 4-manifolds, Springer-Verlag, 1984. [T2) C. H. Taubes, Self-dual connections on [G) R. Gompf, An infinite set of exotic R4 's, manifolds with indefinite intersection matrix, J. J. Diff. Geom. 21 (1985), 283. See also R. Diff. Geom. 19 (1984), 517-560. Gompf, Three exotic R4 'sand other anomalies, J. [T3) C. H. Taubes, Gauge theories on end­ Diff. Geom. 18 (1983), 317-328. periodic 4-manifolds, J. Diff. Geom., to appear. [Ge) H. Georgi, A unified theory of elementary [Ti) R. Ticciati, Singular points in moduli particles and forces, Sci. Amer. 242, 4 (April spaces of Yang-Mills fields, Ph.D. thesis, Harvard 1981), 48-63. University, 1984. [GI) S. Glashow, Quarks with color and flavor, [U) K. K. Uhlenbeck, Connections with LP­ Sci. Amer. 233, 4 (October 1975), 38-50. bounds on curvature, Comm. Math. Phys. 83 ['tH) G. 't Hooft, Gauge theories of the forces (1981), 31. between elementary particles, Sci. Amer. 242, 6 [We] S. Weinberg, Unified theory of elemen­ (June 1980), 104-138. tary particle interaction, Sci. Amer. 231, 1 (July [81) R. Stern, Gauge theories as a tool 1974), 5Q-59. for low-dimensional topologists, Prospectives in [Wi) J. H. C. Whitehead, On simply connected Mathematics, Anniversary of Oberwolfach 1984, 4-dimensional polyhedra, Comment. Math. Helv. Birkhauser-Verlag, 1984. 22 (1949), 48-92. [82) R. Stern, Instantons and the topology of 4-manifolds, Math. Intelligencer 5 (1983), 39--44. [SS) S. Smale, An infinite dimensional version of Bard's theorem, Amer. J. Math. 87 (1965), 861-866.

This article is the fourteenth in the series of Special Articles published in Notices. The author, Clifford Henry Taubes, received his Ph. D. from Harvard University in 1980. From 1980 to 1983, he was a junior fellow at Harvard University. From 1983-1985, he was acting associate professor at the University of California at Berkeley. Since 1985, he h!l.!l been a professor at Harvard University. The series of Special Articles was created to provide a place for articles on mathematical subjects of interest to the general membership of the Society. The Editorial Committee of Notices is especially interested in the quality of exposition and intends to maintain the highest standards in order to assure that the Special Articles will be accessible to mathematicians in all fields. The articles must be interesting and mathematically sound. They are first refereed for accuracy and (if approved) accepted or rejected on the basis of the breadth of their appeal to the general mathematical public. Items for this series are solicited and, if accepted, will be paid for at the rate of $250 per page up to a maximum of $750. Manuscripts to be considered for this series should be sent to Ronald L. Graham, Associate Editor for Special Articles, Notices of the American Mathematical Society, Post Office Box 6248, Providence, Rhode Island 02940.

715 Mathematics: The Unifying Thread in Science

EDITOR'S NOTE: In conjunction with National their research, mathematical modeling of various Science Week, the National Research Council's physiological processes, things of that kind. Board on Mathematical Sciences presented a sym­ So, I think the theme of mathematics as a posium at the National Academy of Sciences on unifying thread is certainly something I see from May 12, 1986. The purpose of this conference was a university perspective. It is a theme that will to highlight the interconnections that exist between be considerably deepened by today's Symposium, mathematics and other sciences. The program and without further ado, I would like to introduce consisted of an opening welcome by Phillip A. our moderator, Professor Isadore M. Singer, John Griffiths, Chairman of the Board on Mathemati­ D. MacArthur Professor of Mathematics at MIT. cal Sciences, followed by presentations from three PROFESSOR SINGER: Welcome. I suppose as Nobel laureates: Allan M. Cormack (Nobel Prize moderator my first task is to warm you up, and in Medicine, 1979}, Herbert A. Hauptman (Nobel I will do so by telling a joke in the form of a Prize in Chemistry, 1985}, and Steven Weinberg riddle. It concerns a guy who parachutes out of {Nobel Prize in Physics, 1979}. Each speaker a plane, ends up in a tree and a fellow walks was introduced by Isadore M. Singer. The fol­ by and he says, "Say, fellow, where am I?" The lowing article is an edited transcription of the fellow looks up and says, "You are in an oak tree symposium, together with the question and answer dangling from a parachute, 10 and a half feet off period that accompanied the lectures, and is pre­ the ground." The riddle is who is the "fellow," sented in narrative form to preserve the informal and the answer is he is a mathematician. The and spontaneous character of the program. reasons are three. First, his answer is concise; PROVOST GRIFFITHS: I would like to welcome secondly, his answer is accurate; and thirdly, his everyone to the Symposium, "Mathematics: The answer is completely irrelevant. Unifying Thread in Science," and thank you for I am kind of relieved to get a laugh out of coming, many of you from a far distance. that joke, but I am also depressed because that I am Phillip Griffiths, a mathematician and is the problem mathematics faces: there seems Chairman of the Board on Mathematical Sciences, to be a prevalent opinion that mathematics is and I am, also, Provost at Duke University. From irrelevant. The purpose of this Symposium is to this perspective, I would like to briefly comment point out that in fact just the opposite is true. on a phenomenon that I have found very striking I had lunch with the speakers and you are all in my two and one-half years as Provost, a phe­ in for a treat. It is going to be great fun. Every nomenon not unrelated to today's Symposium. student in mathematics taking classical analysis The phenomenon I have in mind is that, in my is exposed sooner or later to the beautiful theory role as Provost, I have been very struck by the of the Radon transform. We never dreamed of its fact that mathematics is really pervasive in a important applications. Dr. Allan Cormack did, university. It is certainly true at Duke, and I and society is eternally grateful for that. He is a think this is the case elsewhere. You certainly university professor at Tufts University, a Nobel expect mathematical activity in the computer sci­ laureate in medicine and physiology in 1979 and ence department, physics, engineering, those areas he will explain the use of the Radon transform which you might think of as traditional areas, but, in tomography and variants of it in radiotherapy for example, in biological sciences, evolutionary planning. Professor Cormack? biology, biomechanics, population biology and PROFESSOR CORMACK: Professor Singer, social sciences, mathematical economics, econo­ ladies and gentlemen, first of all I would like metricians, demographics; you have quantitative to state that I am not a mathematician, wasn't people in political science and sociology, public trained as a mathematician, but I like mathemat­ policy, all of these areas. ics, and I and here to give mathematics a hand Even in our business school, there is a large along, if I possibly can. group of people involved with decision sciences What I am going to talk about today is not which basically model how one makes decisions at all profound mathematics. It is really classical under conditions of uncertainty, using Bayesian analysis, and it has been around for at least 80 statistics. In environmental studies we have water years, and what I wanted to try to point out to and resource modeling folks, and in the Medical you is how, as so often happens in mathematics, Center, there is a very large group of people in­ physics, all branches of science, ideas keep popping terested in various applications of mathematics to up and being forgotten and popping up again. I

716 think this is a nice illustration of a number of ago and she is our great, great grandmother or things, and it points out how even an old classical grandaunt or something. The anthropologists are problem can still have some important unsolved constantly revising our bloodlines so I am not bits to it, and so I am going to end my talk with a quite sure exactly where she fits in. Anyhow, problem for Professor Singer and his mathematical what is interesting about her and other fossils colleagues, particularly the analysts. of that period and that species, Australopithecus First of all, I had better explain what I mean Africanus, is that you can get quite detailed CAT by Radon's problem. Suppose that you have a scans of them. It is amazing to me that these domain, and there is some variable quantity F days one can do detailed anatomy on 3 million defined inside-and it doesn't matter what the year old specimens. A CAT scan of Mrs. Pless's quantity is, it could be a density, an absorption ear showed the orientation with respect to the coefficient, a statistical distribution and so on­ vertical of her ear's semicurcular canals, and this and suppose you draw a straight line L across information verified what had been determined the domain. Now you denote by F the average by other means-namely, that this creature was of F along L; that is, we simply measure F at walking around in an upright position. various points on L and take its average. Suppose The absorption of x-rays, of course, dates we do this for a number of lines. There may be from the late 19th century, and in the two or many lines, maybe only a few. The question is, three applications that I am going speak of next, if we know F for many different lines L, can we we will progress from the 1890s when x-rays were infer this variable quantity F? That is Radon's discovered, to the 1930s, when positrons were problem. discovered. One can create radioactive isotopes Putting it in a slightly more mathematical which produce positrons, and they annihilate form, we think of L as being specified by p and 4> and produce photons, which go off in opposite the polar coordinates of the point on L which is directions. If the electrons and positrons in the closest to the origin, and we form the integral of isotope start off at rest, their linear momentum F along that line, giving a sort of average value must be zero, and the great conservation laws of F. Given this function of these two variables of physics tell us that the linear momentum p and , can we find the function F of the polar must stay zero, and that is why you have to coodinates r and 0? The answer is yes, and in have two photons instead of just one: they fact there are several ways of doing this, and I have to have equal but opposite momenta, and won't get into the technicalities of it. that is why they have to go off in exactly What I will do is give you a survey of various opposite directions. This is the basis of PET applications. The applications that have gotten scanning. The two photons define a line, and all the publicity so far have been medical. They by the appropriate equipment, which I won't have been in the field of CAT scanning, PET describe, one can find out how this positron­ scanning and, more recently, NMR scanning, and emitting material is distributed throughout some I am going to go briefly through these to show object. Application of the Radon transform then you how the straight lines appear and how they gives you a distribution of this material. may be used to solve certain problems. Then I Before saying more about PET scanning, I will go on to some nonmedical applications and want to say that the electrons in the substance irr some other generalizations of this problem. to which the isotope has been injected generally do CAT scanning, PET scanning, and NMR have momentum, as do the positrons, so they do scanning are, of course, all very expensive medical not generally start off with nonzero momentum. techniques, and I thought just for kicks I would So, the two photons go off in nearly opposite talk a little about the CAT scanner that I used in directions, but there is a small angle (} between 1963. It cost about 100 dollars, which was a small their lines of motion that tells us something about amount of money even in 1963, and it had two the momentum of the system. It is a very small collimators to form a fine beam or line of x-rays. angle and from the point of view of PET scanning I made a model of a head, and I positioned the it can be totally ignored, but from the point of collimators in various ways around the head so· view of a later application, which I will mention, as to pick out various lines. You can measure to the fact that this angle is in general not zero is average absorptions along these lines, and from the key to the game. From it one can deduce that you can reconstruct a model of the head. In something about the momenta of the electrons 1963, I took two days to get about 256 pieces of in the substance in which the annihilation takes data to do a reconstruction, and I will contrast place. that with a modern and much more expensive An early application of this technique is the device which gets about ·1 million pieces of data uptake of glucose in the human body. To examine in about 5 seconds and gives you, of course, a this, one gives the patient glucose labelled with much better and very valuable picture. fluorine 19, which is a positron emitter, and I would now like to talk about a fossil well­ then uses a PET scanner, a whole array of photon known in the world of paleoanthropology called counters, to catch all these emitted photons. From Mrs. Pless. She lived about 2.9 million years this one can deduce the distribution of the level of

717 metabolic activity in a patient, and a succession of about whether things had changed while you were such pictures shows how this activity varies with taking the data. one time. This is a very powerful method because About 1956, just about the time when I was with can make positron-emitting isotopes combine starting to think about CAT scanning in South gives an awful lot of things. Just carbon alone Africa, Ron Bracewell, an Australian astronomer, to deal with, so you a wide spectrum of chemicals produced a picture of the brightness distribution be done a tremendous amount of physiology can of the moon. Remember this is not modern there are limitations. with PET scanning. But astronomy with the VLA and things of that that you One is the amount of radioactive isotope sort. They just had a series of radar dishes that means you have can pump into a patient, and which they could put up and align, and which are never going to get limited statistics, so you approximates a cylindrical lens, so you get very that you get with a the same kind of resolution good definition in one direction but very bad say, ten CAT scan. But on the other hand, with, definition, in fact, just averaging over the other powerful thousand or so detectors it would be a direction. Radon's problem applies in this case, PET scanning will research tool. But I think and you can determine, as Bracewell did, the tool that, say, never become quite the diagnostic brightness distribution of the moon. CAT scanning and NMR scanning have become. NMR scanning is a more recent technique. The history of this goes back to before 1905, When I was an undergraduate and just starting and I am not quite sure how long before 1905. my research work, magnetic resonance was then a This problem has the smack of a 19th century very new subject, and in order to study NMR you problem, to me at any rate. I thought Cauchy, needed very uniform magnetic fields, so a great or Riemann, or someone had dashed it off in a deal of designing of NMR equipment was devoted weak moment, but the earliest reference I have to getting very uniform fields. It was only about been able to get is dated 1905, and that refers 1972 that Paul Lauterbur said, "Let us apply a to and earlier work of H. A. Lorentz who, for gradient to this magnetic field which will isolate reasons nobody knows, thought of this problem a plane." If you put it on two gradients which in three dimensions where you are concerned with intersect, you isolate a line, and you pick out the averages not over lines in the plane, but over NMR signal from the signal occurring along a surfaces that intersect a three-dimensional region. line, and from that you can use Radon's problem You integrate or average over various sections of to produce a picture. The physician can then use the region. The question is, given this average, these pictures of several slices of the body. Again, can you find the original variable quantity? The the physics of this is irrelevant to us, and I would answer is, sure, and, in fact, it is a lot easier like to go on to some nonmedical applications. in three dimensions than in two dimensions. Of course, one can generalize this to any number Wood's A group of oceanographers from of dimensions, but for odd dimensions it is very and two Hole at MIT, from Scripps Institute, simple and local, whereas for even dimensions it is tomo­ or three other places, applied essntially more complicated-not terribly complicated but graphic means to determining the temperature just more complicated. This work was done by of finding in the ocean. The traditional method Lorentz. He never published it, and we only know square the temperature in, say, an area of 300 about it from a paper by his student in 1905, who kilometers, is to throw a thermometer overboard refers to the problem as being solved by Lorentz forwards on a ship and then go backwards and and gives the solution. The work had something across the area. By the time you get to one end to do with the propagation of light in crystals, if things have changed of the area, you wonder and probably has something to do with partial they since you were at the other end, and indeed differential equations, the field in which Radon's used sometimes have. The method this group problem has probably been better recognized than with compares data taken by traditional methods in any other field of mathematics. data taken by tomographic methods in another and doing so depends on having a source and I could give a long list of occurrences of a detector and measuring the time that a pulse Radon's problem since then. I will mention just takes to go from one to the other. In a variable a couple of highlights. It was rediscovered or medium, what you are measuring, therefore is generalized to arbitrary dimensions by George the integral or the average value of the inv~rse Uhlenbeck in 1925. In fact, it appears in the issue of the velocity, so if you do this for a number of of Physica which follows the famous Uhlenbeck­ sources and detectors, you have your lines. You Gotschmidt paper on electron spin. He doesn't have Radon's problem, and you can find out what say why he did it either, but does say that Paul the veloci~y is due to, and from the velocity, you Ehrenfest had suggested he do it, and I guess can then mfer the temperature. In order to get in those days in Holland if your thesis adviser this picture in the same time that you would take suggested you do a problem, well, you went and for traditional methods would require a ship with did that problem. He was the first to discover the a speed of, oh, I think it is 2,700 knots! So, Fourier treatment, a fairly obviou'l way of looking there is a considerable advantage and less doubt at the problem.

718 In 1936, it was rediscovered in Stockholm as by the polar coordinates of the diameter. Could a statistical problem, in which the function F you then solve the problem? I did the same sort that I talked abut was a statistical distribution. of thing that I did for the straight lines, and I was Radon's problem is known in statistics as the amazed to find that the solutions were remarkably Cramer-Wold theorem. Statisticians always seem similar. to have different names from the rest of us for Since then, in the late seventies, I found well-known things, but that is their privilege. that Radon's problem for circles and for straight In the neighborhood of the sun, there are a lines are very intimately related in a way which large number of stars, and, in the thirties, an I didn't realize at that time. This transform is interesting problem was to determine the velocity in fact called the adjoint of the ordinary Radon distribution of those stars. Now, there is one transform. Apart from just publishing a formal problem with this. It is very easy to measure solution of this as a sideline, I didn't do anything their radial velocities because they can be found about that until a few years ago, and it occurred from the Doppler shift of the spectra. The to me, why do you stop at circles and straight catch is to find what the astronomers call the lines? Surely there must be other things similar proper motion, which is the velocity at right to this. And more recently I discovered that, in angles to the line of sight, and that can be fact, there is a whole family of curves. One can measured for only a few close and fast-moving generalize the straight line problem by varying stars. The problem is, how do you determine a certain parameter, a, which must be greater the true velocity distribution from the knowledge than 0 and less than infinity, but otherwise it of only the radial velocities? This problem can be any number you like. If a equals 2, you was posed by Eddington in 1936, and solved by get a rectangular hyperbola, and if it equals 1/2 the Armenian astronomer Ambartsumian in the you get a parabola. If a is less than 1/2, the same year, and the solution depends on using curves start intersecting themselves one or more Radon's problem in a three-dimensional velocity times, but the point about these curves is that space. Ambartsumian plotted the distribution of they all go off to infinity. Generalizing the case velocities for about 500 stars. This gives the with a circle, one varies another parameter, {3, to lie to the fact that you couldn't do computer get the f3 curves, as I call them, and then the tomography without computers. Ambartsumian corresponding thing happens: they all go into the spells out how you use classical methods of origin. f3 equals 1 is the circle, f3 equals 2 is a numerical integration by drawing sine waves on lemniscate of Bernoulli, I guess, f3 equals 1/2 is graph paper and counting up the numbers of stars a cardioid, and so on. Of course, this problem in strips, and so on. This is, to my knowledge, can be generalized to arbitrary dimensions, and the first real application of Radon's problem to I have done a little bit about that in Euclidean experimental data. space, and real mathematicians have generalized The last application that I want to mention the whole problem for much more complicated goes back to what I said earlier about positron spaces that I know very little about. I cannot annihilation, and when I pointed out that, in even understand their papers, but, anyhow, I do fact, there was a small angle between the lines of know that they have done it and gotten some motion of the two photons which was determined interesting results. by the momentum distributions of the positrons Recently the problem of circles that I have and of the electrons. Now, the momentum referred to has come up in a completely different distribution of electrons and solids is of great connection that I had not thought about, although interest to solid state physicists and others. In if I had been on the ball, I would have thought particular, there is a surface, called the Fermi about it in 1956. It is in connection with surface, that occurs in the momentum space of radiotherapy with x-rays, which is the way most the electrons. This surface used to be determined people get radiotherapy these days. Use of heavy mostly by band structure calculations, but by ions is better, but very few people are able to studying the variations of the small angle that get to the facilities that use them. So we are I mentioned earlier, one can, in fact, determine stuck with the fact that most people who receive the Fermi surface directly. People in the Geneva radiotherapy are going to have radiotherapy with group and elsewhere are actively pursuing this x-rays, and the question now is, how do you research today. give the proper dose of x-rays? Suppose you I have given you a very brief summary of are a physician and need to irradiate a tumor various applications of the Radon problem, and in a patient, and the tumor is near the lungs I would like to go on ·to talk now about some and spine. The lungs and the spine are very generalizations of it. The thought that occurred radiosensitive, so in any treatment that you plan, to me way back in 1956 or 1957, when I first you should really try to keep the dose to these started thinking about this was, well, if you can parts to a minimum, while at the same time do it for straight lines, can't you do it for some hitting the tumor as hard as you want to. That other curves, and how do you specify a curve? is the problem: the tumor is to get a full dose Suppose you specified a circle through the origin and the lungs and spine are to be avoided. So if

719 you drew up an ideal dose distribution, you would some time, being forgotten, then popping up in have a full dose in the neighborhood of the tumor, another entirely different context 40 years later. and zero dose everywhere else. But because the Thank you. x-rays have long tails, any x-ray that comes in PROFESSOR SINGER: Thank you for a most to irradiate the tumor will, of course, leave the beautiful, and, for the mathematicians, a most the full tail outside. What you will get is maybe provocative talk. dose at the tumor, and the x-rays will be smeared received his Ph.D. in out around the tumor. This places an essential Dr. Herbert Hauptman of Maryland after restriction on the kinds of things you can do. mathematics at the University extended service as a teacher, in the Air Force, Now suppose you use rotation therapy, in and then at the Naval Research Laboratory. He which you have an x-ray tube sending a beam of and Jerome Karle received the Nobel Prize in x-rays in one direction with a variable intensity Chemistry in 1985 for their joint work done in which can be adjusted by some kind of arrange­ x-ray crystallography. That work was done in the ment of lead absorbers, and so on. You are fifties, and it took the modern computer to make interested in irradiating a particular point P, so it most effective in using the direct method to you rotate your x-ray tube around the patient. Of quickly determine the three-dimensional models course, you consider only those beams, or lines, of molecules. Dr. Hauptman is research professor which pass through P and they all will be on a of biophysical sciences at the State University of circle which passes through P, and that is how New York in Buffalo and President and Research circles get into it. Director of the Medical Foundation of Buffalo, Now suppose that G is a function describing and he will describe the direct method and its the radiation field. The dose found from this numerous applications in chemistry, medicine, and function is the integral, F, of Gover these circles. biology. Dr. Hauptman? So in this case, the attenuated Radon problem DR. HAUPTMAN: Following in the footsteps that comes up is, how do you go from a desired of Professor Cormack, I think it appropriate distribution F, to a radiation field G, which is the to begin by saying that my total educational thing that the radiotherapist can control? In a background in chemistry consisted of a single certain approximation which I worked out a long course in freshman chemistry taken some 50 years time ago, you can indeed get a complete solution. ago. So, I am not primarily a chemist. I am not If you specify an arbitary dose distribution then sure that I am a mathematician either. However, you can, in fact, calculate this function G, but most of my education is in mathematics, including there is one big problem. my Ph.D. In general, if you have an arbitrary F, Some 2,400 years ago, the Greek philosopher an arbitrary dose distribution will give negative Democritus postulated the existence of molecules, values of G, and as anybody who works in the that is, the existence of a smallest unit of any ma­ science knows, nature is very perverse, and there terial whatsoever. He also guessed that molecules is no such thing as a negative dose of radiation. themselves consist of still smaller particles called Life would be a lot easier if there were. So atoms, the nature and arrangement of which de­ here is a real mathematical problem in analysis: termines the physical and chemical properties of how to get a useful characterization of F, that the substance. Now, this was a remarkable guess. is the dose distribution, which will ensure that So far as I can tell, there could not have been G is nonnegative. As far as I know, this has no any justification for that guess that we would find answer. So here is an old problem of classical acceptable today. So I think that he was lucky, analysis waiting for an answer, and it is a problem and it is too bad that not all philosophers are just which needs solution right now. as lucky. In 1940, the late Professor George Birkhoff To begin with, then, I would like to say published a drawing made of straight lines in that Democritus' conjecture has been confirmed an article entitled, "On Drawings Made Up or only in this century, and this has come about Constructed of Straight Lines." The drawing through the technique which is known as x-ray shows how you can get quite a lot of complexity, crystallography, or the diffraction of x-rays by and what the x-ray problem is. Each straight line crystals. you can think of as being a fine beam of x-rays, A molecular structure may be defined simply producing a dose of x-rays. They all add up in one as the geometrical arrangement of the atoms spot, giving a heavy dose. I only came across this in the structure, and once we have found the article about one year ago. Birkhoff said, "Sure coordinates of the atoms in the molecule with you can make an arbitrarily complicated drawing respect to any reference frame, then, of course, of it," but he permitted himself the freedom we have solved the problem of determining the that artists have, which is to make erasures and molecular structure. Crystals ordinarily have that's the same as having a negative dose of well defined planar surfaces which are simply the radiation. This is a nice example of a piece of manifestations on a macroscopic scale of internal mathematics being discovered in one context at order. Normally, for crystals to be euitable for the

720 x-ray diffraction experiment, the size needs to be several tenths of a millimeter in linear dimension. I would like to describe very briefly what the mathematical description of a crystal is. After all, I am here to talk about the use of mathematics in the development of methods which are used to determine molecular structures in a routine way and it is necessary, first of all, using the lan~uage of mathematics, to define what we mean by crystal. The e~iest way to do th~s is to. start with two dimens10ns, rather than Immediately jumping into three di~ensions, ~hich is ~hat we are really interested m, so I Will descnbe how one may tile the plane by means of congruent FIGURE 1 parallelograms. It is a very simple construction. We have two sets of parallel lines. In each set the When a beam of x-rays is incident on a crystal, distances between consecutive parallel lines are all it scatters in many directions. This effect, which equal, so that the lines decompose the whole plane is called the diffraction or scattering of x-rays into a set of congruent parallelograms, having the by means of crystals, was discovered by two property that there are no spaces in between the German physicists around the year 1912. It was parallelograms, so that we cover the entire plane soon discovered by von Laue and the Braggs, in this way. father and son, that the directions these scattered To arrive at the definition of a two-dimension­ rays travel in and their intensities depend upon al crystal, we imagine that in each parallelogram the molecular structure of the crystal which is we put a few points, and join the points with causing the scattering. If we therefore put up straight lines to form a polygon. These points a photographic plate so that the scattered rays are distributed in exactly the same way in each of are incident on that plate, each scattered ray will these parallelograms, so that each parallelogram darken the plate, and the amount of darkening will and its contents are exactly the same as every depend upon the intensity of the corresponding other parallelogram. If we imagine now that these scattered ray. There may be thousands of these points are the positions of atomic centers, then we scattered rays. Not only was it shown in those define in this way not merely a two-dimensional early days that the intensities and the directions periodic array of points, but we define what we of these scattered rays are determined uniquely by call an electron density function. The electron the crystal or molecular structure, but it was also density function in the two-dimensional case is shown then that the phases of these scattered rays simply a function of position which gives us at were also determined by the molecular structure. each point the number of electrons per unit area. Now, I mentioned phases. I am assuming Clearly, this function of position, the electron that all of you understand that x-rays, like ordi­ density function, is a doubly periodic function. It nary visible radiation or visible light, is a wave repeats. It has two periods corresponding to each motion, and waves, as we all know, have, in of the dimensions of the parallelograms. addition to intensities or amplitudes, a phase. Now, with this background, I think we can The graphs of sine waves may have the same jump immediately to the real crystal which is frequency and amplitude, but differ in phase, so the three-dimensional analog of what I have just that they go through zero at different points­ described. Instead of tiling the plane by means different points in time and different points in of congruent parallelograms, we tile the plane space. So the relative difference measured, say, in by means of congruent parallelopipeds. We degrees, is the phase difference between these two must imagine that, in general, each so-called waves. What was known soon after the discovery ."unit cell," or fundamental unit of a crystal, of the diffraction of x-rays by crystals is that the IS a parallelopiped. Again, we put into each complete nature of the so-called "diffraction pat­ parallelopiped a set of points and join them with tern," which is to say the distribution and relative straight lines, as shown. (See Figure 1.) intensities and relative phases of the scattered If we again imagine that each point is an rays, were uniquely determined by the molecular atom, then surrounding each atom there is again structure. It was also known early on that the a certain cloud of electrons defining an electron converse was also true, and the converse is an density function. Because of the nature of the immediate consequence of the three-dimensional const:uction, this electron density function, which periodicity of the density function. w~ Will call p(r), a function of position only, is I would like to remind you once again that tnply periodic. a crystal from the mathematical point of view What I would like to describe next is the is simply a triply periodic function of position. rela~ionship of the crystal that I have just been I think it is important to state that explicitly talkmg about to the x-ray diffraction experiment. here because the purpose of this meeting is to

721 show what mathematics has to say about physical that p(r) must be everywhere nonnegative. After sciences. It is clear, as has been said before, all, it is a density function, it is the number that mathematics is the language of science, and, of electrons per unit volume. In general, if we of course, the simplest and most elegant way to put in arbitrary values for these phases, we will describe the three-dimensional periodicity of the get a function p(r) which usually is negative electron density function is simply to expand it somewhere, so we cannot put in arbitrary values by means of a Fourier series. for these ~hases. Any periodic function may be represented as Now, as it turns out, imposing the condi­ a Fourier series. The triply periodic density func­ tion of nonnegativity does severely restrict the tion p( r) is representable as a three-dimensional permitted values of the phases, but doing so is or triple Fourier series, not sufficient to determine unique values for those phases. A much stronger restriction is required, but p(r) = ~ L FH exp( -27riH · r), this is also available because it is known that H atoms in a molecule may be approximated by vicin­ where r is the position vector, V is the volume of points. There are regions in the immediate the unit cell of the crystal, and the three compo­ ity of the centers of atoms where p(r) takes on be­ nents of the vector H range over the integers. The rather large positive values, but in the spaces down vectors H are called the reciprocal lattice vectors tween the atoms, the density function falls this con­ and they label the different scattered rays. to very small values. So, if we impose or discreteness of the atoms The coefficients FH in this series are complex dition of atomicity we have made an extremely severe numbers called the structure factors, and they in a molecule, know the can be defined in terms of p( r), as we will soon restriction, because ordinarily we even this see. The magnitudes of these structure factors are number of atoms in a molecule. By exploiting out that obtainable directly from the measured intensities, prior structural knowledge, it then turns by so in everything I say, these magnitudes may by the phases are, in fact, uniquely determined the crystal presumed to be known. We then are left with an the observed magnitudes. Therefore, structures are uniquely determined equation with many variable quantities, but the and molecular alone, only ones unknown to us are the so-called phases by the observable quantities, the intensities ap­ ¢>H of the structure factors, which are defined by and this is in spite of the fact that what pears to be needed in advance, the unmeasurable phases, are not available. To calculate the Fourier coefficients, the structure factors, in terms of p(r), we must evaluate a simple integral, These phases cannot be measured, they are lost. They are also necessary if we are to deduce the crystal structure from the diffraction exper­ FH = fv p(r) exp(27riH · r) dV. iment. It turns out that the intensities of the scattered rays can be measured in the diffraction Well, it is not a simple integral. It is a triple inte­ experiment; they are simply proportional to the gral in this case because we are dealing with three darkening on the photographic plate I mentioned dimensions. The integration is taken over the vol­ a few moments ago. While the intensities can be ume of a fundamental parallelopiped, and it is the measured, the phases themselves are lost. So we resulting equation which relates the structure fac­ lose one-half of the information which is needed tors, the magnitudes of which correspond to the if we are to deduce the crystal structure from the measured intensities in the diffraction experiment, diffraction experiment. to p(r). In the old days, that is to say prior to Obviously, if we go to the extreme, and about 1950 or so, crystallographers argued in the replace the real crystal by an ideal one in which following way. They said that since we can put the atoms are assumed to be points, then the in arbitrary values for these phases, because we integral becomes just a sum over the N atoms in lose them in the diffraction experiment, we can the unit cell of the crystal, calculate almost any density function p(r) which would be consistent with the measured intensities. 1 N Therefore, they argued that in principle, the EH = ~ LZiexp(27riH·rj)· diffraction experiment was not able to determine (}"2 j=l crystal and molecular structures. Zj is the atomic number of the jth atom, r j is Now, there was a fallacy to this argument. its position vector, and a2 is a scaling factor that As I said, prior to 1950, all crystallographers need not concern us. The structure factor is now believed this. The fallacy was very simply this: if replaced by a normalized structure factor, E H, arbitrary values for these phases are put into this which is also a complex number. The equation expression, we get in general density functions defining the phases is now p(r) which are inconsistent with known properties of the function p(r): for example, one "[)roperty is

722 Now, I want to discuss this equation at some not merely its magnitude. That means we have length which will probably not leave me very calculated these phases rPH which, as I empha­ much time to say much more, but that is not sized before, could not be measured. There was too important. Actually, it is really a system of no way that we could measure them directly, but equations because we have one equation for each the only information that has been put into this scattered ray, the direction of which is defined by system of equations is measured magnitudes. So the reciprocal lattice vector, a three-dimensional what is this telling us? This tells us then that vector with integer components. the measured intensities, these magnitudes IEHI, Suppose we take the magnitude of both do, in fact, contain the phase information which sides of this equation. On the left side, we was lost in the experiment. The only problem have the magnitude of the normalized structure that remains-and when I say "only" I am be­ factor, which can be determined once we do our ing a little facetious now-is, how do we extract diffraction experiment and measure the intensities this phase information which is contained in the of the scattered rays, as earlier. On the right measured intensity? side, we have a sum over the number of atoms The reason I have spent so much time is in the unit cell. The only unknowns on the that I want to stress that in coming to this right side are these position vectors rj which conclusion, which all of us know required not really describe the crystal and molecular structure. Let such profound mathematics, the mathematics was us consider the number of equations that we get essential, that prior to about 1950, the conclusion in this way. We have a number of equations equal that we just reached was regarded as ridiculous. to the number of magnitudes for which we have No one believed what we were saying in those measured intensities. In a typical case, there may days. We were regarded, I think, as rather foolish be thousands of these, five, six, seven thousand. for wasting our time trying to solve what was felt And in a typical case, the number of atoms in the to be an unsolvable problem even in principle. molecule is usually pretty small, 60 or 80, maybe The molecular and crystal structure does not 100 or so. Each position vector rj has three determine unique values for the phases, and the components. So we may be talking of something reason for that is that the position vectors depend like two or three hundred unknowns with five, not only on the molecular structure, but also six, seven thousand equations. What does this on the choice of origin. But we can move the tell us? That the position vectors are in general origin around, for there is nothing in the crystal overdetermined by what we can measure. Not which says that the origin must be at such and only is it not true, as had generally been thought such a point, since no point is distinguished from prior to about 1950, that observed intensities any other point. So if we change the origin, we are not sufficient to determine unique crystal change the position vectors r j and we change the structures, but it turns out that the problem is a values of the phases, so that individual phases greatly overdetermined one. are not even uniquely determined by the crystal Of course, we are asking for much less than structure. Clearly, there is no way that we can crystallographers were asking for before. Before hope to extract phase information from observed they were talking about determining the density intensities alone. It is not possible, and this, of function p(r). We do not pretend to determine course, explains why in the diffraction experiment, that. What we are saying is that we can determine the phase information is lost. We have to specify the positions of the atoms which, of course, is in some way or another what our origin should be much less information than the actual density before we can even hope that measured intensities function itself, but ordinarily this is all that we can determine the values of the individual phases. are concerned with. Now, this introduces a complication which Incidentally, it is a very complicated system is most readily resolved by concentrating not on of equations. By no stretch of the imagination the individual phase but on certain special linear can this be regarded, for example, as a system combinations of the phases whose values are, in of linear equations because we have linear com­ fact, uniquely determined by the structure and binations of exponentials, and the unknowns are are independent of the choice of origin. These in the exponent. Furthermore, we have taken the linear combinations of phases are called structure magnitudes of both sides, and that complicates invariants. By suitably specifying the origin, our problem immeasurably. But anyway, let us we can expect that the values of the structure imagine the problem is solved. Let us imagine invariants, since they are linear functions of the that these position vectors rj are now known. phases, will lead to unique values of the phases. They are surely determined by this redundant That program can, in fact, be carried through system of equations. successfully. I will just take the next couple of So, we have a molecular structure, the inten­ minutes to define the structure invariants. sities of the scattered rays of which must surely It turns out that a linear combination of coincide with what has been measured, but we three phases is a structure invariant provided that have done more. If we know the rj, we can the three reciprocal lattice vectors add up to zero. actually calculate the full complex number EH, This is a large class of structure invariants. We can

723 hope to estimate these special linear combinations which are standard in view of the development of of the phases from observed intensities alone. Of mathematical probability in the last 100 years or course, this is only a small subset of all the so. structure invariants, but I have- time to discuss For the case of the three-phase structure here in any detail only the three-phase structure invariant, the conditional probability distribution invariants, defined by of any three-phase structure invariant is given by a simple expression,

1 where H, K and L are reciprocal lattice vectors. P ~ - exp(Acos Ill), K, Once we have estimated the values of a sufficiently large number of these special linear combinations where of the phases and once we have specified the origin, then we can hope to evaluate the individual phases N themselves. an= Ezj, Now, the structure invariants play the im­ j=l portant role of linking the observed magnitude with the phases. Once we have estimated a suffi­ and r;, is a normalizing constant not important to ciently large number of these structure invariants, us now. The magnitudes given in the expression we can specify the origin in some suitable way for A are known, as are a3 and a2. which is not relevant for my talk today, and then This is a distribution which is best described from these estimates we can hope to recover the by means of a picture. In the case that tne value individual phases. of A is rather small, say, 0. 7, the distribution The next important step is what I have called looks like Figure 2. the neighborhood principle. It turns out that, in The most probable value of the three-phase order to estimate the value of any structure invari­ invariant is zero, but the estimate is not very good ant, we do not have to, in general, use the whole in this case because this distribution is relatively set of measured intensities. A small subset of mea­ flat. Its variance is large. The estimate is not sured intensities is sufficient in what I have called reliable, but for larger values of A, say, A = 2.3, favorable cases, which I will identify in a moment, the distribution looks like Figure 3. to estimate the value of any specified structure in­ Again we find that the most probable value variant. This idea is extremely important because of the invariant is zero, and the estimate now is it means that to estimate any structure invariant, good because we have a distribution with a very we just have to look at a few magnitudes or small variance. a few intensities. Those few intensities or few Okay, I see my time has run out. So I will magnitudes are called the neighborhoods of the simply summarize these results by saying that, structure invariant. The first neighborhood of the in the favorable case, these three magnitudes are three-phase structure invariant turns out to con­ large and we find the estimate of the three-phase sist simply of three magnitudes, IEHI, IEKI, IELI, structure invariant is zero. As we have already so that once we know these three magnitudes, we seen, it is the ability to estimate the values of the can, in favorable cases, hope to estimate this par­ structure invariants which allows us to go from ticular linear combination of phases. The value of observed magnitudes or measured intensities to this structure invariant is very insensitive to the the estimates of the invariants, which, in turn, values of thousands of other measured intensities. allows us by fairly routine processes to estimate I want to tell you very briefly what the solution the values of the individual phases. So we have strategy is, how one makes the estimate. What one does is to try to eliminate those troublesome p unknown position vectors ri. The method which was adopted is simply to assume that all possible values of these position vectors are equally likely. In the language of mathematical probability, we make the assumption that the position vectors are random variables which are uniformly and inde­ pendently distributed. Therefore, the normalized structure factor is itself a random variable and, on the assumption that these primitive random variables are uniformly and independently dis­ tributed, we can find the probability distribution of this structure factor. More than that, we ~----~--~----~--~----~---L I]! can find the conditional probability distribution -180 -120 -60 60 120 180 of any structure invariant, assuming as known Degrees the three magnitudes in its first neighborhood. All that work can be carried out by methods FIGURE 2

724 p course, the invention of the calculus itself, at least as far as Newton is concerned, was motivated by problems in celestial mechanics. However, in physics in particular and especially in the sort of physics I do, the physics of elementary particles, the connection between mathematics and science is considerably deeper and considerably more puzzling. It is not only that we use mathematics to solve problems because the phenomena we study are so complicated that we cannot think through them without mathematics; it is not just that we are dealing with complicated phenomena like +---~----4---~~--~--~~--~ w crystals or brains. As Feynman, in particular, has -180 -120 -60 60 120 180 emphasized in the Messenger lectures, the laws Degrees of physics that we are discovering are themselves only understandable in mathematical terms. The indescribable except in the FIGURE 3 universe seems to be language of mathematics. But the connection physics and mathematics is still deeper. found it possible to extract the phase information between from the measured intensities, and as shown The mathematical structures that arise in earlier, using measured intensities with phases the laws of nature, as far as we know them, at which can now be calculated, we are led directly the deepest level that we know them, are often to the crystal and the molecular structure. mathematical structures that were provided for us by mathematicians long before any thought of PROFESSOR SINGER: Thank you, Dr. Haupt­ physical application arose. It is positively spooky man, for a lecture of such great clarity, and the how the physicists finds the mathematician has theorem in Fourier analysis as well. been there before him or her. It is now my great pleasure to introduce a A famous example is provided by the devel­ good friend, Professor Steve Weinberg. We share opment of general relativity by Einstein from 1905 a common enthusiasm, string theory, and I am to 1916. As you probably all know, the essence a great admirer of his, not only for his work on of general relativity is the realization that the unifying electromagnetism with a weak force for phenomenon of gravitation is really just a symp­ which he won the Nobel Prize in Physics in 1979, tom of the curvature of space time. Now, this is but, also, for his books. I am sure many of you Einstein's idea, but Einstein did not develop the have read The First Three Minutes. I would like mathematics of curved space or space time. Of to commend to you his book on gravitation and course, the mathematics of a curved surface, (the cosmology. It is not only a great scientific treatise surface of a sphere, for example), as something but an intellectual treat, as well. It is a real work embedded in an ordinary three-dimensional space of art. is very old. The fact that three-dimensional space Professor Weinberg is a member of the Royal can itself be curved was developed many decades Society. He is Josey Regental Professor of Science before Einstein, in the 19th century, by Riemann, at the University of Texas in Austin, and he following work of Gauss, Bolyai, and Lobachevsky. will discuss the interaction of mathematics and Their motivation had nothing to do with grav­ physics brought about by work on unified field itation; I don't think the idea ever occurred to theory. Riemann or his predecessors that curved space Steve? had anything to do with gravity. The motivation PROFESSOR WEINBERG: Thanks very much. for their work was one internal to mathematics. hardly need to say much more to convince It was an effort to purify the axioms of geometry this audience that mathematics and the natural by seeing which axioms depended logically on sciences depend on each other. You have heard which other axioms. Einstein didn't reinvent the from Professor Cormack and Professor Hauptman mathematics of curved space; he was taught by a of how the scientist uses mathematics to solve friend, and he was delighted to learn that all this problems that arise in the course of his or her had been done and was ready to his hand. work. An even more impressive example, perhaps, The dependence goes the other way, also. is provided by the development of group theory. Many of the mathematical constructions that Group theory was invented by the mathematician mathematicians have developed over the centuries Galois in the early part of the 19th century, were originally inspired by scientific problems. again, as a means of solving problems that were The Fourier transforms that Professor Hauptman internal to mathematics, problems of deciding discussed did not arise from pure mathematics which kinds of equations would have solutions in but from problems in the flow of heat, and of rational terms.

725 As time passed, the physicists discovered to dissolve at this level, and we see that all that group theory. They realized that group theory is really left to us are the symmetries of the provides the mathematics needed for describing underlying laws, and that the universe is just one of the fundamental question of physics, the one very large reducible representation of these question of the symmetries of physical systems. symmetries. A group from the point of view of physicists is Another example that Iz Singer mentioned simply the set of all the transformations which, in his kind introduction of me is the example of when applied to an object or to the equations string theory. This is right now the hot topic of which describe a physical system leave that object theoretical physics. String theory, as you may read or those equations looking the same. Group in journals like Time Magazine and the Atlantic theory came to be incorporated into physics as Monthly, is a new view of what the fundamental the theory of symmetries. constituents of nature are. According to this When this began in the 19th century, the picture, the fundamental constituents of nature kinds of symmetries that physicists were con­ are not, in fact, particles as I have been implying, cerned about were rather incidental aspects of or even fields, but are instead little strings, little the systems under study. For example, everyone elementary rubber bands that go zipping around, knows that ordinary salt, sodium chloride, forms each in its own state of vibration. In these a crystal, a cubic lattice with a certain group of theories what we call a particle is just a string in symmetries. You can rotate your head 90 degrees, a particular state of vibration, and what we call and the crystal looks the same, or you can move a reaction among particles, is just the collision in a certain direction a certain amount, and the of two or more strings, each in its own state of crystal looks exactly the same. That is what you vibration, forming a single joined string which mean by a cubic lattice. However the fact that then later breaks up, forming several independent salt forms a cubic lattice is not the most fun­ strings, each again in its own mode of vibration. damental thing about the elements sodium and It seems like a strange notion for physicists to chlorine. have come to after all these years of talking Another example is the simpler symmetry about particles and fields, and it would take too describing the human body. Our bodies are long to explain why we think this is not an to a good approximation symmetrical under the unreasonable picture of nature, but perhaps I interchange of right and left, but that is surely can summarize it in one sentence: String theories not the most important thing about people. incorporate gravitation. In fact, not only do they On the other hand, when you get down to a incorporate it, you cannot have a string theory really fundamental level of physics, you find that without gravitation. The graviton, the quantum the symmetries are the most important things of gravitational radiation, the particle which is you can say about physical systems. It is not so transmitted when a gravitational force is exerted important to say what the system is composed of between two masses, is just the lowest mode of or even to say what the equations that govern the vibration of a fundamental closed string (closed system are. The most important thing you can meaning that it is a loop). Not only do they say is what are the symmetries of those equations. incorporate and necessitate gravitation, but these To try to explain this, consider how we describe string theories for the first time allow a description elementary particles. Every elementary particle is of gravitation on a microscopic quantum level pretty much like every other elementary particle. which is free of mathematical inconsistencies. All you have to do in order to distinguish one All other descriptions of gravity broke down elementary particle from another is to give a mathematically, gave nonsensical results when certain set of numbers. You have to give the carried to very small distances or very high momentum. You have to give the energy. You energies. String theory is our first chance at a have to give the electric charge, and a few other reasonable theory of gravity which extends from more recondite numbers. Now, the mathematician the very large down to the very small and as such, will recognize that these numbers are simply it is natural that we are all agog over it. the characterization of the way that the particle String theory itself has focused the attention transforms under all the symmetry groups that we of physicists on branches of mathematics that believe are the symmetries of the laws of nature, most of us weren't fortunate enough to have that is that group of transformations which leaves learned when we were students. You can easily the form of the laws of nature unchanged. The see that a string (just think of a little bit of a energy of a state is the way the state of the cord) traveling through space, sweeps out a two­ system changes when we reset our clock, and the dimensional surface. A very convenient (and, in momentum is the way the state of the system fact, perhaps even more fundamental than talking changes when we move our measuring apparatus. about strings) description of string theory is to And likewise for charge, etc. say that it is the theory of these two-dimensional In other words, there really is nothing to say surfaces. about an elementary particle except to say how The theory of two-dimensional surfaces is re­ it transforms under symmetries. Matter begins markably beautiful. There are ways of classifying

726 all possible two-dimensional surfaces according to Apollo landing module found in the lunar dust in their topology, the number of handles on them front of him the footsteps of Jules Verne. A num­ and the number of boundaries, which simply don't ber of authors have wondered about this, though exist in any higher dimension. The theory of two­ perhaps not precisely in the same terms. (There dimensional surfaces is a branch of mathematics is an interesting essay along this line by Eugene that when you get into it is one of the loveliest Wigner called, I believe, The Unreasonable Effec­ things you can learn. It was developed in the 19th tiveness of Mathematics.) One explanation may century, again, I believe starting with Riemann, be that the mathematicians, of course, live in this and further developed by mathematicians working world, and they have countless experiences, both in the late 19th century motivated by problems conscious and unconscious, of the way the world in complex analysis, and then continuing into the is put together, and when they come to do their 20th century. There are mathematicians who have mathematical work, they are profoundly affected spent their whole lives working on this theory of by these experiences. two-dimensional surfaces, who have never heard of That is certainly true in many cases. For string theory (or at least not until very recently). example (as Einstein emphasized) Euclid, though Yet when the physicists started to figure out how he may have thought he was doing the purest to solve the dynamical problems of strings, and of mathematics was, in fact, incorporating into they realized what they had to do was to perform an axiomatic system experimental facts about sums over all possible two-dimensional surfaces in uncurved space. We now regard these as exper­ order to add up all the ways that reactions could imental facts because as Einstein's work showed, occur, they found the mathematics just ready for it is perfectly possible to have a three-dimensional their use, developed over the past 100 years. curved space, if there is a gravitational field. So, String theory involves another branch of Euclid, to an extent that Euclid himself did not mathematics which goes back to the group theory realize was, in fact, a physicist. I was talking about before. The equations which It is hard to accept this explanation in govern these surfaces have a very large group of general. It is really hard to see how the work, symmetries, known as the conformal group. One for example, of Galois on group theory arose out description of these symmetries is in terms of an of any Galois' experience of the kind of physical algebraic structure representing all the possible laws that govern our universe. group transformations, which is actually infinite Another explanation of the ability of math­ dimensional. Mathematicians have been doing ematicians to anticipate physics is based on the a lot of work developing the theory of these assumption that there is some simplicity under­ infinite dimensional algebraic structures which lying the world, that there is some order at the underlie symmetry groups, again without any bottom of things. Mathematics is the science of clear motivation in terms of physics, and certainly order, so perhaps the reason the mathematician without knowing anything about string theory. discovers kinds of order which are of importance Yet when the physicists started to work on it, in physics is that there are only so many kinds of there it was. order. (I am to some extent paraphrasing a talk Speaking quite personally, I have found it by the mathematician Andrew Gleason here.) exhilarating at my stage of life to have to go There are many examples of the limitations back to school and learn all this wonderful math­ of mathematical structures. For example, the ematics. Some of us physicists have enjoyed our Greeks were charmed by the fact that there are conversations with mathematicians, in which we only a certain number of regular polyhedra. (A beg them to explain things to us in terms we can regular polyhedron is just any solid figure with understand. At the same time the mathemati­ flat sides, all of them identical, like a cube or a cians are pleased and somewhat bemused that we triangular pyramid.) There are only five of these are paying attention to them after all these years. so-called "Platonic solids," so this particular kind The mathematics department at the University of order is not unlimited. Kepler thought for the of Texas at Austin now allows the physicists to five Platonic solids had something to do with the use one of their lounges-which would have been five planets that were known in his time. We unlikely in previous years. Unfortunately, I must realize today that that is not the case, but Kepler's admit that there is no experimental evidence yet error was not that he tried to apply fundamental for string theory, and so, if theoretical physicists ideas of symmetry and beauty to science, but that are spending more time talking to the mathemati­ he applied them on the wrong level. The planets cians, they are spending less. time talking to the have nothing directly to do with the fundamental experimentalists, which is not good. laws of nature; they are random agglomerations I have talked about the spooky character of of matter whose orbits reflect the history of the mathematics, that the mathematicians are often solar system. Nevertheless the kind of reasoning there before the physicists, inventing structures that Kepler used is precisely the kind used all the which the physicists then find are relevant to time by elementary particle physicists. the world. It is as if Neil Armstrong when he There are other examples of the limitations first arrived on the moon and stepped out of the of the kinds of structures that are possible in

727 mathematics. I have talked a good deal here afternoon that there is this intimate connection about group theory. Of course, there are an between mathematics and the physical sciences, infinite number of groups, but in fact, all groups and so, it seems to me that of necessity we will find (both finite and infinite) that can exist can that the physical sciences can advance only to the all be built up out of certain simple groups, extent that mathematics is deliberately employed. and although again there are an infinite number I think, also, that a possible explanation of this of simple groups, they form certain well-known connection which you, Steve, were guessing about, families of which there are just a finite number. and which we do not, I think, understand at all is Within each family each group is pretty much like that the laws of physics are connected with each every other, except they get bigger and bigger as other in logical ways, and of course, this is exactly you go up the list. When physicists started to what mathematics is all about. Mathematics think seriously about group theory in connection is the study of logical implication, and the real with the physics of strongly interacting particles world seems to be logical in that sense, and this, I in the early sixties they were absolutely delighted think, is basically the reason that mathematics is to find that the question of which the symmetry just an inescapable part of the physical sciences. group underlies the particles was not one that Now, that is not to say that we will, of required an infinite amount of experimental work. necessity, soon see that the physical scientists will There were only a finite number of possible become strong mathematicians in the near future, families of groups that could be candidates for the if recent past history is any guide. I think it physics of strongly interacting particles. is no doubt true that both the mathematicians I don't know if this explanation of the re­ and the physical scientists are at fault here. The markable ability of mathematicians to get there mathematicians tend to be rather elitist in this ahead of physicists appeals to you. There is a respect, and view the physical scientists as some­ third explanation. It is that mathematicians (or how or other beneath them, that mathematics at least some of them) have sold their souls to can only be worthwhile if it is pure mathematics, the devil in return for advance information about preferably with no applications whatsoever. The what sort of mathematics will be of scientific physical scientists, for example, the physicists, importance. I know many mathematicians, and believe that if it requires mathematics to be un­ of course, none of the ones that I know would I derstood, it cannot be good physics, and I think suspect of anything like that, but who can tell? many chemists feel the same way. This is unfor­ Thank you. tunate. I also think that until these attitudes can PROFESSOR SINGER: Thanks Steve, for a be overcome, we will find that the integration of beautiful and provocative talk. mathematics with physical sciences will be a long QUESTION: I would like to know how these time in coming. I am not very optimistic about gentlemen, who all come from different areas that. which interact with mathematics, view this trend PROFESSOR SINGER: Actually in my expe­ in the last five or so years and if they can rience, at least in a limited area of geometry and give a prognosis as to how the future is going physics, these changes began to take place about to bring this kind of interaction that they have 10 years ago when we began to recognize that been involved with. Do they see the kind of gauge theories in physics were really the same cross-disciplinary discussion that they have been as fiber bundles and connections in mathematics. talking about, do they see that trend accelerating, A number of us jumped in, and there has been and what aspects do they see of education, of the a developing liaison between high-energy physics universities, that are going to affect this? and mathematics since then. PROFESSOR CORMACK: I am not very good What I am concerned about in this aspect about predicting the future so I don't know quite is the education of young people, and although what is going to happen, but the thing that money cannot solve all evils, I think funding bothers me a great deal as a person trained in would help a great deal in this matter. experimental physics is that in undergraduate My experience with young scientists who are training in, say, physics or biology, you learn a interested in this area has been that they don't relatively small amount of mathematics, maybe have enough time to be able to learn both subjects one-half dozen courses to understanding the kinds thoroughly. Secondly, because of the prevailing of things that Steve has just been talking about. attitudes, they have to make decisions as to which There is a long way to go, and a lot of that subject they are going to be in in order to get a mathematics isn't easy. I have tried to understand job and sustain a position. They have to make it from time to time and have usually failed, and that decision very early. I think this is a problem for education in both We are dealing with very sophisticated physics mathematics and physics which may or may not and very sophisticated mathematics, and it takes become more acute as time goes on. time to establish one's self. It takes time to DR. HAUPTMAN: I think I would like to say learn these fields, and therefore we should have something in that connection. There is clearly programs in this area, as well as many others, in no doubt after what all of us have heard this which students, young people, have long periods

728 in which they do not have to make a decision as even understandable to mathematicians who are to which area they are going to go into. They not the graduate students of the author. can establish themselves by research work before QUESTION: I was particularly stimulated by having to be looked upon for tenure appoint­ Professor Weinberg's comments because my un­ ments, etc. In mathematics that is certainly not dergraduate work is in physics, and I want to possible today, and in physics it was possible for propose some educational questions for consider­ a while for people to have fellowships for five ation. I have thought back to what I enjoyed as to seven years, but that is decreasing as well. I was learning mathematics and physics, and it People are making decisions much too early as to seemed to me that the natural path for studying the long-term career that they will have, and the the calculus, which is one of the things that it whole society, the scientific society, is trying to seems traditional to study, no matter which of our force that decision upon them too early. I think fields you are in, is to go through some training in this respect a change in the funding pattern in physics first. I think that the student who and increased funding would help a great deal. hits calculus first and then physics or worse yet, simultaneously both PROFESSOR WEINBERG: I did want to say of them, doesn't learn any physics, and I a word about the communication between math­ question whether he learns any mathematics. It ematicians and physicists. It has been very bad seems to me, for example, that if Newton had in the past, and some of the blame is doubtless to learn physics first, maybe the rest of us generally to be laid on the physicists' shoulders. We tend do, too. The hardest thing about teaching calculus to be very vague, and we don't know what the to freshmen is that they don't know any problem is until we have already seen how to solve physics or they don't know any chemistry or they it. We drive mathematicians crazy when we try don't know any economics. I am not quite sure to explain what our problems are. When we write where to begin with them articles we don't do a good enough job of spec­ to motivate what I consider a solution to the ifying how certain we are about our statements; problem of particle motion. we don't distinguish guesses from theorems. On I have never gotten any enthusiasm for over the other hand, since I have said a lot of nice two decades from anybody, at least in mathe­ things about mathematics, I have to say that the matics departments, about how maybe the pre­ mathematicians carry an even greater burden of requisite for the standard, serious calculus course guilt for these communication problems, largely for scientists ought to be something like physics because of what Professor Hauptman referred to without calculus. I don't even know if they write as their elitism. They often have, it seems to me, physics books that way anymore. as their ideal the savant who is understandable Should we consider that? Is that something only to a few cospecialists and who write articles to be considered seriously in today's world where that one has to spend years to try to fathom. we have a lot of cross-fertilization that we see and When physicists write articles, they generally more that we would like to see? start them with a paragraph saying, "Up until Secondly, understanding that mathemati­ now this has been thought to be the case. Now, cians get carried away with this sort of thing so-and-so has pointed out this problem. In this and bearing in mind that I am a friend of the article we are going to try to suggest a resolution physicists, is there, nevertheless, some value in of this difficulty." That is, they set the stage. On putting a little bit more proof into the early years the other hand, I have seen books on mathemat­ of, say, the lower division mathematics, calculus, ics, not just articles but books, in which the first analytic geometry and such matters? We cannot sentence in the preface was, "Let H be a nilpotent possibly teach people everything they are going to subgroup of ...." These books are written in need because we don't know what five years from what I would call a lapidary style. The idea seems now they are going to need, so maybe we ought to be that there should be no word in the book to teach a little bit more rigor, if we can avoid that is not absolutely necessary, that is inserted doing it in a stuffy sort of way. merely to help the reader to understand what is PROFESSOR WEINBERG: Could I comment going on. I think this is getting much better. that I completely agree. It may be a failing of I find it is wonderful how mathematicians these mine. I am not able to learn any mathematics, days are willing to explain their field to interested unless I can see some problem I am going to solve physicists. At some interdisciplinary meetings, I with the mathematics, and I don't understand how have heard the mathematicians do a better job of anyone can teach mathematics without having a explaining what they are doing than the physi­ battery of problems that the student is going to cists. This situation is improving, partly because be inspired to want to solve and then see that he as Iz Singer mentioned, we realize now that in or she can use the tools to solve. In calculus they certain areas we have much more in common than have been usually geometric problems such as we had thought, but I think a lot more has to calculating areas under curves, but I think it is a be done. There is still too much mathematics great idea (and one that I have tried when I taught written which is not only not understandable to an undergraduate course to nonphysicists) to experimental or theoretical physicists, but is not motivate the teaching of mathematics by showing

729 the kind of physics problem that you can solve if and things, I think it depends at what level you you know the mathema~ics. I don't see why all are going to do it. So the area of a circle is 1rr2 , calculus isn't taught that way. I think that the big deal. If you go too far, you can b!"ing the failure to understand this was one of the things subject into contempt. I don't know where the that was so dumb about the idea of introducing middle ground is. set theory in elementary school. Of course, set DR. HAUPTMAN: On the other hand, when theory is fundamental to all of mathematics from Archimedes discovered the volume of the sphere a certain point of view, but there is no problem and the surface area of the sphere, I believe he that an elementary school student can solve using considered that, and I think justly so, his greatest set theory. Learning set theory becomes a matter discovery. of just learning jargon that the student parrots PROFESSOR SINGER: One of the wonderful back. I don't think you can teach mathematics things about the question that has been raised unless you teach what kind of problem you are is it brings us all back to our own education, going to use the mathematics for. and that is what we are hearing, really: responses DR. HAUPTMAN: I think your remarks are to our own education. I think it depends very generally valid, Steve, but I also think that it much on the subject that you are talking about. depends on the kind of student. There are some The calculus certainly was derived and developed students for whom just teaching the mathematics for applications. I myself remember being a alone without the applications is sufficient, and wisecracking freshman and I didn't think college they love to study logical connections between had anything to offer until in a calculus course I statements. For example, I remember when I saw how Newton had derived Kepler's laws from first learned plane geometry, I thought this was the inverse square law, and I realized that I had the greatest thing that I had ever seen because arrived. Something very special in the intellectual it was the first time that I had been exposed history of mankind had taken place at that to a logical structure which was very beautiful, moment, and I was now at a place where I could and it wasn't necessary to see that there was any understand it. It was so beautiful. The reason I application of this work. But I do agree that most switched was because in the other class I had in people find their motivation from seeing what quantum mechanics some, I might say, fool was kinds of problems can be solved. I think calculus trying to convince me that quantum mechanics in particular is something which, again, purely as could be derived from classical mechanics, which a mathematical structure is extremely beautiful, is not the case at all. but for most people you need the motivation. QUESTION: On the relation between math­ I mean what does the calculus enable you to ematics and the sciences, I would like to ask a do which cannot be done otherwise? At the somewhat similar question, perhaps on a more same time, I think you will find some students, individual level, of Dr. Hauptman. His Ph.D. was and these may be the ones who become pure in pure mathematics, in number theory, and he mathematicians, who couldn't care less about did a lot of very good work in that area. So I what the applications are; for some students just would like to know how he made the transition learning calculus as a beautiful structure in itself from the pure mathematics of number theory to is enough. It is sufficient for them. the applied mathematics of crystallography? PROFESSOR CORMACK: I agree with that. DR. HAUPTMAN: Of course, there is a con­ There are some like that. This is the real problem nection between the two. A crystal is closely of teaching, like teaching even elementary physics. associated with a crystal lattice, and the theory There are some students who are really quite of lattices is closely connected to one aspect of theoretically inclined, and they like to have all number theory which is the Euclidean algorithm the i's dotted and the t's crossed and in a and the division algorithm of number theory. The mathematical sense, and then there are others connection which I found was that the ordinary who are quite different. They run on kind of Euclidean algorithm may be interpreted as a one­ gut feelings about physics. They really feel those dimensional lattice, and the generalization of this forces acting, and it is very hard in teaching idea to three-dimensional lattices, which, after elementary physics to try to keep them apart. I all, are closely related to crystal structures, was agree, also, that it is nice to make things relevant something which had not been done before. So for those who need some kind of relevancy, but there is a connection. Number theory has been you have also got to be careful not to trivialize it. thought of as the one branch of mathematics I remember in junior high school when I learned which surely could have no possible application trigonometry. People were very careful then to to the real world, and I think this was part of make sure that we knew what the trigonometry the reason that number theory attracted some was used for. So, we used to joke in classes. We mathematicians. said, "Oh, yes, we are all going to find the height Actually number theory, like many other of a tree today or a mountain tomorrow" and that branches of mathematics~like, for example, Ga­ kind of thing. If you trivialize it, you bring the lois theory which is certainly one of the most subject into disrespect, and if you calculate areas beautiful branches of mathematics~can stand on

730 its own feet. It does not need to have any and so on. This number is called the partitio relationship to the real world. numerorum. We needed to know its behavior As it turns out though, even number theory at very high integers corresponding to very high does have such a relationship. When I first came mass levels. The problem had been solved for the to work at the Naval Research Laboratory, the partitio numerorum in 1918 by G. H. Hardy and group that I was assigned to, as a matter of his colleague Ramanujan, and it amused me very chance, was interested at that time in electron much to list that as a reference because Hardy diffraction, the electron diffraction of gases. We was notorious as a mathematician who was proud soon recognized, however, that the action was in of the fact that the work he was doing would x-ray diffraction rather than electron diffraction. never have any physical application. The technique of electron diffraction has limited QUESTION: I would like you to brainstorm a application. There is much more information to little on how to promote the communication that be derived from the x-ray diffraction experiment, we all agree is necessary between mathematicians and that is where the major interest was at that and others. Young people are told again and time. This was in 1950. As I mentioned in my again that if you have any aspirations for inter­ lecture, the big problem in those years was the disciplinary work, keep it under your hat until phase problem. The belief that the phase problem you have tenure, and I think it is probably good was an unsolvable one was, for me at least, and I advice, if you want to get tenure. What can think for my colleague as well, a challenge, which be done to enable us to be able to communicate somehow or other we were bound and determined safely without sacrificing our own careers? to overcome. It turned out, maybe by luck, that PROFESSOR WEINBERG: I liked one of it was a problem which was solvable. Singer's ideas of giving mathematicians a little bit I think the immediate answer to your question more time before they have to get on the tenure is that it was chance that exposed the problem track. I have never understood why physicists get in x-ray crystallography to me. My background postdoctoral appointments and mathematicians was, again, I think by chance, peculiarly suited don't. I think it may be just the saintly quality for this problem. So I think the answer to your of mathematicians that they don't apply for that question is simply that it was a chance thing. It kind of money. was a random sort of thing. I will also say that In many respects, we are suffering the faults at that time, when I first came to work at the of our virtues. One of the great strengths of Naval Research Laboratory, I had just come out American universities is the department system. of three years in the Navy. It was the end of the It is what allows us to make judgments about war, and I felt that I had wasted three years in a hiring people because we know their work well certain sense, at least as far as career development enough so that we can make correct judgments, was concerned. I was getting kind of old at the and yet, it is the department system, I think time. I was 29 years old and did not yet have a that makes it so hard for a mathematician to Ph.D., and what I wanted most to do was to get a get any credit toward advancement by talking to Ph.D. and start working in some area of scientific physicists, for example or vice versa. There has research. I wasn't very fussy at that time. I don't to be something in the academic structure to give know if that answers your question. It is the best credit, extra credit for work that goes beyond answer I can think of. the bounds of one's own department, but I don't PROFESSOR SINGER: May I add a footnote know how to set that up. to your comments on applications and number QUESTION: I have an anecdote and a question theory? It would appear that at the one-loop for Dr. Weinberg. I am an engineer who became approximation in super string theory, the van­ a science writer and a broadcaster for the Voice ishing of the cosmological constant amounts to of America. The only script I ever wrote for an the Jacobi identity on theta functions in number international broadcast that was rejected was a theory. very brief piece which I thought was a completely DR. HAUPTMAN: That is remarkable. comprehensible statement about calculating the PROFESSOR WEINBERG: If I can also put in digits of 1r, and it was rejected outright as being my 2 cents: in 1970, in the early days of string totally unacceptable for our listening audience, theory, Kerson Huang and I set out to solve the which includes 42 languages and everyone from problem of calculating the number of states that primitives to sophisticates. I just thought that would arise in a vibrating string at a given level of was curious; but my question for Steven is does mass. (This is important in thermodynamics, for he draw any parallel in his mind between the example, if you want to know the energy density spooky connection between mathematics having of empty space filled with string fluctuations.) We been there before physics and the business about found that this number was very closely related to the observer being central supposedly in quantum the number of ways an integer can be written as mechanics? Is there a parallel effect in his mind a sum of integers. For instance, 2 can be written about this? one way, as 1 plus 1. Three can be written two PROFESSOR WEINBERG: Oh, I am the most ways, either as 1 plus 1 plus 1, or 2 plus 1, philistine of physicists when it comes to the

731 fundamental philosophical problems of quantum to understand or explain ideas like this without mechanics. My more learned friends tell me the use of metaphors, analogies and similes? that the observer has become important in a way PROFESSOR WEINBERG: Actually opinions that he or she never was before, but it seems differ. You have to realize that the subject has to me that quantum mechanics really describes not settled down. If you had asked me a question a perfectly deterministic evolution of the wave about general relativity, I would probably give function, and every once in a while, we human you a clear answer one way or another because beings interfere and make a measurement, and we think we know what general relativity is. then, of course, the results we get depend on the String theory is still in a formative stage, particularly clumsy way that we interfere with and there is quite a division among those who the system, and the importance of the observer is are trying to develop it, between those who see nothing more than that. I tend to resist drawing it in the terms that I was describing as really a any philosophical messages at all from quantum theory of Riemann surfaces in which the strings mechanics, but then, as I said, I am a philistine. are just a metaphor, and other physicists who Your question is a very serious one. I don't mean are trying to make the strings real. From my to pass it off lightly. In fact, John Wheeler, one of point of view, I would prefer a description which the most distinguished physicists in the century, may sound even more unworldly, that what is feels that because the observer is so important in really going on is a two-dimensional field theory quantum mechanics, we have now a breakdown of which we physicists in four dimensions interpret the reductionist idea that everything is based on in terms of observations in a rather indirect way. the very small. As I understand it, he feels that (The reason I said, "Two dimensions" is that the universe has to be such as to allow for the of what I said before because the string sweeps existence of human beings. I am uncertain about out two dimensions as it goes through space). this and, of course, it is a hard point to test. There are other physicists who would really like PROFESSOR WEINBERG: With regard to to understand string theory in terms of the joining your story about the digits of 1r, it is infuri­ and unjoining of physical strings. So, the answer ating to scientists that many journalists (not the isn't it. We don't know whether it is a metaphor science journalists like you and Gina Kolata and or whether the strings will become more and more others, but the everyday journalists, the people real as time passes. who would rather be writing sports news or po­ It is very hard to think of strings as being litical news) feel contemptuous of the public's the fundamental objects out of which the universe interest in science. They will tell you that if they is composed, and I tend to find it easier to cannot relate what you are doing to something of understand string theory in terms of the theory practical importance like a military development of surfaces, and particularly the symmetry groups or a medical cure, then the public will not be that govern the theory of surfaces. I think my own interested. guess is that eventually string theory will be seen as a realization of a particular symmetry group I don't think that is true. I think that called the Virasoro group, and that the strings the public is interested in all sorts of things will become less and less important. However, in fundamental science, including the digits of there are many experts (in fact, physicists who 1r, perhaps, but a lot of people in the news are much more expert than I) who are working business have decided that the public is not night and day trying to make all this into an sophisticated enough to grapple with any sort honest-to-God theory of strings. of science, unless it can be related to practical PROFESSOR SINGER: I must say that I al­ applications. I often fly American Airlines up and ways thought mathematicians were imaginative, down between Austin and the East Coast, and but I am very impressed with the physicists. It they have a program that they have been running doesn't faze them at all that dimension is now 10 called Science in the News. They have about or 26. I think if we were faced with that, we would one-half dozen items in each program that they have simply dropped the subject because space is show to the whole cabin. I have never seen any four dimensional. I can recall the early stages of items that didn't relate to something of practical super gravity when space time was 11-dimensional importance. The people who run that program and I was going to a physics colloquium and hav­ have obviously decided that the public, even the ing one of my physics colleagues essentially say flying public, is not interested in science, except that, to paraphrase it in mathematical terms, where it relates to things of practical, everyday space time was an 11-dimensional fiber bundle concern. I don't believe that is true, but there are where the fibers were seven dimensional and the a lot of professionals who disagree with me. base was four dimensional, and physicists listened QUESTION: I have a question for Professor very closely. I realized had I gotten up and made Weinberg. When you speak of strings, is this a that statement, I would have been laughed out of metaphor you are using in the same sense that in the auditorium. the old days of quantum mechanics, they used the QUESTION: Not so much a question but a planetary metaphor, and in general, is it possible couple of comments. Dr. Weinberg said a while

732 back that calculus and other courses should be relevance of mathematics to the things that had taught with motivation by problems that people been spoken about here to people in Congress felt compelled to solve and suggested that these and to the public at large? I think that is a came primarily from the physical world, and I very real problem and one that mathematicians agree with the need for problems as a source of have compounded by not trying to communicate motivation, but I think that the mathematicians to the public. But there is a certain prejudice feel that certain problems, even those which have in the press and in American culture. The origins in physics are intrinsically compelling. biologists are allowed to use technical terms, and Especially, for example, problems that arise in the physicists are allowed to use terms like quarks number theory which have to do with prime num­ which are effectively metaphors that the public bers or solutions of certain kinds of equations that has no concrete understanding of, and yet a at their outset have no discernible origin in the word like "group," which has appeared in your physical world. Even topics like Boolean algebra discussion of string theory which was implicit in which you lamented having been introduced in the the discussion of crystal structures, which is a curriculum probably are going to return as a basis fundamental notion in mathematics and natural for more combinatorial mathematics and as part science, is a term which we are not at liberty to of the tools that people will need in dealing with use in an article written for a public audience. So, problems that arise in computational science. the basic mathematical vocabulary with which to PROFESSOR WEINBERG: I think you are begin describing something serious is considered conflating two different issues. One is how taboo. mathematicians learn their craft and how they PROFESSOR SINGER: Any other questions or are motivated, and there I completely agree with comments? you. In fact, part of my talk here described how QUESTION: I just have a comment. We mathematicians in a mysterious way get their heard that the mathematics for the physics of direction from internal problems in mathematics. today were invented about 100, almost 200 years The question on which I was commenting though ago. Is it possible that there is a time lag and was how grade school and high school students that what the mathematicians work on today will should be taught, the vast majority of who are not only be relevant to natural sciences 150 to 200 going to be mathematicians. I think very few of years from now? them will be inspired by the ultimate relevance of PROFESSOR SINGER: Steve alluded to the Boolean algebra to the branches of mathematics origins of Riemann surface theory, but he also that you have mentioned. mentioned that people were working actively on QUESTIONER: I agree basically with that. I what is called modulized space of Riemann sur­ am reminded of a remark once made by a mathe­ faces, and that in fact, physicists are using results matician named Jacques Tits, who commented on that are hot off the presses, so to speak, results speculations about the control of fire by primitive that are not more than one or two years old in man, with some anthropologists suggesting that it order to talk about the space over which you have was motivated by a need to control fire for cooking to add all the different possibilities, average over or for heating or warmth. Tits felt that he could them. So that subject, although begun in what we much more easily believe that the control of fire would now call antiquity, is very active in its own came about simply because of people's fascination right, and in fact, the number 26 which occurs with the flame, and I think that that kind of in string theory was independently discovered in esthetic appeal for a problem, much more than algebraic geometry related to modulized spaces. the pragmatic implications of it, are what draws So, I would say that it is pretty up to date, the mathematicians into their work. stuff that is needed. In fact, we need a lot more. Finally, a comment about communication. It is stimulating a great deal of research in the I think someone asked, how do we convey the subject right now.

733 MATHEMATICAL REVIEWS ASSOCIATE EDITORS

The Mathematical Reviews Editorial than special skill, is sought. A reading Committee invites applications and rec­ knowledge of two main foreign languages ommendations for two-year appointments is important, but not essential. (Russian as Associate Editor of MR, to commence and Chinese are especially desirable.) as soon as possible, but no later than Those interested in combining a sab­ the summer of 1987. Applications will be batical or other leave with a part-time welcomed from persons taking leave from or full-time appointment as an Associate other positions, and in particular from Editor should write for further details. tenured faculty members who could take The twelve-month salary is negotiable, leave to come to MR for two years. and will be commensurate with the ex­ The MR office is located in Ann Ar­ perience applicants bring to the position. bor, Michigan, adjacent to the campus Retirement, insurance plans, and other of the University of Michigan, and the fringe benefits are similar to those in editors enjoy many faculty privileges at universities. Of special importance is a the university. At present, MR employs policy providing a study leave after at eleven editors, about a dozen consul­ least two years. This amounts to three tants, and over fifty noneditorial person­ months of full pay for each two years nel. It produces Mathematical Reviews spent as Editor. and Current Mathematical Publications Applications (including curriculum and various indexes, as well as the online vitae, bibliography, data on experience, service Math\Sci. The responsibilities and names and addresses of three refer­ of Associate Editors fall primarily in the ences) and recommendations should be day-to-day operations of classifying arti­ sent to Dr. R. G. Bartle, Executive Ed­ cles and books, assigning these items to itor, Mathematical Reviews, P. 0. Box reviewers, and editing the reviews when 8604, Ann Arbor, MI 48107. Telephone they are returned. Other responsibilities 313-996-5250. Those interested in apply­ evolve in accordance with the individual's ing for this position are urged to inquire experience and capabilities. At this time, immediately. no particular area of mathematical spe­ Mathematical Reviews is an equal cialization is sought, although strength opportunity employer. in applied areas is desirable. Consid­ erable breadth in mathematics, rather International Congress of Mathematicians 1986

The 28th International Congress of Mathemati­ established Nevanlinna Prize to Professor Leslie cians was held in Berkeley, California, August Valiant of Harvard University, a computer scien­ 3- 11, 1986. It was attended by approximately tist who has made a variety of major contributions 3,500 mathematicians from over 70 countries. to the study of the relationship between theoreti­ Slightly more than half of the mathematicians cal computer science and mathematics, especially came from the United States. in complexity theory. Professors Freedman and The Congress began in the open air Greek Donaldson received their medals primarily for Theater on a beautiful Berkeley morning. An­ their work on the topology of 4-manifolds; Pro­ drew Gleason, who had served as Chairman of the fessor Freedman proved the Poincare Conjecture Steering Committee for the Congress, was chosen in dimension four and Professor Donaldson has as its President, and Lars Ahlfors was chosen as shown that there is a differential structure on Honorary President. Professor Ahlfors received R4 which is different from the usual structure. one of the first two Fields Medals, both of which Professor Faltings, a number theorist and native were awarded at the Congress in Oslo, Norway in of West Germany working in arithmetic algebraic 1936, just 50 years ago. Professor Ahlfors pre­ geometry, received his medal primarily for his sented this year's three Fields Medals to Professor proof of Mordell's Conjecture. Simon Donaldson of Oxford University, England, EDITOR'S NOTE: Additional information about to Professor Gerd Faltings of Princeton Univer­ the awards and recipients appears at the end of sity, and to Professor Michael Freedman of the this article. University of California, San Diego. Professor The mathematics of the prize winners was de­ L. D. Faddeev, the Chairman of the Selection scribed to the Congress in four addresses given by Committee and the new President of the Interna­ Professors Barry Mazur of Harvard University on tional Mathematical Union, presented the newly Valiant's work, Michael Atiyah of Oxford Univer-

Fields Medal and Nevanlinna Prize Winners

Photo courtesy of William G. Chinn. From left to right: Nevanlinna Prize Winner, Leslie Valiant and Fields Medal Winners, Michael Freedman, Gerd Faltings, and Simon Donaldson.

735 sity on Donaldson's work, Volker Strassen of the The officers and executive committee of the University of Ziirich on Faltings's work, and John IMU serving through this Congress were: Milnor of Princeton University on Freedman's President Jiirgen Moser (Switzerland) work. These four addresses were exceptionally Past President Lennart Carleson (Sweden) well done and set the tone for the entire Congress. Secretary Olli Lehto (Finland) EDITOR'S NOTE: Summaries of these addresses Vice Presidents L. D. Faddeev (U.S.S.R) are being prepared. The editors hope to publish J. P. Serre (France) them in the next issue of Notices. Members S. Mizohata (Japan) G. D. Mostow (U.S.A.) Something that really stood out about this M. S. Narasimhan (India) Congress was the large number of really remark­ C. Olech (Poland) ably well thought-out and presented addresses. J. Palis (Brazil) (hour-long survey There were 15 plenary addresses The new officers and executive committee of 45-minute talks) and 147 more specialized invited the IMU who will serve through the 1990 Congress for shorter lectures. There were many sessions are: communications as well as a variety of infor­ mally organized special lecture series, and a seven President L. D. Faddeev (U.S.S.R.) session special program organized by the Interna­ Past President Jiirgen Moser (Switzerland) tional Commission on Mathematical Instruction. Secretary Olli Lehto (Finland) Also, the United States National Commission on Vice Presidents Walter Feit (U.S.A.) Mathematical Instruction arranged for four su­ Lars Hormander (Sweden) perb expository lectures in the major areas of Members John Coates (United Kingdom) the Congress which were presented on Saturday Hikosaburo Komatsu (Japan) afternoon, August 2, just before the opening of Laszlo Lovasz (Hungary) the Congress. Jacob Palis (Brazil) C. S. Seshadri (India) The campus setting, the cool comfortable weather, the generally satisfactory lecture halls, Also, Jean-Pierre Kahane of France is the the interesting and nearby city of San Francisco, New President of the International Commission the very carefully thought-out plans, the usu­ on Mathematical Instruction. M. S. Narasimhan ally quite satisfactory living arrangements, the of India is the new Chairman of the Commission hospitable local mathematicians-all made for a on Development and Exchange whose executive cheerful atmosphere and a good setting in which committees were also chosen with proper interna­ to absorb mathematics. The Thursday afternoon tional representations. excursion to San Francisco with a barbecue and a The next International Congress of Mathe­ rodeo at the Cow Palace drew a large crowd. maticians will be held in August of 1990 in Kyoto, The only seriously disturbing feature of the Japan. Congress was the absence of many Soviet math­ Mary Ellen Rudin ematicians. There were only 38 mathematicians Chairman, U.S. National Committee from the U.S.S.R. attending the Congress includ­ for Mathematics ing the 7 who attended the General Assembly of the International Mathematical Union preceding Fields Medals the Congress. Of the 34 U.S.S.R. mathemati­ cians who accepted invitations to speak at the Fields Medals are awarded by the International Congress, only 19 attended. Even one of the Mathematical Union (IMU). Recipients are se­ plenary speakers was absent. Also, those U.S.S.R. lected by a committee appointed by the IMU. mathematicians attending only the Congress did The Fields Medals were established in accordance not arrive until the second day, which made a with the terms of an endowment from the estate number of last minute program changes neces­ of Professor J. C. Fields of the University of sary. Those speakers not attending generally sent Toronto. Professor Fields wished for the awards lecture notes which were read by those present. to recognize both existing work and the promise But there was a feeling of sadness for the lack of of future achievement. Consistent with this wish, free interchange. it was agreed to restrict the medals to mathemati­ The General Assembly of the International cians not over forty years of age. The first two Mathematical Union, whose primary function is medals were presented at the 1936 International the sponsoring of the International Congresses, Congress of Mathematicians in Oslo and have was held July 31-August 2 in Oakland, California. been awarded at all subsequent Congresses. Delegates from 54 nations attended and the spirit Simon Kirwan Donaldson was born on Au­ was one of warm international cooperation. gust 20, 1957, in Cambridge, England. He Chile, Malaysia, the Ivory Coast, and the received his B.A. degree from Pembroke College, People's Republic of China (in a special compro­ Cambridge, in 1979 and his Ph.D. from Worces­ mise with China-Taiwan) have entered the Union ter College, Oxford, in 1983. He was a visiting since the last Congress. member for the Institute of Advanced Study at

736 Princeton University from 1983-1984 and was a Nevanlinna Prize University during the visiting scholar at Harvard The University of Helsinki granted funds to a professor at spring of 1985. He is currently award a Nevanlinna Prize at international con­ Oxford University, England. gresses. The Nevanlinna Prize, which recognizes Gerd Faltings was born on July 28, 1954, the achievements of young mathematicians for in Gelsenkirchen-Buer, West Germany. He re­ work in the mathematical aspects of information ceived his Ph.D. from Universitiit Munster in science, was first presented at the International 1978 and was a visiting research fellow at Harvard Congress of Mathematicians in Warsaw in 1983. and University from 1978-1979. Between 1979 Leslie Valiant, a British citizen, was born on at Universitiit Wuppertal. 1982 he was a professor March 28, 1949, in Budapest, Hungary. He re­ the Danny Heineman In 1983, Faltings received ceived his B.A. degree in Mathematics from King's der Wissenschaften, Prize from the Akademie College, Cambridge, in 1970. From 1970 to 1971, He has been teaching Gottingen, West Germany. he studied at Imperial College, London, where at Princeton University since 1985. he received a diploma in Computing Science, and Michael H. Freedman was born on April 21, went on to receive his Ph.D. from Warwick Uni­ 1951, in Los Angeles, California. He received versity in 1973. Valiant was a visiting assistant his Ph.D. from Princeton University in 1973. professor in the Computer Science Department Freedman has delivered a number of addresses at Carnegie-Mellon University from 1973-1974, for special sessions of the AMS on topics such as lecturer at the Center for Computer Studies at algebraic and geometric topology, Dehn's lemma, Leeds University from 1974-1976, and lecturer Bing topology, infinite procedures, Poin~are's. con­ and reader in the Computer Science Department jecture in dimension four, and four-dimens~on:U at Edinburgh University from 1977-1982. In 1985, manifolds. He was awarded a Sloan Fellowship m Valiant received a 1985-1986 Guggenheim Fellow­ 1980. In 1984, he was named California Scien­ ship. He is currently Gordon McKay Professor of tist of the Year and was elected to the National Computer Science and Applied Mathematics at Academy of Sciences. He received a MacArthur Harvard University. Fellowship in 1985 and the Oswald Veblen Prize in Geometry in 1986. He is currently professor of mathematics at the University of California at San Diego.

737 The University Research Initiative

The Department of Defense (DOD) has estab­ The DOD channeled much of the responsi­ lished a new program called the University bility for the implementation of the URI through Research Initiative (URI) which is designed to four agencies: the Air Force Office of Scientific strengthen the ability of universities to conduct Research (AFOSR), the Army Research Office research and educate graduate and undergraduate (ARO), the Office of Naval Research (ONR), students in areas important to national defense. and the Defense Advanced Research Projects A supplement to existing DOD research funding, Agency (DARPA). Each agency formulated a list the URI has a budget of $110 million for fiscal of specific research areas, called block research year 1986 and will provide for grants, fellowships, programs, and solicited proposals for research in and assistantships for training graduate and un­ these areas. To review the proposals, the agencies dergraduate students; exchanges of personnel and set up evaluation panels for each block research information with other research organizations, program. Each panel was chaired by an offi­ particularly DOD laboratories; and the building cer from the agency who was familiar with the or acquisition of research instrumentation. research area the panel addressed and who was The idea for the URI grew out of the DOD­ ultimately responsible for the recommendations University Forum, a group of DOD officials and and management of that block research program. university faculty that was established to discuss The rest of each panel consisted of representa­ the relationship of the DOD with universities. tives from naval laboratories, the Army, the Air In 1983, a report from the forum recommending Force, DARPA, the National Science Foundation, that the DOD increase its support of university and industrial research organizations. Academic research reached George Keyworth II, then Sci­ representatives could also be included if they had ence Advisor to President Reagan. The report no conflict of interest. prompted Keyworth and Tony Battista, from the Proposals were graded on five criteria: their staff of the House Armed Services Committee, to overall scientific and technical merit; their po­ formulate the idea of the URI and to begin dis­ tential contribution to the defense mission; the cussing its possibilities with the DOD, Congress, qualifications of the investigators; the university's and government agencies. Originally, the budget commitment to the proposed program; and the for the URI was to be $25 million for fiscal year effectiveness of the proposed plan to manage the 1986, but Keyworth and others pressed for $200 research. After reviewing the proposals, the panel million and the URI ended up with $110 million. selected the best applicants and made site vis­ Since then, funding for the URI for fiscal year its to assess the facilities, research equipment, 1987 has been uncertain: the Defense Subcom­ potential for quality graduate education, and will­ mittee of the House Appropriations Committee ingness of the university to support the program. recently recommended cutting the entire URI The panels then presented their recommendations budget for fiscal 1987. The full Appropriations to a Program Selection Board, which passed its Committee subsequently rejected the recommen­ selections on to Ronald Kerber, the Pentagon's dation and restored the full budget of $50 million deputy undersecretary for research and advanced for the URI programs. The legislation is pending technology and the head of the URI, who made action by the House, and if passed will be sent to the final selections. the Senate. One purpose of the URI is to bring together The agencies solicited proposals in January researchers from several disciplines to work on a 1986 and, after receiving a total of 963 propos­ single project. This interdisciplinary approach is als from 175 universities, awarded funds for 86 intended to stimulate the growth of new technolo­ block research programs in 70 universities. In gies based on more than one traditional academic mathematics, the agencies received 120 proposals discipline. By combining scientific and engineer­ of which they funded 9. Some of the grants will ing talents in this way, the DOD hopes to smooth be shared among several institutions, and some the transition from research discovery to practical institutions will conduct more than one program. application. In addition, the DOD intends the Funding for fiscal year 1986 ranged from $170,000 URI to provide more opportunities for contacts to $3 million. Most of the programs run for between universities, industry and DOD laborato­ 5 years, but the funds a program receives may ries to maximize both the military and economic vary from year to year, for they are subject to security derived from such research. congressional funding decisions. Some programs

738 will be more heavily funded during their first year framework for computational combinatorics. The in order to purchase necessary equipment. research will be in the areas of algorithmic graph theory, computational geometry, mathematical aspects of VLSI (Very Large Scale Integration), ONR Programs computational algebra, matroid theory, polyhe­ The Department of Mathematics at the Univer­ dral methods, foundations of heuristics, and com­ putational sity of California, Los Angeles received a 5-year dynamic programming. The grant will provide grant of $3,613,315 for research in a diverse array for visiting faculty and 10 graduate re­ of topics in applied and computational mathemat­ search assistantships. There are 10 investigators ics with Paul Roberts as principal investigator. from the departments of industrial engineering, computer Some of the specific topics to be addressed are science, and mathematics and from the Management numerical analysis; fluid mechanics, including free School at Purdue: S. Abhyankar, M. boundary problems; nonlinear waves; boundary J. Atallah, V. Chandru, C. R. Coullard, G. N. layer analysis of nonlinear diffusion equations; Fredrickson, S. E. Hambrush, R. L. Rardin, D. K. buckling analysis; vorticity methods; and shock Wagner, and R. T. Wong. calculations in fluid dynamics. The group will also conduct research in analytical modeling, in­ ARO Program cluding the structure and stability of solitons in dimensions greater than 1, and in analytical Sanjoy Mitter, head of the Center for the Study of methods for partial differential equations with Intelligent Control Systems at the Massachusetts small parameters. The program includes plans Institute of Technology (MIT), is the principal for research assistantships, fellowships, postdocs, investigator for a consortium program to be con­ and visitors. The co-principal investigators are: ducted through the center involving researchers B. Engquist, C. Lange, and S. Osher. from MIT, Harvard, and Brown Universities. Us­ The Department of Mathematical Sciences ing a theoretical, mathematical approach, the at Clemson University received $3,216,259 over 5 consortium plans to blend the analytical and years to conduct basic research in discrete math­ theoretical insights of control theory and signal ematics and computational analysis. William processing with problems in artificial intelligence. R. Verry is named as the principal investigator Research topics will include information process­ and is primarily involved with administration of ing corresponding to sensory modalities, such as the project. Charles Johnson is the coordinator vision, and the development of algorithms for par­ for the discrete mathematics portion of the pro­ allel and distributed computing. Having a strong gram which will focus on combinatorial matrix interdisciplinary flavor, the research will utilize analysis, network reliability, and the theory of analysis, geometry and numerical analysis, as well linear algorithms. Dan Warner is the coordinator as probability theory, statistics, computer science, for the computational analysis portion, concerned and control theory. The program will also provide with hierarchical systems, large scale distributed for graduate student and postdoctoral support. control systems and parameter estimation and dy­ The co-principal investigators are: D. Bertsekas, namical systems. Plans for the program include an R. Brocket, W. Fleming, R. Gallagher, S. Geman, experimental FPS 16-node hypercube computer. U. Grenander, Y. H. Ho, P. Huber, H. Kussner, T. During the first year of the program several of Leighton, D. McClure, D. Mumford, G. Strang, the researchers will familiarize themselves with and L. Valiant. the computer and write or obtain software for it in order to make it accessible to the entire group. A major feature of the program is educational as AFOSR Programs it provides 43 years of combined graduate student Alan C. Newell of the University of Arizona is support. In addition, the program includes plans the principal investigator for an interdisciplinary for 1-2 postdocs, visiting faculty, and exchanges research program concerned with mathematical with naval research laboratories. modeling of nonlinear phenomena with appli­ Thomas L. Morin of the Department of Indus­ cation to optics and fluids, utilizing analysis, trial Engineering at Purdue University is the prin­ differential equations, numerical analysis, algebra, cipal investigator for a program entitled "Compu­ and topology. The program will receive $2.75- tational Combinatorics" with a 5-year budget of 3 million over five years and will be conducted $4,246,350. The program will develop the mathe­ jointly by the Applied Mathematics Program and matical theory underlying computational aspects the Department of Mathematics at the University of combinatorics, a field with many applications of Arizona. The program provides for 6 graduate from airline scheduling to logistical problems in fellowships and educational workshops to provide the military. The impetus for this program comes a quality learning environment. In addition, from today's technology, which is producing sys­ funds will be used for research workshops and tems of ever increasing complexity that need to be conferences and to support 6-8 visitors per year. controlled, and from breakthroughs in computa­ The co-principal investigators are: G. Baker, H. tional complexity theory, which provide a unifying Flaschka, C. Jones, D. McLaughlin, J. Moloney,

739 C. Neuman, T. Ratiu, M. Wojtkowski, and L. S. problems in fluid dynamics. The research areas Young. will be: numerical simulations for the prediction The Department of Mathematics at New of large scale flow patterns in the ocean; experi­ York University received $390,000 for a program ments and numerical calculations for bluff bodies with Russel Caflisch as the principal investigator. (e.g., wings and ship bodies); fluid dynamics in The program will receive URI funds for one concurrent processing; turbulence at the edge year and it will probably continue after that of rapidly rotating disks (as in computer disk with funds from other sources. The research storage); statistical dynamics of surface waves in will apply the methods of partial differential the ocean; and the nature of the layer between equations (especially nonlinear systems), physical laminar and turbulent flow in combustion. The modeling, and scientific computing to problems grant will provide for 15 graduate fellowships, in nonlinear optics, dynamics of fluid interfaces 5 postdocs, and 2 visiting researcher positions. (including vortex dynamics), composite materials, The co-principal investigators are: H. Aref, R. D. and random media. The grant will provide for Blevins, G. F. Carnevale, A. Ellis, M. Freedman, 2 postdocs, 2-4 fellowships, and several visitors. M. Gharid, P. Libby, K. Lindenberg, J. Miles, M. The co-principal investigators are: C. Flytzanis, Rotenberg, F. Talke, G. Vallis, C. Van Atta, and G. W. Milton, G. C. Papanicolaou, C. Sulem, P. K. Watson. Sulem, and L. Ting. Steven Orszag of Princeton University is the principal investigator in a program entitled "The DARPA Programs Mathematical Theory of Complex Flows in Com­ plex Geometries." The program will be concerned The Applied Mathematical Sciences Program with the prediction and control of turbulent flows, (AMSP) at DARPA will receive a projected $31 with engineering applications. The research will million of URI funds for three 5-year programs focus on the development of mathematical and all concentrating on turbulent flow in fluid dy­ computational tools for mathematical models of namics. While each program has its own focus, problems in transitional and turbulent flows, and the AMSP made an effort to select programs will utilize renormalization group methods for tur­ that would complement each other and plans to bulence, nonlinear stability methods, and spectral coordinate the research to reduce duplication of methods. In addition, about one quarter of the effort and to maximize the gains. The main project will be concerned with the development goal of the DARPA-URI programs is to control of a desktop supercomputer with the computing and predict turbulent flow in fluid dynamics by power of a Cray 2. Funds are provided for several four means: developing new mathematical models graduate fellowships, postdocs, and visitors. The and applying existing models in dynamical sys­ co-principal investigators are: R. Dashen, J. Gol­ tems; performing experiments to verify models; lub, A. Jameson, R. Kraichnan, M. Kruskal, A. developing algorithms to solve nonlinear equa­ Majda, and D. Nosenchuck. tions for describing fluid flow; and simulating Lawrence Sirovich of the Department of three-dimensional flows over complex geometries. Mathematics at Brown University is the principal The expected payoffs include: improving the de­ investigator in a program addressing problems in sign of military vehicles; developing algorithms turbulence. The research will focus on dynamical and software to calculate and simulate flow over systems, chaos theory, and fluid mechanics, and the entire aircraft through different flight regimes; is likely to find a variety of applications, such as and continuing improvement in the Navier-Stokes in aerodynamics, meteorology, and the chemical Solver computer. industry, to name a few. Five postdocs and Henry D. Abarbanel of the University of 15 graduate fellowships will be awarded. The California at San Diego (UCSD) is the princi­ co-principal investigators are: F. Bisshopt, D. pal investigator in a program to be conducted Gottlieb, D.-Y. Hsieh, S. Karlsson, H. Kocak, J. through UCSD's Institute for Nonlinear Science. Liu, J. Mallet-Paret, M. Maxey, K. Sreenvasin, J. This interdisciplinary effort will bring together T. Stuart, and C.-H. Su. investigators from physics, engineering, oceanog­ raphy, chemistry, and mathematics to work on Allyn Jackson AMS, Staff Writer

740 Mathematical Text Processing RichardS. Palais

Introduction complete descripton of a font consists of two related data structures: the font metrics, which This month, as I finally write my long delayed give the dimensions of the boxes into which the review of Macintosh technical word processing symbols fit (as well as some more esoteric things software, I am happy to report that the IBM-PC called ligatures, kerning, and italic corrections), will have its day again soon in this column. The and the actual geometric description of the shapes Boston Computer Society's IBM-PC Technical of the symbols that go in these boxes. We will Working Group has agreed to undertake a reprise consider these one at a time. of their article published in the January 1986 Notices. Their new report, which should appear just a year after the original, will be more Font Metrics discursive and evaluative than the previous one. If you look closely at text typeset in some partic­ In addition to updating their review of TWP ular font you will notice that it consists of rows of software packages for IBM compatibles, they characters that seem to rest on imaginary lines, plan a discussion of the relative merits of using called the baselines of the rows. For the most part WYSIWYG and markup languages on that family the lower case letters extend above the baseline of machines. by a fixed height, called the x-height of the font Before taking up the Macintosh, I want to in question. But there are some exceptions to respond to some reader comments and requests. these generalities: some lower case characters, First, responding to numerous requests for further namely "g," "j," "p," "q," and "y," have pieces technical material on TWP, and in view of the called descenders that extend below the baseline, central role that fonts play, I have tried to give and others, namely "b," "d," "f," "h," "k," "1," a reasonably detailed discussion of that complex and "t," have ascenders that extend above the and interesting concept. Secondly, since it is clear x-height line. The height of the highest ascender from a number of letters I have received that there is called the ascent or height of the font (or some­ are many confusions and misunderstandings about times its cap-height since, in so-called Modern the nature and purpose of 'JEX, I have added some fonts at least, it is also the height of the capitals). further comments about 'lEX in the hope that We shall also take this to be the height of the top it will help clarify matters. And finally, I have of a character box above the baseline. The depth received so many comments (and testimonials) below the baseline of the lowest ascender is called about the question of just what is "the best" the descent of the font. Note that the x-height, way to use a word processor to communicate ascent, and descent are global parameters of a mathematics that I have had to give up my font. original idea of quoting excerpts from readers' In so-called mono-spaced or "typewriter" letters, and instead I will try to summarize as fonts the width of a character box is also global best and as honestly as I can the various points to the font, while in proportionally spaced fonts of view. it is local to each character. Needless to say, it makes it much easier to write a formatting Fonts program if one has only to deal with mono-spaced fonts, and this is a simplification that almost all Fonts are the foundation of word processing; the IBM-PC WYSIWYG formatters take advantage purpose of all WP software is, roughly speaking, of. However, proportionally spaced fonts are more to take symbols from various fonts and to arrange legible, and for this reason the ability to handle them appropriately into lines, paragraphs, and proportionally spaced fonts was built into the pages. However, the concept of a font is so Macintosh ROMs. complex and multifaceted, involving as it does The sum of the ascent and descent of a questions of aesthetics, psychology, history, and font is called its "body size" ; the nominal point technology, that I can only give a brief intro­ size of a font is a somewhat arbitrary dimension duction here and must refer the reader who is with no precise definition, but usually it closely interested in more detail to the excellent writ­ approximates this body size. By the way, since ten version of Don Knuth's 1979 Gibb's lecture we will be using the printers' units of points "Mathematical Typography" (Bull. Amer. Math. and picas, you should know that one inch is Soc. (New Series) 1, March 1979) and the many seventy-two points or six picas. references given there. When a type formatting program makes a In this age of computer WP and typesetting, row of character boxes (i.e., a line of type) it a font is usually understood to mean a set of up aligns their baselines and butts the left edge of to 256 related "symbols" or lettershapes. The one box up against the right edge of the preceding

741 one. Note this means that the intraword spacing CHARACTER BOX is fixed by the font; in fact, normally a character fits flush left in its box so that the space between cap-height it and any next letter in a word is the space Ill ...... N between the rightmost point of the character and ... u ... the right edge of its box. Actually, characters do -,:::: Ill x-height ;::! ... >- not necessarily fit inside their associated boxes! d'O '0 ""'d baseline In particular, characters in italic or slanted fonts Ill Ill 0 0 '0 ...... Q will frequently lean outside the right-hand edge of I descent their boxes. 1:p.h When a typesetting program arranges lines of type into a page, the distance between a pair of 'W'idth adjacent baselines is called leading. Text is easier to read if the leading is very nearly uniform and, FIGURE 1 since it also looks better, an effort is usually made to insure uniform leading. The most primitive way to arrange this is to simply butt the top of one row of character boxes against the bottom of the preceding row. This gives what I shall call the default leading of the font. Notice that g if the bottom edges of character boxes in a font were taken to coincide with the descent line, then the default leading would equal the body size FIGURE 2 and descenders would touch the tops of capitals and ascenders just below them, giving a crowded Figure 3 is a slanted version of the same look. In fact, traditionally leading is taken about character. Notice how it slops a little outside its twenty percent greater than body size, "ten on box! twelve" (i.e., ten point body size with twelve points of leading) or "twelve on fourteen" being two popular choices. For this reason, the bottom of character boxes (the depth) is usually taken one or two points below the descent line in text fonts. However, mathematical symbol fonts are g usually "set solid" (i.e., with leading equal to body size) so that it will be possible, for example, to make large brackets by fitting together pieces FIGURE 3 on adjacent lines. Since all the characters in a slanted font are You may wonder what happens if you mix slanted at the same angle, intraword spacing is several fonts on the same line. The obvious answer not affected; the difficulty comes in the spacing is correct: the top or bottom of the line is given between words where there is a change from by the maximum top or bottom for any font in slanted to nonslanted text. In fonts designed for the line. Similarly, superscripts in a line will move sophisticated typesetters, the font metric file will the top of the line up and subscripts will move the contain so-called italic corrections that tell the bottom down. This means that in mathematical system how much extra interword space to add to typesetting, where font mixing, superscripts, and overcome this problem. In a WYSIWYG system subscripts abound, default leading would lead to the italic correction must be added by hand by uneven vertical spacing that is hard to read, and the user. any good TWP program must use some vertical The algorithm mentioned above for determin­ spacing algorithm more sophisticated than default ing the spacing between pairs of adjacent letters leading. in a word is clearly rather crude; it depends only This may all seem somewhat abstract, so here on the first letter of a pair, and if you have are a few pictures which perhaps will illustrate had any calligraphic experience you know that and clarify the above concepts. First a diagram for good looking text you must also take the of a character box; the point 0 is called the origin shape of the following letter into account. The or reference point and plays a role in the more process of adding a positive or negative correction sophisticated systems like JEX. (See Figure 1.) to the spacing between adjacent letters to make the spacing look more even is called kerning. For Figure 2 is a picture of a particular character example, the word "Type" looks better if the "y" in a particular font (the author's approximation is kerned left under the "T." Most WYSIWYG of a lower case "g" in the Times Roman font). systems consider this correction not worth the We show it alone and in its box. effort, but again the top of the line typesetters

742 will put kerning information in their font metric a quarter inches. Set your left margin at one files for every pair of characters that need them. and a half inches and your right margin at seven And what are ligatures? Look carefully at inches. As for interline spacing, for text with a a fine book and you will notice that certain se­ lot of formulas interspersed I would recommend quences of characters are replaced by a special "double spacing," i.e., about twenty to twenty­ alternate character. These are the pairs "fi," four points of leading. (With reasonable top and "ff" "fl " the triples "ffi" and "ffl " and some­ bottom margins this will give about twenty-seven ti~es the diphthongs "ae" and "oe.'; The quality lines per page.) This may seem like too much typesetters like 'lEX will have such ligature infor­ white space, and I'll agree that one and one­ mation built into their font metric files and take half spacing (fifteen to eighteen points of leading) care of them automatically, while the WYSIWYG looks better (it's what I use for letters) but at that systems almost universally ignore ligatures. spacing a subscript "j" may overlap a superscript Kerning and ligatures may seem like precious "l" below it. A final formatting choice is whether fine points, and clearly whether they are worth to have fully justified text or to use "ragged right" the trouble depends both on the application margins. This is mainly a matter of taste: ragged and on personal taste. They look out of place right tends to be slightly more readable because in an informal memorandum, but on the other of the evenness of the interword spacing, but hand their lack in an expensive new edition of many people prefer the neater appearance of full the complete works of Shakespeare would be justification. unthinkable. Here are a few examples in large type. First "Type" both with and without kerning in Figure 4 (I added extra kerning to what 'lEX Lettershapes gave automatically to get what my eye told me So far we've said very little about the detailed was ideal): lettershapes in a font. In fact, the qualities that make for a beautiful font are remarkably subtle and have been learned painstakingly over Type Type hundreds of years. Also, with each change in the technology of printing, corresponding changes FIGURE 4 must be made in the design of old fonts and whole new families of fonts must be designed to maintain and in Figure 5 are the words "fit," "affine," and an optimal appearance for the text produced by "affluent," both with and without ligatures: the new methods. These are interesting questions but their general consideration would take us too far afield, so I will just say a little about the fit fit problems raised by the recent change from the older cold and hot metal type technologies to the current digital or raster technology of computer typesetting. affine affine The first thing to realize is that the angular resolution of the human eye is approximately one minute of arc. Stated in terms of spatial frequency affluent affluent this means that, at a reading distance of one foot, FIGURE 5 the eye will resolve 250 lines per inch, but will see lines closer together than this as a gray blur. How is all this theory related to the practical It then follows from the Nyquist theorem that to choices to be made in word processing? avoid "aliasing" or "jaggies" in a rasterized image If you use a WYSIWYG word processor, (e.g., the staircase appearance of a diagonal line) a decision you will often have to make is how the spatial sampling frequency must be on the wide to choose your column width and how much order of 500 per inch, and this is indeed the lower leading to use. The main criterion for this choice limit of what is considered acceptable photo­ should of course be the ease and speed of reading typesetting. In fact, quality typesetters have a the resulting text. A lot of experimentation has resolution of at least 1000 lines per inch. This is gone into this question and the general wisdom because of what I call the kite string effect: the is that the width of two copies of the lower case eye can see objects far smaller than it can resolve. alphabet of your principal text font is close to A kite string at one-hundred yards subtends less ideal for a single column of text. In a twelve than one-tenth of a minute of arc, yet is clearly point type this would give you a column width visible, and similarly a good typesetter will set of between four and one-half inches for a very quite visible hairlines far thinner than the eye's tight font to five and three-quarters inches for ability to resolve. Now, for an eight and one-half a loose one. I would recommend the choice of by eleven inch page, 500 pixels per inch translates a proportionately spaced, twelve point, serif font to a total of about twenty-four million pixels, or with a "two alphabets width" of at least five and three megabytes to store one page. And if you

743 want a video monitor which is flicker free (say 60 scans per second, noninterlaced), to show a full page at this resolution would require a processor with a clock speed of about 1400 megahertz, about two orders of magnitude faster than the chips used in current microcomputer technology. Clearly, some compromises must be made in the world of WYSIWYG! One common compromise is to show only a partial page on the screen. Another is to ignore reality and define "high" resolution for a video monitor to be about 75 pixels per inch and for a dot matrix printer as about twice that. Obviously, a great deal of aliasing is going to be inevitable at these resolutions and it takes very careful design to produce an even halfway decent FIGURE 7 screen font. (My hat is off to Charles Maurer for his incredible Boston font for the Macintosh.) in a WYSIWYG word processor, formatting a Fortunately, at the laser printer resolution of page and painting it on the screen are concurrent 300 pixels per inch it becomes possible to build processes, so both the font metrics and the bit some semblance of quality into fonts, and while map images of the fonts must be known and used one misses the crispness of a fine font at high by these programs, adding considerably to their resolution, at least eye strain is less pronounced. complexity. To get some idea of the degradation of quality Traditionally, each type formatting system at various levels of resolution, see Figure 6. I has had some unique method of its own for gener­ have placed my 12 point Times Roman lower case ating, storing, and using fonts. 'IE;X, for example, "g" as it might look if digitized at 2000 pixels has a highly sophisticated sister program called per inch and at 300 pixels per inch (LaserWriter METAFONT that is capable of creating fonts of quality), and have magnified both by a factor of remarkable beauty and flexibility. Using META­ twenty-five. FONT, numerous font experts have collaborated with Knuth over a five-year period to create a font family of great character (the Computer Modern or CM family) that, together with the "plain" format, makes 'lEX a complete system for most simple typesetting tasks. To my mind, one of the major advances of the designers of the Macintosh family of computers was the idea of breaking away from this traditional mold and defining a standard Macintosh font format, thereby making fonts into universal, shareable, system "resources" on a par with windows and menus. A user can create a font resource with a font editor and install it in the System file where it immediately becomes available to all Macintosh graphics and word pro­ FIGURE 6 cessing application programs. This means, in Figure 7 represents how it might look at the particular, that a developer of a Macintosh WP same magnification when digitized as a screen program is relieved of the considerable burden of font with a resolution of 72 pixels per inch. reinventing the font wheel and creating numerous But why make so much of this distinction fonts, saving his efforts for more creative purposes. between font metric information and the letter­ shapes of a font? Simply because it is, in fact, More on not only a crucial logical distinction, but also 'lEX one that is central to understanding how word I have received many letters saying, in effect, processing and formatting software works. All "Using 'lEX I cannot easily make commutative the real complexities of typesetting are inherent diagrams, chemical formulas, tables, ... , so 'lEX is in the font metrics and only the rather trivial clearly not such a great program after all. Why do final stage of physically painting the final typeset you keep pushing it?'' Now 'lEX is certainly not pages involves the actual lettershapes. The 'lEX above criticism, but this particular objection does, program, for example, knows absolutely nothing I believe, miss an essential point: 'lEX was not of letter shapes! Only screen previewing and "de­ designed to be a stand alone, complete typesetting vice driver" software that converts 'lEX's device system for all possible typesetting tasks. So yes, it independent output file into a bit-mapped image is difficult to do complex, specialized typesetting knows or cares what a letter looks like. Of course, tasks in vanilla 'IE;X. If you are using 'lEX only

744 to set simple text, then the "plain" TEX macro they use quite successfully on small scale personal package and the Computer Modern fonts are all computers. I have resisted reviewing these for you need to do highest quality typesetting with several reasons. Chief among them is the feeling ease. But Knuth knew well that any system that few mathematicians would now purchase that tries to do too much ends up doing nothing these obsolescent systems when so much more well, so he purposely limited the functionality powerful and easy to use systems are becoming of the "plain" package. True, this package will standardized and are rapidly dropping in price. not set chemical formulas easily. On the other But I have to admit the fact, pointed out by H. A. hand, it is very easy to add on to "plain" other Watson, that there are an enormous number of macro packages with specialized functions such Commodore 64s around and a new one can be had as setting tables or chemical formulas. It is for under $150. As Watson notes, with only one interesting to see just what Knuth's goals were. of these, a floppy disk drive, an inexpensive dot Addison-Wesley, the publishers of his monumental matrix printer (Epson, Star, or Okidata), and the five-volume work Computers and Typesetting (the WP program Fontmaster II, it is possible to create definitive specification, program listing, and user very respectable mathematical text. However, manuals for TEX and METAFONT), recently there are two important caveats. Due to limited gave a coming out party at the Boston Computer memory the system responds slowly, and without Museum. I would like to quote from Professor a lot of disk swapping it can only handle relatively Knuth's remarks at that event: small documents. Secondly, it is a markup . . . I wanted to design something that language system rather than WYSIWYG, and would still be usable in 100 years. In to use its quite powerful features would require other words my goal was to arrange a considerable investment of time to learn the things so that, if book specifications are complex command language. The program and saved now, our descendants should be excellent documentation are available for $49.95 able to produce an equivalent book in from Xetec, 2804 Arnold Road, Salina, KS 67401, the year 2086. Although I expect that (913) 827-0685. there will be continual developments of From William G. McCallum comes a plea "front ends" to TEX and METAFONT as to amateur designers of technical fonts to take well as a continual development of "back more care with Greek letters. I agree. For ends" or device drivers that operate on example, the lower case Greek letters in most the output of the system, I designed TEX of the Macintosh technical fonts are extremely and METAFONT themselves to serve as sloppily drawn, probably with no study at all of useful fixed points in the middle, solid what they should actually look like. The popular enough to build on and to rely on. Princeton font is one example of this, and even the Boston font, which for such low resolution has When Knuth refers to "front ends" with beautiful Roman characters, has undistinguished respect to 'IE;X, he clearly means not only prepro­ and inaccurate Greek characters. cessor software, such as "Sweet-'IE;X," which feed But in addition to the constant their output into 'IE;X, but also macro pack­ stream of TEX interesting miscellaneous reader comments ages such as U\'IE;X, AMS-'IE;X, and many others I have become used to, that now exist and that were designed precisely I have also been receiving a surprising flood of mail and even to ease the input of special typographic entities phone calls like tables or chemical formulas. on a question that seems to engender passionate feelings and opinions: How can a The implementations of for the Macin­ mathematician TEX best use a word processor tosh (reviewed in detail below) are a particularly to help in his or her research? I have been reading nice example of what Knuth is talking about. and pondering the TEX many opinions expressed itself is one subroutine inside a standard Mac­ on this point and would like to try to summarize them and also intosh main event loop, others being an editor share some of my own conclusions. to prepare the TEX input, a graphics module, a In fact, I am beginning to feel that perhaps the most screen previewer, and a PostScript postprocessor. lasting benefit this column might have Using MacPaint or MacDraw as a "front end," one is to help mathematicians realize that aside from the obvious can create a complex diagram very quickly and use of TWP, to report the final fruits of easily and then, with Mac'IE;X's built in graphics research, there is a different use, and facility, instantly place the resulting "picture" in potentially an even more important one, to which a box anywhere in the TEX input. it can be put. This is to replace, at least partially, the pencil and pad of paper, the very tools that have come to symbolize Readers' Comments the creative mathematical process. Many readers continue to write to me with mis­ cellaneous comments and I would like to include a couple of these which I feel are of general interest. Using a TWP to Write Mathematics As I have said before, a lot of mathematicians Several months ago I would have scoffed at the have written in telling me about WP systems above sentence. Indeed, in an earlier column I

745 stated my view that most mathematicians would and gradually adding the lemmas they need to be ill-advised to sit down with a word proces­ prove them while the "bottom up" people will sor without having first completed a reasonable still prove lots of lemmas first and gradually build handwritten first draft of a paper. Although I them into ever more general theorems. But this was writing what I then felt was an obvious truth, is just conjecture on my part. What is clear if I had said it merely to stimulate a debate no from many of the letters is that the writers all other statement I made could have done the job feel they have made a significant change in the better. Almost immediately I began to get letters method by which they carry on their research, strongly disagreeing with my position and, in fact, that this method is a definite improvement, and often taking a diametrically opposite view. that they would not consider going back to pencil It appears that many mathematicians, as and paper. soon as they have an idea, instead of writing it Two final points that were made by several into a notebook in the traditional way, place it of those who wrote me are that, to use a technical electronically either in a fresh file {if it starts a word processor in this way, it is important to be new circle of ideas) or add it to a file of related able to type rapidly and it is essential to have a ideas, or to a journal file. All sorts of things go good WYSIWYG system. into such files: conjectures, new definitions, equa­ It seems clear then that, for good and suffi­ tions, diagrams, examples and counter-examples, cient reasons, many authors are going to choose lemmas, theorems, and sketches of proofs. When a WYSIWYG system for their basic word pro­ there are enough related ideas in these files, they cessor. On the other hand, there is mounting are reorganized by electronic cut and paste (per­ evidence that 'lEX will become a standard for haps with the help of an outline processor) into the preparation of final camera copy for journal the first draft of a paper. From then on details are publication. The challenge to bridge that narrow fleshed out, the introduction is written, the expo­ but expensive gap between the WYSIWYG ver­ sition is improved, copies are exchanged between sion of a paper and its 'lEX input version is an co-authors, until, at last, the paper is complete. important and interesting problem that cries out "But wait!" you say. "Except for the way for an automated solution. the information is stored, isn't this just the usual way papers have always been written?" In some deep sense the answer must, of course, be yes, but TWP on the Macintosh those writing to me keep stressing some important The current "standard" Macintosh hardware con­ differences: figuration consists of the Macintosh Plus {with • The only new keystrokes needed are those the new 128K ROM, one megabyte of RAM, and that add new material or revise old material. an 800K internal disk drive), an ImageWriter The practice of literally rewriting old mate­ II printer, and an 800K external drive. If this rial, just to keep the copy legible, is gone. describes your system, then you have more than This saves time and energy for more creative sufficient capacity to easily handle any of the activity. software that I will be discussing and you can skip • Making revisions is so easy that it is a the rest of this section. continual process. Every time a change But what if, like the majority of Mac owners, seems appropriate, even a small one, it is you have the old standard configuration, a 512K made immediately, rather than being ignored, RAM "Fat Mac" with the original 64K ROM, or saved up (perhaps to be forgotten) for the an ImageWriter, and 400K internal and external next "draft." drives? How far down the upgrade path is it • If at any stage it seems best to make no­ necessary to go? tational changes, what used to be a task A lot depends on how you plan to be using so difficult and error prone that it was of­ your Macintosh. For most WP applications, ten foregone is now easy and straightforward except 'lEX, the old standard is already sufficient, using "search and replace." and even a stripped down version of 'lEX will run, The important point seems to be the extra albeit slowly, with this configuration. However, continuity TWP gives to the creative process. A the first milestone down the upgrade road, the paper evolves and is written from its inception 128K ROM and 800K internal drive, provides to its completion in many small steps, better so much extra capacity and functionality and at reflecting the way the writer actually carries out such a low price, currently under $250, that I research, rather than in a sequence of discrete, would strongly recommend it to all serious users. separate, and major rewrites that in fact interrupt Already some Mac software is being packaged on the process of research. BOOK disks and soon, no doubt, some will not run My guess is that the fundamental differences dependably with the old ROM. in the way people think, do, and write research The next step, adding additional RAM, in­ are still there. The pure "top down" people will volves more options. If you use Switcher or plan to use an outline processor to sketch out a whole use 'lEX or other large programs, this is definitely paper early on, conjecturing general theorems, a desirable improvement, but here the Apple one

746 megabyte Mac Plus board does not seem to be Having said all this, it might seem that the the best bargain. There are many thoroughly two MacTEX's must be as alike as two peas tested and dependable third party upgrades that from the same pod. But in fact K&S and FTL provide two megabytes for only little more than clearly had different user bases in mind when Apple's one megabyte. I have been completely they created their implementations and this is satisfied with my own choice, the Levco Monster reflected in user interfaces which have a very Mac, but I believe many others are equally good. different feel. K&S created a version of TEX for By the way, that extra megabyte is extremely use­ a fairly naive user who wants to be spared the ful. Besides giving you extra Switcher partitions innards of TEX and wants to get his output with you can use it to run software quietly and with the least fuss possible. If you are a Macintosh spectacular improvements in speed from a large user familiar with the standard way Mac programs RAM disk. (That is how I run TEX.) work and have had a little experience using TEX The final step you might want to consider on a mainframe, then you can slip two disks in is adding more external storage. Here again the drives of a Mac Plus and get going without there are several options. The simplest and ever looking at a manual. Writing the TEX least expensive is to replace your 400K external source, running it through 'lEX, previewing it on drive with the new 800K drive. This currently the screen, printing it on the Image Writer (or costs about $250 and, for most users, is probably on a LaserWriter if one is attached) are all so sufficient. But for about 50% more you can now completely straightforward and natural that it get 2Mbyte external RAM disks, for example the took me some time to appreciate the remarkable DASCH, which, besides their extra capacity, have planning that made this simplicity possible. Aside considerable speed advantages. Finally, for about from the File and Edit menus, there are only three $700 you can buy a 20 Mbyte SCSI hard disk. others: a Typeset menu (you pull it down and The DataFrame 20 has an excellent reputation. click Typeset when you are ready); a View menu that allows you to choose the page of a typeset document that you want to preview and the Mac'f.EX and Mac'f.EX magnification you want to see it at (including one magnification that fits the whole page in the The competition between PersonalTEX and preview window); and a Windows menu that lets PCTEX for the IBM-PC TEX market has not you switch easily between only meant lower prices. It has also been a spur several different TEX source files and to continued improvements in both implementa­ their preview windows. Unless you are doing something tions. Now it seems that the Macintosh world very fancy, the only point which needs a little explanation is how to include will have the same good fortune. Two implemen­ graphics. This is tations of TEX for the Mac have been announced. handled in a very neat, Mac-ish way . . . each TEX One created by Kellerman & Smith of Portland, source file has associated to it a Scrapbook. Using the Oregon, is being marketed by Addison-Wesley for Clipboard as usual, you can put any MacPaint $495. The other is from FTL Systems Inc. of 234 picture or MacDraw diagram into this Scrapbook Eglinton Ave., Toronto, Ontario, Canada M4P and give it a name, for example MyPicture. 1K5, and is priced at $795. Since both have Then, essentially, all you have to do is place the macro chosen MacTEX as their trademark, I will refer \picture{ My Picture} where you want it to appear. to them as K&S TEX and FTL TEX to avoid confusion. FTL 'lEX, on the other hand, is very much a Both versions are full implementations of TEX hacker's implementation, aimed for example TEX, of course. In particular, they have passed at a '!EXpert working for a publishing house. It the suite of tests Knuth has devised for systems has three more menus than K&S, and the menus that want to call themselves TEX. In fact, as far are more complex, with more options. It is not as functionality goes, both systems have taken ad­ the kind of program to use without a manual, vantage of the Macintosh environment to go well and fortunately the manual is written quite well. beyond TEX in rather similar ways. Both have a (The manual is not, however, without its faults. standard Macintosh editing environment built in On page 4, it starts out by saying that to run for the creation and editing of TEX source. Both the program, you will need a Macintosh Plus, have screen previewers and have provisions for a 20Mbyte hard disk, a LaserWriter Plus, and preloading macros and for interspersing MacPaint an ImageWriter with Thunderscanner. This is and MacDraw graphics with text. Both support simply wrong! Two pages later you are informed the Image Writer for proofing, but in addition they that you can, in fact, fit a minimal version of support the LaserWriter fonts, have AppleTalk FTL TEX on a single 800K disk.) All kinds drivers, and permit inclusion of PostScript com­ of things are a little or a lot harder with the mands along with TEX source, so that final camera FTL version. You cannot, for example, start copy can be output not only on the LaserWriter, it up by double clicking on a document it has but also on much higher resolution PostScript created. Selecting Print from the File menu when phototypesetters such as the Linotronic 100 or the Laser Writer is chosen leads to nasty surprises. 300. The top margin is off the top of the LaserWriter

747 page. It seems that 1E;X and the Laser Writer have process of memonzmg one more arbitrary different ideas about where the origin of a page is command syntax. and some special magical incantation not in the • Most computer operating systems only per­ manual is necessary to make the printing come mit a program to act on the output of a out right-not exactly user friendly. Worse still previously run program (the UNIX pipes is the complex method used for getting graphics mechanism is a powerful formalization of this into a 1E;X file which essentially forces the user concept). But, in the Macintosh system, at to do some PostScript programming. Something least two mechanisms allow well-constructed else that I feel is a defect in the FTL version programs to operate as co-processes on the is the primacy given to the LaserWriter fonts same data, interacting across the standard over the Metafont created fonts that usually go Clipboard and Scrapbook interfaces. These, along with TE;X. Not only are the LaserWriter of course, are Switcher and Desk Accessories. fonts the default for the Plain macro package, but • Macintosh fonts are highly flexible, editable, one cannot even use the screen previewer with shared resources. As we shall see, the 'l.]:y'C's native fonts, and those supplied are the ability to create and modify fonts at will is old, preliminary AM series of fonts rather than a powerful tool for overcoming limitations of the much improved CM family. But one should word processing software! not lose sight of the fact that this is a very solid • There are several excellent macro processors program. Those faults it has can be easily fixed, for the Macintosh, MacTracks and Tempo and no doubt will be in response to its very strong being two of the most popular. competition. By the way, one way in which I feel By taking advantage of these features mathe­ FTL 1E;X is better than K&S is in its handling of maticians have been able to collect many different TEX error messages. pieces of software "off the shelf" and fashion them I did a comparative timing test for the two into a tool kit to create their own WYSIWYG 'IE;X's. The test document was four pages of fairly TWP environment. This makes it difficult to dense mathematics and I ran it on a Mac Plus review software in the traditional way. Each with an external BOOK disk drive and output to program must be considered not on its own, but a LaserWriter Plus. K&S 1E;X took one minute on the basis of how it fits in with oth~r software­ and twenty-five seconds to do the typesetting as usually software it was not explicitly designed compared with three minutes and five seconds for to work with. Also, there are many different, FTL. Previewing, with whole page in the window excellent systems that can be fashioned in this magnification, took thirty seconds for K&S versus way and I could hope to tell you, or even know, four minutes and fifteen seconds for FTL. Printing about only a few of them. In fact, many people on the Laser Writer took one minute and forty-two have written to me explaining how they overcame seconds for K&S and three minutes and thirty­ one or another limitation of the basic WYSIWYG five seconds for FTL. By the way, when I ran the word processors. There is a lot of diversity in same test at home on my Levco Monster Mac, detail, but a definite pattern or consensus seems running out of a lMbyte RAM disk, K&S took to be emerging. I have decided to try to explain only forty-nine seconds to typeset and fourteen roughly what I feel this is. seconds to preview. So let's consider what features a technical word processor should have. First, of course, it needs the basic elements of a word processor WYSIWYG TWP on the Mac designed to handle just ordinary, nontechnical text. These elements include insert and delete, WYSIWYG technical word processing on the find and replace, block copy and move, the Macintosh is a paradox. In contrast to the chaotic ability to move easily between different parts profusion of WYSIWYG word processors for the of a document, justification of text, the ability IBM-PC, many with excellent TWP features, the to set page numbers (or folios), headers and Macintosh world currently does not have even one footers, margin setting, tab setting, and the WYSIWYG technical word processor. Neverthe­ ability to change fonts and font sizes. Mac Write less, mathematicians seem to find it remarkably is reasonably satisfactory on this score and I will easy to get excellent results writing their research assume all readers are familiar with it. It does on the Mac. The explanation of this apparent lack the ability to simultaneously open several contradiction can be found in several nice features files (or several windows into the same file), of the Macintosh: and it also lacks such niceties as style sheets, • The intuitive and uniform user interface flexible control of leading, word abbreviation, mail makes it easy for even new users to learn merge, and automatic footnoting. These features, and to integrate a lot of software with almost lacking in Mac Write, do exist in the only other no effort. This contrasts with traditional sys­ Macintosh WYSIWYG word processor, Microsoft tems in which learning to use a new program, Word. They are not particularly necessary for even one almost identical to another in which the sort of work most mathematicians do; so you are expert, can be a long and tedious whether you use Mac Write or MS Word as your

748 basic word processor is not too important. Word, entering an arbitrary symbol from an arbitrary by the way, does have a couple of disadvantages font. But mathematical formulas have a rich, compared to Mac Write. It is copy protected and recursive structure and while it would be possible not completely WYSIWYG. to ignore that structure and paint formulas on This is probably the place to mention two the screen in this haphazard way, the resulting other WYSIWYG word processors for the Mac system would be very difficult and tedious to use that should be available by the time you read this. and unless used with extreme care, it would give One is called MacAuthor and is being developed ugly and uneven results. If you are taking the by Icon Technology Ltd. of Leicester, England. square root of an expression, for example, the The early versions I tested were both slow and size of the radical sign depends on the size of exceedingly buggy, but apparently MacAuthor is the subformula for the expression, and editing gradually being improved. I just received version the subformula can change that size. Similarly, 1.2 which does seem to be faster. It is also when creating a matrix or other tabular array, supposed to cure many of the bugs in earlier it is difficult to align the array elements and to versions, but I have not had time to test it have the enclosing brackets come out the correct extensively. MacAuthor has many nice extra size and in the right place. Anyone who has features, like kerning, but one in particular which tried to use Mac Write or MacDraw to create (see below) can solve a lot of the problems of complex formulas or arrays in this way knows TWP, namely nondestructive backspacing. On how difficult it is to get acceptable and uniform the other hand, MacAuthor makes the process of results. Fortunately several programmers have changing fonts and font sizes so complicated that created "equation processors." A great deal of this alone may kill it as an option for TWP. knowledge about the structure of mathematical The second is Write Now (nee Macauthor), formulas is built into them, taking much of created by Steve Jobs's new company, Next Inc., the frustration and tedium out of formatting and to be marketed by T /Maker. I have not equations. Their use is based on an intuitive had a chance to test it personally and saw stack-based discipline. For example, if you tell it first only a few days ago at the MacWorld the system you want to make a fraction, the Expo. It was being demonstrated by D. John cursor first jumps up to the numerator, and after Anderson, its programmer, who told me that the you type it in and type "Enter," the cursor next original specifications called for it also to have jumps down for the denominator. When you nondestructive backspacing, but that this feature finish this and again type "Enter," the equation was unfortunately dropped. It seems beautifully processor centers the numerator and denominator crafted, fast, and has many nice features. But and puts the correct length solidus between them. what I think may prove most useful for TWP is Similarly, if you ask for an integral, the integral that in Write Now an imported graphic is treated sign is drawn and then the cursor jumps first to as just another character box, so a formula created the lower limit. When that is finished it jumps by an equation processor can be inserted in-line, to the upper limit, and finally to the integrand. rather than having to be displayed. Of course, one of the limits of the integral might But now let's concentrate on the "technical" be a fraction, the denominator of which could be in technical word processing. What characterizes another fraction, etc. One quickly gets the hang mathematical or technical text, as distinguished of using an equation processor. Creating formulas from ordinary text, is that it contains lots of becomes easy and almost enjoyable, and it is hard formulas, or "equations," and diagrams. What to imagine ever doing without one again. distinguishes a technical word processing system At the moment, there is only one fully opera­ from an ordinary one is its ability to handle tional, released equation processor for the Macin­ these extra elements easily. Making good vertical tosh. This is MacEqn, created by Dennis Venable arrows is difficult with a word processor, and and available from Software for Recognition Tech­ slanted ones are almost impossible, so diagrams of nologies, 110 University Park, Rochester, NY the "commutative" type are usually prepared in 14620 for $44.95. Venable has configured MacEqn MacDraw or MacPaint and then transferred either as a Desk Accessory, making it very convenient to directly to MacWrite via the Clipboard (which is use. Of course DA's are more limited in size than a lot faster and easier using Switcher) or are first ordinary Applications, but Venable has shown put in the Scrapbook and later transferred to the great programming skill in squeezing so much Mac Write document. Other types of pictures and functionality into 18K! Early versions of MacEqn diagrams are also easily handled in this way. But had two important defects: you could not edit this is definitely not a satisfactory way to handle an equation after it was created, and you had formulas. Let's see why. access only to the four LaserWriter fonts. I have When one first thinks of how it might be just received the latest version which fixes both of convenient to handle equations in a WYSIWYG these problems. environment, it seems that all that is needed is There are two more equation processors I the possibility of moving the cursor around freely know of under development. One, called Wein­ to an arbitrary location on the screen and then berg, is an ambitious project by Allan Bonadio.

749 The version I have used looks prom1smg but is to some objects or to state some simple relation still in too primitive a state for me to guess how (equality, membership, or inclusion) between two successful it will eventually be. typographically simple objects. Without going The other one, programmed by Ted Sobel, into further detail let me say that a word proces­ is quite polished and is nearly ready for release. sor will be adequate for handling in-line formulas Called Math Writer, it will be sold by Cooke if it can just do the following quickly and easily: Publications, P.O. Box 4448, Ithaca NY 14852 • Switch to any font or font size; for $50. By the way, similar programs for the • Attach to characters on the baseline both IBM-PC sell for five times as much and more! I sub- and superscripts, with the possibility of only received my review copy a few days ago at a superscript over a subscript. Moreover, the Mac World Expo, but I have spent a couple of these first order sub- and superscripts should hours playing with it and am very impressed with themselves admit sub- or superscripting. Fi­ both the program and its documentation. The nally, the baseline characters and at least designers have chosen an almost opposing design the first order sub- and superscripts should philosophy from that of Mad::qn. If Mac~qn is a admit decoration by accents both above and lean, mean racing car of a DA, then Math Writer to the side. is a luxurious sedan of an Application. The For example, it should be easily possible to program is loaded with features that make it include the formula i)i,k 2 within a line of text. easy to use, but also add to its size-a very hefty Now while the latter formula is very easy in 'lEX, 115K compared to Mac Write's 70K. Since it needs and is not hard with an equation processor, it is about 250K of RAM to run, and Mac Write needs enough to make a beginning MacWrite user give a similar amount, it seems that you will need a up TWP in disgust. The problem with Mac Write, Mac Plus to run the two together under Switcher. of course, is that, after you insert a character, Now the bad news! An ideal technical the insertion point moves forward by the width of word processor would have an equation processor the character box, so for example, after typing a built into it, and formulas would be created superscript there is no way to place a subscript "in place." But in today's Macintosh world, just below it. If you hit the Backspace key, an equation processor is implemented either as a then the insertion point will move back, but the Desk Accessory or as a separate Application. This superscript you just inserted will simultaneously means that after a formula has been created in an disappear. In other words, the Backspace key equation processor, it must then be transferred to actuates a "delete and backspace" or "destructive the Clipboard in one of the public data formats backspace" operation. (e.g., as a Pict file) and then "pasted" back into In MacAuthor, though, if you hit option the word processing document. And this causes a backspace after typing a character, the insertion problem! Both MacWrite and MS Word consider point backs up to where it was before that charac­ an imported graphics file to be (in TEX terms) a ter was typed without erasing the character. This "vertical box," i.e., analogous to one or more lines is what is called a "nondestructive backspace." It of text, rather than a "horizontal box," like a means that you can overstrike characters, creat­ character, that can be inserted into a line of text. ing, for example, a "#" symbol by overstriking an This means that an imported formula can not be equal sign with a slash. Much more importantly, pasted into a line of text as an in-line formula, but if you do this trick after typing a superscript, must be pasted between lines of text, i.e., it must then you can type a subscript just below it! Sim­ be a displayed formula. (See the above remark, ilarly, nondestructive backspacing makes it easy however, concerning the forthcoming Write Now). to create simple in-line fractions. It might seem then that equation processors Fortunately, there is a trick for simulating have not solved our basic problem, namely creat­ the effect of nondestructive backspacing in any ing formulas with a nontechnical word processor, Macintosh WYSIWYG word processor. If you since the great majority of formulas in mathe­ recall the above discussion of fonts, particularly matical writing are in-line, and these must still the point about characters not having to be inside be handled by the word processor itself. But their boxes, you can probably guess how it is this reduction of the problem, to one of handling done. Just make a character's box have a width only in-line formulas, turns out to be the key of zero, and let the character jut entirely out to to a simple solution. If you examine a large the left side of its box. When you type such a sample of mathematical text with this "in-line character, the insertion point will stay put, just versus displayed" distinction in mind, you will as if you had followed it with a nondestructive soon detect clear patterns in how mathematicians backspace. If you modify an entire font so that and typesetters decide whether or not to display a all of its characters have zero width, you will, in formula. These decisions are apparently based on effect, be able to do a nondestructive backspace intuitive rules that we all seem to unconsciously after inserting any character from that font. Still, know and use. In-line formulas are generally recall that the major reason we wanted nonde­ short and simple. Most in fact are only a few structive backspacing was to be able to put one symbols long, serving merely to name or refer character over another. We can make this even

750 easier if we create not one but two zero width is $30. Princeton was the first TWP font for the versions of the font, one (for superscripts) with Mac, and probably more mathematical papers the characters shifted up from the baseline by have been written on the Macintosh with it than seven points and one (for subscripts) with the with any other font. It was also the first font that characters shifted down by five points. Finally, I I know of to use zero width characters, although would recommend that you do this with a nine in a much more limited way than that suggested point version of the same font you use as your above. My main objection to it is that it is main text font at twelve point. If you do this with based on the sans serif Geneva font which I find the Boston font (version II), you will have exactly somewhat illegible and lacking in character. the two fonts I have in my System under the There are many TWP fonts advertised and names "up" and "down." With these fonts and a available commercially. I've tried a number of little experimenting, you will soon discover that it them and my feeling is that, for the most part, has suddenly become surprisingly easy not only to they were created following someone's theoretical put superscripts over subscripts, but also to make idea of what technical writers should need rather fractions and two levels of sub- and superscripts. than studying what they use. They are generally A little warning though; you probably should serviceable, but there are none I have tried that not make your fonts really zero width. Certain I can recommend with any enthusiasm. Like any versions of the Image Writer driver can not han­ fine tool, a good font family has usually evolved dle zero width characters, so although your text gradually from the experience and needs of its might look just right on the screen, if you have users. I know this to be the case with the one of these versions of the Image Writer driver in Maine and the Princeton fonts and I am sure that your System Folder, you may find that zero width there are many more such fonts of this same high characters will not print! To be safe, you can quality created by working mathematicians to fill make the characters all one pixel wide instead. Of their own needs. Let's hope some method can course, to be able to do high quality printing on be found to publicize and distribute such fonts. your Image Writer, you should also make two pixel They are a precious resource. wide, eighteen point versions of these fonts. If you have the font editor Fontasic, from Altsys, and The Last Word a little experience in using it, then carrying out this program of font modification should be about I am completing this month's column shortly af­ an evening's work. But there is a better and ter returning from the ICM-86 in Berkeley. I easier option. Bill Fulton and Joe Harris of the soon lost track of the number of people who Mathematics Department at Brown University approached me there and said, "I'm enjoying your have created the best set of fonts I have yet seen column in Notices, but ... ," and then went on for WYSIWYG mathematical word processing on to tell me why they too disagreed with my rec­ the Mac. It will be sent to you, along with a ommendation against starting to write a research short but clear manual, if you make out an eight paper on a word processor without a handwritten dollar check payable to Maine Fonts and mail draft. When the occasion permitted, I pressed it to: Maine Fonts, Dept. of Mathematics, Box these people to tell me just how they were using 1917, Providence, RI 02912. The Maine family word processors as an aid in doing and writing up has a font named Machias as its principal text research. Not surprisingly, none of those I asked font, which has borrowed the marvelous roman claimed to be doing anything nearly as radical lettershapes from Charles Maurer's Bostonii font. as the more extreme methods I described above. But for mathematical text the choice of "extra" And I noticed little uniformity in the point of the characters in Machias is a great improvement over research and write-up cycle at which people claim Boston. Maine also has an excellent new symbol to bring word processors into play. The methods font called Lovell. It implements, in a slightly mathematicians employ to help them create and different way, the zero width, raise and lower communicate concepts and ideas are apparently trick mentioned above. Fulton and Harris are too personal and variable to be easily codified. considering making a Laser Writer version of their In the end, it seems that the only general advice fonts if there is enough interest. By the way, that can be given legitimately on how best to fit the only WYSIWYG LaserWriter technical fonts word processors into the complex creative process that I know of come as part of the new Version of mathematical research is that one should be 3 of the Princeton Fonts of Dougherty and Lam, flexible, imaginative, and open-minded, experi­ available from many user groups, or by writing to: menting with different techniques until finding Princeton Font, Dept. of Mechanical Engineering, one that seems most suitable personally. Princeton, NJ 08544. The suggested contribution

751 Washington Outlook Kenneth M. Hoffman

Public Information: small. The few who have been close to the ef­ The View from Inside the Rocket fort sense greater change and feel the momentum building. A colleague recently asked for my assessment of I'm reminded of two very different perspec­ our public information activities, as we pass the tives on a rocket launching. From the ground, the first anniversary of the Joint Policy Board for darned thing always appears to lift off too slowly Mathematics (JPBM) decision to retain Kathleen at first, as if it would never really get moving. Holmay and Associates to organize and expand This perception of sluggishness can be overcome "public education" efforts on behalf of mathe­ somewhat by cerebral means, e.g., by invoking matics. When I· answered that I thought our one's familiarity with the laws of motion. It's community should be actively celebrating, he much more persuasive, however, to ride in the seemed mildly surprised. But he was either too rocket. No knowledge is required to convince you polite or in too much of a hurry to articulate that the thing is going. the question on his face: Celebrate what? What follows is the answer I didn't get to give. The stepped-up public information effort A well-known science writer for one of the na­ which Kathleen Holmay has coordinated, and tion's most prominent newspapers returned home to which lots of people have contributed, has had in mid-August, following a two-month leave, and a successful lift-off during its first year. For me as phoned Kathleen Holmay to say: "It looks like a passenger, it's been a total pleasure to watch a I really missed it!" The "it" he had missed real professional at work, to see the motion, and was a quantum leap forward in the mathematical to feel the acceleration. c?mmunity's public information activities, par­ ticularly as reflected in the press seminar which MANAGING EDITOR'S NOTE: Over the past Kathleen and Ed Block (Managing Director of SIAM) arranged for the Boston SIAM meet­ few months, the number of news articles about that have appeared in major news­ ing in July and the extensive publicity efforts mathematics for the International Congress of Mathematicians papers and magazines has noticeably increased. (Berkeley, August). What he had missed was not This has been substantially due to the activities sci­ a chance to read the articles but the opportunity of Kathleen Holmay in building a liaison with to access the usable material now being generated ence writers. Holmay is the President of Kathleen for reporters interested in covering mathematics. Holmay and Associates, Inc., a media consulting by the Joint Policy What he was really calling to say was that he and public affairs firm retained wanted in on the action. Board on Mathematics. During a discussion with offered to write an account What a change! this Editor, Holmay of this activity, and it is presented here for the A year ago my oversimplified mental picture information of the mathematical community. of our community's national media coverage was a blend of 75% articles in Science by reporter Gina Kolata and 25% other articles planted by math­ During July and August, mathematics was ematician Ron Graham of Bell Labs, through a reporters network he had built. Although both of these the hottest topic covered by science are still quite important, our network has now across the country. grown considerably, with firmly implanted roots The coverage began when the U.S. team in several leading national newspapers as well as to the International Mathematics Olympiad tied several major science and education publications. with Russia for first place. The six returning It is supported by an expanding group of math­ team members were met at Kennedy Airport by ematicians who are willing to make the effort a reporter and photographer from the New York to explain their subject in terms reporters can Times. The next day they were interviewed understand. Coverage has expanded accordingly by United Press International. Next came an and there is much more to come. ' impressive front page Science Times story (New The fact that many members of our com­ York Times, August 5, 1986) featuring a session munity have not noticed the marked change over at the July 21 SIAM Press Briefing. the last year or so is not surprising. Only the Then, an embarrassment of media riches mathematicians with a strong interest in public stemming from ICM-86, and mathematics in gen­ information tend to monitor what goes on in a eral, followed in early August. A series of articles variety of national publications, and even those in major daily newspapers and significant spe­ who see more than the "usual" number of articles cialty publications dispelled, once and for all an appearing may still feel that the total number is idea prevalent among mathematicians, as well as

752 some media representatives-that mathematics is Fourth, we took advantage of the cumulative "impossible to cover." effect of the Olympiad, the SIAM Press Briefing, An early and conservative estimate indicates ICM-86, and other events, such as the AAAS that our collected clippings over the last two exhibit of fractal art. No one event would months reached about 10 million people, many of have been as significant without the support whom are involved in government, public policy, of the others. Joint efforts within each event, and higher education. such as the ICM Public Information Committee Why the sudden barrage? First, we had working together with the U.C. Berkeley Public a series of newsworthy events, activities that Information Office, worked in the same manner to allowed us to highlight people and their work in give us a larger impact than we otherwise would mathematics. have realized. Second, we have been making media over­ So, what are our priorities at this point, tures-distributing news releases, mailing copies now that we have passed the introductory public of testimony, giving away subscriptions to various information phase? The newly formed Public In­ publications in mathematics, calling and meeting formation Resource Committee is emerging as an reporters and editors, etc.,-in a concentrated important group both as a source for story ideas fashion for about nine months. Hence, mathe­ and as a source of individuals who can effectively matics was fresh in the minds of key reporters. communicate with nonmathematicians. Increas­ Third, we have been building momentum in ing the involvement of this group with ongoing addition to conveying that we are in the public public information activities is mandatory. information business for the long haul. We Further work with reporters is necessary. We haven't expected quick coverage. Instead, we will continue giving useable information to the have been listening to reporters and methodically reporters who know mathematics and we will responding to their individual needs. We have be expanding that number with several initia­ also set the mathematics stage, so to speak, with tives designed to introduce other reporters to the activities like Math Awareness Week in April. excitement, relevance, and creativity of mathe­ matics.

PARTICLE SYSTEMS, RANDOM MEDIA AND LARGE DEVIATIONS Richard Durrett, Editor "The book is an excellent introduction to the exciting recent developments which combine ideas from physics with mathematical techniques from the theory of probability." - Frank L. Spitzer Cornell University This volume of proceedings of the 1984 AMS Summer Research Conference The Mathematics of Phase Transitions provides a handy summary of results from some of the most exciting areas in probability theory today: interacting particle systems, percolation. random media (bulk properties and hydrodynamics), the Ising model and large deviations. Thirty-seven mathematicians. many of them well-known probabilists, collaborated to produce this readable introduction to the main results and unsolved problems in the field. In fact, it is one of the very few collections of articles yet to be published on these topics. To appreciate many of the articles, an undergraduate course in probability is sufficient. The book will be valuable to probabilists, especially those interested in mathematical physics and to physicists interested in statistical mechanics or disordered systems. ISBN 0-8218-5042-3. LC 85-6181. ISSN 0271-4132 391 pages. August 1985 Individual member $19. Institutional member $26. List price $32 To order. please specify CONM/41NA Shipping/Handling: 1st book $2. each add'l $1. $25 max. By air. 1st book $5. each add'l $3. $100 max. PREPAYMENT REQUIRED. Order from American Mathematical Society. PO Box 1571. Annex Station. Providence. Rl 02901-1571. or call 800-556-7774 to use VISA or MasterCard.

753 News and Announcements

Sydney Henry Gould bally, MacArthur Foundation pre~ident, states, 1909-1986 "The award recognizes outstandmgly talented and promising individuals and gives them the Sydney Henry Gould, former Editor of Trans­ freedom to create. Our goal is to remove finan­ lations at the American Mathematical Society cial constraints, allowing Fellows to focus their and former Executive Editor of Mathematical Re­ energies on the issues and problems they find views, died in Providence, Rhode Island, July 7, important or critical in our time." Since May 1986, at the age of 77. 1981, the MacArthur Foundation has named 191 Born in Ilderton, Ontario, Gould received Fellows in nine separate announcements and has a B.A. in mathematics from the University of committed more than $50 million to the Program. Toronto in 1929, a Ph.D. in classics at Yale The Foundation does not accept applications for University in 1933, and an M.A. in mathematics Fellowships. Instead, anonymous nominators rec­ from Columbia University in 1938. He taught ommend Fellowship candidates to an independent classics at Yale University and the University of selection committee. No fixed schedule guides the Toronto before becoming assistant professor of announcement of Fellows. mathematics at Purdue University in 1947, where Benedict Gross has distinguished himself in he remained until 1955. He was professor of his work on number theory. He is best known mathematics at Williams College from 1955-1956. for his contribution to an imaginative solution Gould became the Executive Editor of Math­ of a problem dealing with imaginary quadratic ematical Reviews in 1956, and the Editor of fields that dates back to the nineteenth century. Translations at the AMS in 1962. During this This work was done jointly with Don Zagier of period, he was also a research associate (1956-- the University of Maryland and the Max Planck 1959) and visiting lecturer (1959-1960) at Brown Institute for Mathematics in Bonn. (See "£-Series University, and served on the Committee on the of Elliptic Curves, The Birch-Swinnerton-Dyer Chinese Glossary of the Mathematical Sciences Conjecture, and the Class Number P~oblem of (1963) and the editorial advisory board of the Gauss," D. Zagier, November 1984 Nottces, pages English edition of Encyclopedia of the Mathe­ 739-743.) matics, published by the Mathematical Society Gross was born in South Orange, New Jersey, of Japan (1969-1976). In 1972, Gould retired in 1950. He received his B.A. from Harvard from the AMS to take a position as a research University in 1971, his M.Sc. from Oxford in 197~, associate at the Mathematical Institute of the and his Ph.D. from Harvard in 1978. He begIsrael Gohberg of the School of Mathematical.Sci­ new MacArthur Fellows. The present group ence at Tel Aviv University in Israel has receiVed includes one mathematician, Benedict Gross of the Rothschild Prize for his work in mathemat­ Harvard University. ics. The Rothschild Prizes Organization deemed MacArthur Fellowships begin at $24,000 an­ Gohberg's work "outstanding in its originality, its nually for recipients aged 21 or younger and depth its breadth and its widespread influence in increase by $800 per year-up to $60,000 for the ar~as of both ~ure and applied mathematics." recipients aged 66 or older. Fellowships con­ Gohberg's work spans functional analysis, tinue for five years. The Foundation imposes no operator theory, systems theory,. lin~ar algebra, requirements, restrictions, or performance stan­ combinatorial geometry, and apphcat10ns of these dards on Fellowship recipients. John E. Cor- areas. His work in abstract Fredholm theory

754 topics. Past recipients are Nicholas Varopoulos (1968), Richard Hunt (1969), Yves Meyer (1970), Bieberbach Conjecture: Charles Fefferman (1971), Thomas Komer (1972), Proceedings of the Symposium E. M. Nikisin (1973), Hugh Montgomery (1974), William Beckner (1975), M. R. Herman (1976), on the Occasion of the Proof S. B. Bockarev (1977), Bjorn E. Dahlberg (1978), The photographs of I. M. Milin and C. Loewner Gilles Pisier (1979), Stylianos Pichorides (1980), were exchanged during the production pro­ Peter Jones (1981), Alexei B. Aleksandrov (1982), cess. The proofs approved by the editors were Jean Bourgain (1983), Carlos Kenig (1984), and correct; the err?r was made at a later stage. Th. H. Wolff (1985). The jury consisted of L. Replacements will be sent on return of defective Carleson, Y. Katznelson, Y. Meyer, and E. M. copies to Mary C. Lane, Director of Publica­ Stein. - LMS Newsletter tion, Amer~can Mathematical Society, P.O. Box 6248,_ Provide~ce, Rl 02940. We apologize for Replacement Ballots any mconvemence that this error may have caused. This year ballots for the AMS election will be mailed September 10, 1986, or within a day or two thereafter. The deadline for receipt of ballots recently played ~ key role in the development in Providence is November 10, 1986. of ~oncommutative approximation theory and There has been a small but recurring and dis­ clarified the foundations of new areas in functional tressing problem concerning members who state :malysis. In the area of systems theory, his work that they have not received ballots in the annual mcludes a ~omplete realization theory for linear election. It occurs for several reasons, including systems havmg transfer functions which are monic failure of local delivery systems on university or polyn~mial operators. Gohberg has also proved corporate properties, failure of members to give the eXIstence of stable factors, and has classified timely notice of changes of address to the Provi­ them and demonstrated their importance. His dence office, failures of postal services, and other treatment of problems in linear analytic algebra human errors. has I~ to the solution of many types of integral To help alleviate this problem, the following equatiOns. replacement procedure has been devised: A mem­ The R~t~schild Prize is awarded to recognize ber who has not received a ballot by October excellent ongmal works published since the state 10, 1986, or who has received a ballot but has o!lsrael was established in 1948. In highly unusual accidentally spoiled it, may write after that date Circ';lmstances, the prize is awarded for excellent to the Secretary of the AMS, Post Office Box 6248, service or an outstanding original work which has Providence, Rl 02940, asking for a second ballot. not been published in a scientific journal or book. The request should include the individual's mem­ ber code and the address to which the replacement Makarov Wins Salem Prize ballot should be sent. Immediately upon receipt The Salem Prize for 1986 was awarded to N. of the request in the Providence office, a second G. M~ov of the Steklov Institute (LOMI) ballot, which will be indistinguishable from the o~ Len~ngrad, for his work on the Hausdorff original, will be sent by first class or air mail. It must be returned in an inner envelope, ~ImensiOn of the support of the harmonic measure which will be supplied, on the outside (m_ the case. of g~neral Jordan domains.) The of which is the Pnze, established m 1968, is given every year to following statement to be signed by the member: a young mathematician who is judged to have The ballot in this envelope is the only ballot done outstanding work in the field of interest of that I am submitting in this election. I understand Salem, primarily on Fourier series and related that if this statement is not correct then no ballot of mine will be counted.

Request for Photos signature ~n conjunction with the Centennial of the Amer­ Although a second ballot will be supplied on Ican M~thematical Society, which is to be cele­ request and will be sent by first class or air mail, ~rated m Providence in 1988, the AMS would the deadline for receipt of ballots will not be hke ~o set up an exhibit of group photos from extended to accommodate these special cases. meetmgs and similar items of interest. If anyone has memorabilia of this kind which they would like to give or loan to the 1986 National Chairmen's Colloquium AMS, please. write to William J. LeVeque, The Fourth Annual Chairmen's Colloquium will Ex~utive Director, American Mathematical be held at The Washington Marriott in Wash­ Society, P.O. Box 6248 Providence Rhode ington, DC, on October 10 and 11, 1986. The Island 02940. ' ' activities of this year's Colloquium will comprise

755 a one and one-half day intensive and informative open to professional societies, corporations, insti­ event focusing on several critical topics involving tutions, and individuals who share its interests federal policy and programs in the mathemati­ and objectives. Commissioners (representatives) cal sciences. (See August 1986 Notices, pages are appointed by professional societies and cor­ 626-627.) porations, and elected by individual members. The program will begin on Friday, October CPST is a Participating Organization of the 10, with three concurrent sessions on Mathemati­ American Association for the Advancement of cal Education (Session Organizer: Harvey Keynes, Science (AAAS), and serves as the Manpower University of Minnesota), Statistics (Session Or­ Office for the AAAS. ganizer: Fred Leysieffer, Florida State University), The CPST is concerned broadly with the and Computational Mathematics (Session Orga­ recruitment, training, and utilization of scientific nizer: James Greenberg, University of Maryland­ and technological personnel. The charge in its Baltimore County). Friday evening's activities constitution calls for: will include a reception and a banquet. The the collection, analysis, and publication of banquet speaker will be Ramesh Gangolli, for­ data regarding the manpower resources of mer Chairman at the University of Washington the United States in the fields of science and Chairman of the NSF Mathematics Advisory and technology; the promotion of programs Committee. He will assess the mathematical of education and training of potential scien­ community's progress in redressing the problems tists and technologists; the promotion of the which were so effectively presented in the National proper utilization of scientific and technolog­ Research Council's David Report. ical manpower by educational institutions, Saturday's program will begin with a con­ industry, and government; all devoted toward tinental breakfast for which participants will be aiding the development of our country's sci­ grouped according to region. This will pro­ entific resources for the benefit and welfare vide an opportunity for attendees from similar of our people. regions to become acquainted and discuss mu­ CPST serves as a focus for identifying and seeking tual issues with their neighbors. Following a solutions to scientific manpower problems com­ welcome by Phillip Griffiths, Chairman of the mon to all the scientific disciplines, and cooperates Board on Mathematical Sciences, the morning closely with other organizations in areas of over­ sessions will encompass presentations on funding lapping concern and interest. The purposes of the programs including Department of Defense and CPST are implemented through a wide variety of Other Programs, National Science Foundation's programs. The general program areas include: Division of Mathematical Sciences Programs, and •collecting, correlating, and analyzing enroll­ Mathematics Education Programs. There will ment and degree data to project the potential be an opportunity for discussion following each supply of trained technological manpower in all presentation. areas of science and engineering; Lunch on Saturday will include roundtable •maintaining continuous current information discussions for which seating will be determined on the job market for technological professionals by the participant's preselection of interest areas. and attempting to project potential variance with Experts in the various topic areas will be assigned available supply; to specific tables to facilitate in-depth discussions. •collecting, correlating, and disseminating The afternoon session will outline recent activities availability data on women and minorities in the in the efforts to establish a Washington presence sciences and engineering to provide employers in for the mathematical sciences and will include all sectors with basic information needed to plan a discussion of community involvement in issues and carry out programs of affirmative action; which will impact on federal programs and policy. •collecting and publishing all significant All Department Chairmen and interested col­ salary data for scientists, engineers, and tech­ leagues are encouraged to attend this important nicians from all available sources; event. If you have not yet registered for the Col­ •maintaining information on career guidance loquium and wish to do so, call Frank Gilfeather, materials in science and engineering; Staff Director of the Board on Mathematical Sci­ •assisting industry, educational institutions, ences, at (202)334-2421. Although the deadline government agencies, and members of Congress for registration was mid-September, every effort by providing current oral or written informa­ will be made to accommodate applicants. tion in any of these areas, both in response to specific requests and through a regular series of AMS Joins CPST publications; •conducting symposia to address vital and The AMS has recently become a member society current topics in science, engineering, and tech­ of the Commission on Professionals in Science nology; and and Technology (CPST), a private, nonprofit •representing various governmental and pri­ corporation established in 1953 as the Scientific vate groups dealing with professional manpower Manpower Commission. Membership in CPST is and manpower policy.

756 Special Program/Workshop for Department Chairs Mathematical Sciences Research Institute The Joint Policy Board for the Mathematics Berkeley Committee for Department Chairs will sponsor a As this is being written, ICM is at its height in one day Special Program/Workshop for Chairs in Berkeley. At its conclusion a large, year-long the Mathematical Sciences. The Program will be program in Number Theory with Connections held on January 20, 1986, in San Antonio, Texas, to Algebraic Geometry will get under way at the day before the start of the AMS-MAA Joint MSRI. Approximately 130 participants will be Mathematics Meetings. in residence for varying periods. The chair­ The program will be presented by Dr. John man of the program committee is Benedict Bennett of the American Council on Education Gross, who was recently awarded a MacArthur for Leadership Development. Dr. Bennett has Fellowship. given numerous workshops of this nature for Two workshops will be features of the num­ many universities and is considered to be the ber theory program: Iwasawa Theory and Spe­ best in the field. The agenda planned for the day cial Values of L-functions from January 20 to includes an overview of chairs's responsibility, case 24, 1987, with committee John Coates (Chair­ studies, and faculty development and evaluation. man), Ralph Greenberg, and Barry Mazur, and There will also be opportunities for small group a second workshop from March 23 to 27 with discussions. The sessions are designed to help committee Yasutaka Ihara, Kenneth Ribet, and new department chairs, who may have received Jean-Pierre Serre. little or no training and direction, to better fulfill The first of a series of "microprograms" the administrative aspects of the position. at MSRI runs from August 25 to September Support for this meeting is provided- oy 12, 1986. The topic is Nonlinear Diffusion the registration fee only, and unless sufficient Equations, chaired by James Serrin. A volume confirmed registrations are received by December of proceedings is planned for the MSRI book 1, the program will be cancelled. Full refunds will series, published by Springer. be made if the program is cancelled. A workshop on Nonlinear Hyperbolic Waves For program details, including registration is scheduled for May 25-27, 1987. The orga­ information, contact David Ballew, Department nizing committee consists of Stuart Antman, of Computer Science, 447 Stipes Hall, Western Phillip Colella, Ronald DiPerna, and James Illinois University, Macomb, IL 61455. Glimm (Chairman). The year will conclude with a microprogram on Commutative Algebra, U.S. Team Ties for First Place chaired by Melvin Hochster, with dates June in International Mathematical Olympiad 15-July 2, 1987. Here is a brief summary of future programs: Six American high school students solved their 1987-1988: classical analysis, representa­ way to a tie with the Russians for first place in tions of Lie groups, microprogram on Banach the 27th International Mathematical Olympiad in spaces; Warsaw on July 9 and 10, 1986. The Americans 1988-1989: combinatorial group theory, and Russians each had team scores of 203 out of symplectic geometry; a possible 252. Behind them were teams from 1989-1990: logic and a second program still West Germany (196), China (177), East Germany to be selected. (172), and Romania (171). In all, 37 nations Suggestions for future programs, micropro­ participated in the Olympiad. grams, and workshops are always welcome. The members of the 1986 U.S. team were William Cross (Kalamazoo, Michigan), David Grabiner (Claremont, California), Jeremy Kahn Listed below are the publications of the (New York, New York), Joseph Keane (Pitts­ CPST, with the date of the last issues. burgh, Pennsylvania), Darien Lefkowitz {New •Scientific, Engineering, Technical Manpower York, New York), and John Overdeck (Columbia, Comments, Vol. 23, 1986 Maryland). •Professional Women and Minorities-A Man­ Individual first, second, and third prizes power Data Resource Service, February 1986 were also awarded to deserving team members. •Salaries of Scientists, Engineers and Techni­ Grabiner, Kahn, and Keane received first prizes cians-A Summary of Salary Surveys, November as individuals and the other members of the 1985 U.S. team all won second prizes. Keane also • The International Flow of Scientific Talent: received the only Special Prize that was awarded Data, Policies and Issues, May 1985 at this Olympiad. The Special Prize was awarded • The Technological Marketplace-Supply and for a particularly elegant solution to one of the Demand for Scientists and Engineers, May 1985 Olym}Jiad Qroblems. •Opportunities in Science and Engineering, The Mathematical Olympiad activities are November 1984 sponsored by seven national associations in the

757 mathematical sciences with arrangements made by the Mathematical Association of America. Institute for Mathematics Financial support was provided by IBM, the Army and its Applications Research Office, the Office of Naval Research, and Minneapolis Hewlett-Packard. A program on Applied Combinatorics is being orga­ nized by Victor Klee, Daniel Kleitman, Dijen Ray­ Chaudhuri, and Dennis Stanton for the period from September 14, 1987, to June 25, 1988. During the Errata fall, V. Klee and E. Lawler will lead parallel efforts In the August 1986 Notices 1986 AMS Elec­ and Computational Geometry and in in Discrete tions, p. 634, Marc A. Rieffel's name was Combinatorial Optimization. From December 14 to incorrectly spelled Mark A. Reiffel. April 30, D. Kleitman will lead a period of concen­ 1986 Notices, Reci­ tration on Applied Graph Theory, and G. C. Rota Also in the August and D. Stanton are organizing a period of concen­ procity Agreements did not. include one ,n~w tration on Interactions Between Combinatorics and reciprocity society, the Soc1edad Matematlca Other Parts of Mathematics. D. Ray-Chaudhuri de la Republica Dominicana. See Miscellaneous will lead a period of concentration on Coding The­ in this issue of Notices. ory and Design Theory during May and June.

Long-term visitors will include the Organiz­ :i;:;::;;:::;;;::::;:::;:::======:i::===:::===;i::::;n~::::;:;:!i;;;;: ing Committee as well as E. Bannai, E. Lawler, V. THE BIEBERBACH CONJECTURE: Proceedings Pless, N. Singhi, L. Terlinck, J. H. Van Lint, and R. of the Symposium on the Occasion of the Proof M. Wilson. Albert Baernstein. David Drasin. Peter Duren In each program segment, there will be individ­ and Albert Marden. Editors ual lectures by resident and visiting experts. There {Mathematical Surveys and Monographs. Volume 21) will also be the following workshops: Polyhedral Combinatorics and Geometric Complexity, Septem­ For 70 years. the Bieberbach conjecture has intrigued de Branges by D. Dobkin and M. Grotschel; the mathematical world. In 1977. Louis ber 14-18, organized of this in Space (Tilings, Quasi­ of Purdue University took up the challenge Orderly Dispositions He will be recognized organized by B. Grun­ famous unsolved problem. crystals, etc.), October 19-23, who proved Bieberbach's of Combina­ as the mathematician baum and M. Senechal; Applications And more importantly. his method carne Science (in­ conjecture. torics and Graph Theory to Computer totally unexpected sources: operator theory and and Geo­ from cluding a special session on Combinatorial special functions. metric Problems in VLSI Design), December 14-18, Symposium on the Occasion organized by R. Graham, L. Snyder, and R. Tarjan; This book. based on the the story behind this fascinating Applications of Combinatorics and Graph Theory of the Proof. tells insight into the nature of the Social Sciences, February, or­ proof and offers in the Biological and its history and its proof. A special and F. Roberts; Invariant The­ conjecture. ganized by J. Cohen and of the book is the enlightened personal by G. C. Rota unusual feature ory and Tableaux, March, organized accounts of the people involved in the exciting events and D. Stanton; Design Theory and Applications, surrounding the proof. Especially attractive are the June 12-18; Coding Theory and Applications, June photographs of mathematicians who have made 19-25. significant contributions to univalent functions. the A number of postdoctoral memberships as well area of complex analysis which provides the setting as some support for more senior persons are avail­ for the Bieberbach conjecture. be found in the advertisement at able. Details can Research mathematicians. especially analysts. are the back of this issue. sure to enjoy the articles in this volume. Most A four-week program on Robotics will precede articles require only a basic knowledge of real and this program, and a ten-week program on Signal complex analysis. The survey articles are accessible Processing will follow it. to non-specialists. and the personal accounts of all The 1986-1987 program in Scientific Computa­ who have played a part in this important discovery tion is well underway. Martin Schultz has organized will fascinate any reader. Algorithms for Modern a workshop on Numerical 1980 Mathematics Subject Classifications: 30, 47 Parallel Computer Architectures, which will take ISBN 0-8218-1521-0. LC 86-10843, ISSN 0076-5376 from November 3-7. A minisymposium on 260 pages (hardcover). 1986 place List price $45, Institutional member $36, Individual member $27 Numerical Simulation in Oil Recovery to be held Code SURV /21NA at the beginning of December is being organized by Shipping/Handling: 1st book $2. each add'l $1, $25 Mary Wheeler. max. By air. 1st book $5, each add'l $3. $100 max. Robert Burridge is coordinating a four-week Prepayment required. Order from AMS. P.O. Box 1571. period of concentration on Inverse Problems in Jan­ Annex Station. Providence, Rl 02901-1571. or call uary 1987. 800-556-7774 to use VISA or MasterCard.

758 National Science Foundation News f3 Reports

Increased Support for Mathematics: support with senior investigators. The post­ An Open Letter doctoral fellowships described above helped that situation a bit, as did the availability of a lim­ Dear Colleague: ited number of postdoctoral fellowships at various The most consistent theme in the mathematical institutes. Named instructorships in some cases sciences over the last two years has been the need provided released time from teaching, but still for revitalization of the community along the lines offered no real opportunity for dedicated research described in the David Report. That report has time. had a tremendous impact on the mathematical sciences community and its ability to present its The Division is now attempting to remedy case to the general public. this situation by providing the opportunity for se­ Based in large part on perceptions of the nior investigators to request postdoctoral research needs of the field established by the David Re­ support for their junior colleagues on research port, the Division of Mathematical Sciences has grants. In most instances, these colleagues are had significant budget increases aimed at improv­ individually identified and reviewed. In others, ing the support for graduate students and young the senior investigator is allowed some choice people. Our considerable concern over the short­ in naming a postdoctoral associate. In either age of mathematical scientists has, to a great case, the postdoctoral researcher must have had a Ph.D. extent, determined our policy. This letter de­ (or equivalent) for less than five years, and should scribes accomplishments, goals for the future, and be working in an area closely related to some new areas of activity designed to increase that of the principal investigator. There should be the flow of young people into our field. a mentorial relationship which allows a postdoc­ toral colleague to gain the In FY 1984, we supported 521 graduate benefits of advice and students on grants with an average annual stipend consultation from the senior investigator. This of $4,500. During FY 1986, we expect that these should enable the young researchers to initiate a research program and be better numbers will increase to 750 students with an able to compete for average stipend of about $5,300. Students are research support later in their careers. At present, most postdoctoral support supported in a number of ways, some for the on research grants is for summer research. If summer, some full-time, some with a combination additional funds are forthcoming in future of part-time academic year and full-time summer years, we anticipate gradually adding more academic support, etc. The type of support is largely year support for dictated by the combined needs of the research postdoctoral researchers on grants. project and the institution. Guided by our The current shortage of Ph.D.'s in the math­ Advisory Panel, we will continue to be flexible ematical sciences is likely to continue for some in our response to requests for graduate student time. The situation will not be easy to correct support; however, we expect that there will be because of the sharply falling number of individu­ a gradual shift toward more full-time support, als in the 18-22 age bracket. Further complicating with increased stipends. The availability of such the problem is the fact that, starting about 1995, support at a level competitive with other sciences the rate of retirement of mathematics faculty is important in attracting strong undergraduates will increase sharply. Keeping the number of into study at the graduate level. mathematically trained personnel at the level of Support for postdoctoral researchers in the expected need will require significant effort on the mathematical sciences is another mechanism for part of the entire community to attract young improving the opportunities available to those people into the mathematical sciences. entering the field. The Mathematical Sciences To assist in this effort, the Division of Math­ Postdoctoral Research Fellowship program is now ematical Sciences intends to pursue, more vigor­ well established. The number and quality of the ously than in the past, the option of including applicants for these fellowships continues to be funds on research grants for the support of un­ high. We encourage applications for the next dergraduates or even high school students whose competition. interest in the mathematical sciences might be Of more recent note is the support of post­ furthered by such support. The early interest of a doctoral researchers on individual research grants. senior mathematician in a student can have a de­ There has not been a formal tradition of postdoc­ cisive effect on the student's career choice, and we toral researchers in the mathematical sciences as would like to encourage investigators to do some one finds in fields such as chemistry or physics. active recruiting. Research proposals which con­ Traditionally, new Ph.D.'s in our field have been tain requests for undergraduates should describe forced into head-to-head competition for research the activities of the student during the period of

759 support. While in some cases these young people The CSE initiative is the outgrowth of several can make a contribution to the research effort of years of policy studies and recommendations in the principal investigator, a more typical activ­ the scientific community. In particular, several ity might be a summer study /research program reports were important to the development of the under the direction of the principal investigator. programmatic goals in this area, including: the The activities undertaken should be designed to Report of the Panel on Large-Scale Computing significantly further the student's advance toward in Science and Engineering (the Lax Report) a career as a research mathematician. in December 1982; the Bardon-Curtis Report It is a goal of the Foundation to increase in 1983-1984 (a report internal to the NSF); the participation of women, minorities, and hand­ the David Report of 1984; and more recently icapped individuals in science and engineering. the Report of the Panel on Future Directions The situation in the mathematical sciences is in Computational Mathematics, Algorithms, and particularly unsatisfactory. While the percentage Scientific Software (the Rheinboldt Report) which of Ph.D.'s in the mathematical sciences who are resulted from a workshop in November 1984. women is steadily increasing, we cannot take great From the names of these reports, you can see pleasure from the fact that it has reached only that the mathematical sciences community, led 16%. In 1985, a total of 30 Ph.D.'s in the mathe­ by Peter Lax and Werner Rheinboldt, played an matical sciences were granted to blacks, hispanics, important role in citing the need for the creation and native Americans, and only 9 of these were of a major program in computational mathematics American citizens. Given the shortage of new and algorithms. Ph.D.'s in the mathematical sciences, we must The NSF initiative in Computational Science look to these currently under-represented groups and Engineering incorporates many of the rec­ as sources to be tapped. We will be particularly ommendations of these reports, in particular, the receptive to requests in research proposals for the need to establish new multidisciplinary teams, support of graduate students, postdoctoral re­ to enhance existing centers and groups, and to searchers, and especially undergraduate students provide for equipment and training needs. We which is to be used to recruit talented women, expect strong participation by the Division of minorities, and handicapped individuals into the Mathematical Sciences (DMS). mathematical sciences. Within the DMS, we envision that requests This letter describes several new options will range from support for individuals and small available to grantees. It also leaves open many groups, to support for interdisciplinary teams of questions. If you or any of your colleagues are the type described in the Rheinboldt Report. We interested in exploring these opportunities further, expect that many proposals to the computational please feel free to contact the most appropriate mathematics activity will come from the tradi­ program officer at the Division of Mathematical tional areas of scientific computing, numerical Sciences at (202)357-9764. We would be happy to analysis, and computational statistics, partic­ hear from you. ularly as applied to other disciplines. However, Sincerely, computation is being increasingly used as a tool in John C. Polking geometry, algebra, and other core areas of math­ Division Director ematics. Accordingly, we encourage proposals for Division of Mathematical Sciences support of computational activities from all areas National Science Foundation of the mathematical sciences. The large-scale scientific computing resources currently available allow mathematical scientists to develop more Computational Mathematics experimental techniques in carrying out their tra­ ditional research projects and to take a more Initiative: An Open Letter active role in developing techniques for the use Dear Colleague: of other scientists and engineers. The active The President's FY 1987 Budget to Congress for participation of mathematical scientists in these the National Science Foundation incorporates a computational opportunities will be of long-term request for additional funding on the order of $27 benefit to science and technology. million for an initiative in Computational Science In recent years, it has become increasingly and Engineering (CSE) in order to respond to sig­ difficult to predict the outcome of Presidential nificant opportunities in scientific and engineering Budget Requests. This year does not appear to research. Should this funding be approved, the be an exception. Nevertheless, the Division of initiative will provide new kinds of opportunities Mathematical Sciences, with the concurrence of for mathematical scientists. This letter is being its Advisory Committee, has begun to take steps sent to a broad cross-section of the mathemat­ that will enable us to announce a program in ical sciences community to provide information Computational Mathematics. regarding the nature of the anticipated NSF ini­ During the next few months, NSF will estab­ tiative and the participation of the Division of lish organizational structures that will respond to Mathematical Sciences in it. this research opportunity. I hope you will use this

760 time to develop your own thoughts and projects in award. The award was established by Congress this exciting scientific area. If you wish to inquire in 1975 to mark the twenty-fifth anniversary of about the general progress of this initiative, feel the National Science Foundation and to honor the free to call me at (202)357-9669 or Dr. Melvyn Foundation's first Director, Dr. Waterman. It Ciment at (202)357-9764. As a Program Director is intended to give recognition to an outstanding for Applied Mathematics, Dr. Ciment was instru­ young researcher in any field of science, mathe­ mental in funding the workshop leading to the matics, or engineering and to encourage further Rheinboldt Report and will continue to serve as high quality research. the Division of Mathematical Sciences liaison on In addition to the medal, the recipient re­ CSE issues. The Rheinboldt report provides a ceives up to $100,000 per year for up to three years useful reference to mathematical scientists wish­ of research or advanced study in the mathemat­ ing to participate in the CSE initiative. It was ical, physical, medical, biological, engineering, published by, and is available from, SIAM. It was social, or other sciences at the institution of reprinted in Notices in November 1985. his/her choice. Let me encourage you to discuss these oppor­ Deadline for nominations for the 1987 award tunities with other NSF program directors both to be received by the Award Committee at the inside and outside the Division of Mathematical NSF is December 31, 1986. Announcement of the Sciences. Regardless of the exact organizational award will be made in May of 1987. structure at NSF, it is clear that the management Candidates must be permanent residents or of this effort will require close cooperation between citizens of the United States. They must be various programs in order to review and fund the thirty-five years of age or younger, or not more multidisciplinary aspects of this effort effectively. than five years beyond receipt of the Ph.D. degree It will be important for investigators and program by December 31 of the year in which nominated. directors to maintain communication during the Candidates should have completed sufficient scien­ start-up of this activity. tific or engineering research to have demonstrated, Please feel free to call or write us with your through personal accomplishments, outstanding questions and suggestions. We hope you will feel capability and exceptional promise for signifi­ encouraged to take time to investigate how your cant future achievement. In addition, candidates research can benefit from activities of this kind. should exhibit quality, innovation, and potential Sincerely yours, for discovery in their research. John C. Polking Nominations for the award may be submitted Division Director by the scientific and educational communities, Division of Mathematical Sciences individuals, professional societies, industry and National Science Foundation other appropriate organizations. Six copies of each nomination should be submitted to the Alan T. Waterman Award Committee, National Science Postdoctoral Research Fellowships Foundation, Washington, DC 20550. NSF Postdoctoral Research Fellowships in the The Award Committee is composed of twelve mathematical sciences will be offered only to members appointed by the Director of the N a­ persons who: 1. are U.S. citizens or nationals tional Science Foundation, and four ex officio as of January 1, 1987; 2. will have earned by members. The ex officio members are: Mr. Erich the beginning of their fellowship tenure a doctoral Bloch, Director, National Science Foundation; Dr. degree in one of the mathematical sciences; 3. Frank Press, President of the National Academy will have held the doctorate for no more than of Sciences; Dr. Roland W. Schmitt, Chairman five years as of January 1, 1987; and 4. will not of the National Science Board; and Dr. Robert previously have held any other NSF postdoctoral M. White, President of the National Academy of fellowship. Engineering. For copies of the application brochure or fur­ Additional information and/or a copy of the ther information, contact the Special Projects nomination form may be obtained by contacting Program, Division of Mathematical Sciences, the Exective Secretary for the Committee, Lois National Science Foundation, Washington, DC J. Hamaty, at the National Science Foundation. 20550, (202)357-9764; or the American Mathe­ Telephone: (202)357-7512. matical Society at ( 401 )272-9500. -NSF News Release The deadline for applications is November Grants Awarded for Experimental 15, 1986. Computer Research Facilities Nominations Sought for The NSF has made five-year grants totalling more Twelfth Alan T. Waterman Award than $15,000,000 to five universities to help estab­ lish and operate major experimental computer re­ The National Science Foundation Alan T. Wa­ search facilities. The grants, announced by NSF's terman Award Committee has issued a call for Coordinated Experimental Research ( CER) Pro­ nominations of candidates for the twelfth annual gram, are to establish the research facilities and

761 provide support for major projects involving large Staff for the groups of researchers. Institutions receiving the Division of Mathematical Sciences NSF grants and their projects are: The program directors for the coming year are: Cornell University, $3,606;266: to experiment with Classical Analysis John Ryff large interactive systems involving complex ol>­ Modern Analysis Ken Gross jects such as those in physical systems, formal Geometric Analysis Paul Goodey proofs and programming environments. The Topology systems involved in the research include those & Foundations Ralph Krause associated with robotics, formal-reasoning, pro­ Algebra and William Adams gramming environments, and VLSI design tools. Number Theory Ann Boyle Indiana University, $2,853,313: to expand its ex­ Applied Mathematics Melvyn Ciment perimental computing facility supporting research Andrzej Manitius in the theory and development of programming Frederick Wan languages, the design of functional languages for Statistics & Probability Nancy Flournoy parallel computation, and the implementation of Yashaswini Mittal tools for the production of hardware and software Special Projects Bernard McDonald from formal specifications. Alvin Thaler University of Wisconsin, $3,385,582: to develop The administrative staff includes: a laboratory for research in parallel computing. In particular the design of parallel algorithms Division Director John Polking for application in areas such as: mathematical Deputy Division programming, numerical methods, artificial intel­ Director Judith Sunley ligence, database management systems, robotics, Administrative Officer Fay Childress programming languages, and performance model­ The telephone number for the program direc­ ing and analysis. tors is (202) 357-9764, and for the administrative University of California at Irvine,$3,130,100: to staff it is (202) 357-9669. The permanent staff establish a major experimental facility to support consists of Ciment, Krause, McDonald, Ryff, research in four areas of software engineering: Sunley, and Thaler. The incoming rotators modeling and analysis, software development en­ are Boyle, University of Wisconsin-Milwaukee; vironments, advanced methods and tools, and Flournoy, Fred Hutchinson Cancer Research Cen­ measurement and evaluation. ter; Goodey, University of Oklahoma; Mittal, Virginia Polytechnic Institute and State Univer­ Yale University, $2,301,720: to expand ongoing sity; and Wan, University of Washington. Polk­ investigations into parallel multiprocessor-based ing, Adams, Gross, and Manitius are continuing computers. Basic parallel-processing procedures rotators. will be developed, analyzed, and applied to The program staff consists of six permanent real-world problems utilizing a range of parallel­ staff members and seven rotators. This mix processing architectures. of permanent staff members and rotators allows internal consistency and intimate contact with the universities, both of which are important to the Division's operation.

762 For Your Information

Science and Engineering Doctorates in 1985 • The number of science doctorates was 15,090 in 1985, slightly less ~han in'1984. Non­ This report, "Output of Science and Engineer~n.g U.S. citizens earned 3,120 science doctorates, an Doctorates Stable in 1985, but non-U.S. Citi­ all-time high, increasing their share from 17 per­ zens and Women Increase Their Shares of the cent in 1978 to 22 percent in 1985. The number of Total" is taken from the National Science Foun­ U.S. citizens receiving science doctorates declined datio~ Sciences Resources Studies Highlights. It during this period, a result of fewer presents information from the Survey of Earned awards to U.S. males. Doctorates, conducted annually by the National • S /E doctorates Research Council for the National Science Foun­ were awarded to fore~gn citizens from more than a dation, the National Endowment for the Human­ hundred countnes. An increasing share of the recipients ities the National Institutes of Health, and the are from countries in East Asia. Consistently since Dep~tment of Education. 1 Approximately 95 1960, over one-half the East percent of the 1985 recipients of the Ph.D. and Asians were from China, mostly from Taiwan. 2 similar doctorates, e.g., Ed.D., responded to the • Women survey questionnaire. The data base also in­ continued to increase their repre­ sentation among cludes some information on nonrespondents that S/E doctorate recipients, but not as rapidly as in was obtained from public sources. Recipients the seventies and early eighties. The 4 650 of first-professional degrees are not included (the degrees earned by women in 1985 were only percent more than M.D. degree, for example). For information on 2 in 1984, significantly below the average Graduate Science and Engineering Students and annual increase of about 6 per­ cent achieved Enrollments for the Fall of 1984, see the June during the preceding 10 years. In 1986 issue of Notices, pp. 511-514. 1985, women earned 1,340 more S/E doctorates than in 1978; men, in contrast, earned 135 fewer. The net effect of these divergent trends has been Highlights an increase in the proportion of degrees awarded to women, from 19 percent in 1978 to 25 percent • The number of science and engineering in 1985. Over the years women have earned over (S/E) doctorates awarded in 1985-18,255-was 80 percent of their degrees in psychology, life slightly greater than in 1984 and 7 percent above sciences, and social sciences. the recent low point in 1978. Although S/E doctorates have been increasing, the 1985 total was still 4 percent below the peak reached in Overall Trends in Doctorate Production 1972. There have been three distinct phases in S /E • Non-U.S. citizens earned 4,850 S/E doctor­ doctorate production during the 1960-1985 period ates in 1985, an increase of 10 percent over 1984. (Chart 1). Dramatic growth marked the sixties They received 28 percent of all S /E doctorates with increases for both men and women and awarded in the United States in 1985, up from 25 for both U.S. and foreign citizens in number of percent in 1984, and more than in any previous degrees in each of the major fields. year. Most of these non-U.S. citizens, The annual 81 percent number of S/E doctoral graduates in 1985, held temporary visas. rose from 6,260 in 1960 to a peak of 19,000 in 1972, an increase of • The number of engineering doctorates was over 200 percent. During the second phase, which 3,165 in 1985, a gain of 9 percent over 1984. covered most of the seventies, the total number of Both U.S. and non-U.S. citizens contributed to S/E doctorates declined to a low of 17,050 i?- _1978, the increase. The 1,730 doctoral degrees awarded reflecting the rapid decrease for male U.S. citizens. to foreign citizens in 1985 represented a new high During the third and current phase, gains in the level for this group. A 3 percent increase for U.S. number of doctorates awarded to women and non­ citizens was their second consecutive increase after U.S. citizens have offset the continuing decline for an almost unbroken series of declines following a U.S. male recipients, gradually producing by 1985 peak reached in 1970. Over the 1970-1985 period, an overall 7 percent growth over the 1978 level. the number of engineering doctorates awarded to The annual production of engineering doc­ U.S. citizens fell by one-half, and their share of torates has increased by 31 percent since 1978. the total in this field declined from 7 4 percent to 43 percent.

2 Some respondents report that "China" was their country of citizenship, not allowing a clear distinction to be 1 Additional information is available from the National made between the People's Republic of China and Taiwan. Research Council's annual publication series, Summary Re­ Based on permanent addresses and educational history, port: Doctorate Recipients from U.S. Universiti£s {Washington, however, it is believed that most of those respondents are D.C.) from Taiwan.

763 Chart 1. Long-term trends in doctorate production Table 1. Science/engineering (S/E) doctorates: 1978 and 1985

Thousands Change 35r------~35 F1eld 1978 1985 Number Percent

30 Total S/E degrees 17,048 18,255 1207 7

25 All sciences . 14.625 15,090 465 3 Natural science. 8.562 9,408 846 10 20 Physical sciences . 3.234 3,531 297 9 15 Life sciences 4.369 4,877 508 12 Mathematics 838 689 -149 -18 10 Computer science 121 311 190 157 Non-science/engineering fields

Social and behavioral sciences 6.063 5.682 -381 -6

'65 '70 '75 '80 Social sciences . 3,008 2,607 -401 -13 Year Psychology 3,005 3,075 20 1

SOURCE· NattOnal Sctence Foundalton Engineering . 2,423 3,165 742 31 Non·S/E degrees 13.827 12,946 -881 -6 The 3,165 doctorates in 1985, however, remained below the peak production of 3,500 in 1971 and SOURCE Nattonal Sctence Founaahon in 1972. Within science fields, natural sciences are increasing, while the social and behavioral fields are decreasing. Since 1978, natural sciences Chart 2. Shares of doctorates earned by non-U.S. citizens

gained 10 percent to reach 9,400, while social Percent and behavioral sciences declined by 6 percent. 60 .,.,. ______60 Changes for S/E fields since 1978 are summarized in Table 1. Engineering fields -- -:"" '\. Degrees earned by ~------non·U.S. citizens 40 1978 1985 40 Foreign Citizen Participation Sc1ences 2,383 3,119 1,093 1,728 Doctoral education in the United States continued 30 r- Engineenng - 30 to attract increasing numbers of foreign citizens. All science /engineering fields-...... , The most rapid growth in S/E doctorates earned 20 ...... -_, 20 by non-U.S. citizens occurred in the sixties and ············/····························· Science fields ·-. -· -· -·-·-• early seventies when awards to this group in­ 10 1-·-·-·-·- """' - 10 creased from 960 to more than 4,000. After Non-science/engineering fields I declining to less than 3,500 in 1978, the numbers 0 I I I 0 have again increased rapidly, particularly in the 1978 '80 '82 '84 '85 last three years. In fact, this increase was more Year than sufficient to account for the growth in S/E SOURCE Nal!onal Sttence FoundallOn doctorates; the increase of 1.440 in foreign recipi­ ents with temporary visas since 1978 was greater The trend suggests even more foreign doctorate than the increase of 1,200 in all S/E doctorates. recipients in the years ahead. Foreign citizens earned 4,850 S/E doctorates in 1985, almost 10 percent more than in 1984, and about 21 percent of total S/E doctorates in 1978 Most foreign doctorate recipients have tem­ (Chart 2). This share, showing a definite upward porary visas, which are relatively easy to obtain trend in the last five years, reached 28 percent in and which normally grant them permission to 1985. remain in the United States only while they are The increase in the foreign doctorates since students. This group constitutes an increas­ 1978 follows a substantial increase in the number ing proportion of the non-U.S. recipients. The of full-time foreign S/E graduate students since number of S/E doctorates earned by temporary 1977: 63 percent for science and 71 percent for residents has increased in each year since 1978, engineering between fall 1977 and fall 1984. 3 from 2,500 to 3,950 in 1985. During the same period, the number of S/E doctorates earned by non-U.S. permanent residents declined from 970 3 National Science Foundation, Academic Science/Engi­ to 900. As a result, the proportion of foreign S/E neering: Graduate Enrollment and Support, Fall1984 (Detailed doctorates received by temporary residents rose Statistical Tables) (Washington, D.C., 1986). from 72 percent in 1978 to 81 percent in 1985.

764 Field distribution. For non-U.S. citizens, than for U.S. citizens. Many non-U.S. citizens engineering ha.s been the primary major field with temporary visa.s relied on "other" sources, since 1967, and it has represented a growing which in most ca.ses was support from their home proportion of the awards to non-U.S. citizens countries. since that time. An upturn in the la.st two years World regions. U.S. universities award doc­ ha.s restored the number of U.S. citizens earning torates to students from more than a hundred engineering doctorates to the same level a.s in different countries. For 25 years, East Asia and 1978; however, the doctorates earned by non-U.S. West Asia have each provided about one-fourth citizens have increa.sed over this 7-year period of the foreign recipients, but in the last four by 58 percent, from 1,100 to 1,730. Therefore, years and particularly in 1985, there ha.s been an beginning in 1981 and continuing through 1985, increasing proportion from East Asia. 4 Among non-U.S. citizens have received more than one-half those non-U.S. citizens who reported their country of the engineering doctorates. In 1985, non-U.S. of citizenship in 1985, 32 percent were from East temporary residents received 47 percent, and non­ Asia and 24 percent from West Asia. The recent U.S. permanent residents received 10 percent of increases from East Asia have been provided pri­ the engineering doctorates. marily by China (including Taiwan) and Korea, The engineering disciplines attracting the with increases of 76 percent and 145 percent, re­ most non-U.S. citizens in 1985 were electri­ spectively, since 1981. Doctorate recipients from cal/electronics, mechanical, civil, and chemical Iran reached a peak of 280 in 1984 and declined engineering, in that order. Foreign citizens by 20 percent in 1985 (Table 2). earned between 48 percent and 67 percent of the doctorates in each of these four disciplines. Table 2. Leading countries of origin of foreign citizens Other engineering specialties with high propor­ receiving science/engineering (S/E) doctorates: 1985 tions of doctorates awarded to non-U.S. citizens included aeronautical/ a.stronautical, agricultural, Country of cttizenship S/E doctorates computer, industrial, and ocean engineering. Rep­ resentation of foreign citizens wa.s considerably Total 4,847 lower in some engineering specialties; in bioengi­ Chtna (tncluding Tatwan)' 792 neering, for example, they earned 27 percent of India . 453 the doctorates. Korea. 3t6 Foreign citizens also earned relatively large Iran . 225 Canada. 147 shares of the 1985 doctorates in mathematics Ntgeria. 105 (43 percent), computer sciences ( 37 percent), and Egypt. 100 agricultural sciences (37 percent), but only 17 per­ 100 99 cent of the doctorates in the social and behavioral 95 sciences, and 14 percent in biological sciences. 93 They have increa.sed their shares substantially in Other countries . 1,803 mathematics and computer sciences since 1978. Country not reported . 519 The total number of science doctorates ha.s 'Based on permanent aOdress and educattonal h1story. it is believed ltlat most ol the "Chinese" increa.sed by 465, or 3 percent, since 1978. There respondents are from Tatwan SOURCE Nai!On.ll Scu!!nce Fovndahon wa.s an increa.se of 800 in non-U.S. citizens with temporary visa.s, while the number of U.S. citizens Postdoctoral location. Foreign citizens do and non-U.S. permanent residents declined. As not necessarily leave the United States upon a result, the share earned by non-U.S. temporary receipt of a doctorate from a U.S. institution. residents increa.sed from 12 percent to 18 percent, Temporary visas may be renewed under certain permanent residents declined from 4.5 percent to conditions. Of those S/E recipients in 1985 4.0 percent, and U.S. citizens declined from 83 reporting definite commitments (a signed contract percent to 78 percent. or similar arrangement) and expected location at Support for doctoral study. Foreign graduate the time of graduation,5 97 percent of the U.S. students were more likely than U.S. citizens to citizens, 92 percent of the non-U.S. citizens on obtain their primary support through research permanent visas, and 54 percent of those on and teaching a.ssistantships (Chart 3); Sixty-one temporary visas expected to locate in the United percent of the foreign recipients versus 47 percent States.6 For the latter group, equal numbers of the U.S. citizens received their primary support through these sources. In part, this reflects the limited eligibility of foreign students for direct 4 The countries providing most recipients from East U.S. Federal programs. Those with temporary Asia were China (including Taiwan), Korea, Thailand, status may also face restrictions on employment Japan, and Hong Kong. The principal countries from in the United States because of visa limitations West Asia were India, Iran, Israel, and Turkey. 5 Sixty-seven percent reported definite commitments in or language difficulties, and are less likely to have 1985. working spouses. Family contributions were a 6 Among all foreign 8/E doctorate recipients, 41 per­ more frequent primary source for foreign citizens cent expected to be located in the United States and 31

765 Chart 3. Primary source of support in graduate school of science/engineering doctorate recipients, by citizenship: 1985'

Federal fellowships/ traineeships and other tederal support University related: Research assistantships 1------'

Own earnings 1------'

Spouse's earnings

Family contributions

Other

20 40 60 0 20 40 60 0 20 40 60 Percent

'ExclUdes doctorate rac1ptents whO Okl not report pnmary SOYrce of support. SOURCE: Nat10na1 Science Foonelatlon had commitments for postdoctoral study as for earned by men decreased by 135. Consequently, employment, about 600 each. Recipients of women increased their representation among S /E doctorates in engineering reported higher rates of doctorate recipients, though not as rapidly as in definite commitments and expected locations in the seventies and early eighties. the United States than did the science doctorates. The representation of women among doctor­ The rates in engineering were 96 percent for ate recipients is greater for U.S. than for non-U.S. U.S. citizens, 96 percent for non-U.S. citizens on citizens. Among U.S. citizens, 30 percent of the permanent visas, and 63 percent for those with S/E doctorates in 1985 were earned by women; temporary visas. the comparable figures for non-U.S. citizens with permanent visas were 20 percent, and only 13 Participation of Women percent for those with temporary visas. Field distribution. Traditionally, The increasing participation of women, both U.S. women have earned more doctorates in non-S /E fields than in and non-U.S. citizens, continues to be one of the S/E fields. For men, just the most significant developments in higher education opposite is true. In 1985, 66 percent of the doctorates earned over the last 20 years. The proportion of women by men were in S/E fields, compared with only earning master's degrees (S/E and non-S/E com­ 43 percent for women. Nonetheless, women bined) exceeded one-half of all master's degrees have increased their representation in S/E fields; awarded in the United States for the first time in their 198 engineering degrees accounted for more 1981, and they exceeded one-half of all bachelor's than 6 percent of the total doctorates in that degrees in 1982.7 The proportion earning S/E field. Inasmuch as the full-time enrollment of doctorates reached 25 percent in 1985 (Chart 4). women in undergraduate engineering increased Women earned 4,650 S/E doctorates in 1985, from 34,000 in 1978 to 62,700 in 1984, an 84 only 2 percent more than the previous year, but percent increase, both the number and share of 40 percent more than in 1978. During the last engineering doctorates awarded to seven years women increased their annual number women can be expected to increase. 8 of S/E degrees by over 1,300, while the number Women earned 29 percent of all science doctorates in 1985. They earned 23 percent of percent outside the United States; 29 percent did not report a location. 7 Department of Education, National Center for Edu­ 8 Engineering Manpower Commission, Engineering and cation Statistics, Survey of Degrees and Other Fomwl Awards Technology Enrollment, Fall1984 (New York: American Asso­ Conferred, annual surveys (Washington, D.C.) ciation of Engineering Societies).

766 Chart 4. Shares of doctorates earned by women The field distributions differed for U.S. and foreign women. More than 53 percent of U.S. Percent women chose social and behavioral science fields, 60 60 compared to less than 30 percent of foreign women; this difference resulted primarily from differences in doctorates in psychology, chosen by 37 percent of U.S. women but by only 8 percent of non-U.S. women. Biological sciences attracted nearly equal proportions of the U.S. and foreign women earning S/E doctorates (27 percent), while engineering doctorates were more likely among foreign women (11 percent) than U.S. women (3 percent).

Year

SOURCE. National Sctence Foundatton the degrees in natural sciences and 41 percent in social and behavioral sciences. The greatest proportions of women among doctorate recipients in 1985, as in the last 10 years, were in psychology (49 percent of the doctorates), social sciences ( 32 percent), and life sciences ( 29 percent). The lowest participation rate for women was in computer sciences ( 11 percent).

p.L • 8 ~ ('/ 1-""(t\EMA)'!. .... -~ ~--- c_, ~~~;tW>·~,~~ ; \...: ;:) I • ~ m Ten Lectures on Operator Algebras -'(\ ~ .::;j 0 William Arveson 1Vo"'"'' ""'o (IIVDED- \ $' This book makes available to a wider audience the lectures on nonselfadjoint operator algebras given by the author at a conference in Texas in 1983. The theory of nonselfadjoint algebras has for long been the Cinderella of operator algebra theory but. mainly because of the work of Arveson and his students. it is now beginning to establish itself alongside the ··ugly sisters·· of C *-algebra and von Neumann algebra theory. Those familiar with the author's expository style will not be surprised to find that these ten lectures are beautifully presented. extremely readable and full of new insights. Topics covered include commutative subspace lattices and problems of spectral synthesis. the absorption principle and its application to solve the similarity problem. quasitriangular algebras and applications to single operator theory. There is also a very interesting section on dilation theory and its use in proving the Feynman-Kac formula. - E. Christopher Lance University of Leeds

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767 Letters to the Editor

AMS Discount Policy making Mathematical Reviews available to more mathematicians. Last summer the AMS sent me a letter inviting Bonnie Gold me to join, reminding me that among the ad­ Associate Professor of vantages of membership was that "members are Mathematics also entitled to purchase Society publications at Wabash College a substantial discount from list price." I thought, (Received June 2, 1986) "Great, now I can keep up with what's going on, since I can get Current Mathematical Publications COMMENT BY THE SECRETARY OF THE AMS: and Mathematical Reviews at members prices, The policy established by the Executive Commit­ which I can afford." So I joined and sent away tee and Board of Trustees stated "the following for these publications. I got back a letter from policy was approved with the understanding that the AMS saying that since my institution didn't it may have to be modified somewhat after review subscribe to Mathematical Reviews, I couldn't by the Membership and Sales Department: receive the members discount on that journal. Discounted subscriptions to individuals for all AMS journals will be eliminated, with the This is an outrage! First of all, I doubt following exceptions: the legality of the Society offering something and then withdrawing that offer in this way; a. to individuals at institutions with subscrip­ but more importantly, this is a dreadfully unfair tions to the journal involved; policy. Here I am, at a small college, with b. to existing subscribers in cases where AMS a teaching load of three courses per semester, has verification that the subscription is indeed and many administrative duties as well. It's for personal use only and; hard enough as is with these time constraints to do any mathematical research. Further, being c. by special appeal. at some distance from a major university, I A hardship procedure will be established so don't have the opportunity others have to discuss that smaller universities, colleges, and other ideas with people interested in my field. Now institutions and organizations can subscribe at the big shots at the AMS, all of whom are at reduced rates upon documenting that certain universities large enough to afford Mathematical criteria have been met." Reviews and many other journals, are determined to make it impossible for me even to find out Both special appeals and hardship applications what is happening. I can't afford, on the salary will be handled by a standing AMS Committee they pay at small colleges, to buy Mathematical on Subscriptions. Reviews at the regular price; my institution, There is a need for such a policy because it with at most three people doing anything like is a demonstrable fact that there are individuals active mathematical research, can't justify buying who have subscribed to journals in their own Mathematical Reviews. So the AMS is cutting off name at a greatly reduced price but deposited people like me from research. Perhaps we couldn't them immediately in a library for general use. make any brilliant contributions to mathematics This is not the purpose for which individual in any case-but you folks have some nerve subscriptions are intended. denying us the chance to try. Note that in the United States and Canada there is institutional membership. Institutional I've heard, from talking with people at larger members are entitled to discounts that far exceed institutions, that this policy is an attempt to institutional dues for all but the most produc­ prevent institutions from cheating and buying tive research institutions, particularly if they are Mathematical Reviews at a cheap rate through subscribers to Mathematical Reviews. This is a members. However, a fairer way to accomplish correct route for library subscriptions. this would be to have a sliding scale of institu­ On the other hand, the letter inviting poten­ tional rates-for example, that institutions with tial members to join was deficient in its explana­ fewer than 10 full-time faculty, or fewer than tion of privileges, as Professor Gold has noted. 2,000 students, be able to purchase Mathematical It should say, at a minimum, that "Members Reviews at, say, twice the price charged individ­ are also entitled to purchase Society publications ual members. Were that to happen, my college at a substantial discount from list price under and I expect many others would be able to sub­ prescribed conditions." Perhaps reference should scribe, and then my colleagues and I could read be made to the catalogue of publications. The Mathematical Reviews in the library as people letter is being rewritten. The Society is grateful at larger institutions do. This would have the to Professor Gold for calling attention to this double advantage of stopping the cheating and of problem.

768 Fortunately, it was possible to handle the Professor Coleman's letter. The following is his subscription of Professor Gold as a special appeal. response. It was, of course, by no means my intention The Mordell Conjecture to minimize the significance of the work of Drs. Why in the first paragraph of David Harris's Mordell and Weil. They are among the greatest article, 'The Mordell Conjecture' [Not. AMS 33, mathematicians of the twentieth century. For No. 3 (1986), pp. 443-449], does Harris call the the most part, I must plead that the brevity Mordell Conjecture the "so-called Mordell Con­ forced on my article necessitated not giving many jecture"? The phrase "so-called" usually means people their due. I would hope that the excellent that the name following is inappropriate and it and much longer article by Mazur referred to is usually used in a derogatory way (see the by Professor Coleman would remedy some of_ my OED or Fowler, Modern English Usage). Per­ shortcomings. In particular, I would have hked haps, Harris believes Mordell was anticipated by to mention the work of Coleman described by someone else, if so, by whom? Moreover, he Mazur (p. 226), along with the work of Raynaud, doesn't call the Wei! Conjectures the "so-called Neron-Tate Szpiro Granville & Heath-Brown, Wei! Conjectures," even though they were already Arakelov, Zarhin, ~nd many others. If Weil's conjectured and partially proven for hyperelliptic work in establishing various special cases of the curves by Artin [Quadratische Korper im Gebeite Weil Conjectures was not mentioned, this was not der hohren Kongruenzen, I and II, Math. Zeit. intended as a slight. I did refer the reader to 19 (1924), pp. 153-246], and proven for ellip­ Lang's book, Abelian Varieties, which gives the tic curves by Hasse [Abstrakte begrundung der proper credit (for example, see pp. 246-247). komplexen multiplikation und Riemannsche Ver­ I chose to entitle my article The Mordell mutang in Funktionkorpern, Abstr. Math. Sem. Conjecture. I refer to the conjecture under Univ. Hamburg 10 (1934), pp. 325-348]. It is this name many times. The one appearance of perhaps inconsiderate of Harris not to mention, the phrase "so-called" will serve perhaps as an after he remarks that only the Abelian variety implicit tribute to the insight of Poincare that case of the Wei! Conjectures was needed for Falt­ I did not have the space to spell out. But, ings's proof (§2, p. 445), that their proof in this truthfully, I must confess that I did n?t inten_d case is due to Wei! (not Deligne). The only anything beyond that the Mordell ConJecture 1s mathematician, as far as I know, who saw a the commonly used name of a certain result. In connection between the geometry of the complex the article, I refer to Mordell's 1923 paper as solutions of a system of equations and the ra­ "the seminal paper on the subject" and say "these tional solutions of the system, before Mordell, problems arose out of the work of Mordell." The is Poincare [Sur les proprietes arithmetiques des name "Mordell Conjecture" has been universally courbes algebriques, J. de Liouville (V) 1 (1901), used for sixty years and I am confident will pp. 161-233], who apparently took what is now continue to appear in the literature forever. Wei! Theorem to be self­ known as the Mordell- David Harris evident. Columbia, Maryland If Harris uses the phrase "so-called" because (Received July 7, 1986) Mordell's Conjecture is no longer a conjecture, why does he not apply the same logic to the Wei! Conjectures? It took great insight on the Broadening Sources of Research Funding part of Mordell to see over sixty years into the For almost ten years, the mathematical commu­ mathematical future and I think the Mordell nity has been trying to broaden the sources of Conjecture should always be so called. research funding in the mathematical sciences. Robert F. Coleman The David Report encouraged not only the dou­ Professor of Mathematics bling of such funding, but urged that such support University of California, be sought from government agencies other than Berkeley the NSF. Such efforts are beginning to enjoy some (Received June 23, 1986) success, but this success is creating a new kind of P.S. For more information on the history of the problem. Mordell-Weil Theorem and Mordell's Conjecture, One example is that faculty funded by agen­ see Lang's book, Fundamentals of Diaphantine cies other than the NSF can sometimes "buy Geometry (Springer-Verlag, 1983) and Mazur's back" their time and reduce their teaching; also article, Arithmetic on curves, Bull. AMS 14, No. they can invite visitors, have post-docs, and have 2 (1986), 207-259. For more information on the the possibility of three months summer support. history of the Weil conjectures, see Katz's article, These differences in the funding of the two groups An overview of Deligne 's proof, Proc. Symp. Pure of "haves " those with NSF funding and those Math. AMS 28 (1976). with non-NSF funding, seem to be causing severe morale problems for the former group. EDITOR'S NOTE: David Harris was asked if Second, some mathematicians eligible to ap­ he would like to comment on the points raised in ply for DOD support, who would be eager to

769 "operate" in this style-that is, group research DOD Funding in Mathematics with many graduate students, post-docs, and The good part is that a lot of deserving work in visitors-and whose research would profit from mathematics, and dynamical systems in particu­ such an arrangement, have decided not to because lar will now be supported. The bad part is that of the linkage to Defense. This is the case de­ CIA and the Department of Defense are doing spite the claim that their research would be no th~ the supporting. A priori, such support might not more restricted at DOD (unclassified) than it is be bad, indeed the people whom I've asked had no presently at the NSF. complaints about DOD funding in the fifties. The Thirdly, a mathematician who indicated a issues are political and internal to mathematics desire to apply to DOD, mainly because he would itself. like support for a postdoc, listened when I advised I received my Ph.D. at Berkeley in 1967. I him how to proceed. At the conclusion of my had been active in the student movement there. In conversation advising him to set up a group, he 1968, I applied for NSF and NATO postdoctoral responded that he would be so busy administering fellowships. The NATO program was a State such a grant that he wouldn't have time to do Department program administered by the NSF. research. The main differences, if I am not mistaken, were Lastly, many mathematicians, familiar with that the NSF postdoc paid more and that the the grant application process at the NSF based NATO postdoc application required no loyalty on peer review, are quite upset to learn that other oath or record of arrest, while the NSF did. I agencies operate differently. At these agencies received both fellowships and accepted the NATO the views of program directors and small panels post doc. are often all important in making decisions which In 1969-1970, I used my fellowship to visit affect individual grants. L'Institute des Hautes Etudes Scientifiques out­ side Paris. I mentioned to Grothendieck that I'm not proposing any change in what we've I had a NATO postdoc. He told me that he been doing-namely, attempting to broaden the thought that the military posed a serious threat sources of research funding in the mathematical of destruction to the world and that he thought sciences--only that we must recognize that differ­ it wrong to cooperate with them. Grothendieck ences exist and then try to educate our colleagues on how research is funded by other agencies. In­ was just then resigning his position at the IRES because a small percentage of its funding came dividuals and departments would then be free to from the French military. choose the agencies from which to seek funding. In 1984, my Ph.D. student, Helena Wis­ Since becoming Dean and learning how re­ niewski, took a job with the CIA. While she was search is done and funded in the physical sciences, still considering her new position, I had lunch I have the greatest appreciation of the differences. with her and voiced my disapproval. Several An important one is the support staff (largely months later, Helena called and tempted me with supported by grants) which enables the faculty the possibilities of DARPA-CIA funding. This of a good chemistry or physics department to could have significantly improved my working administer grants and contracts. The lack of conditions and those of people around me at the such support staff makes an enormous difference City University of New York, while providing in what faculty and departments in the mathe­ support for research in dynamical systems and matical sciences can or can not do. This extends geometric complexity theory. There seemed to even to the preparation of grant proposals; since be no strings attached, but when I inquired as proposals take time to prepare and in the math­ to prior publication restraints, the CIA insisted ematical sciences this is usually researcher's time on them. I made up my mind at that point and hence, time taken away from doing research. not to get involved. Ultimately, the universities Let me conclude with an example. I know a accepting the grants negotiated the prior publica­ mathematics department successful in obtaining tion restraints out of the agreement, but I don't a VAX from grant funding, which was forced to regret my decision, even though I have frequently turn it over to the computer science department coveted the support. That kind of money is not because the mathematics department didn't have to be found elsewhere. With working conditions, the support staff to manage it. as they are in CUNY, I am leaving my job and I believe these are problems which the math­ going to IBM in Yorktown. ematical community must begin discussing. I think that the Reagan government has Ronald G. Douglas been escalating the arms race enormously, that Dean of the Division the "star wars" initiative is very dangerous and of Physical Sciences & wasteful, and that DARPA-CIA funding forms a Mathematics continuum with SDI research. My position has State University of come closer to Grothendieck's. Given the current New York at Stony Brook circumstances, I am not going to apply for funding (Received August 13, 1986) from DOD agencies.

770 There are other problems as well. Defense two are not comparable since they serve different agencies are not in the business of promoting purposes. free and open exchange of information. A fair With a WYSIWYG word processor, it is amount of vigilance will be required in monitoring possible to take a manuscript from its first rough or structuring programs to see that they func­ draft to its final (typewriter quality) form without tion appropriately within an academic and free ever rewriting or retyping anything except for the research environment. The National Intelligence precise passages requiring change. To be able to Mathematics and Multiprocessor Project kicked do this, it is absolutely essential that the program off its program with a nonpublicly announced, produce the mathematics (as mathematics, not by invitation only, meeting at CUNY. In some code) on the screen exactly as it is being typed. ways it was no big deal-most people at CUNY With 'lEX, this is not possible. were invited and invitations were easy to get, but 'lEX will be a great boon to printers, but its many people in the New York area who would benefits for the individual mathematician are less have been interested had no way of knowing that obvious. For example, I don't quite understand the meeting was even happening. why (as Palais suggests) a mathematician should This is the only "closed" meeting in dynam­ want to take his finished manuscript and type it ical systems that I am aware of in the twenty into JEX. If he needs to produce a typewritten or so years I have been in the business, and a manuscript himself, a WYSIWYG word processor bad precedent. Some of us posted a public let­ or even an ordinary typewriter will be faster and ter, objecting to this structure. But unlike their easier, and a 'lEX professional will produce a 'lEX universities which resisted the prior publications file with fewer errors and a better appearance. restraints, the program participants did little or Also, I am less than overwhelmed by the nothing about the meeting. It remained to the AMS's offer (same Notices, pp. 299-302) to al­ AMS to pass a resolution opposing closed meet­ low me to typeset my articles myself-at least ings. The lesson may be that an institutional page charges should be forgiven for any article overseer is necessary, but from what institution? submitted in JEX. One of the participants in the NIMMP project One may hope that one day there will be whose funding was channelled through the ONR an efficient WYSIWYG word processor that will instead of the CIA has told me that his grant was produce as its final output a 'lEX file. Then it wiped out by the Gramm Rudman reductions; will be possible to take a manuscript all the way they didn't actually get the money. Someone from its first rough draft to a typeset document slightly more paranoid than I might think that without any superfluous retyping. the whole program of support for mathematics J. S. Milne being organized by the DOD will just be the Professor of Mathematics Gramm Rudman reductions for SDI. University of Michigan, Ann In any case, DOD will have to defend the Arbor legitimacy of its math program sometime if the (Received May 23, 1986) program is to continue. I think that legitimacy will have to be in the form of actual technology transfers to DOD or defense related industries. Ultimately, the work supported by these grants Policy on Letters to the Editor will shift towards applications in DOD's interests. Letters submitted for publication in Natices are reviewed Work tends to follow money. With a lot of by the Editorial Committee, whose task is to determine which ones are suitable for publication. The publication money coming from Defense, we have to wonder schedule requires from two to four months between re­ how it will effect mathematics. Over the long run, ceipt of the letter in Providence and publication of the will their funds provide additional support for earliest issue of Notices in which it could appear. mathematics, or will they provide a· competitive Publication decisions are ultimately made by ma­ market for mathematical talent that will redirect jority vote of the Editorial Committee, with ample pro­ vision for prior discussion by committee members, by it to narrow goals? I know that we need more. mail or at meetings. Because of this discussion period, research support, but I am afraid that the answer some letters may require as much as seven months be­ to the question is the latter. fore a final decision is made. Letters which have been, or may be, published elsewhere will be considered, but the Michael Shub Managing Editor of Notices should be informed of this New York, New York fact when the letter is submitted. (Received June 2, 1986) The committee reserves the right to edit letters. Notices does not ordinarily publish complaints about reviews of books or articles, although rebuttals WYSIWYG Word Processing and correspondence concerning reviews in Bulletin of the American Mathei'IU1I:ical Society will be considered for pub­ 1EX is a great program, and I wish it well, lication. but Palais's article (March 1986, pp. 303~308) Letters should be mailed to the Editor of Notices, in which he compares 'lEX with a what-you-see­ American Mathematical Society, Post Office Box 6248, Providence, Rhode Island 02940, and will be acknowl­ is-what-you-get scientific word processor to the edged on receipt. detriment of the latter is very misleading: the

771 AMS Election 1986 Candidates COLLOQUIUM PUBLICATIONS OFFICERS Vice President (one to be elected) William P. Thurston Karen Uhlenbeck The Geometric Topology of 3-Manifolds Secretary* Everett Pitcher R H Bing Associate Secretaries* The book starts with a treatment of some of the Frank T. Birtel W. Wistar Comfort geometric and topological properties of the plane. (Southeast) (East) This treatment is expanded to study such funda· Treasurer* Franklin P. Peterson mental properties of 3-space as the PL Schoenflies theorem, Dehn's lemma, the loop theorem, and the Associate Treasurer* Steve Armentrout side approximation theorem. Applications of these Member-at-Large of the Council (five to be elected) fundamental results are made to develop further of 3-manifolds. C. Edmund Burgess Linda A. Ness properties is most meaningful to a mathematician David Drasin Marc A. Rieffel This book interested in geometry who has had at least a begin­ H. Blaine Lawson Clarence W. Wilkerson ning graduate course in topology. While a student Howard A. Masur Scott Warner Williams can start the book with less preparation, the chances Carol S. Wood Eric C. Milner are that those with weaker backgrounds will bog Yiannis N. Moschovakis down if they work without the benefit of discussion. Board of Trustees* (one to be elected) The book belongs in both graduate and under­ Ronald L. Graham graduate libraries. It can serve as a useful reference for a graduate student in topology or a researcher Committee to Monitor Problems in topology. A study of the book should provide in Communication* the reader with a better understanding of the phys­ Sheldon Axler Marian B. Pour-El ical properties of Euclidean 3-space-the space in Publication Committees* which we presume we live. The reader should Jearn of some unsolved problems that continue to baffle Richard B. Melrose American Journal researchers. of Mathematics The most profound result in the volume is the Morris W. Hirsch Bulletin side approximation theorem. However, as a reference Colloquium H. Jerome Keisler tool some of the preliminary results and some of the Mathematical Reviews Melvin Hochster applications may be used more frequently. Mathematical Surveys and Monographs Victor W. Guillemin Mathematics of Walter Gautschi Contents Computation John E. Osborn Preface Proceedings Paul S. Muhly 1. Planar complexes Transactions Ralph L. Cohen 2. PL planar maps and Memoirs Ronald K. Getoor 3. The Schoenflies theorem Jerry L. Kazdan 4. Wild 2-spheres David J. Saltman 5. The generalized Schoenflies theorem 6. The fundamental group Robert J. Zimmer 7. Mapping onto spheres NOMINATING COMMITTEE FOR 1987 8. Linking (Preferential Ballot, four to be elected) 9. Separation M. Salah Baouendi Werner C. Rheinboldt Subject Classification: 57 -XX Robert L. Devaney Bruce L. Rothschild 1980 Mathematics Colloquium Publications Paul C. Fife Frank L. Spitzer Volume 40, viii + 240 pages (hard cover) Carl Pomerance William P. Ziemer List price $54, institutional member $43, individual member $32 *Uncontested offices. ISBN 0·8218-1040-5; LC 83-14962 Publication date: October 1983 To order, please specify COLL/40NA

Shipping and handling charges: surface delivery - $2 first book. $1 each additional. maximum $25: air delivery - $5 ftrst book. $3 each additional. maximum $100. Prepayment Is required. Order from American Mathematical Society. PO Box 1571. Annex Station. Providence. Rl 02901-1571 USA.

772 Queries Edited by Hans Samelson and Stuart Antman

QUESTIONS ARE WELCOMED from AMS members regarding mathematical matters such as details of, or references to, vaguely remembered theorems, sources of exposition of folk theorems, or the state of current knowledge concerning published or unpublished conjectures. This is not intended as a problem corner, except for occasional lists of problems collected at mathematical meetings. REPLIES from readers will, when appropriate, be edited into a composite answer and published in a subsequent column. All answers received will be forwarded to the questioner. QUERIES and RESPONSES should be typewritten if at all possible and sent to Queries Column, American Mathematical Society, P. 0. Box 6248, Providence, Rhode Island 02940.

Queries all G-invariant polynomials are generated by the homogeneous components of Pl, ... , Pn? The S60. Melvin Henriksen and Keith Saints (Depart­ answer is positive in some special cases, e.g. com­ ment of Mathematics, Harvey Mudd College, plete graph. Any references and partial results Claremont, CA 91711). Given an origin 0 in are welcomed. the plane and a unit distance 1, the set F of complex numbers x + iy such that ( x, y) can be 362. R. Gurevif (Department of Mathematics, constructed with the aid of a collapsible compass University of Illinois, 1409 West Green Street, and an unmarked straightedge is well known to Urbana, IL 61801). Algebraists use the Zariski be the field of algebraic numbers of degree a topology on the set Spec(·) of prime ideals of a ring power of 2. One may show that an angle of 60° and don't mention M. H. Stone in this connection. cannot be trisected with the aid of these tools A handbook by G. Gratzer: General lattice theory by showing that cos 20° + i sin 20° is not in F. (Akademie-Verlag, Berlin 1978) considers briefly As was known to Archimedes, if two marks are the same construction for lattices, attributes it made on the straightedge, arbitrary angles can be to Stone, and never mentions Zariski. Can one trisected. Is there a known characterization of the clarify this story? Are there recognitions of field G of complex numbers x + iy such that (x, y) this identity in the literature? Not only the is constructible with these augmented tools? If constructions are the same but even Spec(ring) = not, what is known about G? It can be deduced Spec(lattice of its finitely generated ideals). easily from a paper of Robert C. Yates, The angle ruler, the marked rule, and the carpenter's square, 363. Alexander Abian (Department of Mathe­ National Math. Mag. 15 (1940), 61-73, that if matics, Iowa State University, Ames, lA 50011). p(x) is a (real) quartic polynomial in G[x], then Several years ago the following conjecture (Abian) the real roots of p(x) = 0 are in G. appeared in print: Let f be an entire transcen­ dental function. Then there exists an unbounded S61. David Gurarie (Mathematics Department, sequence of complex numbers Cn such that every Case Western Reserve University, Cleveland, OH Cn is a zero of some partial sum of the Taylor series 44106). Let G be the group of automorphisms of a (say, around 0) off and such that limn/( en) = 0. finite graph r c {1, 2, ... 'n} X {1, 2, ... 'n}. The Is anything known about the only unverified group G acts by automorphisms of the polyno­ case: f of infinite order, with 0 as its Picard mial ring C[x1; ... ; Xn], permuting the variables exceptional value? xi, ... , Xn· Denote by Ar = (aii) the adjacency matrix of r: 364. David Aldous (Department of Statistics, Statistical Laboratory, University of California, if i and j are connected by Berkeley, CA 94720). Let A be an n x n positive 1 aij = { an edge ofr, matrix. Let At, .A2, ... , An be its eigenvalues, with 0 otherwise, Re(Ai) nonincreasing. It is well known that A1 is real. There are various conditions, such as and define the matrix B self-adjointness, which imply that all the A's are diag(xt, ... , Xn) + Ar. real. Are there known conditions which imply Let that .A 2 is real, without implying that all the A's are real? I am particularly interested in the case Pi =Xi + · · · + Xn, · · · , Pn = Pn(Xt, · · · , Xn) of stochastic matrices. be the coefficients of the characteristic polynomial 365. Pawel Siwak (Computer Science Centre, Technical University of Poznan, Pl. Sklodowskiej­ Det(.U- B)= An- PiAn-l + · · · + (-1tPn· Curie 5, 60-965 Poznan, Poland). Let G = (V, E) denote a digraph of outdegree 2 with all p vertices Obviously the polynomials Pi, P2, ... , Pn are G­ V labelled by either 0 or 1. For any path the invariant, as well as their homogeneous com­ graph generates a binary sequence (Moore type ponents of degree 1, 2, ... , n. Is it true that automaton), each symbol according to the vertex

773 which G passes through. What is the best possi­ 353(b). (vol. 33, June 1986, Aldo Peretti) Is ble ratio m/p for G, where m denotes the length of the shortest sequence which is nongenerable by 1- = (1 - (1 + - 1-) G? A family of graphs G with p = 3, 5, 7, ... is !e !)2 . 3·2-3 known such that m = .2p- 2. ·(l- 366. Leroy F. Meyers (Ohio State University, 231 4·3·2~4-3+4)···? W. 18th Avenue, Columbus, OH 43210). Is there Reply: This unusual-looking product is actually any way, suitable for presenting to unsophisti­ elementary. The easiest approach probably is to cated students, of defining the determinant of recognize the general term as Sk/Sk-b where a square matrix A which avoids the complica­ tions of the usual definitions (inversions, minors, k ( l)k-1 or alternating multilinear forms)? Here is one sk = L - ' ; proposal. 1 n. If the entries of A belong to a field (or, more so product = lim Sk = 1 - 1/e. (Contributed generally, to a Euclidean domain) D, then A can by E. Pinney, H. W. Gould, L. W. Shapiro, R. be reduced to an upper triangular matrix B by S. Stepleman, D. Coppersmith, M. Newman, M. a succession of shears (replacing one row by that Artzrouni, J. Greene, J. Shallit, R. M. Ziff) row plus a multiple of another row), and the determinant of A is defined to be the product of 358. (vol. 33, p. 629, August 1986, David Halprin) the diagonal entries of B. How can the product be References to the "elastica" curve. Reply: see A. shown to be independent of the sequence of shears Eagle, Elliptic functions, Galloway-Porter, Cam­ used to produce an upper triangular matrix? bridge, 1958, and C. Muses, Destiny and control in human systems: Chronotopology, Kluwer-Nijhoff, Boston, 1985, pp. 62-67. 326. (vol. 32, p. 379, June 1985, Aldo Peretti) Responses dx -? The editors would like to thank all those who sent roo in replies. } 2 x(x2 - l)log x - · 2 353(a). (vol. 33, p. 519, June 1986, Aldo Peretti) Reply: With t = 1/x this becomes Express f1(1 + x; + Yi) as polynomial in the {1/4 dt oo {1/4 tn dt elementary symmetric functions of the x;, say Ui, } 0 (1- t)log t = - } log t; and of the Yi, say Tj. Reply: Putting ~ 0 and tndt p(t) = n(t- Xi) arid q(t) = (t- Yj), I log t = Ei((n + l)log t) (the exponential-integral function), see I. S. this asks for the resultant of p( t - 1) and Gradshteyn-I. M. Ryzhik, Tables of integrals, se­ (-l)mq(-t) in terms of the coefficients of p(t) ries, and products, pp. 204 and 929. (Contributed and q(t). There is the standard determinant by Seung Jin Bang) formula (Sylvester) for the resultant of the two polynomials (e.g., S. Lang, Algebra (1974) Chap­ 343. (vol. 33, p. 53, January 1986, Thomas L. ter V, Par. 10); the step from p(t) and q(t) to Drucker) Reliable birth and death dates of p( t - 1) and ( -1 )nq( -t) is elementary. There mathematicians. Reply: Some sources: Index are other expressions for the resultant; one due biographique de l'Academie des Sciences 1666- to Bezout (probably) [see T. Muir, The theory of 1978, Gauthier-Villars 1979, Paris; A. I. Borodin determinants I, Dover (1960)], and one in terms & A. S. Bugai, Bibiliograficeskie slovar' dejateld of S-functions [see D. E. Littlewood, The theory v oblasti matematiki (Kiev, 1979); Wiadomosci of group characters, Oxford (1950)]. See also matematyczme (Warsaw). Some comments: The I. G. MacDonald, Symmetric functions and Hall mathematician Heinrich Weber (1842-1913) is of­ polynomials, Oxford (1979), p. 37, or A. Las­ ten confuxed with the physicist H. Friedrich We­ coux, Classes de Chern d'un produit tensoriel, C. ber (1843-1912); both did some work on Bessel R. Acad. Sci. Paris 286A (1978) pp. 385-387] functions. In addition there is another Heinrich for the answer to the query. It may be noted Weber, a physicist, born 1839. There are three that the expansion of the nonsymmetrical function Markovs: A. A. (1856-1922), A. A. Jr. (born f1 (1 +Xi+ Yi), i + j :S n, give more general polyno­ 1903), and V. A. (1871-1897), brother of A. A. mials than Schur functions and provide a Newton Finally, there may be confusion between Georg interpolation in several variables (see Seminaire Cantor (1845-1918) and Moritzi B. Cantor (1829-- d'Algebre Malliavin 1984, Springer (L. N. 1146)). 1920) (M. Cantor could be Monsieur or Maurice (Contributed by A. Lascoux, V. Papanicolaou, J. in French). (Contributed by I. H. Anellis, H. R. Stembridge) Cartan, B. Crstici, Z. Semadeni)

774 Logan, October 10-11, Utah State University Program for the 828th Meeting

The eight hundred and twenty-eighth meeting of Geometric topology, DAVID WRIGHT and J. the American Mathematical Society will be held W. CANNON, Brigham Young University. The at Utah State University in Logan, Utah, on speakers are Colin Adams, F. D. Ancel, Mladen Friday and Saturday, October 10-11, 1986. All Bestvina, Philip L. Bowers, John L. Bryant, scientific sessions will take place in the Eccles C. E. Burgess, J. W. Cannon, W. T. Eaton, Conference Center. William J. Floyd, David Gabai, Dennis J. Garity, Matt Grayson, Chi Fai Ho, Jim Hoste, Stephen Invited Addresses Humphries, Louis H. Kauffman, R. C. Lacher, By invitation of the Committee to Select Hour Duane Loveland, Jerome Levine, Vo Thanh Liem, Speakers for Far Western Sectional Meetings, John Luecke, Gordana Matic, Yoav Moriah, Dale there will be two invited one-hour addresses. Rolfson, T. Benny Rushing, Martin Scharlemann, The speakers, the titles of their talks, and the Jonathan Simon, Michael Starbird, K. Wolcott, scheduled times of presentation are: and David G. Wright. PETER LI, University of Utah, Harmonic functions on complete manifolds, 11:00 a.m. Sat­ Contributed Papers urday. There will also be sessions for contributed ten­ KENNETH C. MILLETT, University of Cali­ minute papers. Late papers will not be accommo­ fornia, Santa Barbara, New combinatorial methods dated. in three dimensional topology, 1:30 p.m. Friday. Registration Spedal Sessions The meeting registration desk will be located on By invitation of the same committee, there will the main floor of the Eccles Conference Center. be five special sessions of selected papers. The The desk will be open from 10:00 a.m. to 4:30 topics of these sessions, the names and affiliations p.m. on Friday, and from 8:00 a.m. until noon of the organizers, and final lists of the speakers on Saturday. The registration fees are $10 for are as follows: members of the AMS, $16 for nonmembers, and Applications and computational aspects of $5 for students or unemployed mathematicians. numerical continuation methods, EUGENE ALL­ GOWER, Colorado State University. The speakers Petition Table are Eusebius Doedel, Kurt Georg, R. Baker Kear­ A petition table will be set up in the registration fott, Tien-Yien Li, Hans D. Mittelman, Alexander area. Additional information can be found in a Morgan, Roy Plastock, Aubrey B. Poore, John box in the San Antonio meeting announcement in Tavantzis, Chris Tiahrt, and Layne T. Watson. this issue of Notices. Analysis on manifolds, S. Y. CHENG, Uni­ versity of California, Los Angeles. The speakers Accommodations are Michael Anderson, Hui-Dong Cao, Robert A block of rooms is being held for conference E. Greene, Chong Kyu Han, John Douglas participants at the University Inn. This is a Moore, Vladimir Oliker, Luen-Fai Tam, Andrejs full-service hotel complex located on campus. Treibergs, Johan Tysk, and S. Walter Wei. Other motels are located within 1.5 to 3 miles of Representations of reductive groups, DAVID campus in downtown Logan. Participants should H. COLLINGWOOD, University of Oregon. The make their own reservations directly with the speakers are Brian D. Boe, JenTseh Chang, De­ University Inn or other motel of their choice. vra Garfinkle, Henryk Hecht, Ronald S. Irving, Rates are subject to possible change and do not Joseph F. Johnson, Dragan Mili6c, Ivan Mirkovic, include local taxes. Tomasz Przebinda, Hugo Rossi, Roberto Scara­ University Inn muzzi, Brad Shelton, Peter Trombi, and Joseph Utah State University, Logan, UT 84322 A. Wolf. Telephone: 801-750-1153 Random fields, random measures and appli­ Single $28 Double $32 cations, ED WAYMIRE, Oregon State University. The speakers are Kenneth S. Alexander, Florin Baugh Motel Avram, Rabi N. Bhattacharya, Michael Brennan, 153 South Main Street, Logan, UT 84321 Robert M. Burton, Colleen D. Cutler, R.W.R. Telephone: 801-752-5220 Darling, S. N. Ethier, Marie M. Franzoso, E. Single $26 and up Double $30 and up Gine, Joseph Horowitz, Charles M. Newman, M. Best Western Weston's Lamplighter Ossiander, Richard F. Serfozo, Winfried Stute, 250 North Main Street, Logan, UT 84321 Murad S. Taqqu, Ed Waymire, and Stanley C. Telephone: 801-752-5700 Williams. Single $28 Double $32

775 Holiday House Parking is available in the visitor parking ter­ 447 North Main Street, Logan, UT 84321 race on 700 North, adjacent to the University Inn, Telephone: 801-752-9141 and to the east of campus at 1200 East. Terrace Single $26 and $28 Double $31 and $33 parking will be validated for conference partici­ pants and the University Inn guests. Parking is Food Services permitted virtually anywhere on weekends. The cafeteria in the Taggart Student Center will Regularly scheduled limousine service be­ be open during both days of the meeting. In tween Salt Lake City and Logan is available addition, there are numerous restaurants along through Key North Limousine. Current one-way Main Street; a list will be available at the prices are $36.40 for one person and $25 each for registration desk. two or more persons. Up to six departure times per day are offered weekdays, and four times Travel and Local Information per day on weekends, with additional limousines Logan is located approximately 90 miles north added as needed. Reservations must be made no of Salt Lake City. Salt Lake City is served by later than 24 hours in advance by calling Key Amtrak, Greyhound and major airlines. Most North Limousine, 801-394-77 43. car rental agencies have counters at the airport Logan is at an elevation of 4000 feet. Autumn terminal. days are cool and frost is likely at night. Rain or Persons driving from Salt Lake City to Logan snow is possible, but unlikely. should take I-15 north to Brigham City, exit at 363 and take US highway 89-91 to Logan. In Logan, turn east off Main Street to 400 North and then turn left off 400 North to 700 East to the University Campus.

PARTIALLY ORDERED ABELIAN GROUPS WITH INTERPOLATION K. R. Goodearl (Mathematical Surveys and Monographs. Volume 20)

In the past decade a new branch of ordered Choquet simplices. and an expository discussion of algebraic structures has grown. motivated by various applications of interpolation group theory to K-theoretic applications and mainly concerned with rings and C* -algebras via ordered Ko. A discussion partially ordered abelian groups satisfying the Riesz of some open problems in interpolation groups and interpolation property. This book is the first source dimension groups concludes the book. in which the algebraic and analytic aspects of these Of interest. of course. to researchers in ordered interpolation groups have been integrated into a algebraic structures. the book will also be a valuable coherent framework for general reference. The author source for researchers seeking a background in provides a solid foundation in the structure theory of interpolation groups and dimension groups for interpolation groups and dimension groups (directed applications to such subjects as rings. operator unperforated interpolation groups). with applications algebras. topological Markov chains. positive to ordered K-theory particularly in mind polynomials. compact group actions. or other areas High points of the development include the following where ordered Grothendieck groups might be useful. characterization of dimension groups as direct limits of finite products of copies of the integers: the 1980 Mathematics Subject Classifications: 06, 46, 19, 16 double-dual representation of an interpolation group ISBN 0-8218-1520-2. LC 86-7876, ISSN 0076-5370 continuous real-valued 358 pages (hardcover), 1986 with order-unit via affine List price $68, Institutional member $54. Individual member $41 functions on its state space: the structure of dimension Code SURV /20 NA groups complete with respect to the order-unit norm. as well as monotone sigma-complete dimension groups and dimension groups with countably infinite Shipping/Handling: 1st book $2. each add'! $1. $25 interpolation: and an introduction to the problem max. By air. 1st book $5. each add'! $3. $100 max. of classifying extensions of one dimension group by Prepayment required. Order from AMS. P.O. Box 1571. another. The book also includes a development of Annex Station. Providence. Rl 02901-1571. or call portions of the theory of compact convex sets and 800-556-7774 to use VISA or MasterCard.

776 UTAH STATE UNIVERSITY

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777 Program of the Sessions

All AMS IM!IIIIiooa will take plaee in Eeeles Conference Center. The time limit for e8ch contributed paper in the AMS general sessions is ten minutes. In the special sessions, the time limit varies from session to session and within sessions. To maintain the schedule, time limits will be strictly enforced. Abstracts of papers presented in AMS sessions at this meeting will be found in the October 1986 issue of Abstmcts of 'fJO!Pf'f'S presented to th£ American Math.emalicol Society, ordered according to the numbers in parentheses following the listings below. For papers with more than one author, an asterisk follows the name of the author who plans to present the paper at the meeting.

Friday, October 10, 1986, 8:00 a.m. Speeial Session on Geometric Topology: Knot Theory, I Room 303 8:00- 8:20 (1) Heegard splittings and group presentations. Preliminary report. YOAV MORIAH, University of British Columbia (828-99-91) 8:25- 8:45 (2) Invariants of labeled links. Preliminary report. JIM HOSTE*, Oregon State University, and MARK KIDWELL, United States Naval Academy (828-57-75) 8:50- 9:10 (3) Homotopy linking. Preliminary report. DALE ROLFSON*, University of British Columbia, and ULRICH KOFCHORKE, University of Siegen, West Germany (828-99-90) 9:15- 9:35 (4) Beginning knot tables for graphs in 3-space. JONATHAN SIMON, University of Iowa (828-57-66) 9:40-10:00 (5) Homotopy of classical links. JEROME LEVINE, Brandeis University (828-57-33) 10:05-10:25 (6) Minimally knotted graphs in 8 3. J. SIMON, University of Iowa, and K. WOLCOTT*, University of California, Santa Barbara (828-57-67) 10:30-10:50 (7) The Arf invariant and the link polynomial of Ho. CHI FAr Ho, California Institute of Technology (828-57-42) 10:55-11:15 (8) Signed graphs. Preliminary report. LOUIS H. KAUFFMAN, University of Iowa and University of Illinois, Chicago (828-57-12) 11:20-11:40 (9) Unknotting number, genUB, and companion tori. MARTIN SCHARLEMANN*, University of California, Santa Barbara, and ABIGAIL THOMPSON, Hebrew University, Israel (828-57-77)

Friday, October 10, 1986, 8:00 a.m. Special Session on Geometric Topology: Manifold Theory, II Room 305 8:00- 8:20 (10) Homeomorphisms of Cantor sets in E 3. Preliminary report. DAVID G. WRIGHT, Brigham Young University (828-57-55) 8:25- 8:45 (11) Some homotopy properties of the homeomorphism groups of ROCJ(QOCJ)-manifolds. Vo THANH LIEM, University of Alabama, Thscaloosa (828-54-34) 8:50- 9:10 (12) Congruent double cones touching continua in E 2 and E 3. Preliminary report. DUANE LOVELAND, Utah State University (828-57-51) 9:15- 9:35 (13) Relative embeddings. Preliminary report. T. BENNY RUSHING, University of Utah (828-99-95) 9:40-10:00 (14) Flat 80(3)-connections and rational homology cobordisms. GORDAN A MATIC, Massachusetts Institute of Technology (828-99-94) 10:05-10:25 (15) Menger spaces and inverse limits. Preliminary report. DENNIS J. GARITY*, Oregon State University, JAMES P. HENDERSON, Colorado College, and DAVID G. WRIGHT, Brigham Young University (828-54-70) 10:30-10:50 (16) Dense embeddings of nowhere locally compact separable metric spaces. PHILIP L. BOWERS, Florida State University (828-54-62) 10:55-11:15 (17) Spheres in E 3 with tangent planes, double tangent ball3, or double cones. C. E. BURGESS, University of Utah (828-57-50) 11:20-11:40 (18) Dehn diagrams. Preliminary report. MICHAEL STARBIRD, University of Texas, Austin (828-57-47)

Friday, October 10, 1986, 9:00 a.m. Special Session on Applications and Computational Aspects of Numerical Continuation Methods, I Room 205 9:00- 9:20 (19) HOMPACK: Software for homotopy algorithms. LAYNE T. WATSON, University of Michigan, Ann Arbor (828-65-03) (Sponsored by Eugene L. Allgower) 9:30- 9:50 (20) Forced nonlinear oscillations. JOHN TAVANTZIS, New Jersey Institute of Technology (828-65-20)

778 10:00-10:20 (21) Bifurcation and persistence of solutions in nonlinear parametric optimization. CHRIS TIAHRT, University of Nebraska, Lincoln (828-49-37) 10:30-10:50 (22) A global view of continuation methods. ROY PLASTOCK, New Jersey Institute of Technology ( 828-65-21) 11:00-11:20 (23) The expanded Lagrangian system for constrained optimization problems. AUBREY B . POORE, Colorado State University (828-49-19) (Sponsored by Eugene L. Allgower) 11:30-11:50 (24) Homotopies that respect m-homogeneous structures, with application to solving the inverse kinematics of manipulators. ALEXANDER MORGAN, G. M. Research Labs., Warren, Michigan (828-65-43) (Sponsored by Eugene L. Allgower)

Friday, October 10, 1986, 9:00 a.m. Special Session on Random Fields, Random Measures and Applications, I Room 203 9:00- 9:20 (25) The two-locus genetic model as a measure-valued diffusion. Preliminary report. S. N. ETHIER*, University of Utah, and R. C. GRIFFITHS, Monash University, Australia (828-60-44) 9:30- 9:50 (26) A measure-valued Markov chain induced by random transformations. R. W. R. DARLING, University of South Florida (828-60-02) 10:00-10:20 (27) Exact Hausdorff dimension in random recursive constructions. SIEGFRIED GRAF, University of Erlangen-Nurnberg, Federal Republic of Germany, R. DANIEL MAULDIN, North Texas State University, and STANLEY C. WILLIAMS*, Utah State University (828-60-41) 10:30-10:50 (28) Estimating the dimension of a measure using a random sample. Preliminary report. COLLEEN D. CUTLER*, University of Waterloo, and DONALD A. DAWSON, Carleton University (828-62-60) (Sponsored by Ed Waymire) 11:00-11:20 (29) The central limit theorem for ergodic Markov processes and convergence to Gaussian random fields indexed by functions. RABI N. BHATTACHARYA, Indiana University, Bloomington (828-60-45) 11:30-11:50 (30) A functional central limit theorem for a set-indexed process with Martingale incre­ ments. Preliminary report. P. E. GREENWOOD, University of British Columbia, and M. OSSIANDER*, University of Washington (828-60-46) (Sponsored by Ed Waymire)

Friday, October 10, 1986, 1:30 p.m. Invited Address Auditorium 1:30- 2:30 (31) New combinatorial methods in 3-dimensional topology. KENNETH C. MILLETT, University of California, Santa Barbara (828-57-85)

Friday, October 10, 1986, 2:45 p.m. Special Session on Representations of Reductive Groups, I Room 201 2:45- 3:05 (32) Structure of generalized Verma modules. Preliminary report. BRIAN D. BOE, University of California, Berkeley (828-22-49) 3:10- 3:30 (33) Special K-types and the Beilison-Bemstein realization. JEN- TSEH CHANG, University of California, Berkeley (828-22-39) 3:35- 3:55 Discussion 4:00- 4:20 (34) The generalized tau-invariant. Preliminary report. DEVRA GARFINKLE, University of Utah (828-22-57) 4:25- 4:45 (35) D-module approach to discrete series. Preliminary report. HENRYK HECHT* and DRAGAN MILI

Friday, October 10, 1986, 2:45 p.m. Special Session on Random Fields, Random Measures and Applications, II Room 203 2:45- 3:05 (36) Empirical processes based on general random measures. Preliminary report. JOSEPH HOROWITZ, University of Massachusetts, Amherst (828-60-13) (Sponsored by Ed Waymire) 3:10- 3:30 (37) The central limit theorem for weighted empirical processes indexed by sets. KENNETH S. ALEXANDER, University of Washington (828-60-17) (Sponsored by Ronald Pyke) 3:35- 3:55 (38) Limit theorems for empiricial processes under local conditions. N. T. ANDERSEN, Aarhus University, Denmark, E. GINE*, Texas A&M University, College Station, M. OSSIANDER, British Columbia, and J. ZINN, Texas A&M University, College Station (828-60-18) 4:00- 4:20 (39) New results for conditional empirical processes. Preliminary report. WINFRIED STUTE, Mathematical Institute University of Giessen, West Germany (828-60-32) (Sponsored by Ed Waymire) 4:25- 4:45 (40) Limit theorems for bilinear forms in functions of moving averages. Preliminary report. FLORIN AVRAM*, Purdue University, West Lafayette, and MURAD TAQQU, Boston University (828-60-25) (Sponsored by Ed Waymire)

779 4:50- 5:10 (41) Perturbed smoothing processes. Preliminary report. MICHAEL BRENNAN, Utah State University (828-60-87)

Friday, October 10, 1986, 2:45 p.m. Special Session on Geometric Topology, III Room 305 2:45- 3:05 ( 42) Cusp volumes in hyperbolic 3-manifolds. Preliminary report. COLIN ADAMS, Williams College (828-5 7-24) 3:10- 3:30 (43) Growth functions on surface groups. WILLIAM J. FLOYD*, Virginia Polytechnic Institute and State University, and STEVEN P. PLOTNICK, Columbia University (828-57-56) 3:35- 3:55 (44) Solv geometry and parallel processing networks. Preliminary report. MATT GRAYSON*, University of California at San Diego, La Jolla, and WILLIAM THURSTON, Princeton University (829-99-92) 4:00- 4:20 (45) Surgery on knots in solid tori. DAVID G ABA!, California Institute of Technology (828-57-69) 4:25- 4:45 (46) Solvgroups are not almost convex. J. W. CANNON*, Brigham Young University, W. J. FLOYD, University of Michigan, Ann Arbor, M. A. GRAYSON, University of California at San Diego, La Jolla, and W. P. THURSTON, Princeton University (828-20-83)

Friday, October 10, 1986, 3:00 p.m. General Session, I Room 205 3:00- 3:10 (47) Formulae for the sequence of primes. PANAYIOTIS TSANGARIS, Athens, Greece (828-11-79) 3:15- 3:25 (48) Primes of the form x 2 + (x + 1) 2 Proper divisors of composites of the same form. A. ZACHARIOU* and P. TSANGARIS, Athens, Greece (828-11-80) 3:30- 3:40 ( 49) Generalized inverses of morphisms with kernels. DONALD W. ROBINSON*, Brigham Young University, and ROLAND PUYSTJENS, Rijksuniversiteit, Belgie (828-15-68) 3:45- 3:55 (50) The !-minimal model of a hyperplane complement. M. FALK, Northern Arizona University (828-57-65)

Saturday, October 11, 1986, 8:00 a.m. Special Session on Geometric Topology, IV Room 305 8:00- 8:20 (51) Implications of three level form. W. T. EATON, University of Texas, Austin (828-57-53) 8:25- 8:45 (52) Degenerations of the hyperbolic space. Preliminary report. MLADEN BESTVINA, University of California, Los Angeles (828-57-71) 8:50- 9:10 (53) Extramolecular topological structure in isotropic polymers, I. R. C. LACHER, Florida State University (828-57-48) 9:15- 9:35 (54) Homogeneous ENR 's. JOHN L. BRYANT, Florida State University (828-57-61) 9:40-10:00 (55) Exotic homotopy equivalences of planar surfaces. STEPHEN HUMPHRIES, University of California, Santa Barbara (828-20-82) (Sponsored by David G. Wright) 10:05-10:25 (56) Foliations and finite covers of graphmanifolds. Preliminary report. JOHN LUECKE, Courant Institute of Mathematical Sciences, New York University (828-57-84) 10:30-10:50 (57) The shrinkability of Bing-Whitehead decompositions. Preliminary report. F. D. ANCEL*, University of Wisconsin, Milwaukee, and M. P. STARBIRD, University of Texas, Austin (828-57-31)

Saturday, October 11, 1986, 8:00 a.m. Special Session on Random Fields, Random Measures and Applications, III Room 203 8:00- 8:20 (58) Lattice random fields and short range correlation. ED WAYMIRE, Oregon State University (828-60-30) 8:30- 8:50 (59) Decomposition of binary random fields and zeros of partition functions. CHARLES M. NEWMAN, University of Arizona (828-60-23) 9:00- 9:20 (60) Partitions of point processes: compound Poisson approximations. RICHARD F. SERFOZO, Georgia Institute of Technlogy (828-60-16) 9:30- 9:50 (61) Positive dependence properties of point random fields. Preliminary report. ROBERT M. BURTON and MARIE M. FRANZOSA*, Oregon State University (828-60-40) 10:00- 10:20 (62) Association of vector valued random measures. Preliminary report. ROBERT M. BURTON, Oregon State University (828-60-54) (Sponsored by P.M. Anselone) 10:30-10:50 (63) On renewal processes having stable interrenewal intervals and stable rewards. JOSHUA B. LEVY, State University of New York, Albany, and MURAD S. TAQQU*, Boston University (828-60-15)

780 Saturday, October 11, 1986, 8:30 a.m. Special Session on Applications and Computational Aspects of Numerical Continuation Methods, II Room 205 8:30- 8:50 (64) Multigrid continuation. HANS D. MITTELMANN, Arizona State University (828-65-06) (Sponsored by Eugene L. Allgower) 9:00- 9:20 (65) Numerical solution of a class of deficient polynomial systems. TIEN- YIEN LI, Michigan State University (828-65-10) 9:30- 9:50 (66) Generalized bisection and continuation methods. R. BAKER KEARFOTT, University of Southwestern Louisiana (828-99-93) 10:00-10:20 (67) On the numerical stability of simplex-algorithms. Preliminary report. KURT GEORG*, University of Bonn, West Germany, and RAINER HETTICH, University of Trier, West Germany (828-65-04) (Sponsored by Eugene L. Allgower) 10:30 -10:50 (68) Optimal control of nonlinear systems in the presence of multiplicity and bifurcations. E USEBIUS D OEDEL *, California Institute of Technology, and JEAN PIERRE KERNEVEZ, Compiegne, France (828-65-27)

Saturday, October 11, 1986, 9:00 a.m. Special Session on Analysis on Manifolds, I Room 305 9:00- 9:20 (69) Geometry and the automorphism groups of domains and complex manifolds. ROBERT E. GREENE, University of California, Los Angeles (828-53-86) 9:30- 9:50 (70) Parabolic equations in Kiihler geometry. HUAI-DONG CAO, University of California at San Diego, La Jolla (828-53-74) 10:00-10:20 (71) Stable minimal two-spheres in Riemannian manifolds with nonnegative curvature on totally isotropic two-planes. Preliminary report. MARIO MICALLEF, Oklahoma State University, Stillwater, and JOHN DOUGLAS MOORE*, University of California, Santa Barbara (828-53-76) 10:30-10:50 (72) Regularity of mappings of G-structures of Frobenius type. CHONG KYU HAN, University of Alabama, Tuscaloosa (828-53-07)

Saturday, October 11, 1986, 9:00 a.m. Special Session on Representations of Reductive Groups, II Room 201 9:00- 9:20 (73) Socle filtrations on Verma modules. Preliminary report. RONALD S. IRVING, University of Washington (828-22-14) 9:30- 9:50 (74) Further progress in base change C/R of certain non-tempered representations. Preliminary report. JOSEPH F. JOHNSON, University of Utah (828-22-58) 10:00-10:20 (75) Twisted Harish-Chandra sheaves and representations of semisimple groups with infinite center. HENRYK HECHT and DRAGAN MILICIC*, University of Utah, WILFRIED SCHMID, Harvard University, and JOSEPH A. WOLF, University of California, Berkeley (828-22-52) 10:30-10:50 (76) On the classification of irreducible tempered representations of semisimple groups. Preliminary report. IVAN MIRKOVIC, Brandeis University (828-43-09)

Saturday, October 11, 1986, 11:00 a.m. Invited Address Auditorium 11:00-12:00 (77) Harmonic functions on complete manifolds. PETER LI, University of Utah (828-58-38)

Saturday, October 11, 1986, 1:30 p.m. Special Session on Analysis on Manifolds, II Room 305 1:30- 1:50 (78) Estimates for hyperspheres. Preliminary report. ANDREJS TREIBERGS, University of Utah (828-53-73) 2:00- 2:20 (79) A variational problem connected with the existence problem for hypersurfaces with prescribed Gauss curvature. VLADIMIR OLIKER, Emory University (828-53-72) 2:30- 2:50 (80) The regularity of minimizers. S. WALTER WEI, University of Oklahoma (828-53-29) 3:00- 3:20 (81) Eigenvalue estimates with applications to minimal hypersurfaces. JOHAN TYSK, University of California, Los Angeles (828-53-63) 3:30- 3:50 (82) Minimal surfaces in 3-manifolds of non-negative Ricci curvature. MICHAEL T. ANDERSON, California Institute of Technology (828-99-88) 4:00- 4:20 (83) An analogue of Bernstein's theorem. LUEN-FAI TAM, University of Illinois, Chicago (828-99-89)

781 Saturday, October 11, 1986, 1:30 p.m. Special Session on Representations of Reductive Groups, III Room 201 1:30- 1:50 (84) On Howe's duality correspondence. TOMASZ PRZEBINDA, University of Utah and Yale University (828-43-11) 2:00- 2:20 (85) Singular representations of SU(p,q). C. H. PATTON, Hewlett-Packard, Corvallis, Oregon, and HUGO ROSSI*, University of Utah (828-22-59) 2:30- 2:50 (86) P-spectral type for unitary representations of general linear groups. ROBERTO SCARAMUZZI, Stanford University (828-22-C5) 3:00- 3:20 (87) Highest weight modules for Hermitian symmetric pairs. BRAD SHELTON, University of Oregon (828-22-78) (Sponsored by Theodore Palmer) 3:30- 3:50 (88) LP-harmonic analysis on reductive Lie groups. Preliminary report. PETER TROMBI, University of Utah (828-22-22) 4:00- 4:20 (89) The Schwartz space of a general semisimple group. Preliminary report. REBECCA A. HERB, University of Maryland, College Park, and JOSEPH A. WOLF*, University of California, Berkeley (828-22-26)

Saturday, October 11, 1986, 1:30 p.m. General Session, II Room 205 1:30- 1:40 (90) On the geometrical description of some ideals. ANDREANA S. MADGUEROVA, Sofia, Bulgaria (828-46-01) (Sponsored by Hugo Rossi) 1:45- 1:55 (91) On a geometrical convergence in normed linear spaces. Preliminary report. TECK-CHEONG LIM, George Mason University (828-46-08) (Sponsored by Stanley Zoltek) 2:00- 2:10 (92) Some connections between Pettis integration and operator theory. ELIZABETH M. BATOR and PAUL LEWIS*, North Texas State University, and DAVID RACE, Dartmouth College (828-46-35) 2:15- 2:25 (93) Gateaux differentiability at a dense set of points uniformly in a specified set of directions. RUSSELL G. BILYEU* and PAUL W. LEWIS, North Texas State University (828-46-36) 2:30- 2:40 (94) Group superlogic S0(10)/ SU(8). STEPHEN L. WEINBERG, Berkeley, California (828-81-81) Hugo Rossi Salt Lake City, Utah Associate Secretary

Presenters of Papers Numbers following the names indicate the speakers' positions on the program. •Invited one-hour lecturer *Special session speaker

*Adams, C., 42 * Franzosa, M. M., 61 Lim, T.-C., 91 * Serfozo, R. F., 60 *Alexander, K. S., 37 * Gabai, D., 45 *Loveland, D., 12 *Shelton, B., 87 *Ancel, F. D., 57 *Garfinkle, D., 34 *Luecke, J., 56 *Simon, J., 4 *Anderson, M. T., 82 *Garity, D. J., 15 Madguerova, A. S., 90 *Starbird, M., 18 * Avram, F., 40 *Georg, K., 67 * Matic, G., 14 * Stute, W., 39 * Bestvina, M., 52 * Gine, E., 38 * Mili6c, D., 75 *Tam, L.-F., 83 *Bhattacharya, R. N., 29 *Grayson, M., 44 • Millett, K. C., 31 * Taqqu, M. S., 63 Bilyeu, R. G., 93 *Greene, R. E., 69 * Mirkovic, I., 76 * Tavantzis, J., 20 * Boe, B. D., 32 *Han, C. K., 72 * Mittelmann, H. D., 64 * Tiahrt, C., 21 *Bowers, P. L., 16 *Hecht, H., 35 *Moore, J. D., 71 * Treibergs, A., 78 *Brennan, M., 41 *Ho, C. F., 7 *Morgan, A., 24 * Trombi, P., 88 *Bryant, J. L., 54 *Horowitz, J., 36 *Moriah, Y., 1 Tsangaris, P., 47 *Burgess, C. E., 17 * Hoste, J., 2 *Newman, C. M., 59 * Tysk, J., 81 *Burton, R. M., 62 *Humphries, S., 55 * Oliker, V., 79 *Watson, L. T., 19 *Cannon, J. W., 46 * Irving, R. S., 73 * Ossiander, M., 30 * Waymire, E., 58 * Cao, H.-D., 70 *Johnson, J. F., 74 * Plastock, R., 22 *Wei, S. W., 80 *Chang, J.-T., 33 *Kauffman, L. H., 8 *Poore, A. B., 23 Weinberg, S. L., 94 *Cutler, C. D., 28 * Kearfott, R. B., 66 * Przebinda, T., 84 *Williams, S. C., 27 *Darling, R. W. R., 26 *Lacher, R. C., 53 Robinson, D. W., 49 *Wolcott, K., 6 * Doedel, E., 68 *Levine, J., 5 *Rolfson, D., 3 *Wolf, J. A., 89 *Eaton, W. T., 51 Lewis, P., 92 *Rossi, H., 85 *Wright, D. G., 10 *Ethier, S. N., 25 eLi, P., 77 *Rushing, T. B., 13 Zachariou, A., 48 Falk, M., 50 * Li, T.-Y., 65 * Scaramuzzi, R., 86 *Floyd, W. J., 43 * Liem, V. T., 11 * Scharlemann, M., 9

782 Charlotte, October 17-18, Adam's Mark Hotel Program for the 829th Meeting

The eight hundred and twenty-ninth meeting of Contributed Papers the American Mathematical Society will be held There will also be sessions for contributed ten­ at the Adam's Mark Hotel in downtown Charlotte, minute papers on Friday and Saturday afternoon. North Carolina, on Friday and Saturday, October 17-18, 1986. This meeting will be hosted by the Charlotte Mathematics Conference University of North Carolina at Charlotte. The University of North Carolina, Charlotte, will also host the Charlotte Mathematics Conference Invited Addresses in Analysis and Probability at the Adam's Mark By invitation of the Committee to Select Hour Hotel. The name of the major speaker and title Speakers for Southeastern Sectional Meetings, of the talk to be presented at 10:00 a.m. on there will be three invited addresses. The speakers Friday will be announced at the meeting. Special are as follows: sessions of twenty-minute invited talks will also be PATRICK EBERLEIN, University of North scheduled on Friday and Saturday in the following Carolina, Chapel Hill, Structure of manifolds of areas: Semilinear parabolic systems, organized by non-positive curvature, 1:30 p.m. Saturday. Joel Avrin; Operator theory, organized by Alan STEVEN C. FERRY, University of Kentucky, Lambert; and Probability theory, organized by Closing open manifolds, 8:00 a.m. Saturday. Bob Anderson. All of these organizers are at the FRANK RAYMOND, University of Michigan, University of North Carolina, Charlotte. Any­ Ann Arbor, Recent results in the theory of Seifert one interested in obtaining additional information fiberings, 12:30 p.m. Friday. should contact David Royster, Department of Mathematics, UNCC; telephone 704-547-2649.

Special Sessions Registration By invitation of the same committee, there will The meeting registration desk will be located in be four special sessions of selected twenty-minute the Governor's Foyer at the Adam's Mark Hotel. papers. The topics of these sessions, the names The desk will be open from 10:00 a.m. to 5:00 and affiliations of the organizers, and final lists of p.m. on Friday, and from 9:00 a.m. to noon speakers are as follows: on Saturday. The registration fees are $10 for Singularities and algebraic geometry, GARY members of the AMS, $16 for nonmembers, and KENNEDY, Duke University. The speakers are $5 for students or unemployed mathematicians. James Damon, Sheldon Katz, Henry B. Laufer, A. Libgober, Lee J. McEwan, David R. Morrison, Petition Table Walter Neumann, Richard Randell, Lee Rudolph, Theodore Shifrin, and Roy Smith. A petition table will be set up in the registration area. Additional information can be found in a Geometric topology and continuum theory, box in the San Antonio JOHN MAYER, University of Alabama, Birm­ meeting announcement in this issue of Notices. ingham. The speakers are David P. Bellamy, Beverly L. Brechner, James F. Davis, Jan J. Di­ jkstra, Paul Duvall, Jerzy Dydak, Bruce Hughes, Accommodations James Keesling, George Kozlowski, Piotr Mine, A block of rooms is being held at the Adam's Mark Lee Mohler, Van C. Nail, James T. Rogers, Jr., Hotel. These rooms will be held until October and E. D. Tymchatyn. 3, after which reservations will be accepted on a Group actions on manifolds, DAVID C. space available basis. Participants should make ROYSTER, University of North Carolina, Char­ their own arrangements directly with the hotel lotte. The speakers are Ronald M. Dotzel, Steven and be sure to identify themselves as participants M. Kahn, Hsu-Tung Ku, Kyung BaiLee, Ronnie at the AMS meeting. The rates listed below do Lee, Darryl McCullough, Frank Quinn, David C. not include applicable taxes, and are subject to Royster, Reinhard E. Schultz, Harpreet Singh, change after October 3. Christopher Stark, and Neal W. Stoltzfus. Adam's Mark Hotel (Headquarters) Twistor theory and four-dimensional geom­ 555 South McDowell Street, Charlotte 28204 etry, ALBERT L. VITTER Ill, Tulane Univer­ Telephone: 704-372-4100 or 800-231-5858 sity. The speakers are Charles P. Boyer, N. Single $60 Double $70 P. Buchdahl, Dan Burns, Andrzej Derdzinski, Rooms have not been blocked at the following Jacques Hurtubise, C. N. Kozameh, Claude Le­ locations, but they are included here for informa­ Brun, Y. Sun Poon, Pankaj Topiwala, and Philip tion purposes. Rates do not include applicable B. Yasskin. taxes and are subject to change without notice.

783 Charlotte Marriott City Center (10 blocks) Travel 100 West Trade Street, Charlotte 28202 Major carriers, including Delta, Eastern, People Telephone: 704-333-9000 or 800-228-9290 Express, Piedmont, TWA, United, and USAir, as Single or double $70 up well as several regional carriers, provide regular Quality Inn Downtown (2 blocks) service to Charlotte's Douglas Municipal Airport. 201 South McDowell Street, Charlotte 28204 Most of the downtown hotels provide limousine Telephone: 704-372-7550 or 800-228-5150 service to and from the airport. Single $40 Double $52 Charlotte is on Interstate Routes I-77 and Radisson Plaza Hotel (9 blocks) I-85 South, and there are several exits from I-77 Two NCNB Plaza to the downtown area. From either I-77 North Trade and Tryon Streets, Charlotte 28280 or South, drivers should take the Brookshire Telephone: 704-377-0400 or 800-228-9822 Expressway exit, and from Brookshire Express­ Single or Double $78 to $110 way take the McDowell Street exit, and proceed A number of other accommodations are avail­ approximately nine blocks to the Adam's Mark able at a greater distance from the Adam's Mark Hotel which is on the left. Hotel. Parking Food Service Ample free parking is available to meeting partic­ Two restaurants are located within the Adam's ipants at the Adam's Mark Hotel. Mark Hotel, and there is also a large number of restaurants in the downtown area. A list of restaurants in Charlotte will be available at the meeting registration desk.

"'This book is a must for anyone deeply interested in Geometric Function Theory.·· -A. W. Goodman Univ. of South Florida

"'An interesting and valuable collection of articles . . . The first book to mention the amazing proof of the Bieberbach conjecture by Louis de Branges. Every function-theorist should ... buy this.·· - David A. Brannan The Open University, U. K. Topics in Complex Analysis Dorothy Brown Shaffer, Editor

The unifying theme of the lectures, presented at the AMS meeting in October, 1983, at Fairfield University was Geometric Function Theory. Some of the papers concern: the class L:, its support points and extremal configuration; support points for the class S, Loewner chains and the process of truncation; estimates on the radial growth of the derivative of univalent functions; and a conjecture of Bombieri proved for some cases. Because the proof of the Bieberbach conjecture was not known at the time of preparation of the papers, many of the authors, as well as experts in the field, were interviewed regarding the effect of the proof of the conjecture. Their ideas regarding future trends in research in complex analysis are presented in the epilogue. A graduate level course in complex analysis provides a sufficient background for understanding this material. ISBN 0-8218-5037-7. LC 84-24550. ISSN 0271-4132 March 1985. 142 pages (softcover) Individual member $11. Institutional member $14. List price $18 To order. please specify CONM/38NA Shipping/Handling: 1st book $2. each add"l $1. $25 max. By air. 1st book $5.ieach add"l $3. $100 max. PREPAYMENT REQUIRED. Order from American Mathematical Society. PO Box 1571. Annex Station. Providence. Rl 02901-1571. or call 800-556-7774 to use VISA or MasterCard.

784 DOWNTOWN Q) Adam's Mark Hotel Q) Quality Inn Downtown C H A R L 0 T T E ® Radisson Plaza Hotel @) Marriott City Center

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785 Program of the Sessions

The time limit for each contributed paper in the AMS general sessions is ten minutes. In the special sessions, the time limit varies from session to session and within sessions. To maintain the schedule, time limits will be strictly enforced. Abstracts of papers presented in AMS sessions at this meeting will be found in the October 1986 issue of Abstracts of papers presented to the American Math£matical Society, ordered according to the numbers in parentheses following the listings below. For papers with more than one author, an asterisk follows the name of the author who plans to present the paper at the meeting.

Friday, October 17, 1986, 12:30 p.m. Invited Address Governor's Ballroom 12:30- 1:20 {1) Recent results in the theory of Seifert fiberings. FRANK RAYMOND, University of Michigan, Ann Arbor {829-57-53)

Friday, October 17, 1986, 2:00p.m. Special Session on Singularities and Algebraic Geometry, I Governor's Room 1 2:00- 2:20 {2) Deformations of sections of singularities and Gorenstein surface singularities. JAMES DAMON, University of North Carolina, Chapel Hill {829-14-29) 2:30- 2:50 {3) A topological calculation of number of even and odd {}-characteristics on singular curves. Preliminary report. A. LIBGOBER, University of lllinois, Chicago {829-14-64) 3:00- 3:20 {4) Rational surfaces preserving a cusp singularity. LEE J. MCEWAN, University of North Carolina, Chapel Hill {829-14-59) 3:30- 3:50 {5) Rational double points on K3 surfaces. Preliminary report. DAVID R. MORRISON, Duke University {829-14-51) 4:00- 4:20 {6) Singularities of the Gauss map of a generic hypersurface in P 4 . Preliminary report. THEODORE SHIFRIN*, CLINT MCCRORY and ROBERT VARLEY, University of Georgia {829-14-26) 4:30- 4:50 (7) Gauss maps and deformations of hypersurface singularities. Preliminary report. ROY SMITH* and ROBERT VARLEY, University of Georgia {829-14-54) (Sponsored by Gary Kennedy)

Friday, October 17, 1986, 2:00p.m. Special Session on Geometric Topology (and Continuum Theory), I Governor's Room 3 2:00- 2:20 {8) A three-dimensional prime end theory. Preliminary report. BEVERLY L. BRECHNER, University of Florida {829-57-32) 2:30- 2:50 {9) Embedding surface-like continua up to shape. PAUL DUVALL, University of North Carolina, Greensboro {829-51-11) 3:00- 3:20 {10) Cohomological connectedness of decomposition spaces. Preliminary report. JERZY DYDAK* and JOHN WALSH, University of Tennessee, Knoxville {829-55-18) 3:30- 3:50 {11) Classification of finite-dimensional pseudo-boundaries and pseudo-interiors. JAN J. DIJKSTRA*, University of Washington, JAN VAN MILL, Vrije Universiteit, Netherlands, and JERZY MOGILSKI, Uniwersytet Warszawski, Poland {829-57-40) 4:00- 4:20 {12) Controlled homotopy topological structures, II. Preliminary report. BRUCE HUGHES, Vanderbilt University {829-57-25) 4:30- 4:50 {13) Use of topology in the study of Hausdorff dimension. JAMES KEESLING, University of Florida {829-54-34) 5:00- 5:20 {14) Unstable points and deleted local contractibility. Preliminary report. GEORGE KOZLOWSKI, Auburn University, Auburn {829-54-38)

Friday, October 17, 1986, 2:00p.m. Session on Contributed Papers, I Governor's Room 6 2:00- 2:10 {15) Propositional quantifiers and second-order modal logics. Preliminary report. BILLY JOE LUCAS, Manhattanville College {829-03-46) 2:15- 2:25 {16) Generalized van der Waerden numbers. BRUCE M. LANDMAN, Hofstra University {829-11-61) 2:30- 2:40 {17) Heyting algebras and commutative rings. Preliminary report. ALEXANDRU SOLIAN, University of North Carolina, Charlotte, and T. M. VISWANATHAN*, Unicamp, Brazil, and University of North Carolina, Charlotte {829-13-47)

786 2:45- 2:55 (18) On the group actions of the units on non-units in a compact ring. JO-ANN COHEN* and KWANGIL KOH, North Carolina State University (829-16-48) 3:00- 3:10 (19) A q-analog of the Gauss summation theorem for hypergeometric series in U ( n). STEPHEN C . MILNE, University of Kentucky (829-33-24) 3:15- 3:25 (20) The Mu-continuity problem. Preliminary report. J. C. MAGEE*, State University of New York, College at Potsdam, and W. H. RUCKLE, Clemson University (829-40-44) 3:30- 3:40 (21) Asymptotic expansions for the coefficients of eP(z). Preliminary report. ERIC SCHMUTZ, University of Pennsylvania (829-41-42) 3:45- 3:55 (22) Fixed points for single and multivalued mappings in locally convex spaces. Preliminary report. KANHAYA L. SINGH, Fayetteville State University (829-47-60)

Saturday, October 18, 1986, 8:00 a.m. Invited Address Governor's Ballroom 8:00- 8:50 (23) Closing open manifolds I. STEVEN C. FERRY, University of Kentucky (829-57-02)

Saturday, October 18, 1986, 9:30 a.m. Special Session on Singularities and Algebraic Geometry, II Governor's Room 1 9:30- 9:50 (24) The tangent cone for deformations of two-dimensional quasi-homogeneous singularities. Preliminary report. HENRY B. LAUFER, State University of New York, Stony Brook (829-32-63) 10:00-10:20 (25) Unfoldings of fibered knots (2). Preliminary report. LEE RUDOLPH, Clark University, and WALTER NEUMANN*, Ohio State University, Columbus (829-14-57) 10:30-10:50 (26) Unfoldings of fibered knots (1). Preliminary report. WALTER NEUMANN, Ohio State University, Columbus, and LEE RUDOLPH*, Clark University (829-57-56) 11:00-11:20 (27) Numerical invariants of surface singularities. Preliminary report. RICHARD RANDELL, University of Iowa (829-14-55) 11:30-11:50 (28) The enumeration of rational curves on complete intersection threefolds. SHELDON KATZ, University of Oklahoma (829-14-43)

Saturday, October 18, 1986, 9:30 a.m. Special Session on (Geometric Topology) and Continuum Theory, II Governor's Room 3 9:30- 9:50 (29) Geometric compactifications of Euclidean space. Preliminary report. DAVID P. BELLAMY*, University of Delaware, and BEVERLY DIAMOND, College of Charleston (829-54-14) 10:00-10:20 (30) An atriodic tree-like continuum with positive span which admits a monotone mapping to a chainable continuum. JAMES F. DAVIS*, University of Richmond, and W. T. INGRAM, University of Houston, Houston (829-54-15) 10:30-10:50 (31) Simplicial mappings of graphs that can be factored through an arc. PIOTR MINC, Auburn University, Auburn (829-54-39) 11:00-11:20 (32) Hereditarily decomposable, hereditarily equivalent Hausdorff continua. LEE MOHLER* and LEX G. OVERSTEEGEN, University of Alabama, Birmingham (829-54-35) (Sponsored by John C. Mayer) 11:30-11:50 (33) Decompositions of continua over the hyperbolic plane. Preliminary report. JAMES T. ROGERS, JR., Tulane University (829-54-13) 12:00-12:20 (34) Homeomorphisms of composants in Knaster continua. W. DEBSKI, Silesian University, Poland, and E. D. TYMCHATYN*, University of Saskatchewan (829-54-17)

Saturday, October 18, 1986, 9:30 a.m. Special Session on Group Actions on Manifolds, I Governor's Room 2 9:30- 9:50 (35) Limit sets and root closure in negatively curved manifolds. CHRISTOPHER STARK, University of Florida (829-57-28) 10:00-10:20 (36) Orientation preserving actions of finite Abelian groups on spheres. RONALD M. DOTZEL, University of Missouri, St. Louis (829-57-31) 10:30-10:50 (37) Classification of finite group actions on finite complexes. FRANK QUINN, Virginia Polytechnic Institute and State University (829-57-04) 11:00-11:20 (38) Involutions on odd-dimensional manifolds and the de Rham invariant. STEVEN M. KAHN, Wayne State University (829-57-30) 11:30-11:50 (39) Group actions on handlebodies. DARRYL MCCULLOUGH* and ANDY MILLER, University of Oklahoma, and BRUNO ZIMMERMANN, Ruhr-Universitat Bochum, West Germany (829-57-20) 12:00-12:20 (40) Lustemik-Schnirelmann category and cobordism. HARPREET SINGH, University of North Carolina, Charlotte (829-57-07)

787 Saturday, October 18, 1986, 9:30 a.m. Speeial Session on Twister Theory and Four-dimensional Geometry, I Governor's Room 4 9:30-10:00 (41) Self-dual and anti-self-dual Hermitian metrics on compact complex surfaces. CHARLES P. BOYER, Clarkson University (829-53-33) 10:10-10:40 (42) Twistors and harmonic maps. DAN BURNS, University of Michigan, Ann Arbor (829-99-66) 10:50-11:20 (43) Gauge theory and physics. ANDRZEJ DERDZINSKI, Ohio State University, Columbus (829-99-65) 11:30-12:00 (44) Neighborhoods of holomorphic embeddings. PHILIP B. YASSKIN, Texas A&M University, College Station ( 829-58-23) 12:10-12:40 (45) Holonomy paths and twistors in Yang-Mills theory. C. N. KOZAMEH* and E. T. NEWMAN, University of Pittsburgh, Pittsburgh (829-53-21) (Sponsored by Albert L. Vitter III)

Saturday, October 18, 1986, 1:30 p.m. Invited Address Governor's Ballroom 1:30- 2:20 (46) Structure of manifolds of nonpositive curvature. PATRICK EBERLEIN, University of North Carolina, Chapel Hill (829-53-37)

Saturday, October 18, 1986, 3:00 p.m. Special Session on Group Actions on Manifolds, II Governor's Room 2 3:00- 3:20 (47) Isometric actions, higher Newman numbers and isoperimetric inequalities. HSU-TUNG Ku*, MEr- CHIN Ku and LARRY MANN, University of Massachusetts, Amherst (829-57-19) 3:30- 3:50 (48) Pontryagin numbers and periodic diffeomorphisms of spheres. Preliminary report. REINHARD E. SCHULTZ, Purdue University, West Lafayette (829-57-09) 4:00- 4:20 (49) Finite group actions on P 2 . IAN HAMBLETON, McMaster University, and RONNIE LEE*, Yale University (829-57-01) 4:30- 4:50 (50) Equivariant intersection pairings on covering spaces, the mapping class group of surfaces and the Kaehler geometry of Teichmuller space. Preliminary report. NEAL W. STOLTZFUS, Louisiana State University, Baton Rouge (829-57-10) 5:00- 5:20 (51) There are only finitely many infra-nilmanifolds under each nilmanifold. Preliminary report. KYUNG BAI LEE, University of Oklahoma (829-57-45) 5:30- 5:50 (52) Equivariant fiberings over the circle. DAVID C. ROYSTER, University of North Carolina, Charlotte (829-57-27)

Saturday, October 18, 1986, 3:00 p.m. Special Session on Twister Theory and Four-Dimensional Geometry, II Governor's Room 4 3:00- 3:30 (53) Hypersurfaces in twistor spaces. Preliminary report. N. P. BUCHDAHL, Tulane University (829-32-16) (Sponsored by Albert L. Vitter III) 3:40- 4:10 (54) Kahler surfaces and self-duality. Preliminary report. CLAUDE LEBRUN, State University of New York, Stony Brook (829-53-12) 4:20- 4:50 (55) Small resolutions of double solids as twistor spaces. Y. SUN POON, State University of New York, Stony Brook (829-53-05) 5:00- 5:30 (56) Kiihler-Einstein metrics on K3. PANKAJ TOPIWALA, University of Chicago (829-53-06) 5:40- 6:10 (57) Moduli of instantons and jumping lines. JACQUES HURTUBISE, Universite du Quebec, Montreal (829-51-22) (Sponsored by Albert V. Vitter III)

Saturday, October 18, 1986, 3:00 p.m. Session on Contributed Papers, II Governor's Room 1 3:00- 3:10 (58) Twistor spaces for non-self-dual vacuums? Preliminary report. PETER R. LAW, San Jose State University (829-53-49) 3:15- 3:25 (59) IN-compactness, completeness and the dimension theory of metric spaces. Preliminary report. JOHN S. KULESZA, State University of New York, Binghamton, and Ithaca College (829-54-41) 3:30- 3:40 (60) Partially confluent maps and continua which do not contain n-ods. VAN C. N ALL, University of Richmond (829-54-50) 3:45- 3:55 (61) s-Smith equivalent representations of dihedral groups. Preliminary report. EUNG CHUN CHO, Washington and Lee University (829-57-36) 4:00- 4:10 (62) Submersions, concordances, and pseudoisotopies. Preliminary report. L. CHRISTINE KINSEY, Vanderbilt University (829-57-62) 4:15- 4:25 (63) Finite dimensional simplified regularization. JULIO GUACANEME, University of Puerto Rico, Rio Piedras (829-65-03)

788 4:30- 4:40 (64) Exam mechanics: Equation of motion. STEPHEN L. WEINBERG, Berkeley, California (829-83-52)

Frank T. Birtel New Orleans, Louisiana Associate Secretary

Presenters of Papers Numbers following the names indicate the speakers' positions on the program. •Invited one-hour lecturer *Special session speaker

*Bellamy, D. P., 29 Guacaneme, J., 63 *Lee, R., 49 *Rogers, J. T., Jr., 33 *Boyer, C. P., 41 *Hughes, B., 12 * Libgober, A., 3 *Royster, D. C., 52 * Brechner, B. L., 8 *Hurtubise, J., 57 Lucas, B. J., 15 *Rudolph, L., 26 * Buchdahl, N. P., 53 *Kahn, S. M., 38 Magee, J. C., 20 Schmutz, E., 21 *Burns, D., 42 *Katz, S., 28 *McCullough, D., 39 *Schultz, R. E., 48 Cho, E. C., 61 *Keesling, J., 13 *McEwan, L. J., 4 *Shifrin, T., 6 Cohen, J.-A., 18 Kinsey, L. C., 62 Milne, S. C., 19 *Singh, H., 40 *Damon, J., 2 * Kozameh, C. N., 45 *Mine, P., 31 Singh, K. L., 22 *Davis, J. F., 30 *Kozlowski, G., 14 *Mohler, L., 32 *Smith, R., 7 * Derdzinski, A., 43 * Ku, H.-T., 47 *Morrison, D. R., 5 *Stark, C., 35 * Dijkstra, J. J., 11 Kulesza, J. S., 59 Nail, V. C., 60 *Stoltzfus, N. W., 50 * Dotzel, R. M., 36 Landman, B. M., 16 *Neumann, W., 25 * Topiwala, P., 56 *Duvall, P., 9 *Laufer, H. B., 24 * Poon, Y. S., 55 * Tymchatyn, E. D., 34 * Dydak, J., 10 Law, P. R., 58 *Quinn, F., 37 Viswanathan, T. M., 17 • Eberlein, P., 46 *Lebrun, C., 54 * Randell, R., 27 Weinberg, S. L., 64 44 • Ferry, S. C., 23 *Lee, K. B., 51 • Raymond, F., 1 * Yasskin, P. B.,

789 A Gourmet Guide to Typesetting with the AMS -T EX macro package M. D. SPIVAK, Ph.D.

The Joy of TFiX is the user-friendly user's the finished product, even novice technical guide for A.M.S -'lEX, an extension of 'JEX, typists will find the manual easy to use in Donald Knuth's revolutionary program for helping them produce beautiful technical typesetting technical material. A.M,S -'lEX '!EXt. was designed to simplify the input of math­ This book is designed as a user's guide ematical material in particular, and to for­ to the A.M.S -'lEX macro package and details mat the output according to any of various many features of this extremely useful text preset style specifications. processing package. Parts 1 and 2, entitled There are two primary features of the "Starters" and "Main Courses," teach the 'lEX system: it is a computer system for reader how to typeset most normally en­ typesetting technical text, especially text countered text and mathematics. "Sauces containing a great deal of mathematics; and Pickles," the third section, treats more and it is a system for producing beautiful exotic problems and includes a 60-page dic­ text, comparable to the work of the finest tionary of special 'IE;Xniques. printers. Exercises sprinkled generously through Most importantly, '!EX's capabilities are each chapter encourage the reader to sit not available only to 'JEXperts. While down at a terminal and learn through ex­ mathematicians and experienced technical perimentation. Appendixes list summaries typists will find that 'lEX allows them to of frequently used and more esoteric sym­ specify mathematical formulas with greater bols as well as answers to the exercises. accuracy and still have great control over

PREPAYMENT REQUIRED. Order from American Mathematical Society PO Box 1571 Annex Station Providence, RI 02901-1571 or call 800-556-7774 to use VISA or MasterCard. add'! $1, max. $25; by air, 1st book $5, each add'! $3, max. $100 Denton, October 31- November 1, North Texas State University Program for the 830th Meeting

The eight hundred and thirtieth meeting of the William Ray, R. E. Showalter, Henry A. Warshall, American Mathematical Society will be held at Luther White, and Anton Zettl. North Texas State University in Denton, Texas, Mathematics for large scale computing, JULIO on Friday, October 31, and Saturday, November CESAR DIAZ, University of Oklahoma. The 1, 1986. All scientific sessions will be held in the speakers are R. E. Ewing, Alan Genz, Andreas Union Building on the campus. Griewank, Kirk E. Jordan, S. Lakshmivarahan, Brad Lucier, M. Minkoff, Jorge J. More, Paul Invited Addresses Nelson, and D. C. Sorenson. By invitation of the Committee to Select Hour Computational mathematics, WARREN FER­ Speakers for Central Sectional Meetings, there GUSON, JR., Southern Methodist University. The will be four invited one-hour addresses. The speakers are M. R. Celis, Julio Guacaneme, Tien­ speakers are as follows: Yien Li, Theresa A. Parks, Patrick J. Rabier, PAUL FONG, University of Illinois at Chicago, L. F. Shampine, William W. Symes, Kathryn Modular representations of finite groups, 11:00 Turner, and David Young. a.m. Saturday. Geometric structures on manifolds and Klein­ RAVI S. KULKARNI, Indiana University, ian groups, RAVI S. KULKARNI. The speakers Bloomington, Geometric structures on manifolds, are Mark Baker, Robert Brooks, Mark Feighn, 11:00 a.m. Friday. Frederick P. Gardiner, F. W. Gehring, Jane ALEXANDER NAGEL, University of Wiscon­ Gilman, William M. Goldman, J. Alfredo Jimenez, sin, Madison, Nonisotropic matrices in real and Linda Keen, Adam Koranyi, Bernard Maskit, B. complex algebras, 1:30 p.m. Saturday. Osgood, John Ratcliffe, Richard Skora, Christo­ MARY WHEELER, Rice University, Numeri­ pher Stark, Peter Waterman, and Norbert Wie­ cal simulation of microbial biodegradation of hy­ lenberg. drocarbons in ground water, 1:30 p.m. Friday. Invariant theory, V. LAKSHMIBAI, Texas These addresses will be presented in the A & M University. The speakers are Kaan Lyceum of the Union Building. Akin, Joseph P. Brennan, Vinay Doedhar, Robert M. Fossum, Robert A. Gustafson, Special Sessions Roy Joshua, V. Lashmibai, Andy Magid, Susan Montgomery, By invitation of the same committee, there will be Brian J. Parshall, Robert A. Proctor, John Stem­ seven special sessions of selected twenty-minute bridge, and Bernd Ulrich. papers. The topics of the sessions, the names and affiliations of the organizers, and final lists of Contributed Papers speakers are as follows: There will also be two sessions for contributed Banach spaces and related topics, ELIZABETH ten-minute papers. BATOR, RUSSELL BILYEU and PAUL LEWIS, North Texas State University. The speakers are Registration Dale Alspach, Kevin T. Andrews, N. L. Carothers, The meeting registration desk will be located in S. J. Dilworth, G. A. Edgar, Peter Greim, Nigel the foyer of the Lyceum in the Union Building. J. Kalton, N. T. Peck, R. R. Phelps, Haskell The desk will be open from 7:00 p.m. to 9:00 p.m. Rosenthal, Elias Saab, Paulette Saab, Mark A. on Thursday, from 8:00 a.m. to 2:00 p.m. on Smith, Robert F. Wheeler, and J. H. M. Whitfield. Friday, and from 8:00 a.m. to noon on Saturday. Connections between combinatorics, algebra The registration fees are $10 for members of the and geometry, NEAL BRAND and JOSEPH P. · AMS, $16 for nonmembers, and $5 for students or S. KUNG, North Texas State University. The unemployed mathematicians. speakers will be Brian Alspach, Neal Brand, Joseph P. Brennan, Ronald Gould, Jerrold R. Petition Table Griggs, W. Cary Huffman, Devadatta Kulkarni, A petition table will be set up in the registration Joseph P. S. Kung, Heinrich Neiderhausen, James area. Information concerning placement of peti­ Oxley, T. D. Parsons, Talmage James Reid, David tions can be found in a box in the San Antonio R. Richman, Klaus Truemper, James W. Walker, meeting announcement in this issue of Notices. and Thomas Zaslavsky. Differential equations, ALFONSO CASTRO, Accommodations Southwest Texas State University, and JOHN Special rates have been negotiated at selected NEUBERGER, North Texas State University. The local hotels, all of which are located on the I- speakers are Ilya J. Bakelman, Alfonso Castro, 35 corridor. Both the Royal Hotel Suites and Constantin Corduneanu, Jerome A. Goldstein, the Sheraton are within easy walking distance Hans G. Kaper, Alan C. Lazer, Fred R. Payne, of the campus. Participants should make their

791 own arrangements directly with the hotel of their Parking choice and ask for the special AMS meeting rate. A metered parking garage (50 cents per hour; The rates listed are subject to change and do $2.25 per day) is located immediately east of the not include applicable taxes. The deadline for Union Building. Visitors may also park in the reservations at all locations is October 10. Fouts Field parking lot and take a shuttle (25 Holiday Inn (2.5 miles from campus) cents each way) to the Union Building. Parking in 1500 Dallas Drive, Denton 76205 designated lots is unregulated on Saturday only. Telephone: 817-387-3511 Single $40 Double $45 Travel La Quinta Motor Inn (1.5 miles from campus) Denton is located approximately 35 miles north of 700 Ft. Worth Drive, Denton 76205 Dallas and Ft. Worth and 25 miles north of DFW Telephone: 1-800-531-5900 International Airport, which is served by most Single $28 Double $32 major domestic airlines. Airport shuttle service is available from the DFW International Airport Motel6 (3 miles from campus) to Denton (telephone 817-383-2007), with regular I-35 North of University Drive, Denton 76205 stops at the Holiday Inn, Ramada Inn, LaQuinta, Telephone: 817-387-0571 and Sheraton. However, since one-way fare is $14 Single $20.85 Double $25.25 and only the Royal Hotel Suites and Sheraton Ramada Inn (2 miles from campus) Hotel are within easy walking distance of campus, 820 S I-35E at Teasley Lane, Denton 76205 those flying to DFW should consider renting a car Telephone: 1-800-228-2828 or 817-387-0591 and driving to Denton. Participants driving to Single $36 Double $36 Denton from DFW should take the North Airport Royal Hotel Suites ( .5 miles from campus) Exit to State Highway 121 to I-35E and proceed 1210 I-35E, Denton 76205 north on I-35E to Denton. To reach the campus, Telephone: 817-383-2007 take the Avenue D exit from I-35E and proceed Single $32 Double $36 north on Avenue D one block to Eagle Drive. Sheraton Hotel ( .5 miles from campus) Turn east on Eagle Drive and proceed four blocks 2211 I-35E North, Denton 76205 to Welch Street, then turn north on Welch Street. Telephone: 817-565-8499 The metered parking garage adjacent to the Union Single $49 Double $54 Building is on Welch Street, three blocks north of Eagle Drive. Food Service Many fast food restaurants are located within two blocks of the campus, and more formal dining is available at the Sheraton Hotel and several local restaurants. Food service in the Union will only be available according to the following schedule: Breakfast, lunch, and sandwiches until 5:00 p.m. on Thursday; breakfast and lunch on Friday; and lunch only on Saturday.

792 ---NORTH TEXAS STATE UNIVERSITY----. DENTON, TEXAS

CD Holiday Inn Q) Ramada Inn us 380 G) La Quinta @) Royal Hotel Suites ® Union Building ® Sheraton Hotel SCRIPTURE (j) Motel 6 +-OAK [0] NTSU campus

793 Program of the Sessions

All AMS sessions will take place in the Union Building. The time limit for each contributed paper in the AMS general sessions is ten minutes. In the special sessions, the time limit varies from session to session and within sessions. To maintain the schedule, time limits will be strictly enforced. Abstracts of papers presented in AMS sessions at this meeting will be found in the October 1986 issue of Abstracts of papers presented to the American Maiherrwtical Society, ordered according to the numbers in parentheses following the listings below. For papers with more than one author, an asterisk follows the name of the author who plans to present the paper at the meeting.

Friday, October 31, 1986, 8:00 a.m. Special Session on Geometric Structures on Manifolds and Kleinian Groups, I Room 410 8:00- 8:30 (1) Convex fundamental polyhedra. Preliminary report. JOHN RATCLIFFE, Vanderbilt Univer- sity (830-52-15) 8:35- 9:05 (2) Extremal values and the gradient flow for the determinant of the Laplacian. B. OSGOOD*, R. PHILLIPS and P. SARNAK, Stanford University (830-30-54) 9:10- 9:40 (3) lsospectral surfaces of small genus. ROBERT BROOKS* and RICHARD TSE, University of Southern California (830-53-04) 9:45-10:15 (4) Finiteness conditions for 3-manifolds with boundary. MARK FEIGHN* and DARRYL MCCULLOUGH, University of Oklahoma (830-51-73) 10:20-10:50 (5) Riemannian 4-symmetric spaces. J. ALFREDO JIMENEZ, Pennsylvania State University, Hazleton (830-53-62)

Friday, October 31, 1986, 8:30 a.m. Special Session on Banach Spaces and Related Topics, I Room 411 8:30- 8:50 (6) Complex uniform convexity. G. A. EDGAR, Ohio State University, Columbus (830-46-23) 9:00- 9:20 (7) Constructing unconditional F. D. D.'s. Preliminary report. DALE ALSPACH*, University of Texas, Austin, and NEAL CAROTHERS, Texas A & M University, College Station ( 830-46-43) 9:30- 9:50 (8) Proximinality in operator algebras on L1. KEVIN T. ANDREWS*, Oakland University, and JOSEPH D. WARD, Texas A & M University, College Station (830-46-02) 10:00-10:20 (9) Geometry of vector-valued function spaces. Preliminary report. PETER GREIM, The Citadel (830-46-20) 10:30-10:50 (10) Compact convex sets and complex convexity. NIGEL J. KALTON, University of Missouri, Columbia (830-46-33)

Friday, October 31, 1986, 8:30 a.m. Special Session on Mathematics for Large Scale Computing, I Room 412 8:30- 8:50 (11) The numerical performance of the Gauss-Broyden method. ANDREAS GRIEWANK* and LAIHUA SHENG, Southern Methodist University (830-65-78) (Sponsored by M. F. Wheeler) 9:00- 9:20 (12) Quasi-Newton updates for structured problems. JORGE J. MORE, Argonne National Laboratory, Argonne, Illinois (830-65-79) (Sponsored by M. F. Wheeler) 9:30- 9:50 (13) Preconditioned iterative methods for PDE systems. M. MINKOFF, Argonne National Laboratory, Argonne, Illinois (830-65-80) (Sponsored by M. F. Wheeler) 10:00-10:20 (14) Large-scale computing in fluid flow problems. R. E. EWING, University of Wyoming ( 830-65-81) 10:30-10:50 (15) Performance issues of MIMD, shared-memory multiprocessors. BRAD LUCIER, Purdue University, West Lafayette (830-65-82) (Sponsored by M. F. Wheeler)

Friday, October 31, 1986, 9:00 a.m. Special Session on Connections Between Combinatoric&, Algebra and Geometry, I Room 417 9:00- 9:20 (16) Cayley digraphs and (1,J,n)-Sequencings of the alternating groups An. RONALD GOULD* and ROBERT ROTH, Emory University (830-06-74) 9:30- 9:50 (17) Saturated chains of subsets and a random walk. JERROLD R. GRIGGS, University of South Carolina, Columbia (830-05-11) 10:00-10:20 (18) The Hamilton space of a Cayley graph on a finite abelian group. BRIAN ALSPACH*, Simon Fraser University, STEPHEN C. LOCKE, Florida Atlantic University, and DAVID WITTE, Massachusetts Institute of Technology (830-05-19)

794 10:30-10:50 (19) A decomposition of linear codes and applications. W. CARY HUFFMAN, Loyola University of Chicago (830-94-18)

Friday, October 31, 1986, 9:00 a.m. Special Session on Invariant Theory, I Room 418 9:00- 9:20 (20) Completions of rings of invariants. ANDY MAGID, University of Oklahoma (830-13-31) 9:30- 9:50 (21) Prime ideals and group actions in non-commutative algebras. SUSAN MONTGOMERY, University of Southern California (830-16-68) 10:00-10:20 (22) Properties of the kernel of a nilpotent derivation in characteristic p. ROBERT M. FOSSUM, University of Illinois, Urbana-Champaign (830-13-106) 10:30-10:50 (23) Linkage and rings of invariants. BERND ULRICH, Michigan State University (830-13-58)

Friday, October 31, 1986, 11:00 a.m. Invited Address The Lyceum 11:00-12:00 (24) Geometric structures on manifolds. RAVI S. KULKARNI, Indiana University, Bloomington (~30-53-50)

Friday, October 31, 1986, 1:30 p.m. Invited Address The Lyceum 1:30- 2:30 (25) Numerical simulation of microbial biodegradation of hydrocarbons in groundwater. MARY F. WHEELER, Rice University (830-65-55)

Friday, October 31, 1986, 3:00 p.m. Special Session on Geometric Structures on Manifolds and Kleinian Groups, II Room 410 3:00- 3:30 (26) Dynamics of a class of functions holomorphic on C*. LINDA KEEN, Herbert H. Lehman College, City University of New York (830-30-44) 3:35- 4:05 (27) Inequalities and discrete subgroups of PSL(2,R). JANE GILMAN, Rutgers University, Newark (830-20-40) 4:10- 4:40 (28) Quasiconformal groups and the conical limit set. J. B. GARNETT, University of California, Los Angeles, F. W. GEHRING*, University of Michigan, Ann Arbor, and P. W. JONES, Yale University (830-30-47) 4:45- 5:15 (29) Special parameters for Teichmuller spaces. Preliminary report. BERNARD MASKIT, State University of New York, Stony Brook (830-30-49) 5:20- 5:50 (30) Singular measures invariant for Fuchsian groups. Preliminary report. FREDERICK P. GARDINER, Brooklyn College, City University of New York (830-30-34) 5:55- 6:25 (31) The virtual Z-representability of certain three-manifold groups. MARK BAKER, Brown University (830-57-65)

Friday, Oct9ber 31, 1986, 3:00 p.m. Special Session on Banach Spaces and Related Topics, I Room 411 3:00- 3:20 (32) Counterexamples concerning support theorems for convex sets in Hilbert space. R. R. PHELPS, University of Washington (830-46-25) 3:30- 3:50 (33) Banach sublattices of weak Li. N. T. PECK*, University of Illinois, Urbana-Champaign, and MICHEL TALAGRAND, University of Paris VI, France (830-46-97) 4:00- 4:20 (34) Some Banach spaces that have properties (V) or (V*) of Pelczynski. Preliminary report. GILLES GODEFROY, Universite Paris VI, France, and PAULETTE SAAB*, University of Missouri, Columbia (830-46-24) 4:30- 4:50 (35) Weak uniform rotundity in Banach spaces. Preliminary report. MARK A. SMITH, Miami University, Oxford (830-46-38) 5:00- 5:20 (36) The dual Mackey topology and Banach spaces. Preliminary report. GEORG SCHLli'cHTERMANN, Universitat Miinchen, West Germany, and ROBERT F. WHEELER*, Northern Illinois University (830-46-66)

795 Friday, October 31, 1986, 3:00 p.m.

Special Session on Computational Mathematics, I Room 412 3:00- 3:20 (37) Preconditioned conjugate gradient methods for the solution of large sparse linear systems. DAVID YOUNG, University of Texas, Austin (830-65-13) 3:30- 3:50 (38) Characterization and computation of singular points with maximum rank deficiency. PATRICK J. RABIER* and GEORGE W. REDDIEN, Southern Methodist University (830-65-57) (Sponsored by W. E. Ferguson, Jr.) 4:00- 4:20 (39) The root-solving problem for ODEs. Preliminary report. L. F. SHAMPINE, Southern Methodist University (830-65-06) (Sponsored by W. E. Ferguson, Jr.) 4:30- 4:50 (40) Homotopy method for eigenvalue problems. TIEN- YIEN LI, Michigan State University (830-65-08)

Friday, October 31, 1986, 3:00 p.m. Special Session on Differential equations, I Room 413 3:00- 3:20 ( 41) Recent developments in Thomas-Fermi theory. Preliminary report. JEROME A. GOLDSTEIN*, Tulane University, and GISELE RUIZ RIEDER, Louisiana State University, Baton Rouge (830-99-109) 3:30- 3:50 (42) Periodic solutions of a system of reaction-diffusion equations. Preliminary report. SHAIR AHMAD and ALAN LAZER*, University of Miami (830-99-110) (Sponsored by Alfonso Castro) 4:00- 4:20 ( 43) A Stefan problem with memory. Preliminary report. R. E. SHOWALTER, University of Texas, Austin (830-35-59) 4:30- 4:50 (44) £P norms of a function and its derivatives. ANTON ZETTL, Northern Illinois University (830-39-10) 5:00- 5:20 (45) Radially symmetric solutions to a superlinear Dirichlet problem. ALFONSO CASTRO* and ALEXANDRA KUREPA, North Texas State University (830-35-67) 5:30- 5:50 (46) Variation of parameters formula for abstract Volterra functional differential equations. CONSTANTIN CORDUNEANU, University of Texas, Arlington (830-45-94)

Friday, October 31, 1986, 3:00 p.m.

Special Session on Connections Between Combinatorics, Algebra and Geometry, II Room 417 3:00-- 3:20 ( 47) Transforms and inversion formulas for double series. Preliminary report. HEINRICH NIEDERHAUSEN, Florida Atlantic University (830-05-89) 3:30- 3:50 (48) Very general Menelaean and Cevian theorems. THOMAS ZASLAVSKY, State University of New York, Binghamton (830-51-90) 4:00- 4:20 (49) The "easier" Waring problem for polynomials and Dickson's lemma. Preliminary report. DAVID R. RICHMAN, University of South Carolina, Columbia (830-11-21) (Sponsored by Michael Filaseta) 4:30- 4:50 (50) Isomorphisms of combinatorial objects. NEAL BRAND, North Texas State University ( 830-05-48)

Friday, October 31, 1986, 3:00 p.m. Special Session on Invariant Theory, II Room 418 3:00- 3:20 (51) The invariants of the additive group in characteristic p. Preliminary report. JOSEPH P. BRENNAN, Michigan State University and Seton Hall University (830-20-102) 3:30- 3:50 (52) Three interconnections between orthogonal and symplectic characters. ROBERT A. PROCTOR, University of California, Los Angeles (830-22-42) 4:00- 4:20 (53) Rational actions associated to the adjoint representation. BRIAN J. PARSHALL, University of Virginia (830-20-45) (Sponsored by V. Lakshmibai) 4:30- 4:50 (54) Alternating tableaux and the tensor algebra of gl( n). Preliminary report. JOHN STEMBRIDGE, University of California, Los Angeles (830-05-69) 5:00- 5:30 (55) Bases for the fundamental representations of the exceptional groups E5 and F4. V. LAKSHMIBAI, Texas A&M University, College Station (830-14-105)

796 Friday, October 31, 1986, 3:00 p.m. Session on Analysis, I Room 415 3:00- 3:10 (56) New characterizations of regular open sets, regular closed sets, extremally disconnectedness, and RS-compactness. CHARLES DORSETT, Louisiana Tech University (830-54-09) 3:15- 3:25 (57) Asymptotic integration of a perturbed constant coefficient differential equation under mild integral smallness conditions. WILLIAM F. TRENCH, Trinity University (830-34-72) 3:30- 3:40 (58) On an integral equation approach for the exterior Robin problem for the Helmholtz equation. TZU-CHU LIN, University of Wisconsin, Milwaukee (830-35-37) 3:45- 3:55 (59) Global approximation theorems for Bernstein type operators. Preliminary report. C. JAYASRI, University of Kerala, India, and Y. SITARAMAN*, University of Louisville (830-41-61) (Sponsored by Ibrahim N. Imam) 4:00- 4:10 (60) A theorem for uniform approximation in algebras of differentiable functions. ANDREANA S. MADGUEROVA, Sofia, Bulgaria (830-46-01) (Sponsored by Robert M. Fossum) 4:15- 4:25 (61) The summability and isomorphic structure of BK spaces. JEFF CONNOR, Loyola University (830-46-104) 4:30- 4:40 (62) An error bound for a direct approximation method of nonlinear parameter estimation. FRANK H. MATHIS, Baylor University (830-65-92) 4:45- 4:55 (63) Numerical computation of near points in distance geometry. JIM WELLS* and T. L. HAYDEN, University of Kentucky (830-65-52) 5:00- 5:10 (64) Zeroth law of path-superlogics in axiomatic field theoretic exam. STEPHEN L. WEINBERG, Berkeley, California (830-80-60) 5:15- 5:25 (65) On some fundamental limits and derivatives of trigonometric functions. Preliminary report. S. VERMA, University of Nevada, Las Vegas (830-99-114) (Sponsored by L. J. Simonoff)

Saturday, November 1, 1986, 8:00a.m. Special Session on Geometric Structures on Manifolds and Kleinian Groups, III Room 410 8:00- 8:30 (66) Distinguishing quasi-conformal from conformal. Preliminary report. MICHAEL H. FREEDMAN, University of California at San Diego, La Jolla, and RICHARD SKORA*, Indiana University, Bloomington (830-30-63) 8:35- 9:20 (67) Hyperbolic structures with conical singularities and a component of flat PSL(2,R)-bundles over surfaces. WILLIAM M. GOLDMAN* and WALTER D. NEUMANN, University of Maryland, College Park (830-57-64) 9:25-10:10 (68) Geometric structures on boundaries of symmetric spaces. Preliminary report. ADAM KORANYI, Herbert H. Lehman College, City University of New York (830-53-51) 10:15-10:45 (69) Measured laminations and deformations of Puchsian groups. YOSHINOBU KAMISHIMA, Hokkaido University, Japan (830-57-41)

Saturday, November 1, 1986, 8:00 a.m. Special Session on Differential Equations, II Room 413 8:00- 8:20 (70) Estimation of an elastic coefficient in a static plate. LUTHER WHITE, University of Oklahoma (830-99-111) 8:30- 8:50 (71) Uniqueness for a class of non-linear initial value problems. HANS G. KAPER* and MAN KAM KWONG, Argonne National Laboratory, Argonne, Illinois (830-34-95) 9:00- 9:20 (72) Gateaux differentiable operators and Lagrange multipliers. WILLIAM RAY, University of Oklahoma (830-47-100) {Sponsored by Stanley B. Eliason) 9:30- 9:50 (73) Explicit multidimensional solitary wave solutions to nonlinear evolution equations. HENRY A. WARCHALL, North Texas State University (830-99-112) 10:00-10:20 (74) A new turbulence closure candidate. Preliminary report. FRED R. PAYNE* and FEUNG­ TAI KAO, University of Texas, Arlington (830-45-75) 10:30-10:50 (75) Geometric concepts and methods in the theory of elliptic Euler-Lagrange equations. Preliminary report. ILYA J. BAKELMAN, Texas A&M University, College Station (830-99-113)

Saturday, November 1, 1986, 8:30 a.m. Special Session on Banach Spaces and Related Topics, III Room 411 8:30- 8:50 (76) On spaces having the B*RNP and the W*RNP. Preliminary report. ELIAS SAAB, University of Missouri, Columbia (830-46-29) 9:00- 9:20 (77) On the strucuture of non-dentable closed bounded convex sets. HASKELL ROSENTHAL, University of Texas, Austin (830-46-99) 9:30- 9:50 (78) Equidistributed random variables in Lp,q· N. L. CAROTHERS*, Texas A&M University, College Station, and S. J. DILWORTH, University of Texas, Austin (830-46-16)

797 10:00-10:20 (79) Subspaces of Lp,q· N. L. CAROTHERS, Texas A&M University, College Station, and S. J. DILWORTH*, University of Texas, Austin (830-4&-17) 10:30-10:50 (80) Banach space properties of Ciesielski-Pol's C(K) space. J. H. M. WHITFIELD, Lakehead University (830-4&-05)

Saturday, November 1, 1986, 8:30 a.m. Special Session on Mathematics for Large Seale Computing, II Room 412 8:30- 8:50 (81) Parallel processing of a domain decomposition method. R. C. Y. CHIN and G. W. HEDSTROM, Lawrence Livermore National Laboratory, Livermore, California, J. S. SCROGGS, University of Illinois, Urbana-Champaign, and D. C. SORENSEN*, Argonne National Laboratory, Argonne, Illinois (830-65-83) (Sponsored by M. F. Wheeler) 9:00- 9:20 (82) On fast computations of an elastic wave model. KIRK E. JORDAN, Exxon Research and Engineering Company, Annandale, New Jersey (830-65-84) 9:30- 9:50 (83) A class of hypercube based architecture and its applications. S. LAKSHMIVARAHAN, University of Oklahoma (830-65-85) (Sponsored by M. F. Wheeler) 10:00-10:20 (84) Hypercube implementation of parallel shooting. HERBERT B. KELLER, California Institute of Technology, and PAUL NELSON*, Texas Tech University (830-65-86) 10:30-10:50 (85) Parallel implementation of an adaptive algorithm for multiple integrals. ALAN GENZ, Washington State University (830-65-87) (Sponsored by M. F. Wheeler)

Saturday, November 1, 1986, 9:00 a.m. Special SeBSion on Connections Between Combinatories, Algebra and Geometry, III Room 417 9:00- 9:20 (86) Strict refinement for graphs and digraphs. JAMES W. WALKER, University of South Carolina, Columbia (830-05-88) 9:30- 9:50 (87) Vector representations of graphs. Preliminary report. T. D. PARSONS, California State University, Chico (830-05-28) 10:00-10:20 (88) The doubleton (3,3)-rounded sets. TALMAGE JAMES REID, Louisiana State University, Baton Rouge (830-05-14) (Sponsored by James G. Oxley) 10:30-10:50 (89) Growth rates of matroids not containing M(K4) or U2,q+2 as minors. Preliminary report. JOSEPH P. S. KUNG, North Texas State University (830-05-12)

Saturday, November 1, 1986, 9:00 a.m. Special Session on Invariant Theory, III Room 418 9:00- 9:20 (90) Vanishing of odd-dimensional intersection cohomology. Preliminary report. ROY JOSHUA, Clark University (830-14-03) 9:30- 9:50 (91) Parabolic analogue of Kazhdan- Lusztig polynomials. VINAY DEODHAR, Indiana University, Bloomington (830-20-77) 10:00-10:20 (92) Resolutions of representations. Preliminary report. KAAN AKIN, University of Oklahoma (830-20-91) 10:30-10:50 (93) Invariant theory and special functions. ROBERT A. GUSTAFSON, Texas A&M University, College Station (830-99-108)

Saturday, November 1, 1986, 9:00 a.m. Session on Algebra Room 415 9:00- 9:10 (94) Three new theorems on Fermat primes. C. MUSES, Mathematics and Morphology Research Centre, Miramonte, California (830-11-53) (Sponsored by K. Demys) 9:15- 9:25 (95) Invertible ideals in some subrings of Q[X]. Preliminary report. SCOTT T. CHAPMAN, North Texas State University (830-13-07) 9:30- 9:40 (96) Elliptic surfaces with q = 1. Preliminary report. PETER F. STILLER, Mathematical Sciences Research Institute, Berkeley, California (830-14-98) 9:45- 9:55 (97) Medial near-rings. Preliminary report. HENRY HEATHERLY, University of Southwestern Louisiana (830-1&-56) 10:00-10:10 (98) Differentiable semigroups on manifolds with boundary. J. P. HOLMES, Auburn University, Auburn (830-20-32) 10:15-10:25 (99) On the ideals of the group rings on the complex field. Preliminary report. MORTEZA SEDDIGHIN, Blackburn College (830-20-76) 10:30-10:40 (100) Distributive and medial elements of a groupoid. Preliminary report. GARY F. BIRKENMEIER, University of Southwestern Louisiana (830-20-93)

798 Saturday, November 1, 1986, 11:00 a.m. Invited Address The Lyceum 11:00-12:00 (101) Blocks of characters in finite Chevalley groups. PAUL FONG, University of Illinois, Chicago {830-20-70)

Saturday, November 1, 1986, 1:30 p.m. Invited AddreBB The Lyceum 1:30- 2:30 {102) Nonisotropic metrics in real and complex analysis. ALEXANDER NAGEL, University of Wisconsin, Madison (830-42-27)

Saturday, November 1, 1986, 3:00 p.m. Special Session on Geometric Structures on Manifolds and Kleinian Groups, IV Room 410 3:00- 3:30 {103) Root-closure for negatively curved manifold fundamental groups. CHRISTOPHER STARK, University of Florida {830-57-26) 3:35- 4:05 {104) A generalization of Jorgensen's inequality to higher dimensions. Preliminary report. PETER WATERMAN, Northern Illinois University (830-20-22) 4:10- 4:40 (105) On presentations of some Bianchi groups. Preliminary report. NORBERT WIELENBERG, University of Wisconsin, Parkside {830-57-36) 4:45- Panel D~cussion

Saturday, November 1, 1986, 3:00p.m. Special Session on Computational Mathematics, II Room 412 3:00- 3:20 ( 106) A dogleg algorithm for nonlinear equality constrained optimization. M. R. CELIS*, IMSL, Inc., Houston, Texas, and J. E. DENNIS, JR. and R. A. TAPIA, Rice University (830-65-39) {Sponsored by Warren E. Ferguson, Jr.) 3:30- 3:50 (107) Constrained separable nonlinear least squares problems. TERESA A. PARKS, Schlumberger Well Services, Houston, Texas {830-65-71) 4:00- 4:20 {108) A variable metric variant of the Karmarkar algorithm for linear programming. J. E. DENNIS, JR. and KATHRYN TURNER*, Rice University, and A. M. MORSHEDI, Shell Development Company, Houston, Texas (830-49-46) (Sponsored by Warren E. Ferguson, Jr.) 4:30- 4:50 (109) Simplified regularization. Preliminary report. JULIO GUACANEME, University of Puerto Rico, Rio Piedras {830-65-96) 5:00- 5:20 (110) Sources of ill-conditioning in a model reflection seismology problem. WILLIAM W. SYMES, Rice University (830-99-107) (Sponsored by Warren E. Ferguson, Jr.)

Saturday, November 1, 1986, 3:00 p.m. Special Session on Connections Between Combinatoric&, Algebra and Geometry, IV Room 417 3:00- 3:20 {111) The Cohen-Macaulay type of a determinantal ideal. Preliminary report. JOSEPH P. BRENNAN, Michigan State University and Seton Hall University (830-13-101) 3:30- 3:50 (112) Fixing elements in matroid minors. JAMES OXLEY*, Louisiana State University, Baton Rouge, and DON Row, University of Tasmania, Australia (830-05-35) 4:00- 4:20 {113) On Hilbertian ideals. SHREERAM ABHYANKAR, Purdue University, West Lafayette, and DEVADATTA KULKARNI*, Oakland University {830-05-30) 4:30- 4:50 (114) A fast algorithm for testing total unimodularity of matrices. KLAUS TRUEMPER, University of Texas, Dallas (830-99-115) {Sponsored by Joseph P. Kung)

Robert M. Fossum Urbana, Illinois Associate Secretary

799 Presenters of Papers Numbers following the names indicate the speakers' positions on the program. •Invited one-hour lecturer *Special session speaker

*Akin, K., 92 *Goldstein, J. A., 41 * Maskit, B., 29 Sitaraman, Y., 59 *Alspach, B., 18 *Gould, R., 16 Mathis, F. H., 62 * Skora, R., 66 *Alspach, D., 7 *Greim, P., 9 *Minkoff, M., 13 *Smith, M. A., 35 *Andrews, K. T., 8 * Griewank, A., 11 *Montgomery, S., 21 *Sorensen, D. C., 81 * Bakelman, I. J., 75 *Griggs, J. R., 17 *More, J. J., 12 *Stark, C., 103 *Baker, M., 31 * Guacaneme, J., 109 Muses, C., 94 *Stembridge, J., 54 Birkenmeier, G. F., 100 *Gustafson, R. A., 93 • Nagel, A., 102 Stiller, P. F., 96 *Brand, N., 50 Heatherly, H., 97 *Nelson, P., 84 *Symes, W. W., 110 *Brennan, J. P., 51, 111 Holmes, J.P., 98 * Niederhausen, H., 47 Trench, W. F., 57 *Brooks, R., 3 *Huffman, W. C., 19 *Osgood, B., 2 * Truemper, K., 114 *Carothers, N. L., 78 *Jimenez, J. A., 5 *Oxley, J., 112 *Turner, K., 108 *Castro, A., 45 *Jordan, K. E., 82 *Parks, T. A., 107 *Ulrich, B., 23 *Celis, M. R., 106 * Joshua, R., 90 *Parshall, B. J., 53 Verma, S., 65 Chapman, S. T., 95 * Kalton, N. J., 10 *Parsons, T. D., 87 *Walker, J. W., 86 Connor, J., 61 * Kamishima, Y., 69 *Payne, F. R., 74 * Warchall, H. A., 73 * Corduneanu, C., 46 * Kaper, H. G., 71 *Peck, N. T., 33 *Waterman, P., 104 * Deodhar, V., 91 *Keen, L., 26 *Phelps, R. R., 32 Weinberg, S. 1., 64 *Dilworth, S. J., 79 * Koranyi, A., 68 *Proctor, R. A., 52 Wells, J., 63 Dorsett, C., 56 *Kulkarni, D., 113 * Rabier, P. J., 38 • Wheeler, M. F., 25 *Edgar, G. A., 6 • Kulkarni, R. S., 24 * Ratcliffe, J., 1 *Wheeler, R. F., 36 *Ewing, R. E., 14 *Kung, J. P. S., 89 *Ray, W., 72 *White, 1., 70 * Feighn, M., 4 * Lakshmibai, V., 55 *Reid, T. J., 88 *Whitfield, J. H. M., 80 • Fong, P., 101 * Lakshmivarahan, S., 83 *Richman, D. R., 49 * Wielenberg, N., 105 *Fossum, R. M., 22 * Lazer, A., 42 * Rosenthal, H., 77 *Young, D., 37 *Gardiner, F. P., 30 * Li, T.-Y., 40 * Saab, E., 76 * Zaslavsky, T., 48 *Gehring, F. W., 28 Lin, T.-C., 58 * Saab, P., 34 * Zettl, A., 44 * Genz, A., 85 *Lucier, B., 15 Seddighin, M., 99 *Gilman, J., 27 Madguerova, A. S., 60 *Shampine, L. F., 39 *Goldman, W. M., 67 *Magid, A., 20 *Showalter, R. E., 43

800 San Antonio Meetings, January 21-24, 1987 Preliminary Announcement

The January 1987 Joint Mathematics Meetings, in­ 93rd Annual Meeting of the AMS cluding the 93rd Annual Meeting of the AMS, the January 21-24, 1987 70th Annual Meeting of the Mathematical Associa­ tion of America, and the 1987 annual meetings of Sixtieth Josiah Willard Gibbs Lecture the Association for Symbolic Logic, Association for The 1987 Gibbs Women in Mathematics, and the National Associa­ lecture will be presented at 8:30 tion for Mathematicians, will be held January 21- p.m. on Wednesday, January 21, by THOMAS C. 24 (Wednesday-Saturday), 1987, in San Antonio, SPENCER of the Courant Institute of Mathemat­ Texas. MAA will cosponsor a session on Wednesday, ical Sciences, New York University. The title of January 21, with the National Council for Teachers of his lecture will be announced later. Mathematics {NCTM). Sessions will take place in the San Antonio Convention Center and the San Antonio Marriott Hotel. Colloquium Lectures The members of the Local Arrangements Com­ mittee are Donald F. Bailey, Robert M. Fossum There will be a series of four Colloquium Lectures (ex-officio), William J. LeVeque (ex-officio), Kenneth presented by PETER D. LAX of the Courant A. Ross (ex-officio), Gregory P. Wene (chairman), Institute of Mathematical Sciences, New York Lawrence R. Williams, and Bennir A. Zinn. University. The title of this lecture series is Uses of the non-Euclidean wave equation. The lectures will be given at 1:00 p.m. daily, Wednesday WHERE TO FIND IT PAGE through Saturday, January 21-24. PREREGISTRATION AND HOUSING 802-803 ANNUAL MEETING OF THE AMS 801 Gibbs Lecture, Colloquium Lectures, Prizes, Invited Prizes Addresses, Special Sessions, Contributed Papers, Other Sessions, Council, Business Meeting The 1986 Leroy P. Steele Prizes and the 1987 AMS SHORT COURSE 805 Frank Nelson Cole Prize in Number Theory will ANNUAL MEETING OF THE MAA 806 be awarded at 4:25 p.m. on Thursday, January Invited Addresses, Minicourses, Contributed Papers, 22. Other Sessions, Business Meeting, Board of Governors, Section Officers, Films JOINT AMS-MAA SESSIONS 810 Invited Addresses OTHER ORGANIZATIONS 811 ASL, AWM, ICEMAP, JPBM, NAM, NCTM, NSF, By invitation of the Program Committee, there RMMC will be seven fifty-minute invited addresses. The TIMETABLE 813 names of the speakers, their affiliations, the dates OTHER EVENTS OF INTEREST 812 and times of their talks, and some of the titles Book Sales, Exhibits, Information Table, Petition Table follow: EMPLOYMENT REGISTER 826 MARC CULLER, Rutgers University, Free ACCOMMODATIONS 812 groups, trees and their automorphisms, 2:15 p.m. Hotels Saturday; REGISTRATION AT THE MEETINGS 818 RONALD J. DIPERNA, University of Cali­ Fees, Dates, Locations, Times, Services fornia, Berkeley, title to be announced, 10:05 a.m. MISCELLANEOUS INFORMATION 824 Wednesday; Child Care, Local Information, Parking, Social Event, RICHARD T. DURRETT, Cornell University, Travel, Weather Crabgrass, measles, and gypsy moths: An intro­ MAP 814 duction to modern probability, 10:05 a.m. Friday; IMPORTANT DEADLINES ROBERT M. HARDT, University of Min­ nesota, Minneapolis, Regularity and singularity AMS Abstract& for energy minimizing maps, For cOD8l.deratlon for special sessions Expired 3:20 p.m. Thursday; or contributed papers October 15 ROBERT J. MCELIECE, California Institute MAA Abstract& of Technology, title to be announced, 9:00 a.m. or contributed papers Expired Wednesday; Employment Reglater (AppHcanta It Employers) November 15 DAVID J. SALTMAN, University of Texas at EARLY Prereglatratlon and Houalng October S1 Austin, Noether's problem, Galois theory and the Prereglatratlon and Houalng November 15 Brauer group, 9:00 a.m. Friday; MAA Mlnlcourae Prereglatratlon November 15 Motions for AMS BualneBI Meeting December 22 LESLEY M. SIBNER, Polytechnic Institute Preregistration cancellations (50% refund) January 18 of New York, Analytic aspects of gauge field theory, 2:15p.m. Thursday.

801 Preregistration

Preregistration for these meetings and the not intended to include any person who has Mathematical Sciences Employment Register voluntarily resigned or retired from his or her must be completed by November 15, 1986. latest position. Those wishing to preregister must complete The emeritus status refers to any person the form which appears at the back of this who has been a member of the AMS or MAA issue and submit it together with the appropri­ for twenty years or more, and is retired on ate preregistration fee(s) to the Mathematics account of age from his or her latest position. Meetings Housing Bureau in Providence by A $5 charge will be imposed for all invoices November 15. Please note that a space has prepared when Preregistration/Housing Forms been provided on the Preregistration/Housing are submitted without accompanying payment Form if one wishes to have his /her nickname for the preregistration fee(s) and room de­ printed on the meeting badge. posits, or are accompanied by an amount Preregistration fees do not represent an insufficient to cover the total due. Preregistra­ advance deposit for lodgings. One must, tion/Housing Forms received well before the however, preregister for the meetings in order deadline of November 15 which are not accom­ to obtain hotel accommodations through the panied by correct payment will be returned to Mathematics Meetings Housing Bureau, as the participant with a request for resubmission outlined on the facing page. and full payment. This will, of course, delay Preregistration fees may be paid by check the processing of any housing request so that payable to the American Mathematical Society it will be unlikely that the participant's first (Canadian checks must be marked for payment choices will still be available. in U.S. funds), or by providing a VISA or A 50 percent refund of the preregistration MASTERCARD credit card number on the fee(s) will be made for all cancellations re­ Preregistration/Housing Form. Please be sure ceived in Providence no later than January 16. to give the name and number exactly as they No refunds will be granted for cancellations appear on the credit card, and to include the received after that date, or to persons who do expiration date. not attend the meetings. Those who preregister for the Joint Math­ The only exception to this rule is someone ematics Meetings pay fees which are 30 percent who preregisters for the Joint Mathematics lower than those who register at the meetings. Meetings only in order to attend an MAA The preregistration fees are as follows: Minicourse, and is too late to obtain a slot in the Minicourse. In this case, full refund will AMS Short Course be made of the Joint Mathematics Meetings Student/Unemployed $10 preregistration fee, provided the preregistrant All Others $35 has checked the box on the MAA Minicourse Preregistration Form that this was his or her Joint Mathematics Meetings intent. Individuals who preregister for both the Member of AMS, ASL, MAA, NCTM $59 Joint Meetings and a Minicourse and who in­ Emeritus Member of AMS, MAA $16 tend to participate in the Joint Meetings, even Nonmember $90 if the Minicourse is not available, should not, of Student/Unemployed $16 course, check the box on the MAA Minicourse Employment Register Preregistration Form. In this case the Joint Meetings preregistration will be processed. Employer $75 Those who wish to preregister for the Em­ Applicant $15 ployment Register should read carefully the Employer posting fee $10 special article titled "Mathematical Sciences There will be no extra charge for members of Employment Register" which follows this an­ the families of registered participants, except nouncement of the San Antonio meetings. The that all professional mathematicians who wish attention of applicants is particularly directed to attend sessions must register independently. to the section regarding the December issue of All full-time students currently working Employment Information in the Mathematical toward a degree or diploma qualify for the Sciences. student registration fees, regardless of income. Please read the facing page titled Housing The unemployed status refers to any per­ carefully before completing the Preregistra­ son currently unemployed, actively seeking tion/Housing Form. employment, and who is not a student. It is

802 Housing

Speeial Bonus for Early Preregistrants! Participants who preregister before the early preregistration deadline of October 31 will be eligible for a complimentary room in San Antonio. (Multiple occupancy of these rooms is permissible.) Winners will be randomly selected from the names of all who preregister by October 31 and these lucky individuals will be notified by mail by January 16. So, preregister early! Acknowledgment Form Participants will receive an acknowledgement of their preregistration, room deposit, and hotel assignment from the Mathematics Meetings Housing Bureau, which will be followed by a confir­ mation of the room reservation from the hotel to which they have been assigned.

The Preregistration/Housing Form for request­ either of the latter two circumstances apply, the ing hotel accommodations will be found at the back hotel will then charge your credit card account for of this issue. Use of the services offered by the one night's occupancy. Please read the section on Mathematics Meetings Housing Bureau requires Hotels carefully regarding deposits. preregistration for the meetings. Persons desiring Housing assignments are made on a first-come, confirmed hotel accommodations should complete first-served basis, so participants desiring low-cost the form, or a reasonable facsimile, and send it to accommodations are urged to submit their housing the Mathematics Meetings Housing Bureau, Post requests in as early as possible. Participants should Office Box 6887, Providence, Rhode Island 02940, also be aware that the special rates being offered so that it will arrive no later than November 15, in the section titled Hotels may not be available 1986. Housing requests received after the deadline after December 31. of November 15 most surely cannot be honored. Participants are strongly urged to rank each ho­ All reservation requests must be received in tel on the housing form in the order of preference, writing and will be processed through the Housing and circle the type of room and the rate desired. Bureau in Providence. Telephone requests will not Reservations will be made in accordance with pref­ be accepted. Please do not contact the hotels erences indicated on the reservation form insofar directly. Blocks of rooms and special rates have as this is possible. If not all hotels are ranked, and been set aside for the Housing Bureau, and the all rooms have been filled at the ranked hotels, hotel will either refer you back to the Housing the assignment will be made at an unranked hotel Bureau, or give you a room outside of the block, with the next lowest rate. which may be at a higher rate. Please note that Participants who are able to do so are urged the room occupancy tax in San Antonio is 11 to share a room whenever possible as this proce­ percent. dure can be economically beneficial. The housing Please read carefully the section on Hotels before form should be fully completed to ensure proper completing the form. Forms sent to the wrong assignment of rooms. Participants planning to address and thus incurring delay in delivery to the share accommodations should provide the name( s) Housing Bureau until after the deadline cannot be of the person(s) with whom they plan to occupy a accepted and will, therefore, be returned. room. Each participant should, however, complete Participants requesting hotel accommodations a separate Preregistration/Housing Form. In in San Antonio are required to submit housing order to avoid confusion, parties planning to share deposits or credit card information when prereg­ rooms should send their forms together in the istering. Deposits may be paid by check payable same envelope. The participant requesting the to the AMS (Canadian checks must be marked for room should submit the deposit and will be the payment in U.S. funds), or by providing a VISA recipient of the hotel confirmation. or MASTERCARD credit card number on the Pre­ Please make all changes to or cancellations of registration/Housing Form. Please be sure to give hotel reservations with the Housing Bureau in the name and number exactly as they appear on Providence before January 12, 1986, by calling the credit card, and to include the expiration date. 401-272-9500, extension 239. After that date, Please note that when you provide a credit card changes or cancellations should be made directly number in lieu of a $50 check as a guarantee, no with the hotel assigned. charge against your account will be processed by Please read the facing page titled Preregis­ the hotel unless you fail to claim the reserved room tration carefully before completing the Preregis­ on your given arrival date, or if you fail to cancel tration/Housing Form. your reservation directly with the hotel/motel 48 Please be sure to send housing deposit or credit hours in advance of your given arrival date. If card information with Preregistration/Housing Form.

803 Special Sessions anyone contributing an abstract for the meeting By invitation of the same committee, there will be who felt that his or her paper would be particu­ twelve special sessions of selected twenty-minute larly appropriate for one of these sessions should papers. The topics of these special sessions, have indicated this clearly on the abstract, and the names and affiliations of the mathematicians should have submitted it by September 24, 1986, arranging them, the dates and times they will three weeks earlier than the normal deadline for meet, and some partial lists of speakers, are as contributed papers, in order that it be considered follows: for inclusion. Geometric methods in group theory, ROGER ALPERIN, University of Oklahoma, Thursday 8:00 a.m., Friday 1:00 p.m., and Saturday 8:00a.m. Contributed Papers Nonlinear partial differential equations, There will be sessions for contributed papers RONALD J. DIPERNA, University of Califor­ Wednesday afternoon, Thursday morning, Friday nia, Berkeley, Wednesday 2:15 p.m., Thursday afternoon, and Saturday morning. 8:00 a.m., and Friday 1:00 p.m. Abstracts should be prepared on the stan­ Classical real analysis, MICHAEL J. EVANS, dard AMS form available from the AMS office North Carolina State University, Raleigh and in Providence or in departments of mathematics, PAUL D. HUMKE, St. Olaf College, Thursday and should be sent to Abstracts, Editorial De­ 8:00 a.m., Friday 1:00 p.m., and Saturday 8:00 partment, American Mathematical Society, Post a.m. Office Box 6248, Providence, Rhode Island 02940, Brauer groups and Galois theory, BURTON so as to arrive by the abstract deadline of October FEIN, Oregon State University, DAVID J. SALT­ 15, 1986. A charge of $16 is imposed for retyping MAN, University of Texas at Austin and MURRAY abstracts that are not in camera-ready form. SCHACHER, University of California, Los Ange­ Late papers will not be accepted. les, Thursday 8:00 a.m., Friday 1:00 p.m., and Saturday 8:00 a.m. Geometric inequalities, MICHAEL GAGE, Other AMS Sessions University of Rochester, and ERWIN LUTWAK, Polytechnic Institute of New York, Wednesday Mathematics and Government Speaker 2:15 p.m., Thursday 8:00 a.m., and Friday 1:00 The AMS Committee on Science Policy will spon­ p.m. sor a Special Invited Address on Wednesday, Stochastic processes and analysis, JOSEPH January 21 from 7:15 p.m. to 8:15 p.m. The GLOVER, University of Florida and A. 0. PIT­ name of the speaker and the title of this address TENGER, University of Maryland, Thursday 8:00 will be announced later. a.m., Friday 1:00 p.m., and Saturday 8:00 a.m. F. Knight, R.W.R. Darling, B. Davis, J. Glover, Committee on the Agenda for BusineBB Meetings T. McConnell, M. Pinsky, L. Pitt, and P. March, The Society has a Committee on the Agenda for Busi­ A. 0. Pittenger. ness Meetings. The purpose is to make Business Meet­ Combinatorics and group representations, ings orderly and effective. The committee does not have PHILIP J. HANLON, University of Michigan, legal or administrative power. It is intended that the Wednesday 2:15 p.m., Thursday 8:00 a.m., and committee consider what may be called "quasi-political" motions. The committee has several possible courses Friday 1:00 p.m. of action on a proposed motion, including but not re­ Theoretical optimization, LYNN MCLINDEN, stricted to University of Illinois, Urbana-Champaign and JAY (a) doing nothing; S. TREIMAN, West Michigan University, Friday (b) conferring with supporters and opponents to 8:00 a.m. arrive at a mutually accepted amended version to be 1:00 p.m. and Saturday circulated in advance of the meeting; Orthogonal polynomials and the moment (c) recommending and planning a format for de­ problem, PAUL G. NEVAI, Ohio State University, bate to suggest to a Business Meeting; Friday 1:00 p.m. and Saturday 8:00 a.m. (d) recommending referral to a committee; results in gauge field theory and Rie­ (e) recommending debate followed by referral to a Recent committee. mannian geometry, LESLEY M. SIBNER, Poly­ There is no mechanism that requires automatic technic Institute of New York, Wednesday 2:15 submission of a motion to the committee. However, if a p.m., Thursday 8:00 a.m., and Friday 1:00 p.m. motion has not been submitted through the committee, Mathematical physics, GREGORY P. WENE, it may be thought reasonable by a Business Meeting to refer it rather than to act on it without benefit of the University of Texas, San Antonio, Saturday 8:00 advice of the committee. a.m. A. Bohm, L. C. Biedenharn, J. R. Klauder, The committee consists of M. Salah Baouendi, Ev­ R. H. Oehmke, and R. 0. Wells, Jr. erett Pitcher (chairman), and Carol L. Walker. Geometrical variational problems, BRIAN In order that a motion for the Business Meeting Friday 1:00 p.m. of January 22, 1987 receive the service offered by the WHITE, Stanford University, committee in the most effective manner, it should be in and Saturday 8:00 a.m. the hands of the secretary by December 22, 1986. Most of the papers to be presented at these Everett Pitcher, Secretary special sessions will be by invitation; however,

804 American Mathematical Society Short Course Series Introductory Survey Lectures on Moments in Mathematics San Antonio, Texas, January 20-22, 1987

The American Mathematical Society, in conjunction with its ninety-third Annual Meeting, will present a Short Course titled Moments in Mathematics on Tuesday, Wednesday, and Thursday, January 20, 21, and 22, at the San Antonio Convention Center. The program is under the direction of Henry Landau of AT&T Bell Laboratories. Six lectures are planned, and it is anticipated that proceedings will be published in the series Proceedings of Symposia in Applied Mathematics. Function theory, spectral decomposition of operators, probability, approximation, electrical and mechanical inverse problems, prediction of stochastic processes, and the design of algorithms for signal-processing VLSI chips are among a host of important theoretical and applied topics illuminated by the classical problem of moments, a problem fascinating in its own right because of its elegance and simplicity. The aim of this course is to survey some of these ramifications and the current research which derives from them. The course will consist of six lectures. The titles, speakers, and schedule for the talks are as follows: January 20, afternoon: "The Classical Background," Henry J. Landau, AT&T Bell Laboratories. "Geometry of the Moment Problem," J.H.B. Kemperman, Rutgers University. January 21, afternoon: "Moment Problems and Operators in Hilbert Space," Donald Sarason, University of California, Berkeley. "Signal Processing Applications of Some Moment Problems," Tom Kailath, Stanford University. January 22, morning: "The Multidimensional Moment Problem and Semigroups," Christian Berg, University of Copenhagen, Denmark. "Moment Problems in Probability and Statistics," Persi Diaconis, Stanford University. While no specialized background is required of participants, a familiarity with the basic notions of real and complex analysis will generally be assumed. Synopses of the lectures and accompanying reading lists appear in this issue of Notices. Complete lecture notes will be mailed to those who preregister for the course and will be available at the Short Course registration desk for those registering for the course on site. All who wish to participate in the Short Course may do so upon payment of a $35 advance registration fee ($45 on site). There are reduced fees for students and unemployed individuals. Please refer to the sections titled Preregistration, Housing and Registration at the Meetings for details. Those who plan to also register for the Joint Annual Meeting should take note of a special session entitled "Orthogonal polynomials and the moment problem," organized by Paul G. Nevai of Ohio State University. For more information, see the Special Session section of the San Antonio meeting announcement in this issue of Notices. The Short Course was recommended by the Society's Committee on Employment and Educational Policy (CEEP), whose members are Stefan A. Burr, Edward A. Connors, Philip C. Curtis, Jr., Gerald J. Janusz, Donald C. Rung (chairman), and Audrey A. Terras. The Short Course series is under the direction of the CEEP Short Course Subcommittee, whose members are Stefan A. Burr (chairman), Lisl Novak Gaal, Gerald J. Janusz, Robert P. Kurshan, and Barbara L. Osofsky.

805 Panel Discussion on Social Importance Minicourses of Mathematics Fifteen Minicourses are being offered by the MAA. The AMS Committee on Science Policy is spon­ The names and affiliations of the organizers, the soring a panel discussion on The problem of Star topics, the dates and times of their meetings, and Wars software reliability, organized by DAVID the enrollment limitations of each are as follows: EISENBUD of Brandeis University at 7:00p.m. on Minicourse #1: A microcomputer linear al­ Friday, January 23. Speakers will include DAVID gebra course using LIN-KIT is being organized PARNAS, Queens University. by HOWARD ANTON. Part A is scheduled from 9:00 a.m. to 10:55 a.m. on Wednesday, January 21, and Part B from 2:15 p.m. to 4:15 p.m. on Council Meeting Thursday, January 22. Enrollment is limited to The Council of the Society will meet at 5:00 p.m. 30. on Thesday, January 20. LIN-KIT is a powerful microcomputer pack­ age which can do linear algebra operations in Business Meeting either exact rational arithmetic (without the distraction of round-off error) or floating point The Business Meeting of the Society will take arithmetic (facilitating study of computational place immediately following the award of the aspects). Its data storage and retrieval capabili­ Steele and Cole Prizes at 4:25 p.m. on Thursday, ties lend themselves to self-paced courses. January 22. The secretary notes the following The Minicourse will consist of ( 1) a "hands­ resolution of the Council: Each person who on" session on the use of LIN-KIT, (2) a problem­ attends a Business Meeting of the Society shall solving session, making application of LIN-KIT, be willing and able to identify himself as a ( 3) a session on design of courses to meet various member of the Society. In further explanation, needs, and (4) a summary session devoted to it is noted that each person who is to vote discussion and perhaps design of a new computer­ at a meeting is thereby identifying himself as based linear algebra course. Microcomputers will and claiming to be a member of the American be used extensively by participants, but prior Mathematical Society. For additional information experience is not required. on the Business Meeting, please refer to the box Minicourse #2: Introduction to computer titled Committee on the Agenda for Business graphics is being organized by JOAN P. WYZ­ Meetings. KOSKI, Fairfield University. Part A is scheduled from 9:00 a.m. to 10:55 a.m. on Wednesday, of the MAA January 21, and Part B from 2:15 p.m. to 4:15 70th Annual Meeting p.m. on Thursday, January 22. Enrollment is January 21-24, 1987 limited to 30. Invited Addresses Graphs and illustrations of geometrical ob­ jects are useful tools in the teaching of mathemat­ There will be seven invited fifty-minute addresses. ics. Computer graphics simplifies the production The names of the speakers, their affiliations, the of these teaching aids. This Minicourse will dates and times of their talks, and the titles present some of the mathematical techniques used follow: to produce realistic pictures on graphics display STEVEN J. BRAMS, New York University, devices. Some of the topics to be discussed are Game theory, nuclear deterrence, and Star Wars, curve and surface sketching, 2D and 3D trans­ 2:15p.m. Friday; formations, perspective drawing, and hidden line ANDY DESESSA, University of California, removal. Suggestions will be given for the use Berkeley, Artificial worlds and real mathematics, of these techniques to complement mathematics 3:20 p.m. Wednesday; instruction. Since personal computers will be DANIEL H. GOTTLIEB, Purdue University, available for demonstrations and in-class imple­ Algebraic topology and robots, 2:15 p.m. Wednes­ mentations, programming experience is necessary. day; Minicourse #3: The teaching of applied RICHARD K. GUY, University of Calgary, mathematics is being organized by W. GILBERT The strong law of small numbers, 10:05 a.m. STRANG, Massachusetts Institute of Technology. Saturday; Part A is scheduled from 9:00 a.m. to 10:55 a.m. PETER D. LAX, Courant Institute of Mathe­ on Wednesday, January 21, and Part B from 2:15 matical Sciences, New York University, Euclidean/ p.m. to 4:15 p.m. on Thursday, January 22. non-Euclidean wave equation, 9:00a.m. Saturday; Enrollment is limited to 80. FRANK T. LEIGHTON, Massachusetts Insti­ The organizer will discuss one possible frame­ tute of Technology, Networks, parallel computa­ work for an introduction to modern applied math­ tion, and VLSI, 9:00 a.m. Thursday; ematics. After basic courses in calculus and linear JOHN W. MILNOR, Institute for Advanced algebra, there is an important need that is not Study, Self-similarity and hairiness in the Man­ met by the traditional advanced calculus. The delbrot set, 10:05 a.m. Thursday. course should include both discrete and contin-

806 uous problems, and numerical and combinatorial DAVID A. SMITH, Benedict College and DAVID algorithms, bringing out their analogies and devel­ P. KRAINES, Duke University. Part A is sched­ oping the mathematical ideas that are shared by uled from 4:30 p.m. to 6:30 p.m. on Wednesday, different applications. The organizer is convinced January 21, and Part B from 7:00 p.m. to 9:00 that this syllabus is also the right way to orga­ p.m. on Thursday, January 22. Enrollment is nize the mathematics needed by engineers and limited to 30. computer scientists; that subject does not have to Selected IBM-compatible commercial soft­ be old-fashioned and boring. Topics from several ward packages will be demonstrated, and ways areas will be presented exemplifying this unifying to use them in teaching and supplementing tra­ approach. Participants will be invited to discuss ditional calculus courses will be presented. Par­ effective ways to teach applied mathematics. ticipants will have "hands-on" use of the selected Minicourse #4: Interesting applications of materials. Handouts will provide information elementary mathematics is being organized by about other calculus materials available for IBM, JOANNE S. GROWNEY, Bloomsburg University. Apple II series, and Macintosh computers. Part A is scheduled from 9:00 a.m. to 10:55 a.m. There is no Minicourse #7. on Friday, January 23, and Part B from 7:00 p.m. Minicourse #8: Computer simulation of dis­ to 9:00 p.m. on Friday, January 23. Enrollment crete systems is being organized by ZAVEN A. is limited to 40. KARIAN, Denison University. Part A is scheduled Students with modest mathematical back­ from 9:00 a.m. to 10:55 a.m. on Friday, January grounds often are unaware of the ways that the 23, and Part B from 1:00 p.m. to 3:00 p.m. on arithmetic and logic that they already know can Saturday, January 24. Enrollment is limited to be used in many ways: to organize and un­ 30. derstand information, to make decisions, and to Computer simulations are particularly useful solve problems. This Minicourse will introduce in situations where: (a) complete mathemati­ to participants a variety of interesting and signif­ cal formulation of a problem is not possible or icant applications that can be used in "general available; (b) available analytic methods require education" courses designed to develop mathe­ simplifying assumptions which distort the true matics appreciation and quantitative reasoning nature of the problem; (c) available methods are skills in students in the humanities and other so complex that they become impractical; (d) it non-quantitative fields. is too complex or too expensive to conduct real­ Mathematical topics will be presented in the world experiments; (e) it is necessary to change context of solving particular problems. Prob­ the time scale to study the dynamics of a system. lem types to be considered include: organizing The objective of this course is to provide an information, scheduling time, individual decision understanding of the design, implementation, and making, group decision making, achieving a goal, analysis of discrete-event computer simulations. analysis of a fad, and simulation. The emphasis will be on the computational issues Treatment of topics will include strategies associated with the implementation of simulations for development of student skills in divergent through GPSS and/or SIMSCRIPT 11.5, the two thinking and evaluation as well as convergent most widely used discrete-event simulation pro­ thinking (problem solving). Lists of references for gramming languages. In the second se~sion, there further reading will be supplied. will be an opportunity to work with some mod­ Minicourse #5: Discrete mathematics us­ els on IBM-PC compatible systems using these ing difference equations is being organized by languages. JAMES T. SANDEFUR, JR., Georgetown Uni­ Minicourse #9: Recurrence relations is be­ versity. Part A is scheduled from 4:30 p.m. to ing organized by MARGARET BARRY COZZENS, 6:30p.m. on Wednesday, January 21, and Part Northeastern University. Part A is scheduled B from 7:00 p.m. to 9:00 p.m. on Thursday, from 9:00 a.m. to 10:55 a.m. on Friday, January January 22. Enrollment is limited to 30. 23, and Part B from 1:00 p.m. to 3:00 p.m. on Difference equations provide a non-standard Saturday, January 24. Enrollment is limited to structure to discrete mathematics, permitting 30. standard topics such as linear algebra and prob­ Problems where the behavior of a system ability to be interspersed with interesting models can be expressed in terms of the behavior of a including Markov processes and predator-prey re­ system at a previous stage or time can often be lationships. The Minicourse will introduce both solved using a recurrence relation. In addition, linear and nonlinear difference equations and recurrence relations are used to model population provide illustrative applications of each. Mi­ growth, heating and cooling, radioactive decay, crocomputers will be used to calculate solutions the spread of information and disease, and the for mathematical models and generate graphical time to run computer algorithms. This Mini­ output. Prior programming experience is not course will show how recurrence relations can be required. included in the curriculum of a wide variety of Minicourse #6: Using microcomputer soft­ courses, from advanced high school courses, to ware in teaching calculus is being organized by finite math courses, to calculus, and to discrete

807 structures courses. It will show how models based January 23, and Part B from 3:30 p.m. to 5:30 on recurrence relations lead in a natural way p.m. on Saturday, January 24. Enrollment is to models based on differential equations, and limited to 30. Prerequisite: none, although some therefore can and should be integrated in calculus familiarity with BASIC would be helpful. courses. Microcomputers can significantly enrich in­ Microcomputers will be available to enhance troductory calculus in a number of ways. This the understanding of recurrence relations and the Minicourse will illustrate one such way, which problems studied. uses a powerful, fully structured version of BA­ Minicourse #10: Integrating history into SIC developed at Dartmouth by the inventors of undergraduate mathematics courses is being orga­ BASIC, John Kemeny and Thomas Kurtz. Writ­ nized by JUDITH V. GRABINER, Pitzer College. ing programs in this language can both teach Part A is scheduled from 9:00 a.m. to 10:55 a.m. precise, logical thinking akin to that needed to on Friday, January 23, and Part B from 1:00 p.m. construct proofs and also provide impressive con­ to 3:00p.m. on Saturday, January 24. Enrollment crete illustrations of basic mathematical concepts. is limited to 50. Participants will experience the ease of use and The history of mathematics can help us teach power of True BASIC by entering and running students to understand mathematical ideas better several simple programs designed for student cre­ by understanding how those ideas actually came ation, and will also use programs that numerically to be. This Minicourse will take an in-depth look illustrate limits, differentiation, implicit differenti­ at two examples from the calculus: the calculus ation, optimization, root approximation, integra­ as algorithm, and the foundations of the calculus. tion, sequences, series, Taylor polynomials, and It will include careful study of selections from numerical solution of differential equations. Con­ the work of men like Fermat, Leibniz, Newton, siderable attention will be devoted to True BA­ and Cauchy. Brief accounts (and supporting SIC's machine-independent graphics, which afford materials) for other examples, from probability easy plotting of functions, parametric equations, and statistics and from computer science, will be and polar coordinate equations. touched on as well. Finally, guidance will be given Minicourse #13: For all practical purposes is on how to learn more, what materials are most being organized by SOLOMON A. GARFUNKEL, helpful, and how to develop other examples on COMAP, Inc. Part A is scheduled from 7:00 p.m. one's own. to 9:00 p.m. on Friday, January 23, and Part B Minicourse #11: Teaching mathematical mod­ from 3:30 p.m. to 5:30 p.m. on Saturday, January eling is being organized by FRANK R. GIOR­ 24. Enrollment is limited to 40. DANO, U.S. Military Academy and MAURICE D. This course deals with introducing contempo­ WEIR, Naval Postgraduate School. Part A is rary applications throughout the undergraduate scheduled from 9:00 a.m. to 10:55 a.m. on Friday, curriculum. Materials presented will include tapes January 23, and Part B from 1:00 p.m. to 3:00 from the soon-to-be-released PBS telecourse For p.m. on Saturday, January 24. An optional all practical purposes as well as print modules third session, Part C, will use the microcomputer from the UMAP series. Applications will cover facility and is scheduled from 3:30 p.m. to 5:30 a wide variety of fields with special emphasis on p.m. on Saturday, January 24. Enrollment is discrete mathematics and applications to manage­ limited to 40. ment science and decision making. The MAA Committee on the Undergradu­ Minicourse #14: Applications of discrete ate Program in Mathematics recommended in mathematics is being organized by FRED 1981 that "Students should have an opportunity STEPHEN ROBERTS, Rutgers University. Part to undertake 'real world' mathematical modeling A is scheduled from 7:00 p.m. to 9:00 p.m. on projects ... " as part of the common core curricu­ Friday, January 23, and Part B from 3:30 p.m. to lum for all mathematical science majors. This 5:30 p.m. on Saturday, January 24. Enrollment because many applications of problems in science, is limited to 60. industry, and government are best approached One of the reasons that discrete mathematics using mathematical modeling techniques. has become so important is the enormous variety This Minicourse provides an introduction to of applications of the subject. This Minicourse the modeling process, to several topics under­ will explore these applications. The emphasis lying the construction of mathematical models will be on several simple and traditional discrete and addresses issues related to the design of an techniques: basic counting rules of combinatorics, undergraduate course in modeling. the principle of inclusion and exclusion, the notion The optional third session will consist of of graph coloring, and the concept of eulerian demonstrations and "hands-on" running of models path. These techniques will be quickly reviewed on microcomputers. (though prior knowledge of combinatorics or graph Minicourse #12: True BASIC in freshman theory will not be necessary). Applications will calculus is being organized by JAMES F. HUR­ include switching functions in computer science, LEY, University of Connecticut. Part A is sched­ DNA chains in genetics, power in simple games uled from 7:00 p.m. to 9:00 p.m. on Friday, in economics and political science, scheduling

808 and operations research, engineering problems • Experiences with computer support for ser­ involving telecommunications and mobile radio vice courses, CAROL JONES, University of transmission, urban sciences, computer graph Houston-Downtown, Wednesday morning. plotting of electrical networks, and keypunching • Retaining and recruiting undergraduate wo­ errors in computing. men in mathematics courses: Aspirations Minicourse #15: Constructing placement and experiences, PATRICIA C. KENSCHAFT, examinations is being organized by JOHN W. Montclair State College, Saturday afternoon. KENELLY, Clemson University. Part A is sched­ The deadline for submitting papers for these uled from 7:00 p.m. to 9:00 p.m. on Friday, sessions was September 30. Late papers will not January 23, and Part B from 3:30 p.m. to 5:30 be accepted. p.m. on Saturday, January 24. Enrollment is limited to 40. Other MAA Sessions Lectures and workshops will take partici­ pants, step-by-step, through the entire process MAA-NCTM Panel Discussion of constructing and implementing placement ex­ A panel discussion on Reform in mathematics ams, including: preliminary planning, writing test education is being jointly sponsored by the MAA items, designing a test for establishing cut-off and the National Council of Teachers of Mathe­ scores, and evaluating the test. Placement testing matics (NCTM); it is scheduled from 9:00a.m. to problems of participants' own institutions will be 10:55 a.m. on Wednesday, January 21. The mod­ discussed during question and answer periods. erator is JOHN DOSSEY, president of NCTM, and Participants interested in attending any of the panel members are DONALD L. CHAMBERS, the Minicourses should complete the Minicourse Department of Public Instruction (Madison, WI); Preregistration Form and send it directly to the F. JOE CROSSWHITE, past-president of NCTM; MAA Office at the address given on the form so as PAUL FOERSTER, Alamo Heights High School to arrive prior to the November 15 deadline. DO (San Antonio); CAROL GREENES, Boston Uni­ NOT SEND THIS FORM TO PROVIDENCE. versity; SHIRLEY A. HILL, Mathematical Sci­ The Minicourses are open only to persons ences Education Board; and JACK PRICE, Palos who have registered for the Joint Mathematics Verdes Peninsula School District (Rolling Hills, Meetings and paid the Joint Meetings registration CA). fee. If the only reason for registering for the Joint Software Session Meetings is to gain admission to a Minicourse, this A session on The leading edge of software has should be indicated by checking the appropriate been organized by WARREN PAGE, New York box on the Minicourse Preregistration Form. City Technical College (CUNY). The session is Then, if the Minicourse is fully subscribed, full scheduled from 8:30 a.m. to 10:55 a.m. on refund can be made of the Joint Mathematics Wednesday, January 21. fee. Otherwise, the Meetings preregistration Two-Year College Reception Joint Meetings preregistration will be processed, and then be subject to the 50 percent refund The Committee on Two-Year Colleges is spon­ rule. PREREGISTRATION FORMS FOR THE soring an informal reception for two-year college JOINT MEETINGS SHOULD BE MAILED TO faculty from 4:30 p.m. to 6:00 p.m. on Wednes­ PROVIDENCE PRIOR TO THE DEADLINE day, January 21. OF NOVEMBER 15. The Mathematical Competition in Modeling The registration fee for Minicourses #1, #2, BENJAMIN A. FUSARO, Salisbury State College, and #12 is $35 each. The #5, #6, #8, #9, has organized a presentation on The mathematical registration fee for the other Minicourses is $25 in modeling {MCM) from 2:15p.m. to each. competition 4:15p.m. on Thursday, January 22. Introductory remarks by the organizer will be followed by three Contributed Papers winning solution papers from the contest, which Contributed papers were accepted on five topics will be presented by the winning student teams. in collegiate mathematics. The topics, organizers, their affiliations, and days they will meet are: ICME-6 Panel Discussion • Remedial mathematics: Issues and innova­ A panel discussion titled Post-secondary mathe­ tions, GEOFFREY R. AKST, Borough Man­ matics at ICME-6: What are the major issues? hattan Community College CUNY, Friday is scheduled from 2:15 p.m. to 3:45 p.m. on morning. Thursday, January 22. The organizers are JOHN • The history of mathematics, DUANE BLUM­ M. MACK, University of Sydney, and LYNN A. BERG, University of Southwestern Louisiana, STEEN, St. Olaf College. The purpose of this Thursday afternoon. session is to obtain "an American perspective," • New methods of teaching calculus, WADE identifying the issues most needing debate in ELLIS, JR., West Valley College, San Jose, 1988 on mathematics education in colleges and Wednesday morning. universities.

809 MAA-ACM-IEEE Panel Discussion Prize Session and Business Meeting The Joint MAA-ACM-IEEE Task Force on Teach­ The MAA Prize Session is scheduled from 3:20 ing Computer Science in Mathematics Depart­ p.m. to 4:30 p.m. on Friday, January 23. The ments is sponsoring a panel discusion in order Chauvenet Prize, the Award for Distinguished to exchange views with members of MAA. This Service to Mathematics, and six Certificates of panel is organized by ZAVEN A. KARIAN, Deni­ Meritorious Service will be presented. The 1986 son University, and is scheduled from 2:15 p.m. Carl B. Allendoerfer, Lester R. Ford, and George to 4:00 p.m. on Thursday, January 22. Three P6lya Awards for expository writing will also be members of the Task Force will make brief state­ presented. ments and the remaining time will be devoted to The Business Meeting of the MAA will take an open discussion. place at 4:40 p.m. following the Prize Session. Some bylaw changes will be submitted for mem­ bership approval. This meeting is open to all Mathematics as a Humanistic Discipline members of the Association. ALVIN WHITE, Harvey Mudd College, is orga­ Board of Governors nizing a session on Mathematics as a humanistic The MAA Board of Governors will meet at 9:00 discipline which is scheduled from 8:00 a.m. to a.m. on Tuesday, January 20. This meeting is 10:55 a.m. on Friday, January 23. Partici­ open to all members of the Association. pants will include the organizer; DONALD W. BUSHAW, Washington State University; UBI­ Section Officers RATAN D'AMBROSIO, Univ Estadual de Camp­ There will be a Section Officers' meeting at 7:00 inas (Brazil); PHILIP J. DAVIS, Brown Univer­ p.m. on Tuesday, January 20. sity; A. GARDINER, University of Birmingham (England); SHIRLEY A. HILL, Mathematical Sci­ Films ences Education Board; ANNELI LAX, Courant The MAA Film Program will take place on Friday, Institute of Mathematical Sciences, New York January 23, at 7:30p.m. The program will include University; DAVID B. MEREDITH, San Francisco the films Planar double pendulum and Fly Lorenz. State University; ROBERT OSSERMAN, Stanford University; FRANCES A. ROSAMOND, Ithaca; Joint AMS- MAA Sessions and SHERMAN K. STEIN, University of Califor­ nia, Davis. AMS-MAA Invited Addresses By invitation of the AMS-MAA Joint Program Committee, (Judith V. Grabiner, chairman, Paul CCIME Panel Discussion R. Halmos, F. Reese Harvey, and W. Gilbert The Committee on Computers in Mathematics Strang), two speakers will address the joint meet­ Education (CCIME) is sponsoring a panel dis­ ing of the AMS and MAA on the history and cussion on The use of computers in teaching development of mathematics. The names of the differential equations. The panel will be chaired speakers, their affiliations, the titles, dates, and by HOWARD LEWIS PENN, U.S. Naval Academy, times of their talks follow: and is scheduled from 9:00 a.m. to 10:30 a.m. on EDWARD N. LORENZ, Massachusetts Insti­ Friday, January 23. tute of Technology, Strange attractors: Are they still strange?, 11:10 a.m. Friday. UTA C. MERZBACH, National Museum of Project 2061 Mathematics Panel American History, Algebraic traditions on two continents, 11:10 a.m. Thursday. The Mathematics panel report of AAAS Project 2061 is scheduled from 2:15 p.m. to 4:15 p.m. AMS-MAA Symposium on Saturday, January 24. Among the panelists The AMS and MAA are cosponsoring a symposium will be LEON HENKIN, University of California, on The role of mathematicians in pre-college edu­ Berkeley. cation at 7:00p.m. on Thursday, January 22. This symposium has been organized by PHILIP WA­ GREICH of the University of Illinois at Chicago. Statistics Presentation Speakers include C. HERBERT CLEMENS, Uni­ There will be a presentation titled Working with versity of Utah; LEON HENKIN, University of statistics: Statistical process control {SPC) tech­ California, Berkeley; HARVEY KEYNES, Univer­ niques from 2:15 p.m. to 4:15 p.m. on Saturday, sity of Minnesota, Minneapolis; PAUL J. SALLY, January 24. Speakers will include BARBARA JR., University of Chicago; and PHILIP WA­ ASHLEY and ANALISA L. FRANCE, Jefferson GREICH. This symposium will deal with various Community College (Louisville, KY). The presen­ questions related to the involvement of math­ tation will give an overview of the ways in which ematicians in pre-college education. Does it elementary statistics are being used to improve benefit education? How does it affect mathemati­ quality in business and industry. cians? How to get financial support? Is there

810 moral and financial support from mathematics Activities of Other Organizations departments? Is it possible to be involved in pre-college education and continue to do research The Association for Symbolic Logie (ASL) Council in mathematics? Members of the panel will give will meet on Thursday and Friday, January 22- brief presentations describing projects with which 23, from 8:00 p.m. to 11:00 p.m. Sessions will be they are involved. Open discussion will follow. on Friday and Saturday, January 23-24. There will be an ASL reception on Friday, January 23, AMS-MAA-MSEB Forum on the from 5:30 p.m. to 7:00 p.m. For more details on K-12 Curriculum the ASL program, see the Timetable. The Over the next two decades, the nation's schools Association for Women in Mathematics must make a dramatic transition in their math­ (AWM) will sponsor the seventh annual Emmy ematics programs, with emphasis shifting from Noether Lecture at 1:00 p.m. on Friday, January drill in paper-and-pencil computations to expe­ 23, by JOAN BIRMAN. The title of her talk will rience in using the conceptual, analytical, and be announced later. problem-solving techniques of mathematics. This The AWM will also sponsor a panel discussion transition will involve fundamental changes in on Responses to the David Report: Initiatives for content, modes of instruction, teacher education, women and minorities on Friday, January 23 at and methods of assessing student progress. The 9:00a.m. impact on collegiate mathematics will be substan­ The AWM Business Meeting will be held at tial. In this forum, members of the Mathematical 10:00 a.m. on Friday, January 23. Sciences Education Board of the National Re­ A reception is being planned by AWM at 6:00 search Council will present their current thinking p.m. on Friday, January 23. about how to bring about the transition, and The Interagency Commission for Extramural will seek reactions and advice from the forum's Mathematics Programs (ICEMAP) will present a participants. Individuals intending to participate session at 4:25 p.m. on Wednesday, January in the forum may wish to review materials on the 21. The program will focus on current topics K-12 curriculum prepared by MSEB. To obtain in federal mathematical support. Presentations these materials in advance of the forum, contact: will be made by the National Science Foundation, MSEB, National Research Council, 2101 Con­ Department of Energy, and the Department of stitution Avenue, NW, Washington, DC 20418; Defense agencies discussing a variety 202-334-3294. Copies will also of new and be available at continuing programs the forum, which will and opportunities for federal take place at 9:30 a.m. on funding. Friday, January 23. The Joint Policy Board for Mathematics (JPBM) Committee for Mathematics Department AMS-MAA Workshop Heads has organized a National Meeting of De­ The AMS and MAA are cosponsoring a workshop partment Heads at 7:00 p.m. on Thursday, Jan­ at 4:30 p.m. on Wednesday, January 21. This uary 22. This session will feature a program workshop is being organized by JOEL SCHNEI­ conducted by DAVID P. ROSELLE, Provost at DER, Director of Content of the new series Square Virginia Polytechnic Institute and State Uni­ One TV, produced by the Children's Television versity, on Accreditation for mathematics depart­ Workshop. This exciting new series will pre­ ments. This will be followed by Birds-of-a-Feather miere on public television on January 26 and sessions on The evaluation of instruction /or large is targeted to children ages eight to fourteen. schools by DONALD W. BUSHAW; The evalua­ Highlights include "Mathnet" (a take-off on Drag­ tion of instruction for small schools by DAVID net), which uses a detective metaphor to teach W. BALLEW; Master's degrees in the mathe­ problem-solving, and a musical approach to infin­ matical sciences for large schools by RICHARD ity. Children from the local San Antonio school HABERMAN; and Master's degrees in the mathe­ system will be invited to view this program along matical sciences for small schools by BENJAMIN with mathematicians and press. A question ail.d A. FUSARO. answer session will follow the showing of the The Committee for Department program. Chairs of the JPBM will also cosponsor a workshop for department chairs with the American Council on AMS-MAA Panel Discussion Education (ACE) on Tuesday, January 20 from The AMS and MAA are cosponsoring a panel 9:00 a.m. to 5:00 p.m. Participants who are discussion at 7:00p.m. on Thursday, January 22 interested in this workshop should see News and on What makes news in mathematics? Speakers Announcements in this issue. will include media representatives from the west The National Association of Mathematicians coast, east coast, and the San Antonio area, and (NAM) will receive the William W. S. Claytor two mathematicians currently involved in public Lecture at 1:00 p.m. on Saturday, January 24. information in mathematics. A question and The name of the speaker and the title of this answer session will follow. address will be announced later.

811 The NAM Business Meeting will take place at 10:00 a.m. on Saturday, January 24. Petition Table NAM will also sponsor a panel discussion on At the request of the AMS Committee on Human Saturday, January 24 at 9:00 a.m. Rights of Mathematicians, a table will be made The National Science Foundation (NSF) will available in the meeting registration area at which sponsor a session on Federal support for math­ petitions on behalf of named individual mathemati­ ematics education at 5:30 p.m. on Wednesday, cians suffering from human rights violations may January 21, organized by JOHN A. THORPE, be displayed and signed by meeting participants Deputy Division Director, Division of Materi­ acting in their individual capacities. size may be displayed at the and Informal Science Signs of moderate als Development, Research table, but must not represent that the case of the Education, National Science Foundation. individual in question is backed by the Committee The NSF will also be represented at a booth on Human Rights unless it has, in fact, so voted. in the exhibit area. NSF staff members will be Volunteers may be present at the table to provide available to provide counsel and information on information on individual cases, but notice must NSF programs of interest to mathematicians. The be sent at least seven (7) days in advance of the booth will be open the same days and hours as meeting to the Meetings Department in Providence the exhibits. (telephone 401-272-9500). Since space is limited, The Rocky Mountain Mathematics Consor­ it may also be necessary to limit the number of tium (RMMC) Board of Directors will meet on volunteers present at the table at any one time. on Human Rights may delegate January 22 from 2:15p.m. to 4:15p.m. The Committee Thursday, a person to be present at the table at any or all times, taking precedence over other volunteers. Other Events of Interest Any material which is not a petition (e.g., advertisements, resumes) will be removed by the Book Sales staff. When registration closes, any material on Books published by the AMS and MAA will be the table will be discarded, so individuals placing sold at discounted prices somewhat below the cost petitions on the table should be sure to remove for the same books purchased by mail. These them prior to the close of registration. discounts will be available only to registered participants wearing the official meeting badge. be aware that, when ma­ VISA and MASTERCARD credit cards will be Participants should in any large city, additional accepted for book sale purchases at the meeting. jor conventions occur especially at night. The book sales will be open the same days and safety problems are created, the meetings alone, or hours as the exhibits and are located in the North Those who are attending to and from Banquet Hall. who are concerned about walking the meetings after dark, are encouraged to choose EXhibits a hotel in close proximity to the San Antonio The book and educational media exhibits will be Convention Center. Participants are also urged located in the North Banquet Hall and will be to read the "Words to the Wise" in the local open Wednesday through Friday, January 21-23 information insert in the program they receive at from 9:00 a.m. to 5:00 p.m. All participants the meetings. to visit the exhibits during the are encouraged Reservations at these hotels cannot be made visiting the exhibits will be meeting. Participants by calling the hotel directly until after January meeting badge or acknowl­ asked to display their 12. Also, after that date, the rates below may from the Mathematics edgment of preregistration not apply. Meetings Housing Bureau in order to enter the exhibit area. In all cases "single" refers to one person in one bed; "double" refers to two persons in one Mathematical Sciences Employment Register bed; "twin" refers to two persons in two twin Those wishing to participate in the Employment beds; and "twin double" refers to two persons in Register at the San Antonio meetings should read two double beds. A rollaway cot for an extra carefully the important article about the Register person can be added to a room; however, not all which follows this meeting announcement. hotels are able to do so. Accommodations Participants should be aware that it is general hotel practice in most cities to hold a nonguaran­ Hotels teed reservation until 6:00 p.m. only. When one The rates listed below are subject to an 11 percent guarantees a reservation by paying a deposit or hotel/motel tax. The number in parentheses after submitting a credit card number as guarantee in the name of the hotel is the number it carries on advance, however, the hotel usually will honor this the map. The estimated walking distance from reservation up until checkout time the following the hotel to the San Antonio Convention Center day. If the individual holding the reservation has is given in parentheses following the telephone not checked in by that time, the room is then number. released for sale, and the hotel retains the deposit

812 TIMETABLE

The purpose of this timetable is to provide assistance to preregistrants in the selection of arrival and departure dates. The program, as outlined below, is based on information available at press time.

AMERICAN MATHEMATICAL SOCIETY SHORT COURSE SERIES

TUESDAY, January 20 MOMENTS IN MATHEMATICS

10:00 a.m. - 3:00 p.m. REGISTRATION (Short Course Only) 3:00 p.m. - 4:15 p.m. The classical background Henry J. Landau 4:45 p.m. - 6:00 p.m. Geometry of the moment problem J. H. B. Kemperman

WEDNESDAY, January 21

2:15 p.m. - 3:30 p.m. Moment problems and operators in Hilbert space Donald Sarason 4:00 p.m. - 5:15 p.m. Signal-processing applications of some moment problems Tom Kailath

THURSDAY, January 22

8:30 a.m. - 9:45 a.m. The multidimensional moment problem and semigroups Christian Berg 9:45 a.m. - 11:00 a.m. Moment problems in probability and statistics Persi Diaconis

JOINT MATHEMATICS MEETINGS Mathematical Association of America TUESJAY, January 20 American Mathematical Society and other Organizations 9:00 a.m. - 4:00 p.m. MAA - BOARD OF GOVERNORS' MEETING

4:00 p.m. - 8:00 p.m. REGISTRATION 5:00 p.m. - 10:00 p.m. COUNCIL MEETING 7:00 p.m. - 9:00 p.m. MAA - SECTION OFFICERS' MEETING

WEDNESDAY, January 21 AMS MAA and other Organizations

8:00 a.m. - 10:55 a.m. MAA - CONTRIBUTED PAPER SESSION New methods of teaching calculus Wade Ellis, Jr. 8:00 a.m. - 10:55 a.m. MAA - CONTRIBUTED PAPER SESSION Experiences with computer support for service courses Carol Jones 8:00 a.m. - 5:00 p.m. REGISTRATION 8:30 a.m. - 11:00 a.m. MAA - SESSION IThe leading edge of software Warren Page 9:00 a.m. - 9:30 a.m. EMPLOYMENT REGISTER ORIENTATION SESSION 9:00 a.m. - 9:50 a.m. INVITED ADDRESS Title to be announced Robert J. McEl!ece 9:00 a.m. - 10:55 a.m. MAA-NCTM- PANEL DISCUSSION Reform in mathematics education Donald L. Chambers F. Joe Crosswhite John Dossey (moderator) Paul Foerster Carol Greenes Shirley A. Hill Jack Price 9:00 a.m. - 10:55 a.m. MAA - MINICOURSE ill (Part A) A microcomputer linear algebra course using LIN-KIT Howard Anton

813 I Downtown San Antonio TO AIRPORT j[_j - ~ ~.•

HOTEL 1. San Antonio Marriott - Headquarters Hotel 2. The Crockett Hotel 3. Hilton Palacio del Rio Hotel 4. Holiday Inn - Downtown at Market Square 5. Hyatt Regency San Antonio Hotel 6. La Mansion del Rio Hotel 7. La Quinta Convention Center Motor Inn 8. La Quinta Market Square Motor Inn 9. Travelodge on the River

814 WEDNESDAY, January 21 American Mathematical Society Mathematical Association of America and O:her Organizations 9:00 a.m. - 10:55 a.m. MAA - MINICOURSE #2 (Part A) Introduction to computer graphics Joan P. Wyzkoski 9:00 a.m. - 10:55 a.m. MAA - MINICOURSE #3 (Part A) The teaching of applied mathematics W. Gilbert Strang 9:00 a.m. - 5:00 p.m. EXHIBITS 9:00 a.m. - 5:00 p.m. AMS EXHIBIT AND BOOK SALE MAA BOOK SALE 9:30 a.m. - 4:00 p.m. EMPLOYMENT REGISTER REGISTRATION 10:05 a.m. - 10:55 a.m. INVITED ADDRESS Title to be announced Ronald J. DiPerna 1:00 p.m. - 2:00 p.m. COLLOQUIUM LECTURE I Uses of the non-Euclidean wave equation Peter D. Lax 2:15p.m. - 3:05p.m. MAA - INVITED ADDRESS Algebraic topology and robcts Daniel H. Gcttlieb SPECIAL SESSIONS 2:15 p.m. - 6:00 p.m. Nonlinear partial differential equations I 2:15 p.m. - 6:00 p.m. Geometric inequalities I 2:15 p.m. - 6:00 p.m. Combinatorics and group representations I 2:15 p.m. - 6:00 p.m. Recent results in gauge field theory and Riemannian geometry I 2:15 p.m. - 6:00 p.m. SESSIONS FCR CONTRIBUTED PAPERS 3:20 p.m. - 4:10 p.m. MAA - INVITED ADDRESS Artificial worlds and real mathematics Andy deSessa 4:25 p.m. - 5:30 p.m. Interagency Commission for Extramural Mathematics Programs (ICEMAP) Speakers to be announced 4:30 p.m. - 5:30 p.m. AMS-MAA WORKSHOP Square Ole TV Joel Schneider 4:30 p.m. - 6:00 p.m. MAA- Committee on Two-Year Colleges CASH BAR SOCIAL HOUR AND DISCUSSION 4:30 p.m. - 6:30 p.m. MAA - MINICOURSE #5 (Part A) Discrete mathematics using difference equations James T. Sandefur, Jr. 4:30 p.m. - 6:30 p.m. MAA - MINICOURSE #6 (Part A) Using microcomputer software in teaching calculus David P. Kraines David A. Smith 5:30 p.m. - 6:30 p.m. National Science Foundation SESSION: Federal support for mathematics education John A. Thorpe 7:15p.m. - 8:15p.m. Committee on Science Policy SPECIAL INVITED ADDRESS Speaker and title to be announced 8:30 p.m. - 9:30 p.m. JOSIAH WILLARD GIBBS LECTURE Title to be announced Thomas C. Spencer

THURSDAY, January 22 AMS MAA and O:her Organizations

SPECIAL SESSIONS 8:00 a.m. - 10:55 a.m. Geometric methods in group theory I 8:00 a.m. - 10:55 a.m. Nonlinear partial differential equations II 8:00 a.m. - 10:55 a.m. Classical real analysis I 8:00 a.m. - 10:55 a.m. Brauer groups and Galois theory I 8:00 a.m. - 10:55 a.m. Geometric inequalities II 8:00 a.m. - 10:55 a.m. stochastic processes and analysis I 8:00 a.m. - 10:55 a.m. Combinatorics and group representations II 8:00 a.m. - 10:55 a.m. Recent results in gauge field theory and Riemannian geometry II

815 or applies one night's room charge to the credit Singles $54 card number submitted. Doubles $54 If you hold a guaranteed reservation at a Triples $59 hotel, but are informed upon arrival that there is Quads $59 no room for you, there are certain things you can Suites $150 request the hotel do. First, they should provide There is no charge for children 18 years of age for a room at another hotel in town for that and under. The Crockett Hotel is a full-service evening, at no charge. (You have already paid for hotel. the first night when you made your deposit.) They Hilton Palacio del Rio (3) should pay for taxi fares to the other hotel that 200 South Alamo evening, and back to the meetings the following San Antonio, Texas 78205 morning. They should also pay for one telephone Telephone: 512-222-1400 (1 block) toll call so that you can let people know you are Singles $60 not at the hotel you expected. They should make Doubles $60 every effort to find a room for you in their hotel Triples $66 the following day, and, if successful, pay your taxi Triples $66 (with cot) fares to and from the second hotel so that you Quads $70 can pick up your baggage and bring it to the There is no charge for children. The Hilton first hotel. Not all hotels in all cities follow this is a full-service hotel. practice, so your request for these services may Holiday Inn Downtown at Market Square (4) bring mixed results, or none at all. 318 West Durango Please make all changes to or cancellations of San Antonio, Texas 78204 hotel reservations with the Mathematics Meetings Telephone: 512-225-3211 (10 blocks) Housing Bureau in Providence before January 12, Singles $47 1987. The telephone number in Providence is Doubles $47 401-272-9500 (extension 239). After that date, Triples $47 changes should be made directly with the hotel. Triples $47 (with cot) Cancellations must be made directly with the Quads $47 hotel 48 hours prior to date of arrival in order to Quads $47 (with cot) receive refunds of deposits. There is no charge for children 17 years of The following hotels/motels accept American age and under. The Holiday Inn is a full-service Express, MASTERCARD, VISA, Carte Blanche, hotel. Diners Club credit cards, personal checks with Hyatt Regency San Antonio (5) identification, and travelers' checks in payment for room charges. 123 Losoya Street San Antonio, Texas 78205 Please note that public transportation is Telephone: 512-222-1234 (3 blocks) limited on Saturdays in San Antonio. Participants Singles $62 who have been assigned to the Holiday Inn Doubles $68 Downtown-Market Square and the La Quinta Triples $75 Market Square Motor Inn should be prepared Triples $90 (with cot) to use taxicabs to and from the San Antonio Quads $75 · Convention Center on that day. Quads $90 (with cot) San Antonio Marriott (1) Suites $150-$411 Headquarters Hotel There is no charge for children 18 years of age 711 E. Riverwalk and under. The Hyatt Regency is a full-service San Antonio, Texas 78205 hotel. Telephone: 512-224-4555 (1 block) La Mansion del Rio Hotel (6) Singles $60 112 College Street Doubles $60 San Antonio, Texas 78205 Triples $66 Telephone: 512-225-2581 (4 blocks) Triples $66 (with cot) Singles $60 Quads $70 Doubles $70 Quads $70 (with cot) Triples $80 Suites $125-$300 Triples $80 (with cot) There is no charge for children 12 years of Quads $80 age and under. The San Antonio Marriott is a Quads $80 (with cot) full-service hotel. Suites $275- $425 The Crockett Hotel (2) 320 Bonham There is no charge for children 18 years of San Antonio, Texas 78205-2083 age and under. The La Mansion is a full-service Telephone: 512-225-6500 (3 blocks) hotel.

816 Mathematical Association of America THURSDAY, January 22 American Mathematical Society and Clher Organizations 8:00 a.m. - 10:55 a.m. SESSIONS FOR CONTRIBUTED PAPERS 8:00 a.m. - 4:00 p.m. REGISTRATION 9:00 a.m. - 9:50 a.m. MAA - INVITED ADDRESS INetworks, parallel computation, and VLSI Frank T. Leighton 9:00 a.m. - 5:00 p.m. EXHIBITS 9:00 a.m. - 5:00 p.m. AMS EXHIBIT AND BOOK SALE \ MAA BOOK SALE 9:00a.m. EMPLOYMENT REGISTER DISTRIBUTION OF SCHEDULES 9:30 a.m. - 5:30 p.m. EMPLOYMENT REGISTER INTERVIEWS 10:05 a.m. - 10:55 a.m. MAA - INVITED ADDRESS Self-similarity and hairiness in the Mandelbrot set John W. Milnor 11:10 a.m. - noon AMS-MAA INVITED ADDRESS Algebraic traditions on two continents Uta C. Merzbach 1:00 p.m. - 2:00 p.m. COLLOQUIUM LECTURE II Uses of the non-Euclidean wave equation Peter D. Lax 2:15 p.m. - 3:05 p.m. INVITED ADDRESS Analytic aspects of gauge field theory Lesley M. Sibner 2:15 p.m. - 3:45 p.m. MAA - PANEL DISCUSSION: Post-secondary mathematics at ICME-6: What are the major issues? John M. Mack Lynn A. Steen 2:15p.m. - 4:00 p.m. MAA-ACM-IEEE Task Force on Teaching Computer Science in Mathematics Depart­ ments- PANEL DISCUSSION Zaven A. Karian 2:15p.m. - 4:15p.m. MAA- PRESENTATION: The mathemati- cal competition in modeling (MCM) Benjamin A. Fusaro 2:15p.m. - 4:15p.m. MAA- CONTRIBUTED PAPER SESSION The history of mathematics Duane Blumberg 2:15 p.m. - 4:15 p.m. MAA - MINICOURSE ill (Part B) A microcomputer linear algebra course using LIN-KIT Howard Anton 2:15p.m. - 4:15 p.m. MAA- MINICOURSE 112 (Part B) Introduction to computer graphics Joan P. Wyzkoski 2:15 p.m. - 4:15 p.m. MAA - MINICOURSE 113 (Part B) The teaching of applied mathematics W. Gilbert Strang 2:15 p.m. - 4:15 p.m. Rocky Mountain Mathematics Consortium BOARD OF DffiECTORS' MEETING 3:20p.m. - 4:10 p.m. INVITED ADDRESS Regularity and singularity for energy minimizing maps Rooert M. Hardt 4:25 p.m. - 6:00 p.m. STEELE & COLE PRIZE SESSION AND BUSINESS MEETING 7:00p.m. - 9:00 p.m. NATIONAL MEETING OF DEPARTMENT HEADS 7:00p.m. PANEL DISCUSSION: Accreditation for mathematics departments David P. Roselle 8:00 p.m. SESSION: Birds-of-a-Feather 8:00 p.m. The evaluation of instruction for large schools Donald W. Bushaw 8:00 p.m. The evaluation of instruction for small schools David w. Ballew 8:00 p.m. Master's degrees in the mathematical sciences for large schools Richard Haberman 8:00 p.m. Master's degrees in the mathematical sciences for small schools Benjamin A. Fusaro 817 La Quinta Convention Center Motor Inn (7) this method over an increase in the registration 1001 E. Commerce Street fees. San Antonio, Texas 78205 Telephone: 512-222-9181 (2 blocks) Registration at the Meetings Singles $46 Doubles $56 Meeting preregistration and registration fees only Triples $61 partially cover expenses of holding meetings. All Triples $61 (with cot) mathematicians who wish to attend sessions are Quads $66 expected to register, and should be prepared to Quads $66 (with cot) show their meeting badge, if so requested. Badges There is no charge for children 18 years of are required to enter the exhibit area, to obtain age and under. The La Quinta Convention Center discounts at the AMS and MAA Book Sales, Motor Inn is a full-service Inn. to cash a check with the meeting cashier, and La Quinta Market Square Motor Inn (8) to attend all sessions scheduled in the Theatre 900 Dolorosa Street in the San Antonio Convention Center. (If a San Antonio, Texas 78207 preregistrant should arrive too late in the day Telephone: 512-271-0001 (10 blocks) to pick up his/her badge, he/she may show the acknowledgment of preregistration received from Singles $37 Doubles $45 the Mathematics Meetings Housing Bureau as proof of registration.) The fees for Joint Meetings Triples $49 registration at the meeting listed below are 30 Quads $49 percent more than the preregistration fees. There is no charge for children 17 years of age Participants wishing to attend sessions for and under. There are no food services available one day only may take advantage of the one-day in this Inn. fees listed below. These special fees are effective TraveLodge on the River (9) daily January 21 through 24, and are available at 100 Villita Street the meeting to members and nonmembers only. San Antonio, Texas 78205 These one-day fees are not applicable to student, Telephone: 512-226-2271 (4 blocks) unemployed, or emeritus participants, whose fees Singles $43 for registration at the meetings are listed below. Doubles $46 Triples $49 Joint Mathematics Meetings Triples $49 (with cot) Member of AMS, ASL, MAA or NCTM $ 77 Quads $52 Emeritus Member of AMS, MAA $ 21 Quads $52 (with cot) Nonmember $117 Suites $90 Student/Unemployed $ 21 There is no charge for children 17 years of age and under. The TraveLodge is a full-service hotel, One Day Fee and provides free transportation to and from the Member of AMS, ASL, MAA or NCTM $ 40 airport baggage claim areas. Nonmember $ 61 The AMS-MAA Joint Meetings Committee Employment Register always endeavors to obtain the lowest possible sleeping room rates for participants at annual Employer $100 meetings. The Committee is also responsible Applicant $ 20 for maintaining a sound fiscal position for these Employer Posting fee $ 15 meetings, and, until recently, has been able to AMS Short Course keep the deficits at a reasonable level, while still providing the very best meeting facilities available Student/Unemployed $ 15 to the participants. All Other Participants $ 45 As the meetings have grown in scope and MAA Minicourses complexity over the years, however, it has been (if openings available) necessary to find larger facilities with more and Minicourses 1, 2, 5, 6, 8, 9, or 12 $ 35 more session rooms. Unfortunately, the cost of # Minicourses 3, 4, 10, 11, 13, 14, or 15 $ 25 these facilities is higher than can be covered by the # registration fees, and the Committee has arranged Registration fees may be paid at the meetings in for all of the hotels in San Antonio to collect an cash, by personal or travelers' check, or by VISA extra $2 per room per night from participants, or MASTERCARD credit card. Canadian checks which will be used to offset the rental cost of must be marked for payment in U.S. funds. the San Antonio Convention Center. (The rates There is no extra charge for members of the above include this extra charge.) The Committee families of registered participants, except that all hopes that these extra funds will not be necessary professional mathematicians who wish to attend at future annual meetings, and therefore chose sessions must register independently.

818 THURSDAY, January 22 American Mathematical Society

7:00 p.m. - 9:00 p.m. MAA - MINICOURSE 115 (Part B) Discrete mathematics using difference equations James T. Sandefur, Jr. 7:00 p.m. - 9:00 p.m. MAA - MINICOURSE 116 (Part B) Using microcomputer software in teaching calculus David P. Kraines David A. Smith 7:00 p.m. - 9:30 p.m. AMS-MAA- PANEL DISCUSSION What makes news in mathematics? 7:00 p.m. - 9:30 p.m. AMS-MAA- SYMPCSIUM The role of mathematicians in pre-college education C. Herbert Clemens Leon Henkin Harvey Keynes Paul J. Sally, Jr. Philip Wagreich (organizer) 8:00 p.m. - 10:00 p.m. No-H03T COCKTAIL PARTY 8:00 p.m. - 11:00 p.m. Association for Symbolic Logic COUNCIL MEETING

FRIDAY, January 23 AMS MAA and O;her Organizations

morning ASL - SESSIONS 8:00 a.m. - 10:55 a.m. MAA - CONTRIBUTED PAPER SESSION Remedial mathematics: Issues and innovations Geoffrey R. Akst 8:00 a.m. - 10:55 a.m. MAA - SESSION Mathematics as a humanistic discipline Donald w. Bushaw Ubiratan D'Ambrosio Philip J. Davis A. Gardiner Shirley A. Hill Anneli Lax David B. Meredith Robert Osserman Frances A. Rosamond Sherman K. Stein Alvin White (organizer) 8:00 a.m. - 4:00 p.m. REGISTRATION 9:00 a.m. - 9:50 a.m. INVITED ADDRESS Noether's problem, Galois theory, and the Brauer group David J. Saltman 9:00 a.m. - 10:00 a.m. Association for Women in Mathematics PANEL DISCUSSION: Reponses to the David Report: Initiatives for women and minorities 9:00 a.m. - 10:30 a.m. MAA - Committee on Computers in Mathe­ matical Education- PANEL DISCUSSION: The use of computers in teaching differential equations Howard Lewis Penn 9:00 a.m. - 10:55 a. m. MAA - MINICOURSE il4 (Part A) Interesting applications of elementary mathematics JoAnne S. Growney 9:00 a.m. - 10:55 a.m. MAA - MINICOURSE 118 (Part A) Computer simulation of discrete systems Zaven A. Karian 9:00 a.m. - 10:55 a.m. MAA - MINICOURSE 119 (Part A) Recurrence relations Margaret Barry Cozzens 9:00 a.m. - 10:55 a.m. MAA - MINICOURSE 1110 (Part A) Integrating history into undergraduate mathematics courses Judith V. Grabiner

819 All full-time students currently working to­ Baggage and Coat Check ward a degree or diploma qualify for the student Inquire at the meetings registration desk. registration fees, regardless of income. The unemployed status refers to any person Check Cashing currently unemployed, actively seeking employ­ The meeting cashier will cash personal or travel­ ment, and who is not a student. It is not intended ers' checks up to $50, upon presentation of the to include any person who has voluntarily resigned official meeting registration badge, provided there or retired from his or her latest position. is enough cash on hand. Canadian checks must Persons who qualify for emeritus membership be marked for payment in U.S. funds. It is in either the Society or the Association may advisable that participants bring travelers' checks register at the emeritus member rate. The with them. When funds are low the meetings emeritus status refers to any person who has been cashier will not be able to cash checks and trav­ a member of the AMS or MAA for twenty years or elers' checks can be easily cashed at local banks, more, and is retired on account of age from his or restaurants, or hotels. her latest position. Local Information Nonmembers who preregister or register at the meeting and pay the nonmember fee will This section of the desk will be staffed by members receive mailings from AMS and MAA, after the of the Local Arrangements Committee and other meeting is over, containing information about a volunteers from the San Antonio mathematical special membership offer. community. Registration Dates, Times, and Locations Lost and Found See the Joint Meetings cashier. AMS Short Course Outside Room 102, San Antonio Convention Mail Center All mail and telegrams for persons attending the Tuesday, January 20 10:00 a.m. to 3:00 p.m. meetings should be addressed as follows: Name Joint Mathematics Meetings of Participant, c/o Joint Mathematics Meetings, [and MAA Minicourses (until filled)] P. 0. Box 2277, San Antonio, TX 78298. Mail North Banquet Hall, San Antonio Convention and telegrams so addressed may be picked up at Center the mailbox in the registration area during the Tuesday, January 20 4:00p.m. to 8:00p.m. hours the registration desk is open. U.S. mail Wednesday, January 21 8:00 a.m. to 5:00 p.m. not picked up will be forwarded after the meeting Thursday, January 22 to the mailing address given on the participant's through 8:00 a.m. to 4:00 p.m. registration record. Saturday, January 24 Personal Messages Registration Desk Services Participants wishing to exchange messages during the meeting should use the mailbox mentioned Assistance, Comments, and Complaints above. Message pads and pencils are provided. A log for registering participants' comments or It is regretted that such messages left in the complaints about the meeting is kept at the Trans­ box cannot be forwarded to participants after the parencies section of the registration desk. All meeting is over. participants are encouraged to use this method of helping to improve future meetings. Comments Telephone Messages on all phases of the meeting are welcome. If A telephone message center is located in the a written reply is desired, participants should registration area to receive incoming calls for furnish their name and address. participants. The center is open from January Participants with problems of an immediate 20 through 24, during the hours that the Joint nature requiring action at the meeting should see Mathematics Meetings registration desk is open. the Director of Meetings, who will try to assist Messages will be taken and the name of any them. individual for whom a message has been received will be posted until the message has been picked Audio- Visual Assistance up at the message center. The telephone number A member of the AMS/MAA staff will be available of the message center will be announced later. to advise or consult with speakers on audio-visual usage. Speakers having unusual audio-visual Transparencies requirements such as slide or film projectors Speakers wishing to prepare transparencies in should make their requests prior to the beginning advance of their talk will find the necessary of the meeting. materials and copying machines at this section of Rooms where special sessions and contributed the registration desk. A member of the staff will paper sessions will be held are equipped with an assist and advise speakers on the best procedures overhead projector and screen. Blackboards will and methods for preparation of their material. not be available. There is a modest charge for these materials.

820 FRIDAY, January 23 American Mathematical Society Mathematical Association of America and O:her CEganizations 9:00 a.m. - 10:55 a.m. MAA - MINICOURSE #11 (Part A) Teaching mathematical modeling Frank R. Giordano Maurice D. Weir 9:00 a.m. - 5:00 p.m. EXHIBITS 9:00 a.m. - 5:00 p.m. AMS EXHIBIT AND BOOK SALE IMAA BOOK SALE 9:00 a.m. EMPLOYMENT REGISTER DISTRIBUTION OF SCHEDULES 9:30 a.m. - 10:55 a.m. A MS-MAA- Mathematical Sciences Education Board FCRUM: The K-12 Curriculum 9:30 a.m. - 5:30 p.m. EMPLOYMENT REGISTER INTERVIEWS 10:00 a.m. - 10:55 a.m. AWM- BUSINESS MEETING 10:05 a.m. - 10:55 a.m. INVITED ADDRESS Crabgrass, measles, and gypsy moths: An introduction to modern probability Richard T. Durrett 11:10 a.m. - noon AMS-MAA INVITED ADDRESS Strange attractors: Are they still strange? Edward N. Lorenz afternoon ASL- SESSIONS 1:00 p.m. - 2:00 p.m. COLLOQUIUM LECTURE III Uses of the non-Euclidean wave equation Peter D. Lax 1:00 p.m. - 2:00 p.m. AWM - EMMY NOETHER LECTURE Title to be announced Joan Birman SPECIAL SESSIONS 1:00 p.m. - 4:15 p.m. Geometric methods in group theory II 1:00 p.m. - 4:15p.m. Nonlinear partial differential equations III 1:00 p.m. - 4:15p.m. Classical real analysis II 1:00 p.m. - 4:15p.m. Brauer groups and Galois theory II 1:00 p.m. 4:15p.m. Stochastic processes and analysis II 1:00 p.m. - 4:15 p.m. Geometric inequalities III 1:00 p.m. - 4:15 p.m. Combinatorics and group representations III 1:00 p.m. - 4:15p.m. Orthogonal polynomials and the moment problem I 1:00 p.m. - 4:15 p.m. Recent results in gauge field theory and Riemannian geometry III 1:00 p.m. - 4:15 p.m. Theoretical optimization I 1:00 p.m. - 4:15 p.m. Geometrical variational problems I 1:00 p.m. - 4:15 p.m. SESSIONS FCR CONTRIBUTED PAPERS 2:15 p.m. - 3:05 p.m. MAA - INVITED ADDRESS Game theory, nuclear deterrence, and Star Wars Steven J. Brams 3:20 p.m. - 5:40 p.m. MAA - PRIZE SESSION & BUSINESS MEETING 5:30 p.m. - 7:00 p.m. ASL -RECEPTION 6:00 p.m. - 8:00 p.m. AWM -RECEPTION 7:00 p.m. - 9:00 p.m. MAA - MINICOURSE #4 (Part B) Interesting applications of elementary mathematics JoAnne S. Growney 7:00 p.m. - 9:00 p.m. MAA - MINICOURSE #12 (Part A) True BASIC in freshman calculus James F. Hurley 7:00 p.m. - 9:00 p.m. MAA - MINICOURSE #13 (Part A) For all practical purposes Solomon A. Garfunkel 7:00 p.m. - 9:00 p.m. MAA - MINICOURSE #14 (Part A) Applications of discrete mathematics Fred Stephen Roberts 7:00 p. m, - 9:00 p.m. MAA - MINICOURSE #15 (Part A) Constructing placement examinations John W. Kenelly 7:00 p.m. - 9:30 p.m. Committee on Science Policy PANEL DISCUSSION: The problem of Star Wars software reliability David Eisenbud (organizer) David Parnas

821 SPECIAL AIRFARES 1-800-826-6011 RECEIVE A FREE WIN A FREE ARRIVAL TRANSFER AIRLINE TICKET

SAVE 40% TO 60% ON YOUR AIRFARE TO THE JOINT MATHEMATICS MEETING IN SAN ANTONIO, JANUARY 21-24, 1987. MICA, the official travel management firm for the San Antonio meetings, has arranged savings of 40% to 60% off regular American and Eastern Airlines coach fares and 30% off Eastern Airlines first class fare. • Win a free roundtrip airline ticket good for travel within the U.S., including Hawaii! Simply make your airline reservations through the toll-free number and your name will be entered into a drawing to be held at the San Antonio meeting. For additional savings ... with all tickets purchased through the toll-free number you will receive a free transfer from the airport to the hotel. Sample Airfares to San Antonio (Quoted 6/24/86 and subject to change)

Originating Coach MICA Discounted City Fares Fares Boston $814.00 $489.00 Chicago $660.00 $396.00 Los Angeles $500.00 $300.00 New Orleans $198.00 $118.00 New York $758.00 $454.80 Philadelphia $740.00 $444.00

You may qualify for even greater savings. The lowest fares require a Saturday night stay and may be subject to an airline change/cancellation penalty. Make your reservations today! MICA reservationists can guarantee you receive the lowest available airfare on any airline. The earlier you call, the greater your savings! You may pay by credit card or ask to be invoiced. We urge you to purchase your airline tickets without delay using your credit card. This will confirm your reservation, the current airfare and protect you against later fare increases. Remember, these special discounts are available only through MICA's toll-free number. *Canadian attendees can save 25% to 60% off regular coach fares, additional restric­ tions may apply. Call Today - Toll-Free 1-800-826-6011 And Save In Connecticut and outside the Continental U.S. call (203) 678-1040 Monday- Friday, 9:00a.m.- 5:30 p.m. E.S.T. ··.·~··· AVA American 1-\1-\ Airlines EASTERN We earn our wings every day

822 Mathematical Association of America FRIDAY, January 23 American Mathematical Society and Other Organizations 7:30 p.m. - 10:00 p.m. MAA- FILM PROGRAM Planar double pendulum Fly Lorenz 8:00 p.m. - 11:00 p.m. ASL - COUNCIL MEETING

SATURDAY, January 24 AMS MAA and Other Organizations

morning ASL - SESSIONS SPECIAL SESSIONS 8:00 a.m. - 10:55 a.m. Geometric methods in group theory III 8:00 a.m. - 10:55 a.m. Classical real analysis III 8:00 a.m. - 10:55 a.m. Brauer groups and Galois theory III 8:00 a.m. - 10:55 a.m. Stochastic processes and analysis III 8:00 a.m. - 10:55 a.m. Orthogonal polynomials and the moment problem II 8:00 a.m. - 10:55 a.m. Theoretical optimization II 8:00 a.m. - 10:55 a.m. Mathematical physics 8:00 a.m. - 10:55 a.m. Geometrical variational problems II 8:00 a.m. - 10:55 a.m. SESSIONS FCR CONTRIBUTED PAPERS 8:00 a.m. - 4:00 p.m. REGISTRATION 9:00 a.m. - 9:50 a.m. MAA- INVITED ADDRESS Euclidean/non-Euclidean wave equation Peter D. Lax 9:00 a.m. - 10:00 a.m. National Association of Mathematicians PANEL DISCUSSION: Title and speakers to be announced 10:00 a.m. - 10:55 a.m. NAM - BUSINESS MEETING 10:05 a.m. - 10:55 a.m. MAA - INVITED ADDRESS The strong law of small numbers Richard K. Guy afternoon ASL - SESSIONS 1:00 p.m. - 2:00 p.m. COLLOQUIUM LECTURE IV Uses of the non-Euclidean wave equation Peter D. Lax 1:00 p.m. - 2:00 p.m. NAM- WILLIAM W. S. CLAYTCR LECTURE Speaker and title to be announced 1:00 p.m. - 3:00 p.m. MAA - MINICOURSE 118 (Part B) Computer simulation of discrete systems Zaven A. Karian 1:00 p.m. - 3:00 p.m. MAA- MINICOURSE #9 (Part B) Recurrence relations Margaret Barry Cozzens 1:00 p.m. - 3:00 p.m. MAA - MINICOURSE 1110 (Part B) Integrating history into undergraduate mathematics courses Judith V. Grabiner 1:00 p.m. - 3:00p.m. MAA - MINICOURSE illl (Part B) Teaching mathematical modeling Frank R. Giordano Maurice D. Weir 1:00 p.m. - 5:00 p.m. MAA- CONTRIBUTED PAPER SESSION Retaining and recruiting undergraduate women in mathematics courses: Aspirations and experiences Patricia C. Kenschaft 2:15 p.m. - 3:05 p.m. INVITED ADDRESS Free groups, trees, and their automorphisms Marc Culler 2:15 p.m. - 4:15 p.m. MAA- PANEL DISCUSSION: Mathematics panel report of AAAS Project 2061 Leon Henkin 2:15 p.m. - 4:15 p.m. MAA- PRESENTATION Working with statistics: Statistical process control (SPC) techniques Barbara Ashley Analisa L. France

(continued on page 825)

823 through the center of the city. Located on the Information Table river are several of the major hotels, as well as The information table at Joint Meetings of the restaurants, shops, craftsmen, and art galleries. AMS and MAA is set up in the registration Small river taxis ply the river, and one may area for the dissemination of information of go from one end of the river to the other for a nonmathematical nature of possible interest a nominal fee. The banks of the river are to the members. The administration of the beautifully landscaped, and a walkway known as information table is in the hands of the AMS­ the "River Walk" extends the length of the river MAA Joint Meetings Committee, as are all with frequent egresses to hotels and shops away arrangements for such joint meetings. The from the river. The Convention Center is at one following rules and procedures apply. end of the Paseo del Rio. 1. Announcements submitted by partici­ Among the many places of interest in San pants should ordinarily be limited to a single Antonio are the Alamo (including the Alamo Mu­ seum); Brackenridge Park, which has a Chinese sheet no more than 8~ 11 x 1411• 2. A copy of any announcement proposed Sunken Garden, a one-fifth scale model of a diesel train, and the San Antonio Zoo, ranked as one for t~e table is to be sent to: H. Hope Daly, Amencan Mathematical Society, Post Office of the finest in the world; the Hertzberg Cir­ Box 6248, Providence, Rhode Island 02940 to cus Collection; La Villita and the Arneson River arrive at least one week before the first day of Theater, a small historic Mexican village and an the scientific sessions. open air theater; the five Missions of San Antonio 3. The judgment on the suitability of an founded between 1720 and 1731; and the Spanish announcement for display rests with the Joint Governor's Palace. Meetings Committee. It will make its judg­ Parking ments on a case by case basis to establish Parking is available at all hotels as follows: precedents. San Antonio Marriott-Valet parking is $5 4. Announcements of events competing in per night; self-parking in adjacent lot is approxi­ time or place with the scheduled scientific mately $2.25. program will not be accepted. The Crockett Hotel-Valet parking is $6 per 5. Copies of an accepted announcement for night. the table are to be provided by the proponent. Hilton Palacio del Rio-Valet parking is $7.50 Announcements are not to be distributed in any per night. other way at the meeting (for example, not by Holiday Inn Downtown at Market Square­ posting or personal distribution of handbills). There is no charge for parking. 6. It may be necessary to limit the number Hyatt Regency San Antonio-Valet parking of events or the quantity of announcements is $7 per night. distributed at a meeting. La Mansion del Rio Hotel-Valet parking is 7. At the close of registration the table $6 per night. will be swept clean. A proponent wishes ~ho La Quinta Convention Center Motor Inn­ the return of extra copies should remove them. There is no charge for parking. La Quinta Market Square Motor Inn-There Visual Index is no charge for parking. TraveLodge on the River- There is no charge An ~lphabetical list of registered participants, in­ cludmg local addresses and arrival and departure for parking. dates, is maintained in the registration area. Although most hotels offer valet parking, there are also self-parking lots in the area. Further Miscellaneous Information information on parking will be available later. Child Care Smoking The Marriott Hotel has babysitting services avail­ Please note that smoking is not allowed in any of able which can be arranged through the Concierge the session rooms in the San Antonio Convention desk or by calling North Side Sitters at 341-9313. Center or the San Antonio Marriott Hotel. The current rates are $4 per hour for one family and $5.50 per hour for more than one family in Social Event the same room. There is a four-hour minimum There will be a no-host cocktail party in the and an additional $7 transportation and parking Arcade in the San Antonio Convention Center at fee for the babysitter. A list of local babysitters is 8:00 p.m. on Thursday, January 22. available at the Local Information section of the Travel registration desk. In January, San Antonio is on Central Standard Local Information Time. There is regular airline service to the San The Paseo del Rio, an arm of the San Antonio Antonio International Airport by several major River, extends for about two-and-one-half miles airline carriers. The airport is a little over eight

824 miles from the city center, which takes about Weather ten minutes to reach by taxi ($10) or by airport The location of San Antonio on the edge of limousine ($5). Most major car rental agencies the Gulf Coastal Plains results in a modified maintain desks at the airport. Amtrak has thrice­ subtropical climate, predominantly continental weekly service from New Orleans to the east, and during the winter months. The average daily high from Los Angeles to the west, and from St. Louis temperature for January is 62 degrees F, and the to the northeast. daily low 42 degrees F. Robert M. Fossum Urbana, Illinois Associate Secretary

(continued from page 823)

SATURDAY, January 24 Mathematical Association of America

3:30 p.m. - 5:30 p.m. MAA - MINICOURSE #11 (Part C - Qp;ional) Teaching mathematical modeling Frank R. Giordano Maurice D. Weir 3:30 p.m. - 5:30 p.m. MAA - MINICOURSE #12 (Part B) True BASIC in freshman calculus James F. Hurley 3:30 p.m. - 5:30 p.m. MAA - MINICOURSE #13 (Part B) For all practical purposes Solomon A. Garfunkel 3:30 p.m. - 5:30 p.m. MAA - MINICOURSE #14 (Part B) Applications of discrete mathematics Fred Stephen Roberts 3:30 p.m. - 5:30 p.m. MAA - MINICOURSE #15 (Part B) Constructing placement examinations John W. Kenelly

825 Mathematical Sciences Employment Register January 1987 Meeting in San Antonio

The Mathematical Sciences Employment Register (MSER), held annually at the Joint Mathemat­ ics Meetings in January, provides opportunities for mathematical scientists seeking professional employment to meet employers who have positions to be filled. Job listings (or descriptions) and resumes prepared by employers and applicants are displayed at the meeting for the participants so that members of each group may determine which members of the other group they would like to have an opportunity to interview. A computer program assigns the appointments, matching re­ quests to the extent possible, using an algorithm which maximizes the number of interviews which can be scheduled subject to constraints determined by the number of time periods available, the numbers of applicants and employers, and the pattern of requests. The report below outlines the operation of the register, indicating some of the procedures involved for the benefit of those not familiar with its operation. The Mathematical Sciences Employment Register is apparently unique among employment ser­ vices offered by professional organizations in the sciences, engineering and the humanities. The computer programs used are constructed around a matching program, devised by Donald R. Mor­ rison, and based on an algorithm described in his paper "Matching Algorithms" in Jaurnal of Comhinatorial Theory, volume 6 (1969), pages 20 to 32; see also "Matching Algorithms" (abstract) Notices, August 1967, page 630. The number of interviews arranged by the program is signifi­ cantly greater than the number possible at the employment registers of other organizations, in many cases greater by an order of magnitude.

1987 Employment Register in San Antonio choice of session (day, and morning or afternoon). Such requests cannot be accommodated. The Employment Register will take place in the Requests for interviews taking place during the South Banquet Hall, San Antonio Convention Center two sessions on Thursday MUST BE SUBMITTED on Wednesday, Thursday, and Friday, January 21, 22, on Wednesday between 9:30 a.m. and 4:00 p.m. and 23, 1987. A short (optional) orientation session Requests for interviews to take place during the will be conducted by the AMS-MAA-SIAM Commit­ Friday sessions must be submitted on Thursday before tee on Employment Opportunities at 9:00 a.m. on 4:00 p.m. Those who fail to do so cannot be Wednesday, January 21. The purpose of the orien­ included in the pool of available participants when tation session is to familiarize participants with the the matching program which schedules the interviews operation of the Register and with the various forms is run "on the computer that night. This applies to all involved. Following orientation, participants should employers and applicants both preregistered and on­ pick up their material for participating in the Em­ site registrants. Forms submitted with preregistration ployment Register. Computer-scheduled interviews achieve registration for the Employment Register will be held on Thursday and Friday, January 22 and only. These forms do not automatically include the 23. No interviews will be held on Wednesday. participant in the interviewing process. The interview Fifteen-minute intervals are allowed for inter­ request forms handed out at the Employment Register views, including two or three minutes between suc­ must be turned in before the 4:00 p.m. deadline in cessive interviews. The interviews are scheduled in order to receive a computer printed schedule the next half-day sessions: Thursday morning and afternoon, day. and Friday morning and afternoon, amounting to four half-day sessions for interviews. There are ten time On Thursday and Friday mornings at 9 a.m. periods (9:30-11:45 a.m.) in which interviews can be all schedules for applicants and employers for the scheduled in the morning and fourteen time periods day (both morning and afternoon sessions) will be (1:15-5:00 p.m.) in the afternoon. It is possible that available for distribution in the South Banquet Hall. an applicant or employer may be scheduled for the The Friday afternoon session is the annual "em­ maximum number of interviews in a session. Requests ployers' choice" session. For this session interviews for interviews will be accommodated depending on will be scheduled on the basis of requests made by em­ the availability of participants. The scheduling pro­ ployers. Applicants do not submit specific interview gram does not have a provision allowing participants requests for this session; but, in order to participate to specify particular times for interviews beyond the they must indicate their availability for the session by returning the Interview Request Form for Friday, indicating that they will attend the afternoon session that day. Background of Applicants Applicants should be aware of the fact that Statistics from previous Employment Registers have interviews arranged by the Employment Register shown employers sought to fill approximately 180 represent only an initial contact with employers, and positions, 10 of which were nonacademic jobs. For that hiring decisions are not ordinarily made during 98% of the positions, holders of doctoral degrees or immediately following such interviews. Applicants were preferred, for 65% of the positions only appli­ are advised to bring a number of copies of their cants with doctorates were acceptable, for 30% of vitae or resumes 80 that they may leave them with the positions, holders of masters degrees were con­ prospective employers. sidered eligible. Few of the nonacademic employers The Mathematical Sciences Employment Regis­ indicated an interest in holders of bachelors degrees ter is sponsored by the American Mathematical So­ in mathematics. ciety, the Mathematical Association of America, and the Society for Industrial and Applied Mathematics;

826 "t is operated by members of the AMS staff under accompanied by the Preregistration/Housing Form. ~he general supervision of the joint AMS-MAA-SIAM This registration fee for employers covers the cost Committee on Employment Opportunities. of a copy of the December Issue of Employment Anyone with questions about the Employment Information in the Mathematical Sciences (ElMS). This Register should contact Carole Kohanski at the Amer­ publication contains printed copies of the resumes of ican Mathematical Society at 401-272-9500, extension applicants who preregistered prior to the deadline; it 286. The telephone number to be used after the also contains a copy of the Winter List of Applicants. Register begins will be announced later. Participants It is requested that employers submit both employer should note that this number will be for those who and Preregistration/Housing Forms with appropriate will be participating in the Employment Register and fees in the same envelope. It would also be helpful is not for contacting participants or taking messages. if the names of cointerviewers would be listed on the Those who wish to leave messages should call the employer form. If possible, these individuals should message center telephone number found in the San also preregister at the same time. Antonio meeting announcement. It is the policy of some institutions to pay for employer fees. These payments do not always Preregistered Employers/ Applicants accompany the preregistration forms but are sent in Preregistration for the Mathematical Sciences Em­ after the deadline has p888ed, or when the meeting ployment Register must be completed by November is over. It is important that the institution's fiscal 15, 1986. Applicants and employers (including all department indicate the name of the participating interviewers) who wish to preregister for the Em­ employer so that proper credit can be made in ployment Register must also register for the Joint Providence. Mathematics Meetings. Forms for preregistration, Employers are encouraged to provide more than housing, the applicant resume form, and the employer one interviewer, when they are able to do so, in form are located in the back of this issue. Preregis­ order to increase the number of interviews which may tration for the Employment Register, in addition to be scheduled. Please take care to indicate on the permitting inclusion in the printed winter lists of Ap­ form the number and names of interviewers for whom plicants and Employers, has the advantage of reduced simultaneous interviews may be scheduled. Note that fees and the services of the Mathematics Meetings all interviewers are expected to register for the Joint Housing Bureau, and has the further advantage of Meetings. (If all interviewers will be interviewing for helping to reduce waiting times at the meeting in San the same position, or for the same set of positions, Antonio. only one form should be submitted and only one Employer or Applicant forms received after the employer code number will be assigned; therefore, each November 15 deadline cannot be included in the interviewer would then receive a separate computer printed lists. For details on registration and pre­ schedule and separate table number.) More than one registration for the San Antonio Joint Mathematics employer code will be required if some interviewers Meetings, please refer to the information on these will not interview for all positions. Thus, if there are subjects which may be found elsewhere in this issue. two disjoint sets of positions, two forms are required Employers and applicants who have preregistered and two employer codes will be assigned. for the Employment Register may pick up their MSER A coded strip at the bottom of the form summa­ material after 9:30 a.m. on Wednesday, January 21, rizes the information on each form. All employers are in the South Banquet Hall. (This material includes required to complete the Summary Strip. This is used the interview request forms which are handed out to prepare a computer-printed list of preregistered at the meeting only.) These are not the forms that employers for distribution to the applicants. are submitted with preregistration. Material for the Employment Register will not be mailed in advance. Nonpreregistered Applicants and Employers Employers and applicants who wish to participate in Preregistered Applicants the Register who have neither preregistered nor paid In addition to the Joint Meetings preregistration the Employment Register fee must first go to the fee, there is an applicant fee of $15 payable prior Joint Mathematics Meetings registration desk in the to the November 15 deadline. These fees must be North Banquet Hall, San Antonio Convention Center accompanied by the Preregistration/Housing Form. in order to complete their registration. No provision Applicants' resumes will be made available to will be made to handle cash transactions at the site employers at the Employment Register in printed of the Employment Register. Registration for the form, so that they may be studied carefully at leisure. Joint Meetings is required for participation in the The December issue of Employment Information in Employment Register. It is also required that all the Mathematical Sciences (ElMS) will contain phO-. participating employer interviewers register for the tographic reproductions of the resumes of applicants Joint Mathematics Meetings. who have preregistered by November 15. Forms .not Onsite registration for the Employment Register received in time cannot be included in this issue. See is $100 for employers and $20 for applicants. This the section on preparation of resumes elsewhere in registration fee for employers covers the cost of a copy this announcement. of the December Issue of Employment Information in Employers' job listings and applicants' resumes the Mathematical Sciences (ElMS). This publication will be posted at the meeting, so that applicants and contains printed copies of the resumes of applicants employers may review them. who preregistered prior to the deadline and a copy of the Winter List of Applicants. Preregistered Employers After registration has been completed, applicants In addition to the Joint Meetings preregistration fee and employers should come to the South Banquet Hall there is a $75 fee for employers, if paid prior to to fill out the forms necessary to participate in the the November 15 deadline. These fees must be Employment Register.

827 Nonparticipating Employers project of the American Mathematical Society (pub­ Employers who do not plan to participate in the lisher), the Mathematical Association of America, and Employment Register, but wish to display job de­ the Society for Industrial and Applied Mathematics. scriptions, may obtain special forms from Carole The December issue of ElMS contains resumes of Kohanski, MSER, P. 0. Box 6248, Providence, RI persons seeking professional positions in the mathe­ 02940. These job descriptions, subject to approval, matical sciences. Resumes of applicants taking part must be received in the Providence office by November in the Employment Register and those not attending 15 in order to qualify for the reduced fee of $10. There will be included in the December 1986 issue provided is a $15 fee for listings received after the November they were received before the November 15 deadline 15 deadline. and are in satisfactory condition. Other mathemat­ Employers who attend the Joint Mathematics ical scientists who wish to be included may have Meetings but do not want to interview, can post their resumes printed if the same deadline is observed job descriptions, subject to approval, at the Employ­ and if the copy supplied meets the same technical ment Register. Postings will not be allowed in the requirements described in the following section. Joint Meetings registration area. A fee of $15 will be Copies of the December issue of ElMS will be charged payable to the cashier at the Joint Mathemat­ distributed in San Antonio to the employers who ics Meetings registration desk. Participants should be participate in the Employment Register. sure to inform the cashier that they would like to post Job applicants planning to participate in the a job description but are not planning to interview Employment Register in San Antonio are therefore and obtain the proper receipt in order to receive strongly urged to preregister so that their resumes the form necessary for posting at the Employment can appear in the December issue. Register desk. Please note that the December issue of ElMS contains the Winter List of Applicants, but does not Applicants Not Planning to Attend contain the Winter List of Employers. Additional copies of the December Issue of ElMS Applicants for professional positions in the mathemat­ and ical sciences, who do not plan to attend the meeting will be available for sale at the AMS Exhibits at the meeting. Prices at the meeting in San Antonio and participate in the Employment Book Sale Any copies Register, may also submit resumes for publication are $7 each for the December issue. from the in the December issue if they use the MSER Form remaining after the meeting will be available office of the Society for $12. for Applicants at the back of this issue and observe Providence the deadline of November 15. (It is, of course, not necessary to preregister for the meeting or pay the Preparation of Applicants' Resumes Employment Register registration fee if one is not for the December issue of ElMS attending the meeting. Resumes will not be posted The December issue of ElMS will be printed using at the Employment Register if the participant is not photographic reproductions of forms completed and attending the meeting.) submitted by applicants. For this reason, special care must be exercised by those who prepare the forms in Winter Lists of Applicants and Employers order to assure that the results are of good quality, have been The Winter List of Applicants, which is a summary and will be clear and legible after they of the resumes of preregistered applicants, will be photographed, reduced in size, and printed. available for sale at the AMS Exhibits and Book Sale Because an employer's first impressions of an at the meeting. The price at the meeting is $4 applicant are likely to be based on the appearance of each. Any copies remaining after the meeting will be the printed form, applicants are strongly advised to available from the Providence office of the Society for study the suggestions given below before the forms $6 each. are filled out, so that the original copy will be neither The Winter List of Employers consists of sum­ marred nor damaged. maries of the position listings submitted by the The forms must be carefully typed using a new employers who preregistered for the meeting; it will black ribbon. The best results are obtained by using a be distributed to the applicants participating in the modern typewriter with a carbon-coated polyethylene Register. Others may purchase the Winter List of film ribbon, but satisfactory results may be obtained Employers at the AMS Exhibits and Book Sale at with a ribbon made of nylon or other woven fabric the meeting or from the Providence office after the if suitable care is exercised. It is important that the meeting. The prices are the same as stated in the keys be clean and make a sharp, clear impression, previous paragraph. which must be a uniform dark black. Gray, blue, or Please note that these lists will not be updated other colors will not reproduce and should, therefore, with onsite employers or applicants after the Employ­ not be used. Do not use an eraser, as it will cause Use a ment Register has concluded. smudges which reproduce when photographed. correcting typewriter, or correction tape or fluid, if necessary. December Issue of Employment Only an original copy of the form should be Information in the Mathematical Sciences submitted, a photocopy or xerographic reproduction will not reproduce as well and may not he accepted for For several years the periodical Employment Informa­ publication. It is therefore important to exercise care tion in the Mathematical Sciences (ElMS) has published in order to assure that the renults are satisfactory. six issues per year listing open positions in academic, Submission of copy of good quality is entirely governmental and industrial organizations, primarily the responsibilty of the applicant. The Society (which in North America, along with a few listings from will print this material) must be the final judge of countries in other parts of the world. ElMS is a joint what copy is capable of being reproduced adequately,

828 and therefore of what is acceptable for inclusion in the printed booklet. The Society will not correct or replace inadequate copy, and cannot prepare original Pseudodiflerential Operators and copy. In the event the quality of a resume, submitted by an applicant participating in the Employment Applications Register, does not meet the necessary conditions for Fran~ois Treves, Editor inclusion in the December issue, the resume will be (Proceedings of Symposia in Pure Mathematics. returned if time allows; otherwise the resume will be Volume 43) posted at the Employment Register in San Antonio, along with those of the other participants. Forms This volume gathers together nineteen papers received past the deadline will be returned. devoted to microlocal analysis. which has emerged as one of the most powerful tools of linear analysis since distribution theory. New classes of pseudodifferential operators and Fourier integral List of Retired Mathematicians operators were introduced in the 1970s and used Available for Employment to tackle many classical and outstanding problems The annual List of Retired Mathematicians will be in partial differential equations. Recently. in a included in the December and January issues of the very promising area of research. pseudodifferential publication Employment Information in the Mathe­ operators have been used to tackle nonlinear matical Sciences. Retired mathematicians who are differential problems. interested in being included in the list may send the Contents following information to the Mathematical Sciences S. Allnhec and G. Metivier, Propagation of local Employment Register, American Mathematical So­ analyticity for the Euler equation; J. L. Antonlano ciety, P. 0. Box 6248, Providence, Rhode Island and G. A. Uhlmann, A functional calculus for a class 02940. of pseudodifferential operators with singular symbols; M. S. Baouendi, Uniqueness in a class of nonlinear 1. Full Name Cauchy problems; Michael Beals, Propagation of 2. Mailing Address smoothness for nonlinear second-order strictly hyperbolic 3. Highest degree, year, university differential equations; R. Beals and R. R. Colfman, 4. Most recent employment: institution Multidimensional inverse scatterings and nonlinear partial differential equations; R. R. Colfman and Yves Meyer, 5. Type of position desired Nonlinear harmonic analysis and analytic dependence; 6. Academic or industrial employment preferred H. 0. Cordes, On some c• -algebras and Frechet-algebras 7. Date available for employment (month/year) of pseudodifferential operators; G. Eskin, Boundary-value 8. Geographic location preferred problems for second-order elliptic equations in domains with comers; P. C. Greiner, Imbedding C" in Hn; The deadline for receipt of this information is VIctor Gulllemin, On some results of Gelfand in November 15. Offprints of the list will be available integral geometry; Lars HOrmander, The propagation from the Mathematical Sciences Employment Reg­ of singularities for solutions of the Dirichlet problem; ister, American Mathematical Society, P.O. Box M. Kaahiwara and P. Schapira, Applications of the m~rolocal theory of sheaves to the study of Ox; C. E. 6887, Providence, Rhode Island 02940. Kenlg, Recent progress on boundary-value problems on lipschitz domains; J. J. Kohn, Estimates for 8b on compact pseudoconvex CR manifolds; Yves Meyer, Recent analysis and operator theory; L. P. Rothschild, Integrability and holomorph~ extendibility for rigid CR structures; Johannes SJO.trend, Multiple wells and tunneling; N. K. Stanton and D. S. TertekofJ, The real analytic and Gevrey regularity of the heat kernel for D b; M. E. Taylor, Fefferman-Phong inequalities in diffraction theory

1980 Mathematics Subject Classifications: 22E30, 32F99, 35-XX, 42620, 43A80 and others ISBN 0-8218-1469-9, LC 85-1419 ISSN 0082-0717 viii + 301 pages (hardcover), August 1985 List price S44, Institutional member 135, Individual member $26 ~-iliEMAz,!;f To order, please specify PSPUM/43NA "'~~ ~~h~n~ ~~.~ • "O{;.voEO '~ Shipping/Handling: 1st book $2, each add'l $1, $25 max. By air, 1st book $5, each add'l $3, $100 max. Prepayment required. Order from AMS, P.O. Box 1571, Annex Station, Providence, Rl 02901-1571, or call 800-556-7774 to use VISA or MasterCard.

829 American Mathematical Society Short Course Series Moments in Mathematics (January 20-22, 1987) Synopses and Reading Lists

The following synopses are arranged in the Geometry of the Moment Problem (J. H. B. order of presentation as currently scheduled. Kemperman). When only a finite number of The final schedule will be available at the Short moments is given, a problem for both theory Course registration desk. and practice is to describe the convex set M of measures having these moments, or at least to The Classical Background (H. J. Landau). The give accurate bounds for the integrals of partic­ moment problem stands at the source of many ma­ ular functions J f(x) d11(x) with respect to the jor streams in analysis. From it flow developments measures 11 of M. In questions of this type, one in function theory, in spectral representation of often has to add additional constraints such as operators, .in probability and statistics, in Fourier bounds on the density of these measures. An­ analysis and the prediction of stochastic processes, other variant may be that some of the moments, in approximation and numerical methods, in in­ instead of being prescribed exactly, are required verse problems and the design of algorithms for only to lie in certain ranges. Typically, the task is simulating physical systems. The purpose of this made delicate by numerical instability. Moment lecture is to sketch the ideas in past work and to problems of this type occur in great abundance describe how these interconnections come about. in both pure and applied mathematics, especially The moment problem asks when a given in probability and statistics, numerical analy­ sequence of numbers represents the successive sis and approximation theory, tomography, and moments J zk d11(z), k = 0, 1, ... , of a nonnega­ crystallography. tive measure 11, defined either on the line (the This discussion will introduce a geometric Hamburger, or power, moment problem) or on approach to these problems which involves a cer­ the unit circumference (the trigonometric moment tain natural duality method. The corresponding problem). An obvious necessary condition is the theory leads to the notions of contact set and crit­ positivity of a certain quadratic form defined by ical measure, and covers many results from the the moments. The first object is to prove that theory of finite and infinite linear programming. this condition is likewise sufficient; one then asks Other applications include measures with given whether 11 is unique. marginals, closely related to so-called transporta­ Positivity suggests convexity, one large theme tion problems, as well as dilations or approximate in this subject. Thus, Caratheodory, Herglotz, dilations relative to a cone of functions, which and Schur connected the problem with a convex generalize the usual stochastic partial ordering of family of analytic functions, for which they de­ measures. vised integral representations. M. Riesz exploited 1. S. Karlin and W. Studden, Tcheby- convexity by linear space methods, introducing cheff Systems: With Applications in Analysis and an extension procedure for positive functionals, Statistics, Interscience, New York, 1966. while M. G. Krein, applying the geometry of 2. H. G. Kellerer, Duality theorems for convex bodies, generalized broadly to create the marginal problems, Zeitschrift Wahr., 1984. Chebyshev spaces of approximation theory. 3. J. H. B. Kemperman, On the role of Alternatively, the positive quadratic form can duality in the theory of moments, in Semi-infinite be used to define a scalar product for polynomials. Programming and Applications, A. V. Fiacco This generates orthogonal polynomials, which sat­ and K. 0. Kortanek (eds.), Lecture Notes in isfy recursion relations that are discrete versions Economics and Mathematics Systems, vol. 215, of Sturm-Liouville differential operators. It also Springer, New York, 1983, pp. 63-92. yields orthogonal decompositions and expansions 4. V. Strassen, The existence of probability that lead to spectral representation, to a variety measures with given marginals, Ann. Math. Stat. of electrical and mechanical inverse problems, and 36 (1965), pp. 423-439. to statistical prediction theory. Moment Problems and Operators in Hilbert 1. N. I. Akhiezer, The classical moment Space (D. Sarason). The connections between problem, Hafner, New York, 1965. classical moment problems and the theory of

830 operators in Hilbert space were recognized early nice feature of the topic is that the traffic has been in the development of the latter subject. The two-way. The mathematical results have provided indeterminate case of the Hamburger moment new algorithms and new hardware structures for problem, for example, can be viewed as part of various applications while, in turn, the applica­ the theory of self-adjoint extensions of symmetric tions have yielded some new mathematical results operators with defect numbers (1,1). An extensive and suggested some new mathematical problems. discussion is found in the famous book of M. H. We shall attempt to present a glimpse of Stone. both directions of this interplay. Starting with More recently, operator theory has interacted the so-called linear predictive coding methods strongly with the trigonometric moment problem used to reduce the bandwidth required for storage and its relatives, such as the Nevanlinna-Pick and transmission of speech, we will describe the interpolation problem. One approach here, due role of the Szego polynomials, orthogonal on the to V. M. Adamjan, D. Z. Arov, and M. G. Krein, unit circle, in providing a nice hardware (lattice is a close analogue of the operator treatment of filter) implementation. Next, the issue of parallel the Hamburger problem. Another is connected implementation-inspired by the rapidly falling with the unilateral shift operator and is part of cost of integrated circuits-will bring in an al­ the theory of unitary dilations of contractions ternative algorithm,· traceable to the work of I. developed by B. Sz.-Nagy and C. Foias; their Schur in 1917 on power series bounded in the unit commutant-lifting theorem plays a key role. Both circle. This will lead us to transmission lines and approaches involve the class of Hankel operators. to one-dimensional inverse scattering problems, In this lecture, the theory of unbounded self­ such as arise, for example, in the "layered earth" adjoint operators will first be briefly reviewed, models used in the analysis of seismic data from after which the operator-theoretic treatment of oil-prospecting rigs. We shall combine some of the Hamburger problem will be sketched. At­ the previous ideas to obtain a new class of digital tention will then shift from the real line to the filters with excellent robustness to finite-precision unit circle. Hankel operators will be introduced implementations. In turn, the transmission line and their connection with interpolation problems models lead back to mathematics by suggesting explained. The Adamjan-Arov-Krein approach fast algorithms for the Cholesky and QR fac­ will be indicated. Finally, a bit of the theory torization of the much-studied Toeplitz moment of unitary dilations will be presented and tied to matrices. Moreover, the physical derivations will interpolation problems. show that the Toeplitz results are more naturally The items in the reading list below are not to understood in terms of those applying to a whole be interpreted as prerequisites for understanding family, having what we have called "displacement the lecture, but rather as sources where one can structure." encounter the main ideas to be presented. 1. E. A. Robinson, Spectral approach to 1. N. I. Akhiezer, The Classical Moment geophysical inversion by Lorentz, Fourier and Problem, Hafner, New York, 1965. Radon transforms, Proc. IEEE 70 (9) (1982), pp. 2. S. C. Power, Hankel Operators on Hilbert 1039-1504. Space, Pitman, Boston-London-Melbourne, 1982. 2. T. Kailath, S. Y. Kung, and M. Morf, 3. M. Rosenblum and J. Rovnyak, Hardy Displacement ranks of matrices and linear opera­ Classes and Operator Theory, Oxford University tors, Bull. Amer. Math. Soc. 68 (2) (1979), pp. Press, New York, 1985. 395-407. 4. D. Sarason, Operator-theoretic aspects 3. H. Lev-Ari and T. Kailath, Lattice filter of the Nevanlinna-Pick interpolation problem, in parametrization and modeling of nonstationary Operators and Function Theory, S. C. Power processes, IEEE Trans. lnf. Thy. IT-39 (1) (1984), (ed.), D. Reidel, Dordecht, Holland, 1985, pp. pp. 2-16. 279-314. 4. T. Kailath, A. Bruckstein, and D. Morgan, 5. M. H. Stone, Linear Transformations Fast matrix factorization via discrete transmission in Hilbert space, American Mathematical Society, lines, Linear Alg. and Appl. 75 (1986), pp. 1-25. New York, 1932. 6. B. Sz.-Nagy and C. Foias, Harmonic The Multidimensional Moment Problem and Analysis of Operators on Hilbert Space, North­ Semigroups (C. Berg). The two-dimensional Holland, Amsterdam, 1970. moment problem consists in finding necessary and sufficient conditions for a double sequence Signal-Processing Applications of Some Mo­ s(m,n),m,n = 0,1, ... , to be a moment se­ ment Problems (T. Kailath). Trigonometric mo­ quence, namely, to be representable as s(m, n) = ment problems are connected with positive defi­ J JxmyndJ.t(X, y) for a nonnegative measure J.l nite matrices, orthogonal polynomials, and clas­ on R2 • An obvious necessary condition is sical function theory (Caratheodory, Schur)-all the positive definiteness of s on the semigroup subjects that have numerous applications in a {(m, n), m, n = 0, 1, ... }, cf. below. In contrast to variety of signal processing problems: linear pre­ the one-dimensional theory, this condition is no diction, inverse scattering, digital filtering, etc. A longer sufficient, mainly because there exist non-

831 negative polynomials in two variables which are 1. C. Berg, J. P. R. Christensen, and P. Res­ not a sum of squares. A simple example of such a sel, Harmonic Analysis on Semigroups. Theory of polynomial is p(x, y) = x2 y2 (x2 + y2 - 1) + 1. If Positive Definite and Related Functions, Gradu­ s is a moment sequence, one studies the question ate Texts in Mathematics, vol. 100, Springer, New of determinacy, i.e., is the representing measure York, 1984. unique? If s is a positive definite double sequence, 2. A. Devinatz, Two-parameter moment one can introduce an inner product on the space problems, Duke Math. J. 24 (1957), pp. 481-498. of polynomials and consider it as a dense subspace 3. B. Fuglede, The multidimensional moment of a Hilbert space. Multiplication by x and y problem, Expo. Math. 1 (1983), pp. 47-65. defines two unbounded (symmetric) operators A, Moment Problems in Probability and Statis­ and Ay in this Hilbert space. If one can find tics (P. Diaconis). Chebyshev and Markov devel­ commuting self-adjoint extensions of these opera­ oped the theory of moments to provide rigorous tors, then s is a moment sequence because of the arguments for the basic theorems of probability. spectral theorem for such operators. Thus the From then till now, statisticians have used mo­ lack of validity of Hamburger's theorem in two ment estimators as a basic component of applied or higher dimensions also reflects the fact that work. I will review the history and accomplish­ extensions do not always commuting self-adjoint ments of moment theory in these fields. exist. I will also discuss some recent novel examples The theory of the moment problem may be that help make the subject alive: independent viewed as part of harmonic analysis on semi­ random variables X and Y determined by their groups. If S is an abelian semigroup with moments, but X+ Y not; the surprising fact that neutral element, a real valued function f on S is the cube of a standard Gaussian variable is not called positive definite if the matrix with entries determined; and the strange identity {f(s; + Sj)} is a positive semidefinite n x n matrix for any choice of n elements s 1, ... , Sn E S. A principal question is to decide if a positive defi­ nite function j is a moment function, i.e., has the form f(s) = J p(s)d!l(p), where 11 is a nonnegative measure on the dual space S* of semicharacters for k = 0, 1, 2, ... , 2n + 1, the first integral being p on S. For S = {0, 1, 2, ... }, the semicharac­ over the n-dimensional orthogonal group. ters are the sequences (xn)n>o, parametrized by 1. M. G. Krein, The ideas of P. L. Cebysev X E R, so one may identify s· and R. Similarly, and A. A. Markov in the theory of limiting values {(n,m),n,m = 0,1, ... }* may be identified with of integrals and their further development, Amer. R2 . It follows from Choquet theory that bounded Math. Soc. Transl. Ser. 2, 12 (1959), pp. 1-122. positive definite functions are always determinate moment functions. If this is true for every positive definite function, the semigroup is called perfect. The class of perfect semigroups has many nice properties.

832 Invited Speakers Gregory P. Wene, Mathematical physics Brian White, and Special Sessions Geometrical variational problems Invited Speakers at AMS Meetings March 1987 Meeting in Honolulu Far Western Section The individuals listed below have accepted invi­ tations De.adhne for organizers: October 15, 1986 to address the Society at the times and Deadhne for consideration: January 12, 1987 places indicated. For some meetings, the list of Steven Bleiler, Low dimensional topology speakers is incomplete. George Csordas, Complex function theory Heiner Doverman, Algebraic topology San Antonio, January 1987 Kent R. Fuller, Rings and modules Jerry Griggs, Combinatorics Marc Culler Robert J. McEliece Thomas Jech, Set theory and its applications Ronald J. DiPerna David J. Saltman Peter Li, Richard Schoen, and S.-T. Yau, Differ- RichardT. Durrett Lesley M. Sibner ential geometry Robert M. Hardt Thomas C. Spencer Peter D. Lax (Gibbs Lecturer) (Colloquium Lecturer) April 1987 Meeting in Kent Central Section Honolulu, March 1987 Deadline for organizers: October 15, 1986 Deadline for consideration: January 14, 1987 Martin Scharlemann April 1987 Meeting in Newark Newark, April 1987 Eastern Section Robert V. Kohn Birgit Speh Deadline for organizers: October 15, 1986 Rodolfo Rosales Lars S. Wahlbin Deadline for considemtion: January 16, 1987

Organizers and Topics Spring 1987 Meeting of Special Sessions Southeastern Section No meeting will be held The list below contains all the information about Special Sessions at meetings of the Society avail­ able at the time this issue of Notices went to Information for Organizers the printer. The section below entitled Infor­ mation for Organizers describes the timetable for Special Sessions at Annual and Summer meetings announcing the existence of Special Sessions. are held under the general supervision of the Program Committee. They are administered by the Associate Secretary in January 1987 Meeting in San Antonio charge of the meeting with staff assistance from the Society office in Associate Secretary: Robert M. Fossum Providence. De.adl:i:ne for organizers: Expired Some Special Sessions arise from an invita­ Deadline for considemtion: Expired tion to a proposed organizer issued through the Roger Alperin, Geometric methods in group theory Associate Secretary. Others are spontaneously Ronald J. DiPerna, Nonlinear partial differential proposed by interested organizers or participants. equations Such proposals are welcomed by the Associate Michael J. Evans and Paul D. Humke, Classical Secretaries. real analysis The number of Special Sessions at a Summer Burton Fein, David J. Saltman, and Murray or Annual Meeting is limited to twelve. Proposals, Schacher, Brauer groups and Galois theory invited or offered, which are received at least nine Michael Gage and Erwin Lutwak, Geometric in­ months prior to the meeting, are screened for equalities suitability of the topic and of the proposed list of Joseph Glover and A. 0. Pittenger, Stochastic speakers, and for possible overlap or conflict with processes and analysis other proposals (specific deadlines for requesting Philip J. Hanlon, Combinatorics and group repre­ approval for Special Sessions at national meetings sentations are given above). If necessary, the numerical Paul G. Nevai, Orthogonal polynomials and the limitation is enforced. moment problem Proposals for Special Sessions should be sub­ Lesley M. Sibner, Recent results in gauge field mitted directly to the Associate Secretary in theory and Riemannian geometry charge of the meeting (at the address given in the Lynn McLinden and Jay S. Treiman, Theoretical accompanying box). If such proposals are sent optimization to the Providence office, addressed to Notices,

833 or directed to anyone other than the Associate Abstracts of papers submitted for considera­ Secretary, they will have to be forwarded and may tion for presentation at a Special Session must be not be received before the quota is filled. received by the Providence office (Editorial De­ In accordance with an action of the Executive partment, American Mathematical Society, Post Committee of the Council, no Special Session may Office Box 6248, Providence, RI 02940) by the be arranged so late that it may not be announced special deadline for Special Sessions, which is in Notices early enough to allow any member usually three weeks earlier than the deadline for of the Society, who wishes to do so, to submit contributed papers for the same meeting. The an abstract for consideration for presentation in Council has decreed that no paper, whether in­ the Special Session before the deadline for such vited or contributed, may be listed in the program consideration. of a meeting of the Society unless an abstract of Special Sessions are effective at Sectional the paper has been received in Providence prior Meetings and can usually be accommodated. to the deadline. They are arranged by the Associate Secretary under the supervision of the Committee to Select Hour Speakers for the section. The limitation on the number of sessions depends on the space Send Proposals for Special Sessions to the and time available. The same restriction as for Associate Secretaries national meetings applies to the deadline for an­ The programs of sectional meetings are arranged by the nouncing Special Sessions at sectional meetings: Associate Secretary for the section in question: no Special Session may be approved too late for Far Western Section (Pacific and Mountain) its announcement to appear in time to allow a Hugo Rossi, Associate Secretary reasonable interval for members to prepare and Department of Mathematics submit their abstracts prior to the special early University of Utah Salt Lake City, UT 84112 deadline set for consideration of papers for Special (Telephone 801-581-8159) Sessions. Central Section The Society reserves the right of first refusal Robert M. Fossum, Associate Secretary for the publication of proceedings of any special Department of Mathematics session. These proceedings appear in the book University of Illinois series Contemporary Mathematics. 1409 West Green Street Urbana, IL 61801 (Telephone 217-333-3975) Information for Speakers Eastern Section W. Wistar Comfort, Associate Secretary A great many of the papers presented in Special Department of Mathematics Sessions at meetings of the Society are invited Wesleyan University papers, but any member of the Society who Middletown, CT 06457 wishes to do so may submit an abstract for (Telephone 203-347-9411) consideration for presentation in a Special Session, Southeastern Section Frank T. Birtel, Associate Secretary provided it is received in Providence prior to the Department of Mathematics special early deadline announced above and in Tulane University the announcements of the meeting at which the New Orleans, LA 70118 Special Session has been scheduled. Contributors (Telephone 504-865-5646) should know that there is a limitation in size of a As a general rule, members who anticipate orga­ single special session, so that it is sometimes true nizing Special Sessions at AMS meetings are advised to seek approval at least nine months prior to the sched­ that all places are filled by invitation. Papers not uled date of the meeting. No Special Sessions can be accepted for a Special Session are considered as approved too late to provide adequate advance notice to ten-minute contributed papers. members who wish to participate.

834 Symposium on Some Mathematical Questions in Biology Models in Population Biology Chicago, Illinois, February 18, 1987

The twenty-first annual Symposium on Some The AMS-SIAM Committee on Mathemat­ Mathematical Questions in Biology will be held ics in the Life Sciences serves as the Organizing on Wednesday, February 18, 1987, in the Acapulco Committee for the symposium. The commit­ Room of the Hyatt Regency Chicago in conjunc­ tee consists of Gail A. Carpenter (Northeastern tion with the annual meeting of the American University); Kenneth L. Lange (Massachusetts Association for the Advancement of Science. The Institute of Technology); Hans G. Othmer (Uni­ symposium is sponsored by the American Mathe­ versity of Utah); Alan S. Perelson (Los Alamos matical Society, the Society for Industrial and Ap­ National Laboratory); Richard E. Plant, chair­ plied Mathematics, and Section A (Mathematics) man (University of California, Davis); and John of the American Association for the Advancement Rinzel (National Institutes of Health). of Science. The theme of the symposium is Models in Details regarding registration, local arrange­ Population Biology. There will be two half-day ments, and program information for the AAAS sessions, each including three one-hour lectures. meeting will appear in the November 21 issue of Science.

PROGRAM Chairman: Alan Hastings

9:00 a.m. Some Mathematical Questions in Biology-Models in Population Biology Presiding: ROBERT BOYD, University of California, Los Angeles Infectious inheritance and the maintenance of transposal genetic elements. BRUCE LEVIN, University of Massachusetts, Amherst Diffusion model for migration and selection. THOMAS NAGYLAKI, University of Chicago Multilocus population genetics models. ALAN HASTINGS, University of California, Davis

2:30 p.m. Some Mathematical Questions in Biology-Models in Population Biology Presiding: ALAN HASTINGS, University of California, Davis Evolutionary models of social learning. ROBERT BOYD, University of California, Los Angeles Interactions between environment and competition: Community structure in a variable environment. PETER CHESSON, Ohio State University Detecting order in the chaos of nature. WILLIAM SCHAFFER, University of Arizona

835 Joint Summer Research Conferences in the Mathematical Sciences University of Colorado, Boulder, June 14 to July 25 and Cornell University, July 12 to August 15, 1987

The 1987 Joint Summer Research Conferences in 3. Mailing address the Mathematical Sciences will be held at the 4. Telephone number & area code for office and University of Colorado, Boulder, from June 14 to home July 25, and at Cornell University, from July 12 5. Your scientific background relevant to the to August 15. It is anticipated that the series of topic of the conference conferences will be supported by grants from the 6. Financial assistance requested. Please esti­ National Science Foundation and other agencies. mate cost of travel. There will be five conferences in five different 7. Indicate if support is not required, and if areas of mathematics to be held at each institu­ interested in attending even if support is not tion. The topics and organizers for the conferences offered. were selected by the AMS-IMS-SIAM Committee The deadline for receipt of applications is on Joint Summer Research Conferences in the March 2, 1987. After that date the Organiz­ Mathematical Sciences from proposals submitted ing Committee for each conference will consider by individuals and topics suggested by committee the requests (selection of the participants and the members. The committee considered it important allocation of support is made by the Organizing that the conferences represent diverse areas of Committee.) You will be notified by the AMS mathematical activity, with emphasis on areas of the committee's decision no later than May 1, currently especially active, and paid careful at­ 1987. Funds available for these conferences are tention to subjects in which there is important limited and individuals who can obtain support interdisciplinary activity at present. from other sources should do so. Women and The conferences are similiar in scientific struc­ members of minority groups are encouraged to ture to those held throughout the year at Ober­ apply and participate in these conferences. wolfach. They are intended to complement the Any questions concerning the scientific por­ Society's program of annual Summer Institutes tion of the conference should be directed to the and Summer Seminars, which have a larger atten­ chairman or any member of the Organizing Com­ dance and are substantially broader in scope. The mittee. conferences are research conferences, and are not The Joint Summer Research Conferences in intended to provide an entree to a field in which the Mathematical Sciences are under the direc­ a participant has not already worked. tion of the AMS-IMS-SIAM Committee on Joint It is expected that support will be available Summer Research Conferences in the Mathemat­ for a limited number of participants in each con­ ical Sciences. The following Committee members ference. Others, in addition to those funded, will chose the topics for the 1987 conferences: William be welcome, within the limitations of the facilities B. Arveson, Ronald L. Graham, Benedict H. of the campus. In the spring a brochure will Gross, Malcolm R. Leadbetter, Angus J. Mac­ be mailed to all who are invited to attend the intyre, Jerrold E. Marsden (Chairman), John conferences. The brochure will include informa­ R. Martin, James McKenna, Tilla Klotz Milnor, tion on room and board rates, the residence and Evelyn Nelson, Katusumi Nomizu. dining hall facilities, travel and local information and a Residence Housing Form to use for on­ campus housing accommodations. Information University of Colorado, Boulder on off-campus housing will also be included in June 14 to June 20 the brochure. Participants are required to make Categories in computer science and logic their own housing and travel arrangements. Each JOHN W. GRAY (University of Illinois at participant will be required to pay a fee of $25 to Urbana-Champaign), Chairman cover the cost of social events and refreshments Category theory has had important uses in served at breaks, in addition to a $15 registration logic since the invention of topos theory sixteen fee. years ago, and logic has always been an important Those interested in attending one of the component of theoretical computer science. What conferences should send the following information is new is the increasing direct interaction between to Carole Kohanski, Summer Research Conference category theory and computer science. The aim Coordinator, American Mathematical Society, P. of this conference is to bring together researchers 0. Box 6248, Providence, RI 02940. who are working on the interconnections between Please type or print the following: category theory and computer science or between 1. Title and dates of conference desired category theory and logic. The conference will 2. Full name emphasize how the general machinery developed

836 in category theory can be applied to specific ques­ .June 28 to July 4 tions and utilized for category-theoretic studies of Graphs and algorithms concrete issues. JOE BUHLER (Reed College), Co-Chairman There is a kernel of topics relevant to all PHYLLIS CHINN (Humboldt State Univer­ three fields. It includes, for example, algebraic sity), Co-Chairman theories, Horn logic, lambda calculus, normal In the last several years there have been sig­ form reductions, and categorical models for com­ nificant advances in the study of graph-theoretic putability theory. Such topics will be the central algorithms. These ideas have been intertwined focus of the conference, but time will also be with the increasing connections between graph provided for category-theoretic topics presently theory and various areas of computer science. related either to logic or to computer science. On Some examples include the graph isomorphism the logic side, these include semantical (algebraic problem, self-adjusting data structures, the recent or topos-theoretic) approaches to proof-theoretic proof of Wagner's conjecture, new uses of the questions, problems concerning the internal prop­ ellipsoid algorithm, and work on graph embed­ erties of specific objects in (pre-) topoi and their ding and labelling problems motivated by VLSI representations, and categorical sharpening of design questions. A study of these problems, model-theoretic notions. Category theory is use­ and the related algorithms, often hinges on sur­ ful in studying the proof theory and model theory prisingly sophisticated mathematics. One goal of of various non-classical logics as well as classical this conference is to foster communication among first order logic. At the base of many uses of topos computer scientists and mathematicians pursuing theory is the fact that certain involved mathe­ these questions. matical objects (frequently described in terms of The conference will be organized around talks equivariance or continuity in additional parame­ by mathematicians and computer scientists in the ters) may be viewed as the interpretation, in a forefront of these new discoveries in algorithmic suitable category, of objects of a simpler nature. graph theory. On the computer science side, it has recently been Members of the Organizing Committee: Jean recognized that category theory is appropriate for Larson (University of Florida), Eugene Luks (Uni­ formalizing many aspects of computer program­ versity of Oregon), R. Bruce Richter (United ming and program design. One reason for this is States Naval Academy), Bruce L. Rothschild that in computer science it is necessary to consider (University of California at Los Angeles), and many different structures at the same time. These Robert E. Tarjan, (AT & T Bell Laboratories). structures must be viewed from different aspects and the interactions between them are a central July 5 to July 11 component of program design. Category theory is Geometry of group representations specifically designed to deal with this kind of a sit­ WILLIAM GOLDMAN (University of uation on an abstract level. Mary­ Specific areas where land), Co-Chairman active research is going on include: semantics of ANDY MAGID (University programming languages, data type of Oklahoma), specifications, Co-Chairman categorical programming, and categorical logic. By the "geometry of group representations" Members of the Organizing Committee: A. is meant the general principle of parameterizing Blass (University of Michigan), M. Makkai (McGill the representations of a finitely generated discrete University), A. Pitts (University of Sussex), A. group in a given Lie group by the points Scedrov (University of Pennsylvania), of and D. a space, the space being Scott (Carnegie-Mellon University). the set of tuples of elements of the Lie group (with as many entries as the discrete group has generators) which satisfy June 21 to June 27 all the relations of the discrete group. This is Hamiltonian dynamical systems actually a moduli space in case the Lie group KENNETH MEYER (University of Cincin­ is algebraic: there is a natural way to represent nati), Co-Chairman .the discrete group in the points of the algebraic DON SAARI (Northwestern University), Co­ group with entries in the coordinate ring of the Chairman parameter space, giving a universal representation This conference will concentrate on the recent which specializes to each of the representations in developments in the theory and applications of the parameter space. Hamiltonian systems of ordinary differential equa­ This is a subject and a technique which has tions. Some subfields which will be discussed are been in use explicitly for thirty years, but recent 1) fixed point theory for annulus maps or other developments have brought it back into the spot­ symplectic maps, 2) applications of symplectic light. In particular, representation varieties and geometry to Hamiltonian differential equations, their algebraic geometry are of current interest to 3) bifurcation theory, 4) applications to celestial algebraists, while topologists and differential ge­ mechanics, and 5) integrable systems. New and ometers have found important new applications. interesting developments in other related subfields In addition, there is a substantial body of work will be considered for presentation. and current activity on the analogous subjects for

837 rings and modules. One of the main purposes The object of this conference will be to of this conference is to bring together workers in present to a wide variety of specialists the different these different areas to share common methods directions of mathematical research arising from applicable to their various goals. Karmarkar's algorithm and related algorithms. Topics to be covered include: space of group It will provide opportunity for interdisciplinary representations, applications of spaces of group interaction between mathematicians, operations representations in topology and geometry, de­ researchers, and computer scientists. formations of representations, generic representa­ Members of the Organizing Committee: Vic­ tions, and varieties of representations as transfor­ tor Klee (University of Washington), and Stephen mation spaces. Smale (University of California, Berkeley). Member of the Organizing Committee: Hy­ July 19 to July 25 man Bass (Columbia University). Geometry of random motion July 19 to July 25 RICHARD DURRETT (Cornell University), The connection between infinite dimensional and Co-Chairman finite dimensional dynamical systems MARK PINSKY (Northwestern University), BASIL NICOLAENKO (Los Alamos National Co-Chairman Laboratories), Chairman This conference will be concerned with cer­ In the near future the large-scale computa­ tain connections between analysis, probability and tions in fluid and continuum mechanics will bring geometry centering around the common theme of us closer to the solution of turbulent flows. The diffusion processes. Any second order, semielliptic possible description of the long time behavior of differential operator gives rise to such a Markov such flows by finite dimensional models requires a process with continuous path functions. There are deeper insight into the connection between finite many interesting and important connections be­ and infinite dimensional systems. tween the sample path behavior of these processes There is considerable progress in unravel­ and properties of solutions of the corresponding ing the interactions between spatial structures elliptic and parabolic equations. In the case of and temporal chaos for perturbed Hamiltonian a Laplace-Beltrami operator these solutions can systems and strongly dissipative PDE's; and in further be related to the geometry and topology structuring the maze of bifurcations leading to of the manifold. chaos through group symmetry reductions. The The flow of ideas between probability and seminar will bring together specialists from vari­ geometry goes in two directions. On the one ous fields to clarify and deepen such cross-linkages hand, probabilistic techniques have been used between finite and infinite dimensional dynamical to give new proofs of results in geometry. On systems. the other hand, developments in geometry and Members of the Organizing Committee: Cip­ analysis have inspired developments in probability rian Foias (Indiana University), and Roger Temam theory and related infinite dimensional analysis. (University of Paris-Sud). The conference will emphasize increasing com­ munication between different viewpoints in this Cornell University rapidly expanding field. Members of the Organizing Committee: K. July 12 to July 18 D. Elworthy (University of Warwick), N. Ikeda Mathematical developments arising from linear (University of Osaka), P. Malliavin (University of programming Paris), M. Sharpe (University of California, San JEFFREY C. LAGARIAS (AT&T Bell Labo­ Diego), and D. Stroock (Massachusetts Institute ratories), Co-Chairman of Technology). MICHAEL TODD (Cornell University), Co­ Chairman July 26 to August 1 This conference will be devoted to current Crystal growth and pattern formation m phase research in linear and nonlinear programming and transitions its relations to other areas of mathematics, in­ STUART P. HASTINGS (SUNY at Buffalo), cluding convexity theory, geometry and dynamical Co-Chairman systems theory. The most exciting recent devel­ NICHOLAS D. KAZARINOFF (SUNY at Buf­ opment is N. Karmarkar's interior point linear falo), Co-Chairman programming algorithm. Karmarkar's algorithm There has in the last three years been a burst has stimulated a great deal of research since it of research papers on models of dendritic crystal was discovered in 1984. Aside from many imple­ growth and on global models for phase transitions. mentations and variants of the algorithm, it has A number of mathematicians has been stimulated stimulated research in: (1) other interior point to work on mathematical problems arising from algorithms in linear and nonlinear programming, this and related work. (2) connections to geometry and dynamical sys­ The experimental work includes dendritic tems, (3) developments in numerical linear algebra growth in Hele-Shaw cells, photographs and and sparse matrix computations. movies of dendritic birth from the lip of a capillary

838 tube in super-cooled pure water and subsequent Iteration of automorphisms of en growth and pattern selection. There are theo­ Complex analytic differential equations in retical models of moving-interface dynamics: a en' with emphasis on polynomial equations in geometrical approach leading to a 3-dimensional e2. nonlinear dynamical system, and a boundary­ There will be a sequence of lectures pre­ layer approach leading both to partial differential senting the fundamental results about iteration equations and dynamical systems. Continuum of functions of one complex variable. More spe­ models are stressed by some as bringing impor­ cialized topics will be the subject of individual tant global effects into play, e.g. global and lectures. free energy effects at interfaces (the Cahn-Hilliard Members of the Organizing Committee: Rob­ model). ert Devaney (Boston University), and Adrien Worthy of investigation are numerical simu­ Douady (University of Paris-Sud). lations of these models, mathematical problems of existence of heteroclinic orbits, stability of various August 9 to August 15 solutions, and pattern selection, etc. The object Statistical inference from stochastic processes of this seminar is to bring together engineers, NARAHARI U. PRABHU (Cornell Univer­ physicists and mathematicians interested in the sity), Chairman subjects sketched above so as to explore them in In recent years there has been considerable depth. In particular, it is hoped that progress will activity in statistical inference from stochastic be made in deciding which among the many mod­ processes. This research was first aimed at solv­ els that have been proposed are most promising ing specific problems in hand, but has resulted for future research. in major contributions to the conceptual frame­ Members of the Organizing Committee: Mor­ work of the subject as well as the associated ton Gurtin (Carnegie-Mellon University), and techniques. It also demonstrated the importance John Ockendon (Math. lnst., Oxford University). of concepts and tools such as martingales and partial likelihood. The purpose of this conference August 2 to August 8 is to establish lines of communication between Complex analytic dynamics probabilists and statisticians. JOHN H. HUBBARD (Cornell University), Members of the Organizing Committee: Law­ Chairman rence D. Brown (Cornell University), Nicholas M. Topics to be considered include: Kiefer (Cornell University), Michael J. Phelan Iteration of polynomials (Cornell University), Victor Solo (Cornell Univer­ Iteration of rational functions sity), and Lionel I. Weiss (Cornell University). Iteration of entire functions

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839 Special Meetings

THIS SECTION contains announcements of meetings of interest to some segment of the mathematical public, including 1111 hoc, local, or regional meetings, and meetings or symposia devoted to specialized topics, as well as announcements of regularly scheduled meetings of national or international mathematical organizations. (Information on meetings of the Society, and on meetings sponsored by the Society, will be found inside the front cover.) AN ANNOUNCEMENT will be published in Notices if it contains a call for papers, and specifies the place, date, subject (when applicable), and the speakers; a second full announcement will be published only if there are changes or necessary additional information. Once an announcement has appeared, the event will be briefly noted in each issue until it has been held and a reference will be given in parentheses to the month, year, and page of the issue in which the complete information appeared. IN GENERAL, announcements of meetings held in North America carry only date, title of meeting, place of meeting, names of speakers (or sometimes a general statement on the program), deadlines for abstracts or contributed papers, and source of further information. Meetings held outside the North American area may carry more detailed information. In any case, if there is any application deadline with respect to participation in the meeting, this fact should be noted. All communications on special meetings should be sent to the Editor of Notices, care of the American Mathematical Society in Providence. DEADLINES for entries in this section are listed on the inside front cover of each issue. In order to allow participants to arrange their travel plans, organizers of meetings are urged to submit information for these listings early enough to allow them to appear in more than one issue of Notices prior to the meeting in question. To achieve this, listings should be received in Providence SIX MONTHS prior to the scheduled date of the meeting.

1985-1986. Special Year in Differential Geometry, Uni­ 30-December 6. Eigenwertaufgaben in den lngenieurwis­ versity of lllinois, Urbana-Champaign, lllinois. (October senschaften und ihre Numerische Behandlung, Chat'rm.en: 1985, p. 674) J. Albrecht, Clausthhal-Zellerfeld; L. Collatz, Hamburg; W. Velte, Wurzburg; W. Wunderlich, Bochum. 1985-1986. Special Year in Singularities and Algebraic Geometry, University of North Carolina, Chapel Hill, December North Carolina. (June 1985, p. 397, note change in years 7-13. Combinatorical Optimization and its Relations to since previous listings.) Other Mathematical Areas, Chairmen: R. Burkard, Graz; M. Grotschel, Augsburg. 1985-1986. Year in Mathematical Logic, University of 14-20. Arbeitstagung des Ehemaligen Baerschen Seminars, Notre Dame, Notre Dame, Indiana. (October 1985, p. 674) Chairman: H. Salzmann, Tubingen. October 1985-0ctober 1986. Material Instabilities in January 1987 Continuum Mechanics, Heriot-Watt University, Edinburgh, 4-10. Fraktale und Ihre Bedeutung in Mathematik und Scotland. (June 1985, p. 397) Naturwissenschaften, Chairmen: B. Mandelbrot, York­ town Heights; H. Peitgen, Bremen; P. Richter, Bremen. 1986-1987. Academic Year Devoted to Algebraic Geome­ 11-17. Mathematical Economics, Chairmen: G. Debreu, try, Mittag-Leffler Institute, Djursholm, Sweden. (January Berkeley; W. Hildenbrand, Bonn; D. Sondermann, Bonn. 1986, p. 130) 18-24. Mathematische Theorien der Fluide, Chairmen: W. 1986-1987. Special Year in Modern Analysis, University Burger, Karlsruhe; I. Miiller, Berlin. of lllinois at Urbana-Champaign, Urbana, lllinois. 25-31. Quantenstochastik, Chairmen: L. Accardi, Rom; Program: Synthesis of Modern and Classical Analysis. W. von Waldenfels, Heidelberg. Invited Spealrers: G. Weiss, Y. Meyer, A. Pelczynski, E. February Hewitt, P. Enflo, J. Kahane, B. Beauzamy, R. Fefferman, 1-7. Algorithmen in der Kombinatorischen Geometrie, A. Chang, J. Arazy, J. Lindenstrauss, P.W. Jones, N. Chairmen: A. Dress, Bielefeld; R. Graham, Murray Hill; Varopoulos, T. Wolff, W. Johnson, W. Davis, J. Shapiro. H. Lenstra, Amsterdam. InfO'I'T'IUJii.on: Special Year in Modern Analysis, University 8-14. Flachen in der Geometrischen Datenverarbeitung, of lllinois at Urbana-Champaign, Urbana, lllinois 61801. Chni:rmen: R. Barnhill, Salt Lake City; W. Bohm, Braun­ October 12, 1986-December 13, 1987. Mathematisc:hes schweig; J. Hoschek, Darmstadt. Forschungsinstitut Oberwolfach (Weekly Conferences), 15-21. Operations-Research, Chairmen: K. Neumann, Karl­ Federal Republic of Germany. sruhe; D. Pallaschke, Karlsruhe. InfO'I'T'IUJii.on: Institute Director, Albertstrasse 24, D-7800 22-28. Medizinische Statistik: Design and Analysis of Clin­ Freiburg, Federal Republic of Germany. ical Trials, Chairmen: A. Neiss, Innsbruck; M. Schu­ October macher, Dortmund. 12-18. Geometrie der Banachraume, Chni:rmen: H. Konig, March Kiel; J. Lindenstrauss, Jerusalem; A. Pelczynski, 1-7. Mathematics in Industry, Chainnan: H. Neunzert, Warschau. Kaiserslautern. 19-25. Finite Elemente in der Kontinuumsmechanik, Chnir­ 1-7. New Foundations, Chairmen: M. Bofta, Mons; E. men: D. Braess, Bochum; E. Stein, Hannover. Specker, Ziirich. Nuve:rnher 8-14. Mathematische Stochastik, Chairmen: P. Bickel, 2-8. Affine Differentialgeometrie, Chainnan: U. Simon, Berkeley; F. Gotze, Bielefeld. Berlin. 15-21. Mathematische Modelle in der Biologie, Chairmen: 9-15. Komplexitatstheorie, Chni:rmen: C. Schnorr, Frank­ K. Hadeler, Tubingen; W. Jager, Heidelberg. furt; A. Schonhage, Tubingen; V. Strallen, Zurich. 22-28. Gewohnliche Differentialgleichungen, Chni:rmen: H. 16-22. Stochastik im Mathematikunterricht, Chni:rmen: Knobloch, Wiirzburg; J. Mawhin, Louvain-la-Neuve; R. R. Ineichen, Fribourg; H. Kindler, Bremen; H. Schupp, ReiBig, Bochum; K. Schmitt, Salt Lake City. Saarbrucken. 29-April 4. Anwendungen der lnfinitesimalmathematik, 23-29. Statistik Stochastischer Prozesse, Chni:rmen: W. Chairmen: S. Albeverio, Bochum; D. Laugwitz, Darm­ Fieger, Karlsruhe; N. Keiding, Kopenhagen. stadt; W. Luxemburg, Pasadena.

840 April Navemher 5-11. Reelle Algebraische Geometric, Chairmen: E. Becker, 1-7. Mathematische Methoden der VLSI und Distributed Dortmund; L. Brocker, Munster; M. Knebusch, Regens­ Computing, Chairmen: M. Fischer, New Haven; M. burg. Fontet, Paris; G. Hotz, Saarbriicken. 12-18. Arbeitsgemeinschaft Geyer-Harder. 8-14. Numerische Integration, Chairmen: H. Brai3, Braun­ 19-25. Mathematische Logik, Chairmen: W. Felscher, schweig; G. Hiirnmerlin, Munchen. Tubingen; H. Schwichtenberg, Munchen. 15-21. Mikrolokale Analysis und Partielle Differentialgle­ 19-25. Mathematical Methods in the Study of Natural and ichungen, Chairmen: G. Bengel, Miinster; P. Schapira, Computer Languages, Chairmen: J. Barwise, Stanford; Villetaneuse; J. Sjostrand, Paris. J. Fenstad, Oslo; H. Kamp, Austin; M. Richter, Aachen. December 26--May 2. Geschichte der Mathematik, Chairman: M. 13-19. Convergence Structures in Topology and Analysis, Folkerts, Munchen. Chairmen: E. Binz, Mannheim; H. Herrlich, Bremen; G. Preufi, Berlin. May 3-9. Gruppentheorie, Chairmen: K. Gruenberg, London; 0. Kegel, Freiburg. OCTOBER 1986 1D--16. Dynamische Systeme, Chairmen: J. Moser, Zurich; E. Zehnder, Bochum. 1-4. Coneurrenee Imparfaite et Modeles de Marehe, 17-23. Elliptische Operatoren auf Singularen und Nichtkom­ Luminy, France. (June 1986, p. 559) pakten Mannigfaltigkeiten, Chm"rmen: J. Bruning, Augs­ 3-4. The Fourteenth Annual Mathematies and Statisties burg; R. Melrose, Cambridge. Conferenee, Miami University, Oxford, Ohio. (June 1986, 24-30. Kommutative Algebra und Algebraische Geometric, p. 559) Chairmen: E. Kunz, Regensburg; H. Nastold, Munster; 5-9. International Symposium on Information Theory, L. Szpiro, Paris. University of Michigan, Ann Arbor, Michigan. (March 31-June 6. Darstellungstheorie Endlicher Gruppen, Cha:i:r­ 1986, p. 363) men: B. Huppert, Mainz; G. Michler, Essen. 5-12. Numerische Behandlung steifer DlfFerentialgleiehun­ June gen, Dusseldorf-Himmelgeist, Federal Republic of Ger­ 7-13. Diskrete Geometric, Chairmen: G. T6th, Budapest; many. (August 1986, p. 652) A. Florian, Salzburg. 21-27. Partielle Differentialgleichungen, Chairmen: S. 9-11. Interdiseiplinary Conferenee on Inferenee, Ohio Hildenbrandt, Bonn; P. Lax, New York; K. Uhlenbeck, University, Athens, Ohio. (June 1986, p. 559) Chicago. 13-November 29. Topology and Number Theory, Univer­ July site de Montreal, Quebec, Canada. (March 1986, p. 369) 12-18. Mathematische Probleme in Stromungen und Plas­ 15-18. Convegno su Ipergruppi, Altre Strutture Multi­ men, Chairmen: K. Kirchgii.ssner, Stuggart; J. Marsden, voche e Applieazioni, Udine, Italy. (June 1986, p. 559) Berkeley. 17-18. Fifteenth Annual Midwest DifiTerential Equations 19-25. Differentialgeometrie im Grossen, Chairmen: J. Conferenee, Marquette University, Milwaukee, Wisconsin. Bourgignon, Palaiseau; W. Klingenberg, Bonn; (March 1986, p. 369) R. Schoen, San Diego. 26--August 1. Darstellungstheorie Reduktiver Lie-Gruppen, 2D--24. Les Sondages, Marseille, France. (June 1986, p. 559) Chairman: B. Speh, Ithaca; D. Vogan, Cambridge. 24-25. Seeond Eastern Small College Computing Confer­ August enee, University of Scranton and the Hilton at Lackawanna 2-8. Einhiillende Algebren, Darstellungstheorie und Differ­ Station, Scranton, Pennsylvania. (March 1986, p. 369) entialoperatoren, Chairmen: W. Borho, Wuppertal; M. 24-25. The Sixth Southeastern-Atlantie Regional Con­ Dufio, Paris; J. Jantzen, Hamburg; R. Rentschler, Paris. ferenee on DifFerential Equations, Clemson University, 9-15. Konstruktive Verfahren in der Komplexen Analysis, Clemson, South Carolina. (June 1986, p. 559) Chairmen: D. Gaier, GieBen; P. Henrici, Zurich; R. Varga, Kent. 25-26. Twentieth Midwest Partial DifFerential Equations 16--22. Numerische Methoden fiir Anfangs-Und Anfangs­ Seminar, Purdue University, West Lafayette, Indiana. (Au­ Randwertprobleme, Chairmnn: H. Kreiss, Pasadena. gust 1986, p. 652) 23-29. Partial Differential Equations in Complex Analysis, 27-28. Fifth Annual Paeifie Northwest Computer Graph­ Chairmen: K. Diederich, Wuppertal; J. Kahn, Princeton; ies Conferenee, Eugene, Oregon. I. Lieb, Bonn. Infarrnation: P. Katz, Conference Manager, University of 3D--September 5. Topologie, Chairmen: M. Kreck, Mainz; Oregon Continuation Center, 1553 Moss Street, Eugene, A. Ranicki, Edinburgh; L. Siebenmann, Paris. Oregon 97403, 503-686-3537 September 28-31. Eighth International Conference on Pattern Reeog­ 6--12. Homotopietheorie, Chairmen: M. Mahowald, nUion, Paris, France. (August 1986, p. 652) Evanston; L. Smith, Giittingen. 29-31. Twenty-first Actuarial Research Conference, Ohio 13-19. Hysteresis-Phiinomene: Madelle, Theorie und Ver­ State University, Columbus, Ohio. (August 1986, p. 652) fahren, Chairmen: K. Hoffmann, Augsburg; J. Sprekels, Augsburg. 31-1 November. Eighth Midwest Probability Colloquium, Northwestern University, Evanston, Illinois. (August 1986, 13-19. Nukleare Frechet-R.aume, Chairmen: E. Dubinsky, p. 653) Potsdam; R. Meise, Dusseldorf; D. Vogt, Wuppertal. 2D--26. Risikotheorie, Chairmen: H. Buhlmann, Zurich; W. Jewell, Berkeley. NOVEMBER 1986 27-0ctober 3. Funktionalanalysis: C* -Algebren, Cha:i:r­ men: A. Connes, Paris, J. Cuntz, Marseille; R. Nagel, 2...U. Fall Joint Computer Conference, Dallas, Texas. Tubingen. (March 1986, p. 370) October 3-7. Workshop on Numerical Algorithms for Modern Par­ 11-17. Arbeitsgemeinschaft Geyer-Harder. allel Computer Architeetures, Institute for Mathematics 18-24. Geometric, Chairmen: D. Ferns, Berlin; K. Voss, and its Applications, University of Minnesota, Minneapolis, Zurich. Minnesota. (March 1986, p. 370) 25-31. Multigrid Methods, Chairmen: D. Braess, Bochum; 9-15. Austrian Symposium on History of Mathematics, W. Hackbusch, Kiel; U. Trottenberg, Koln. Neuhofen an der Ybbs, Austria. (June 1986, p. 559)

841 1Q-15. Workshop on Artin £-functions and Related 19-30. Twenty-seventh Summer Research Institute of the Topics, Universite de Montreal, Quebec, Canada. (March Australlan Mathematical Society, University of New Eng­ 1986, p. 370) land, Armidale, N.S.W., Australia. (August 1986, p. 653) 1Q-28. Workshop on Representation Theory of Lie Groups, 2Q-24. Workshop on Iwasawa Theory and Special Values International Centre for Theoretical Physics, Trieste, Italy. of £-functions, Mathematical Sciences Research Institute, (June 1986, p. 560) Berkeley, California. (August 1986, p. 653) 18-22. Didactique des Mathematiques, Marseille, France. 29-31. Utah State University Department of Mathe­ (June 1986, p. 560) matics Second Conference on Matrix Theory, Utah State University, Logan, Utah. (August 1986, p. 653) 19-21. Cours Modulef: Une Blblioth~que Modulaire d'tlements Finis, Rocquencourt, France. (August 1986, p. 653) FEBRUARY 1987 24-26. Trois Journ6es sur le Codage, Cachan, France. February. Seminaire International Grands Calculateurs (August 1986, p. 653) Scientiflques, Paris, France. (August 1986, p. 653) 24-28. Geometrie Symplectique, Marseille, France. (June 8-12. Applied Mathematics Conference, Wairakei, New 1986, p. 560) Zealand. Program: The Conference aims to provide a forum for DECEMBER 1986 presentation and discussion of papers in all areas of Applied Mathematics. 1-5. Minisymposium on Numerical Simulation in Oil Infarmntion: Conference Secretary, Department of Theo­ Recovery, Institute for Mathematics and its Applications, retical and Applied Mechanics, University of Auckland, University of Minnesota, Minneapolis, Minnesota. (August Private Bag, Auckland 1, New Zealand. Telephone: 649- 1986, p. 653) 737-999. 2-5. Second International Conference on Artificial Intel­ 13-18. Annual Meeting, American Association for the ligence, Marseilles, France. (August 1986, p. 653) Advancement of Science, Chicago, Illinois. (March 1986, 3-5. Forty Second Annual Conference on Applied Statis­ p. 370) tics, Atlantic City, New Jersey. (March 1986, p. 370) 16-20. Workshop on Materials with Nonlinear Constitu­ tive Laws, Institute for Mathematics and its Applications, 3~. Theory of Robots Symposium, Vienna, Austria. (January 1986, p. 134) University of Minnesota, Minneapolis, Minnesota. (August 1986, p. 653) 11-12. International Conference on the Numerical Mod­ 17-19. Association for Computing Machinery Computing eling of Semiconductors, Los Angeles, California. Science Conference, St. Louis, Missouri. lnfarmntion: M. Maher, Receptive Services, 8921 Sepul- Injarmntion: A. DeKock, Conference Chair, or G. Zobrist, veda Boulevard, Suite 209, Los Angeles, California 90045, 213-670-8050 or P. McKeever, Conference Man­ Program Chair, CSC'87, MCS 325, University of Mis­ agement Services, Post Office Box 5, 51 Sandycove souri, Rolla, Missouri 65401, 314-341-4492. Road, Dun Laoghaire, Co. Dublin, Ireland. Telephone: ( +353-1)808025. MARCH1987 15-17. Cryptography and Coding, Cirencester, Glos., England. (June 1986, p. 560) 8-11. Computer Science and Statistics: 19th Sympo­ sium on the Interface, Temple University, Philadelphia, 15-19. International Workshop on Multivariate Approxi­ Pennsylvania. mation, Santiago, Chile. lnfarmntion: R. Heiberger, Department of Statistics, School lnjarmntion: F. Utreras, Department of Applied Mathemat­ of Business and Management, Temple University -006- ics, University of Chile, Casilla 170/3 Correa 3, Santiago, 00, Philadelphia, Pennsylvania 19122, 215-787-6879. Chile. 16-20. Seminaire de Probabilites XXII, Laboratoire ****** d'Automatique et d'Analyse des Systemes, Toulouse, France. (August 1986, p. 653) JANUARY 1987 22-25. Institute of Mathematical Statistics Central Re­ gional Meeting, Dallas, Texas. (March 1986, p. 370) January-May. Nonlinear POE's, Brigham Young Univer­ 22-29. Fifth International Conference on Geometry, Uni­ sity, Provo, Utah. (June 1986, p. 560) versity of Haifa, Haifa, Israel. 5-7. First International Symposium on Domain De­ Program: Foundations of Geometry, Combinatorial Struc­ composition Methods for Partial Differential Equations, tures, Convexity. Special Section "Geometry and School" Rocquencourt, France. (meeting simultaneously and at synchronized times). Infarmntion: Institut National de Recherche en Informatique Call far Papers: Send to address below by December 31, et en Automatique, domaine de Voluceau, rocquencourt, 1986. Boite Postale 105, 78153 Le Chesnay Cedex, France. lnfarmntion: R. Artzy or J. Zaks, Department of Mathe- Telephone: (1) 39-63-55-11. matics, University of Haifa, 31999 Haifa, Israel. 7-9. American Statistical Association Winter Conference: 23-27. Workshop on Galois Groups over Q and Re­ Statistics in the Information Age, Orlando, Florida. lated Topics, Mathematical Sciences Research Institute, lnfarmntion: American Statistical Association, 806 15th Berkeley, California. Street, Northwest, Washington, District of Columbia Program: This is the second of two workshops being held at 20005. the Mathematical Sciences Research Institute during the 1986-1987 program on Number Theory with Connections 12-14. Conference on Industry-University Collaborations to Algebraic Geometry. in the Mathematical Sciences, Claremont, California. (Au­ Infarmntion: K. Ribet, Mathematical Sciences Research gust 1986, p. 653) Institute, 1000 Centennial Drive, Berkeley, California 12-16. Seminar on Approximation and Optimization, 94720. Havana, Cuba. (August 1986, p. 653) 23-27. Workshop on Scientific Software, Institute for 19-23. Structures for Documents, Aussois, France. (Au­ Mathematics and its Applications, University of Min­ gust 1986, p. 653) nesota, Minneapolis, Minnesota. (August 1986, p. 654)

842 23-30. NSF-CBMS Conference on Mathematical Statis­ MAY1987 tics, Ohio State University. (June 1986, p. 560) 25-27. Conference on Information Sciences and Systems, 11-15. Thirty-first Annual Meeting of the Australian Johns Hopkins University, Baltimore, Maryland. Mathematical Society, Deakin University, Geelong, Victo­ Call for Papers: A regular or short designation, title, and ria, Australia. (August 1986, p. 654) summary are to be submitted by January 14, 1987 to 18-21. Eighth Symposium on Computer Arithmetic, the address below. Como, Italy. 111/orm.o.tion: 1987 Conference on Information Sciences and Call for Papers: For more information, write to below Systems, Department of Electrical Engineering and address. Computer Science, The John Hopkins University, Balti­ lnform.ation: M. Irwin, Department of Computer Science, more, Maryland 21218. 333 Whitmore Laboratory, Pennsylvania State Univer­ 3Q-15 April. Workshop and Conference on Number The­ sity, University Park, Pennsylvania 16802, 814-865-1802. ory and Dynamical Systems, University of York, York, 2Q-23. Combinatorial Matrix Analysis Conference, Uni­ United Kingdom. versity of Victoria, Victoria, British Columbia, Canada. Inform.ation: M. Dodson, Department of Mathematics, Uni­ Program: Ten invited papers, contributed papers which are versity of York, York, YOl 5DD or J. Vickers, De­ requested. partment of Mathematics, University of Southampton, Invii.ed Speokers: A. Berman, R. Brualdi, D. Carlson, D. Highfield, Southampton S09 5NH. Hershkowitz, C. Jeffries, C. Johnson, V. Klee, J. Maybee, H. Schneider, and R. Varga. APRIL 1987 Inform.ation: P. van den Driessche, Department of Mathe­ matics, University of Victoria, Victoria, British 6-8. Conference on Combinatorial Optimisation, Univer­ Columbia, Canada, V8W 2Y2. sity of Southampton, United Kingdom. 25-29. The Ninth International Symposium on Noise Topics: Topics will include integer programming, complex­ in Physical Systems, Universite de Montreal, Quebec, ity theory, analysis of algorithms, polyhedral combina­ Canada. (March 1986, p. 370) torics, applications to coding theory and cryptography, 26-28. Workshop on Nonlinear Hyperbolic Waves, Math­ parallel and sequential computing, and telecommunica­ ematical Sciences Research Institute, Berkeley, California. tions. (August 1986, p. 654) 111/orm.o.tion: C. Potts, Faculty of Mathematical Studies, University of Southampton, Southampton S09 5NH, 28-1 June. Fourth Southeast Asian Conference on Math­ United Kingdom. ematical Education, Singapore. (August 1986, p. 654) 8-10. Third Workshop on the Mathematical Founda­ tions of Programming Semantics, Thlane University, New JUNE1987 Orleans, Louisiana. Invii.ed Speakers: G. Plotkin, N. Jones, S. Brookes, J. Gray, 1-10. Second Annual Meeting ofthe International Work­ J. Lawson. shop in Analysis and Ita Applications, Dubrovnik-Kupari, Call for Papers: Deadline for sumission is November 17, Yugoslavia. 1986. More information on topics can be obtained by Inform.ation: C. Stanojevic, Department of Mathematics contacting either program committee chairmen below. and Statistics, University of Missouri, Rolla, Missouri Inform.ation: M. Main, Computer Science Department, CB 65401. 430, University of Colorado, Boulder, Colorado 80309, 303-492-7579 or A. Melton, Computer Science Depart­ 8-19. Singapore Group Conference Theory, National ment, Kansas State University, Manhattan, Kansas University of Singapore, Republic of Singapore. (March 66506, 913-532-6350. 1986, p. 370) 21-25. Conference on Nonlinear Numerical Methods 11-13. Congreas on Educational Computing in Mathe­ Baaed on the Use of Rational Functions, University of matics, Universita di Roma I, Roma, Italia. (August 1986, Antwerp, Belgium. (August 1986, p. 654) p. 654) 23-24. The Eighteenth Annual Modeling and Simulation 15-16. NASECODE V SHORT COURSE: The Interfaces Conference, Pittsburgh, Pennsylvania and Integration of Process, Device and Circuit Models­ Call for Papers: Two copies of titles, authors, all author's An Introduction, Trinity College, Dublin, Ireland. addresses, abstracts and summaries should be submitted Call for Papers: Abstracts of 20-minute contributed papers by January 31, 1987. The abstract should be approxi­ should be submitted to the NASECODE Secretariat by mately 50 words in length and the summary should be of February 15, 1987. sufficient length and detail to permit careful evaluation. Inform.ation: NASECODE V, Conference Management Ser­ Identify one author as the correspondent for the paper. vices, Post Office Box 5, 51 Sandycove Road, Dun Direct all correspondence to the address below. Laoghaire, Co Dublin, Ireland. Telephone: 353-1-808025. Inform.ation: W. Vogt or M. Mickle, Modeling and Simu­ 15-July 3. Microprogram on Commutative Algebra, Math­ lation Conference, 348 Benedum Engineering Hall, Uni­ ematical Sciences Research Institute, Berkeley, California. versity of Pittsburgh, Pittsburgh, Pennsylvania 15261. (October 1985, p. 679) 27-30. Workshop on Mathematical Modeling in Combus­ 17-19. NASECODE V Conference: The Fifth International tion and Related Topics, Ecole Centrale de Lyon, France. Conference on the Numerical Analysis of Semiconductor (August 1986, p. 654) Devices and Integrated Circuits, Trinity College, Dublin, 28-May 1. The Mathematics of Finite Elements and Ireland. Applications 1987, Uxbridge, United Kingdom. Inform.ation: NASECODE V, Conference Management Ser­ Inform.ation: The Secretary, The Institute of Computational vices, Post Office Box 5, 51 Sandycove Road, Dun Mathematics, Brunei University, Uxbridge, Middlesex Laoghaire, Co Dublin, Ireland. UB8 3PH, United Kingdom. 23-26. Sixth IMACS International Symposium on Com­ 29-May 8. AMS-SIAM Summer Seminar on Computa­ puter Methods for PDE's, Lehigh University, Bethlehem, tional Aspects of VLSI Design with an Emphasis on Semi­ Pennsylvania. (January 1986, p. 134) conductor Device Simulation, Institute for Mathematics 23-27. International Conference on Generalized Func­ and its Applications, University of Minnesota, Minneapo­ tions, Convergence Structures and Their Applications, lis, Minnesota. (August 1986, p. 654) Dubrovnik, Yugoslavia. (June 1986, p. 560)

843 29--July 3. Preml~re Comm-ence Internationale sur lea ~22. Harmonie Analysis on Real and p-adie Groups, Mathmnatiques AppllquMII et Industrielles, Paris, France. Bowdoin College, Brunswick, Maine. (August 1986, p. 654) Organizers: W. Barker (chairman), J. Bernstein, R. Herb, 2~July 4. Joint IMA-GAMM-SIAM-SMAI Conference H. Rossi, P. Sally, and J. Wolf. on Fint Joint International Conference on Industrial Program: Several series of talks will be scheduled, speakers and Applled Mathematics, Paris, France. (January 1986, including W. Schmid and D. Vogan. Numerous hour p. 134) lectures will also be delivered. Infarmation: W. Barker, Department of Mathematics, Bow- 3Q-July 4. Third International Seminar on Random doin College, Brunswick, Maine 04011. Graphs and Probabilistic Methods in Combinatorics, Poznan, Poland. 1Q-13. Sixth International Conference on Mathemati­ lnfarmation: M. Karonski, Institute of Mathematics, Adam cal Modelling: An Interdisciplinary Integrative Forum for Mickiewicz University, Matejki 48/49, 60-769 Poznan, Researchers and Educators in Engineering, Economies, Poland. Biological, Medical, Environmental, Social and other Sci­ ences, Washington University, St. Louis, Missouri. Call far Papers: Authors are invited to submit papers in JULY 1987 three categories: full length (20 minute presentation), short reports (10 minute presentation), or papers for July-August. Low Dimensional Topology Symposium, poster sessions. Send inquiries to the address below. University of Sussex, Brighton, Great Britain. (August lnfarmation: E. Rodin, Department of Systems Science 1986, p. 654) and Mathematics, Washington University, Campus Box 6-10. Third Gregynog Symposium on Differential Equa­ 1040, One Brookings Drive, St. Louis, Missouri 63130, tions, University of Wales, United Kingdom. 314-889-5806. lnfarmation: The symposium is financed by the Science 16-21. Sixteenth Conference on Stochastic Processes and and Engineering Research Council, and participation is their Applications, Stanford University, Stanford, Califor­ by invitation. Contact N. Lloyd, Department of Mathe­ nia. (August 1986, p. 654) matics, The University College of Wales, Aberystwyth, Dyfed SY23 3BZ, United Kingdom. 17-20. International Conference on Rings, Modules, and Radicals, Hobart, Tasmania. (June 1986, p. 560) 6-16. A Research Symposium in Complex Analysis, Imperial College, South Kensington, London, England. 24-28. Conference on DifFerential Equations "EquadifF Topics: Value Distribution Theory, Univalent and Multi- '87", Democritus University of Thrace, Greece. (January valent Functions, Potential Theory, and Boundary Be­ 1986, p. 134) haviour. 24-28. International Conference on Web Geometry and Infarmation: I. Baker or T. Kiivari, Imperial College of Related Fields, Szeged University, Szeged, Hungary. (June Science and Technology, South Kensington, London, 1986, p. 560) England. 24-28. Second International Conference on Combinatorial 13-17. Automata, Languages, and Programming, Karl­ Mathematics and Computing, Canberra, Australia. (March sruhe, Federal Republic of Germany. (August 1986, p. 654) 1986, p. 370) 13-17. lnequallties: Fifty Yean on From Hardy, Little­ 24-28. Sixth National Conference on Artificial Intelli­ wood and Polya, University of Birmingham, Birmingham, gence, Seattle, Washington. (March 1986, p. 370) England. 24-29. Meeting on Geometry of Banach Spaces, Mons, Program: Many lectures will be given on areas associated Belgium. (August 1986, p. 655) with individual chapters in the book lnequdities by Hardy, Littlewood and Polya which was first published by Cambridge University Press in 1934. SEPTEMBER 1987 lnfarmation: The Organising Secretary, Inequalities Confer­ ence, Department of Mathematics, University of Birm­ ~12. Internationale Konferenz iiber Anwendungen und ingham, Post Office Box 363, Birmingham B15 2TT, Modellbildung im Mathematikunterrieht, Kassel, Federal England, United Kingdom. Republic of Germany. (June 1986, p. 560) 1~24. Conference on Potential Theory, Charles Univer­ 13-19. JoumMII Arithmetiques 1987, Ulm, Federal Re­ sity, Prague, Czechoslovakia. (June 1986, p. 560) public of Germany. (June 1986, p. 560) 2Q-26. DMV-Jahrestagung 1987, Berlin, Federal Republic AUGUST1987 of Germany. (June 1986, p. 560)

1--£. Fifth Conference on Graph Theory of China, People's OCTOBER 1987 Republic of China. (August 1986, p. 654) ~ 15. International Conference on Abelian Groups, Perth, October. 87 ICAR-Intemational Conference on Advanced Western Australia. (June 1986, p. 560) Robotics, Paris or Nice, France. (August 1986, p. 655) October. JoumMII Methodes Numeriques en Meehanique des Fluides, Sophia-Antipolis, France. (August 1986, p. 655)

844 NEW AMS PUBLICA liONS

SOME MATHEMATICAL COMBINATORIAL SYMMETRIES QUESTIONS IN OF THE m-DIMENSIONAL BALL BIOLOGY-PLANT BIOLOGY Lowell Jones Louis J. Gross and Robert M. Miura, Editors (Memoirs of the AMS, Number 352) (Lectures on Mathematics in the Life Sciences, Volume 18) The author provides a complete classification of those subsets of the m-dimensional ball that can occur as fixed point sets of a PL semi-free cyclic Distinguishing itself among other books on group action on the ball. Conversely, given a subset mathematics in plant biology, this book is unique of the ball, the author classifies up to concordance because it presents a broad overview of how plant the PL semi-free cyclic actions on the ball having biologists are currently utilizing mathematics in the given set as its fixed point set. Previously their research, and is the only one to particularly these results had only been proven for subsets of emphasize plant ecology. Each article is unified by the ball with codimension at least 6. an attempt to tie models at one level of organization to an understanding at other levels. This approach 1980 Mathematics Subject Classifications: between theoretical 57S17, 57R65, 57R67 strengthens the connections ISBN 0-8218-2414-7, LC 86-17500 development and observable biology, facilitating the ISSN 0065-9266 testing of new predictions. 128 pages (softcover). September 1986 Individual member 111. List price $18, Intended for mathematicians, plant biologists and Institutional member $14 ecologists alike, this book requires only a basic To order, please specify MEM0/352N knowledge of differential equations, linear algebra and mathematical modeling; a knowledge of plant biology is helpful. Readers will gain a perspective on what types of biological systems can benefit HENSTOCK INTEGRATION IN from mathematical treatment and an appreciation THE PLANE of the current important problems in plant biology. Krzysztof M. Ostaszewski Contents (Memoirs of the AMS, Number 353) Karl J. Niklas, Computer simulations of branching-patterns and their implications on the Historically the subject of the generalization of the evolution of plants Lebesgue integral in the plane has lacked unity. Richard H. Rand and J. L. Ellenson, Dynamics This book clarifies the picture by comparing the of stomate fields in leaves Lebesgue, Perron, Kempisty, Pfeffer and Chelidze John H. M. Thornley and I. R. Johnson, integrals, and relating them to the Henstock Modelling plant processes and crop growth integral. In addition, the author indicates how Louis J. Gross, Photosynthetic dynamics and using Henstock integration can improve and plant adaptation to environmental variability generalize some theorems in classical analysis-e.g., Hal Caswell, Life cycle models for plants Green's or the divergence theorem, and the Fubini Jonathan Roughgarden, Models of population Theorem. Directed to graduate students and processes for plants research mathematicians interested in classical real 1980 Mathematics Subject Classification: analysis and integration theory, the book requires a 92-06. 92A17. 92A15 familiarity with measure theory and the Lebesgue ISBN 0-8218-1168-1, LC 80-646696 ISSN 0075-8523 integral. 280 pages (softcover), September 1986 Contents Individual member 122. List price $36, Institutional member $29 Henstock integral To order, please specify LLSCI/18N Derivation bases on the plane Generalized Fubini Theorem (continued)

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845 The integral of Kempisty Approximate derivation bases THE BERGMAN BILATERAL 1980 Mathematics Subject Classification: 26A39 SHIFT ISBN 0-8218-2416-3, LC 86-17399 Gregory T. Adams ISSN 0065-9266 116 pages ( softcover), September 1986 (Memoirs of the AMS, Number 355) Individual member 18, List price $14, Institutional member $11 space is the Hilbert space of square To order. please specify MEM0/353N The Bergman integrable analytic functions on the unit disk with normalized area measure, and the Bergman bilateral shift is the compression of multiplication by the independent variable to this space. This memoir SUBNORMAL OPERATORS AND studies the Bergman bilateral shift, and explores REPRESENTATIONS OF the interesting structure of its commutant. ALGEBRAS OF BOUNDED Contents ANALYTIC FUNCTIONS AND The Bergman Toeplitz operators The operator Cq, OTHER UNIFORM ALGEBRAS Some Besov spaces Thomas l. Miller, Robert F. Olin The commutant of the Bergman bilateral shift and James E. Thomson The algebra { Cz }' (Memoirs of the AMS, Number 354) 1980 Mathematics Subject Classification: 47B37 This memoir is aimed at researchers in operator ISBN 0-8218-2417-1, LC 86-17404 and/or function theory; however, it requires, as ISSN 0065-9266 the fundamental concepts 80 pages (softcover), September 1986 a prerequisite, only Individual member 17, List price $11, contained in a standard course in functional Institutional member $9 analysis. Its main purpose is to establish a To order, please specify MEM0/355N one-to-one correspondence between the minimal measures of a compact space Y and certain representations 1r of C( Y) into L00 (J.t), where J.t is a measure on a compact set X. The authors give CURRENT PROBLEMS OF many applications of this result; in particular, they apply it to various representations of algebras of MATHEMATICS. subnormal operators and problems on boundary MATHEMATICAL ANALYSIS, values of algebras of bounded analytic functions on ALGEBRA, TOPOLOGY a given domain. A. A. Logunov, Editor-in-chief Contents (Proceedings of the Steklov Institute, Volume 167) Introduction Uniqueness of representations This collection spans a wide range of areas in Continuity properties of unital representations mathematics: differential equations, mathematical Spectral mapping theorems analysis, optimal control theory, approximation Representations of H00 (G) into L 00 (J.t) theory, geometry, algebra and topology. It is the Representations of H00 (G) into L 00 (J.t) that are second in a two-volume publication dedicated to isometries Academician L. S. Pontryagin. Partially subordinate representations Contents A generalization (of the results in chapter 5) D. V. Anosov, On geodesic flows satisfying condition 1980 Mathematics Subject Classifications: (U) 47B20, 46J15, 47A67, 30E25, 47(99 S. M. Aseev, Quasilinear operators and their ISBN 0-8218-2415-5, LC 86-17381 application in the theory of multivalued mappings ISSN 0065-9266 F. P. Vasil'ev, On regularization of Newton's 136 pages (softcover), September 1986 Individual member Ill, List price $19, method when the initial data is imprecise Institutional member $15 A. G. Vitushkin, V. V. Ezhov and N. G. To order, please specify MEM0/354N Kruzhilin, Extension of holomorphic mappings along real-analytic hypersurfaces

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846 v. G. Danilov and V. P. Maslov, The Pontryagin V. B. Giner, l. R. Podoshev and M. l. Sodin, duality principle for computing a Cherenkov type Addition of lower indicators of entire functions effect in crystals and difference schemes. II V. E. Katsnel'son, On the theory of entire v. I. Denisov, A. A. logunov, M. A. functions of Cartwright class Mestvirishvili and Vu. V. Chugreev, On the A. E. Eremenko, On deviations of meromorphic relation between inertial and gravitational masses of functions of finite lower order an extended body in metric theories of gravitation Yu. M. Dyukarev and V. E. Katsnel'son, V. K. lsaev, The Pontryagin maximum principle Multiplicative and additive classes of Stieltjes and controlled Hermitian interpolation processes analytic matrix-valued functions. and interpolation A. A. Karatsuba, On the zeros of the Riemann problems associated with them. I zeta function on the critical line 0. V. Ivanov and G. D. Suvorov, Stable Yu. N. Kiselov, The linear time-optimal problem conformally invariant bicompact extensions and the unit sphere of initial values of the conjugate P. M. Tamrazov, A. A. Gonchar's extremal variable problem on capacities of condensers. The method S. N. Kruzhkov and V. l. Kamynin, On passage of mixing charges to the limit in quasilinear parabolic equations I. D. Machavariani, On lower and upper Yu. S. ledyaev, Regular differential games with nontangential limits of the logarithmic potential of mixed constraints on the controls a double-layer distribution A. F. leont'ev, Conditions for the representability 1980 Mathematics Subject Classifications: of functions in convex domains by generalized 30B40, 30Cxx, 30Dxx, 30E05, 31Axx, 54035 series of exponentials ISBN 0-8218-3106-2, LC 86-20550 Yu. I. Merzlyakov, The family of canonical matrix ISSN 0065-9290 128 pages (hardcover), September 1986 representations of an a-powered group Individual member 126, List price $44, K. I. Oskolkov, Subsequences of Fourier sums of Institutional member $35 integrable functions To order, please specify TRANS2/131N A. I. Subbotin, Conditions for optimality of a guaranteed outcome in game problems 1980 Mathematics Subject Classifications: 11M26, 20C15, 26E25, 28B20, 30B50, 32-02, THIRTEEN PAPERS IN ALGEBRA 34A60, 35K20, 42A20 and others ISBN 0-8218-3095-3, LC 86-20640 I. G. Dmitriev et. al. ISSN 0081-5438 (American Mathematical Society Translations, 260 pages (softcover), September 1986 Series 2, Volume 132) Individual member 168, List price $113, Institutional member $90 To order, please specify STEKL0/167N The topics in this collection range over several diverse areas, such as graph theory, continued fractions, complex manifolds and group theory. Contents TEN PAPERS IN ANALYSIS I. G. Dmitriev, Characterization of the class of V. A. Zmorovich et. al. k-trees (American Mathematical Society Translations, S. Kh. Darbinyan, Directed cycles of any length Series 2, Volume 131) in digraphs with large semidegrees A. V. Kostochka, On the minimum of the This collection focuses primarily on topics in Hadwiger number for graphs with a given mean the area of the theory of functions of a complex degree of vertices variable_ 0. G. Rudenskaya, An upper bound on the number of edges of a hypergraph with a given Contents diameter V. A. Zmorovich and V. A. Pokhilevich, On T. M. Zuparov, On a theorem from the metric the order of starlikeness of the class of a-convex theory of continued fractions functions for a > 0 V. V. lliev, Surfaces with pg = 3 and K2 = 3. V. I. Ryazanov, Some questions of convergence Part I and compactness for quasi-conformal mappings V. V. lliev, Surfaces with pg = 3 and K 2 = 3. V. N. Ginzburg and G. A. Sharshanova, Part II On singularities of analytic Hermitian-positive D. A. leites, A formula for the characters of functions irreducible finite-dimensional representations of Lie superalgebras of series C (continued)

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847 V. Ya. Golodets and N. I. Nessonov, Free de quelques idees geometriques tres simples, sans groups. the T -property, and 11 1 -factors with faire appel a beaucoup d'analyse. different countable fundamental groups I. S. Bezverkhnyaya, On the conjugacy of finite En plus des exemples donnes dans le corps du sets of subgroups in a free product of groups texte, une substantielle conclusion et plusieurs V. A. Grinblat, On normalizers of Artin groups des appendices permettront au lecteur de mieux A. l. VishnevetskiT, Groups of class 2 and apprecier les implications de nos resultats et Ia exponent p with commutator subgroup of order p2 richesse du sujet. N. A. Vavilov, Subgroups of the full linear 1980 Mathematics Subject Classifications: group over a ring which contain a group of block 53, 35, 58 442 pages (softcover), 1986 triangular matrices Individual member 126, List price $37 1980 Mathematics Subject Classifications: To order, please specify AST /138/139N 05Cxx, 11K50, 17A70. 20E06, 20Fxx, 20G35, 46L35 ISBN 0-8218-3107-0, LC 86-17254 ISSN 0065-9290 112 pages {hardcover), September 1986 Individual member 124, List price $40, GEOMETRIE ET ANALYSE Institutional member $32 To order, please specify TRANS2/132N MICROLOCALES J. L. Brylinski (Asterisque, Number 14Q-141)

L'analyse microlocale est un outil puissant dans SOCIETE MATHEMATIQUE DE FRANCE, Ia theorie des E. D. P. lineaires (propagation des ASTERISQUE singularites, formes normales pour des systemes d'E. D. P., developpements asymptotiques). Ses liens avec Ia geometrie algebrique, Ia topologie The AMS distributes Asterisque only in the U.S .. Canada. algebrique et Ia theorie des representations se sont and Mexico. Orders from other countries should be sent to the SMF. B.P. 126-05. 75226 Paris Cedex 05. France. eclaircis ces dernieres annees. Ce volume contient or to OFFILIB. 48 rue Gay-Lussac. 75240 Paris Cedex deux articles de Teresa Monteiro-Fernandes et un 05. France. Individual members of either AMS or SMF article de Jean-Luc Brylinski. are entitled to the member price. (ISSN 0303-1179) Un article de Teresa Monteiro-Fernandes utilise des methodes 2-microlocales pour demontrer GEOMETRIE DIFFERENTIELLE un theoreme general de Cauchy-Kowalewska ET SINGULARITES DE pour les systemes microdifferentiels. II en SYSTEMES DYNAMIQUES resulte un theoreme de propagation pour les solutions, a travers une hypersurface non Marc Chaperon "1-microcaracteristique." Un deuxieme article (Asterisque, Number 138-139) du meme auteur demontre, sans une hypothese geometrique, Ia constructibilite des solutions d'un Cet ouvrage, issu de cours donnes a I'Ecole systeme micro-differentiel. Normale Superieure et a I'Universidade Federal de Sao Carlos (Bresil), poursuit un double but; Un article de Jean-Luc Brylinski etudie un analogue d'une part introduire quelques outils fondamentaux topologique des transformations de contact de Ia geometrie differentielle Uets, theoreme quantifiees, et ses applications geometriques de prolongements de Whitney, theoreme de (theorie de Lefschetz). II contient aussi une transversalite de Thorn, methode du chemin, etude de Ia transformation de Fourier formelle structure symplectiques et de contact) sous une pour les modules sur l'algebre de Weyl, qui forme adaptee aux applications; d'autre part, est reliee a Ia transformation de Fourier a Ia employer ces outils pour etudier les singularites Sato-Malgrange. Quelques applications aux d'actions differentiables (formes normales, enonces sommes trigonometriques concluent !'article. divers du theoreme de Sternberg, linearisation 1980 Mathematics Subject Classifications: d'actions de groupes abeliens, varietes integrales 35. 58 singulieres d'equations aux derivees partielles, 252 pages (softcover), 1986 Individual member 115, List price S21 etc.) Un de nos objectifs etait de montrer que To order, please specify AST /140/141N l'on pouvait obtenir des resultats frappants et relativement profonds sur les actions, avec l'aide

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848 Miscellaneous

Personal Items Albert D. Otto of Illinois State University has been appointed Acting Chair of the Department John Frank Adams of Cambridge University, of Mathematics at that institution. Cambridge, England, has been awarded an honorary doctorate by Naturwissenschaftliche Doraiswamy Ramachandran of California Gesamtfakultiit der Universitiit Heidelberg at a State University, Sacramento, has been pro­ ceremony held May 31, 1986. moted to Professor of Mathematics at that Shiing-Shen Chern of the University of Cali­ institution. fornia, Berkeley, was elected an Honorary Member Murad S. Taqqu of Cornell University has of the London Mathematical Society at its meeting been appointed Professor of Mathematics at on May 16, 1986. Boston University. Burton H. Colvin, Director of the Center Robert W. Thomason, Associate Professor at for Applied Mathematics at the National Bureau The Johns Hopkins University, has been promoted of Standards, has been appointed Director of to Professor at that institution. Academic Affairs in the Office of the Director. Dr. Francis E. Sullivan will be the new Director Boris Weisfeiler of Pennsylvania State Uni­ of the Center for Applied Mathematics. versity was reported missing by the Chilean government in January 1985. Weisfeiler disap­ Frank C. Hoppensteadt of the University of peared while hiking in a remote area of southern Utah has been named Dean of the College of Chile. No further information concerning his Natural Science at Michigan State University. whereabouts has been discovered and there is Fritz John, Professor Emeritus of the Courant reason to believe that he died while on this trip. Institute of Mathematical Sciences, has been awarded an honorary doctorate by N aturwis­ senschaftliche Gesamtfakultiit der Universitii.t Deaths Heidelberg at a ceremony held May 31, 1986. Paul R. Beesack of Carleton University, Ot­ Donald E. Knuth of Stanford University has tawa, Canada, died on October 16, 1985, at the been awarded an honorary Doctor of Science age of 61. He was a member of the Society for 30 degree by Muhlenberg College at its May 18, years. 1986, commencement exercises. William M. Borgman, Jr., of Redford, Michi­ Kenneth 0. Kqrtanek of Carnegie-Mellon gan, died on March 20, 1986, at the age of 82. He University has been appointed Murray Research was a member of the Society for 51 years. Professor of The Management Sciences in the Charles C. Bramble of McLean, Virginia, Department of Management Sciences at The died on June 26, 1986, at the age of 95. He was a University of Iowa. member of the Society for 67 years. Steven G. Krantz of Pennsylvania State Uni­ Geoffrey J. Butler of the University of Al­ versity has accepted a position of Professor of berta, Alberta, Canada, died on July 13, 1986, at Mathematics at Washington University in St. the age of 42. He was a member of the Society Louis, Missouri. for 15 years. Robert A. Leslie of Agnes Scott College, Decatur, Georgia, has been appointed Chair William F. Eberlein, Professor Emeritus of of the Department of Mathematics at that the University of Rochester, died on June 13, institution. 1986, at the age of 68. He was a member of the .Society for 47 years. Dale W. Lick, President of Georgia Southern College, has been appointed President of the Joseph L. Ercolano of City University of New University of Maine. York, Bernard M. Baruch College, died on May 21, 1986, at the age of 54. He was a member of Cathleen Morawetz, Director of the Courant the Society for 22 years. Institute of Mathematical Sciences, was awarded an honorary Doctor of Science degree by Princeton Hyman Ettlinger, Professor Emeritus of the University. University of Texas, died on June 8, 1986, at the age of 96. He was a member of the Society for 72 Mervin E. Muller, Professor and Chairman years. of the Department of Computer and Information Science at The Ohio State University, has been Istvan Fary of the University of California, named the Robert M. Critchfield Professor in Berkeley, died on November 2, 1984, at the age of Engineering at that institution. 62. He was a member of the Society for 15 years.

849 Sydney H. Gould of Brown University died Thomas G. Room of St. Ives, Australia, died on July 7, 1986, at the age of 77. He was a on April 2, 1986, at the age of 83. He was a member of the Society for· 50 years. (See the member of the Society for 38 years. News and Announcements section in this issue of J. G. Semple, Professor Emeritus of the the Notices). University of London, died on October 23, 1985, Banesh Hoffmann, Professor Emeritus at at the age of 81. He was a member of the Society Queens College of City University of New York, for 45 years. died on August 5, 1986, at the age of 79. He was Hermann Simon, Associate Professor at the a member of the Society for 49 years. University of Miami, died on June 3, 1986, at the Friedrich Huckemann of the Technical Uni­ age of 52. He was a member of the Society for 23 versity of Berlin died on January 24, 1986, at the years. age of 59. He was a member of the Society for 30 Oscar Zariski, Professor Emeritus of Harvard years. University, died on July 4, 1986, at the age of 87. Ralph C. Huffer, former Professor of Mathe­ He was a member of the Society for 58 years and matics and Astronomy at Beloit College, Wiscon­ was the President of the Society in 1969 and 1970. sin, died on April 15, 1986, at the age of 90. He (An article is being prepared for publication in was a member of the Society for 41 years. Notices. The editors hope it will be ready in time Donald E. Kibbey, Professor Emeritus, for­ to appear in the November issue.) mer Department Chairman and former Vice President for Research and Graduate Affairs at Reciprocity Agreement Syracuse University, died on June 30, 1986, at Supplementary the age of 74. He was a member of the Society for 48 years. Dominican Republic Paik Kee Kim of Iowa State University died Sociedad Matematica de Ia Republica Dominicana on July 2, 1986, at the age of 41. He was a Apply to: Secretary, Sociedad Matematica de Ia member of the Society for 15 years. Republica Dominicana, Apartado 797-2, Santo Lillian Lieber of the Galois Institute of Domingo, Dominican Republic. Mathematics and Art, New York, died on July Dues: US $10; payable to Sociedad Matematica 11, 1986, at the age of 100. She was a member of de Ia Republica Dominicana. the Society for 50 years. Privileges: Right to receive Notimat (bimonthly George J. Minty of Indiana University died newsletter) and Revista Matematica Dominicana on August 6, 1986, at the age of 56. He was a (twice a year). member of the Society for 29 years. Officers: P. A. Suarez (President), E. Luna (Vice­ David K. Pickard of Queen's University, President), C. Perez (Treasurer), and D. Castillo Canada, died on July 28, 1986, at the age of 41. (Secretary). He was a member of the Society for 2 years.

semisimple Lie algebras by W. Bohro, LIE ALGEBRAS AND a course on Kac-Moody Lie algebras by RELATED TOPICS I. G. Macdonald, and a course on formal groups by M. Hazewinkel. D. J. Britten, F. W. Lemire, and R. V. Moody, Editors 1980 Mathematics Subject Classifications: 17-xx, 22-xx As the Proceedings of the 1984 Canadian ISBN 0-8218-6009-7, LC 85-26818 Mathematical Society's Summer Seminar, ISSN 0731-1036 392 pages (softcover), February 1986 these papers focus on some recent List price $44, Institutional member $35, advances in the theory of semisimple Individual member $26 Lie algebras and some direct outgrowths To order, please specify CMSAMS/5NA of that theory. Of particular interest are notes for several courses presented at the meeting: an important survey Shipping/Handling: 1st book $2, each add'l $1, $25 max. By air, 1st book $5, each add'l $3, $100 max. article by R. Block and R. Wilson on Prepayment required. Order from AMS, P.O. Box restricted simple Lie algebras, a survey 1571, Annex Station, Providence, RI 02901-1571, or of universal enveloping algebras of call 800-556-7774 to use VISA or MasterCard.

850 Visiting Mathematicians (Supplementary List)

The list of visiting mathematicians includes both foreign mathematicians visiting in the United States and Canada, and Americans visiting abroad. Note that there are two separate lists.

American Mathematicians Visiting Abroad Name and Home Country Host Institution Field of SQecial Interest Period of Visit Shafer, DouglasS. (U.S.A.) Limburgs Universitair Centrum, Smooth Dynamical Systems 9/86- 5/87 Belgium

Visiting Foreign Mathematicians Albano, Albert (Italy) Institute for Advanced Study Algebraic Geometry 9/86- 4/87 Araki, Huzihiro (Japan) University of Rochester Operator Algebras and 3/87 Mathematical Physics Arguedas, Vernor (Costa Institute for Advanced Study Topological Algebras 9/86- 4/87 Rica) Balslev, Erik (Denmark) Institute for Advanced Study SchrOdinger Operator 9/86- 4/87 Blonde!, Corrine (France) University of Iowa Group Representations 8/86- 5/87 Bojanov, Borislav Texas A&M University Approximation Theory 9/86- 5/87 (Bulgaria) Bowditch, Brian (England) Institute for Advanced Study Low-dimensional Topology and 9/86- 4/87 Geometry Castroconde, Miriam E. Texas A&M University Finite Group Theory 9/86- 5/87 (Guatemala) Chari, Vyjayanthi (India) Institute for Advanced Study Infinite-dimensional Lie Algebras 9/86- 4/87 and Representation Chen, Shu-Mei (Taiwan) Memphis State University Statistics 9/86- 8/87 Ciliberto, Ciro (Italy) Brown University Algebraic Geometry 10/86- 6/87 Dwilewicz, Roman (Poland) Texas A&M University Geometry and Nonlinear Analysis 9/86- 5/87 Group Eng!, Heinz (Austria) Univeristy of Cincinnati; Inverse and Ill-posed Problems 2/87 University of Delaware Fathi, Albert (France) Institute for Advanced Study Topology and Dynamical Systems 9/86- 4/87 Francia, Paolo (Italy) Institute for Advanced Study Algebraic Geometry 1/87- 4/87 Fuchs, Peter (Austria) Texas A&M University Algebra Group 9/86- 5/87 Gohoussoub, Nassif (France) Texas A&M University Banach Lattices 9/86- 5/87 Gutierrez, Cristian University of Rochester Harmonic Analysis 9/86- 5/87 (Argentina) Gyarfas, Andras (Hungary) Memphis State University Graph Theory 9/86- 8/87 Harder, Giinter (West Institute for Advanced Study Cohomology, Automorphic Forms 9/86- 4/87 Germany) Hernandez, R. (Spain) Brown University Algebraic Geometry 9/86- 5/87 Hsu, Leetsch C. (People's Texas A&M University Approximation Theory 9/86- 1/87 Republic of China) Huebschmann, Johannes Institute for Advanced Study Algebraic Topology 9/86- 4/87 (West Germany) Jakubczyk, Bronislaw Rutgers University Control Theory 9/86- 6/87 (Poland) Jevtic, Miroljub University of Complex Variables and Harmonic 8/86- 1/87 (Yugoslavia) Wisconsin-Madison Analysis Kawamata, Yujiro (Japan) Institute for Advanced Study Algebra 9/86- 4/87 Kossak, Roman (Poland) City University of New York, Logic, Model Theory 9/86- 6/87 Bernard M. Baruch College Kroo, Andras (Hungary) University of South Florida Approximation Theory 8/86- 12/86 Kupiainen, Antti (Finland) Institute for Advanced Study Mathematical Physics 1/87- 4/87 Lannes, Jean (France) Johns Hopkins University Algebraic Topology 3/87- 5/87 Ledoux, Michel (France) Texas A&M University Probability and Linear Analysis 9/86- 5/87 Lengyel, Tamas (Hungary) Beloit College Computer Science 8/86- 5/87 Lutkebohmert, Werner Institute for Advanced Study Algebraic Geometry 10/86- 1/87 (West Germany) Madsen, lb Henning Institute for Advanced Study Topology 9/86- 4/87 (Denmark)

851 Name and Home Country Host Institution Field of SQecial Interest Period of Visit Muller, Hans-Georg (West University of California, Davis Nonparametric Functional 7/86- 6/87 Germany) Estimation-Biomedical Statistics Murai, Takafumi (Japan) Yale University Classical Analysis 9/86- 5/87 Mycielski, Jan (Poland) University of Hawaii Universal Algebra; Mathematical 1/87- 6/87 Logic Neubauer, Andreas University of Cincinnati Integral Equations and Ill-posed 9/86- 6/87 (Austria) Problems Oertel, Ulrich (West Institute for Advanced Study Low-dimensional Topology 9/86- 4/87 Germany) Ohtake, Koichiro (Japan) University of Iowa Ring Theory 8/86- 5/87 Otal, Jean-Pierre (France) Institute for Advanced Study Topology, Hyperbolic Geometry 9/86- 4/87 Palfy, Peter (Hungary) University of Hawaii Universal Algebra 8/86- 1/87 Palka, Zbigniew (Poland) City University of New York, Graph Theory 9/86- 6/87 Bernard M. Baruch College Pardoux, Etienne (France) Institute for Advanced Study Probability Theory, Stochastic 9/86- 4/87 Processes Parthasarathy, R. (India) Institute for Advanced Study Representations of Semi-simple 9/86- 4/87 Lie Groups and Lie Algebras Pintz, Janos (Hungary) Rutgers University Number Theory 7/86- 6/87 Poguntke, Detlev (West Yale University Harmonic Analysis on Lie Groups 9/86- 12/86 Germany) Prakasa Rao, B.L.S. (India) University of California, Davis Asymptotic Theory-Inference in 7/86- 6/87 Stochastic Processes Rajeswari, Kota N. (India) Texas A&M University Algebra Group 6/86- 8/88 Ramanathan, K. G. (India) Institute for Advanced Study Algebraic Number Theory 9/86- 4/87 Rao, Marepalli B. (United University of Pittsburgh Probability and Statistics 9/86- 5/87 Kingdom) Rosay, Jean-Pierre (France) University of Complex Variables 8/86- 5/87 Wisconsin-Madison Ruppert, Wolfgang (West Brown University Algebraic Geometry 9/86- 5/87 Germany) Sarkosy, Andras (Hungary) City University of New York, Number Theory 9/86- 6/87 Bernard M. Baruch College Shokranian, Salahoddin Institute for Advanced Study Modular Forms 9/86- 4/87 (Iran) Shorey, Tarlok Nath (India) Institute for Advanced Study Number Theory 9/86- 4/87 Straube, Emil (Switzerland) University of Pittsburgh Analysis 9/86- 8/87 Tarantello, Gabriella (Italy) Institute for Advanced Study Nonlinear P.D.E. 9/86- 4/87 Teicher, Mina (Israel) Institute for Advanced Study Algebraic Geometry 9/86- 4/87 Teixidor, Montserrat Brown University Algebraic Geometry 9/86- 5/87 (Spain) Telgarsky, Rastislav University of Texas, El Paso Topology, Game Theory, 9/86- 6/87 (Poland) Programming Languages Tsuji, Tadashi (Japan) Rutgers University Siegel Domains 9/86- 6/87 Tuncel, Selim (Turkey) Institute for Advanced Study Ergodic Theory 9/86- 4/87 Ventura, Belisario A. Texas A&M University Operator Theory 9/86- 8/87 (Guatemala) Wallin, Hans E. (Sweden) University of South Florida Approximation Theory 1/87- 6/87 Wang, Ping (People's University of Maryland, Partial Differential Equations 6/86- 6/87 Republic of China) Baltimore County Wang, Shengwang (People's University of Pittsburgh Functional Analysis 9/86- 6/87 Republic of China) Weissman, lshay (Israel) University of California, Davis Asymptotic Theory of Order 7/86- 6/87 Statistics Whiteman, John (Great Texas A&M University Numerical Analysis 9/86- 1/87 Britain) Wodzicki, Mariusz (Poland) Institute for Advanced Study Global Analysis 9/86- 4/87 Xiao, Gang (People's Institute for Advanced Study Algebraic Geometry 9/86- 4/87 Republic of China) Zabrodsky, Alexander University of Rochester Topology 9/86- 1/87 (Israel) Zippin, Morry (Israel) Texas A&M University Linear Analysis Group 7/86- 1/87

852 Application Deadlines for Grants and Assistantships

Many fellowship programs have deadlines for re­ December 1 ceipt of applications. These deadlines are noted American Philosophical Society [PS] in news items and in the Stipends Section of the * Lady Davis Fellowship Trust [TSA] December Notices. They are listed below for your Lady Davis Visiting Professorships [TSA] convenience, and as a reminder since many of Royal Norwegian Council for Scientific and these deadlines occur before the publication date Industrial Research (Postdoctorate Fellow­ of the special December issue on Assistantships ships) [TSA] and Fellowships. Dates taken from the 1985 special issue have been updated with information Sigma Delta Epsilon, Graduate Women in Sci­ received in preparation for the December 1986 ence (Eloise Gerry Fellowship) [GS] issue. For information about the various pro­ University of California, San Diego (S. E. grams, the reader is referred to the appropriate Warschawski Assistant Professorship) [PS] part of the Stipends Section of the December 1985 Notices as follows: [GS] = Graduate Support Sec­ tion; [PS] =Postdoctoral Support Section; [TSA] December 1 = Travel and Study Abroad Section; [SFN] = AMS Research Fellowships [PS] Study in the U.S. for Foreign Nationals. * Information from the December 1985 issue not yet confirmed for this year. December 15 • Refers to a news item in this issue of Notices. * Los Alamos National Laboratory ( J. Robert Oppenheimer Research Fellowship) [PS] October 1 American Philosophical Society [PS] December 31 John Simon Guggenheim Memorial Foundation Fellowships [PSJ Institute for Advanced Study Memberships [PS] Massachussetts Institute of Technology (C. L. E. Moore Instructorships in Mathematics) [PS] October 15 * University of Wisconsin, Madison (Van Vleck Bunting Institute of Radcliffe College (Science Assistant Professorship in Mathematics) [PS] Scholar Fellowships) [PS]

October 24 January 1 Kennedy Scholarships [SFN] R H Bing Faculty Fellowship [PS] Brown University (Jacob David Tamarkin As­ sistant Professorships) [PS] November 1 Courant Institute (Instructorships in Mathe­ American-Scandinavian Foundation [TSA] matics) [PS] Fannie and John Hertz Foundation Fellowships Courant Institute (Postdoctoral Visiting Mem­ [GS] berships) [PS] North Atlantic Treaty Organization (Postdoc­ Harvard University (Benjamin Peirce Lecture­ toral Fellowships) [TSA] ships) [PS] * Indiana University, Bloomington (V aclav November 14 Hlavaty Research Assistant Professorships) NSF Graduate Fellowships [GS] [PS] NSF Minority Graduate Fellowships [GS] Mathematical Sciences Research Institute [PS] University of California, Los Angeles (Earle November 15 Raymond Hedrick Assistant Professorships in Mathematics) [PS] Kosciuszko Foundation [SFN] University of Chicago (Leonard Eugene Dickson NSF Mathematical Sciences Postdoctoral Re­ Instructorships in Mathematics) [PS] search Fellowships [PS] * Weizmann Institute of Sciences (Feinberg Grad­ uate School Postdoctoral Fellowships) [TSA] January 6 * University of Michigan, Ann Arbor (T. H. November 30 Hildebrandt Research Assistant Professor­ North Atlantic Treaty Organization [TSA] ships) [PS]

853 January 15 February 1 AAAS Science, Engineering and Diplomacy AAAS Summer Fellowship [GS] Fellowships [PS] American Philosophical Society [PS] Dartmouth College (Joh:n Wesley Young Re­ * American Society for Engineering Education search Instructorships)· [PS] (NASA-ASEE Summer Faculty Fellowships) * IBM Thomas J. Watson Research Center (Math­ [PS] ematical Sciences Department Postdoctoral * American Society for Engineering Education and Junior Faculty Research Fellowships) (Navy- and DOE-ASEE Summer Faculty Re­ [PS] search Programs) [PS] * Institute for Mathematics and its Applications * American Society for Engineering Education [PS] (ONR Graduate Fellowship Program( [GS] Kosciuszko Foundation [GS] Sigma Delta Epsilon, Graduate Women in Sci­ * Kosciuszko Foundation (Graduate and Post­ ence (Grants-in-Aid) [GS] graduate Exchange with Poland) [TSA] University of Cincinnati (Charles Phelps Taft National Research Council (Research Associ­ Postdoctoral Fellowships) [PS] ateship Programs) [PS] * Natural Sciences and Engineering Research February 11 Council of Canada (Visiting Fellowships) California State Graduate Fellowships [GS] [TSA] February 15 Rice University (Griffith Conrad Evans Instruc­ torships) [PS] University of California, Irvine (Visiting Irvine * Rutgers University (Hill Assistant Professor­ Lectureship) [PS] ships) [PS] February 28 * Smithsonian Institution (Predoctoral Fellow­ Australian Institute of Nuclear Science and ships) [GS] Engineering (Research Fellowships) [PS] * Smithsonian Institution (Postdoctoral Fellow­ ships) [PS] March 1 * University of Pittsburgh (Andrew Mellon Post­ American Philosophical Society [PS] doctoral Fellowships) [PS] March 15 January 16 * Weizmann Institute of Sciences (Feinberg Grad­ * California Institute of Technology (Harry Bate­ uate School Postdoctoral Fellowships) [TSA] man Research Instructorships) [PS] Fulbright Program (Collaborative Research March 31 Grants) [TSA] Hubert H. Humphrey Doctoral Fellowships [GS] North Atlantic Treaty Organization [TSA] January 17 * National Research Council (Postdoctoral Fel­ April1 lowships for Minorities) [PS] American Philosophical Society [PS] January 19 June 15 Committee on Institutional Cooperation (Mi­ Indo-American Fellowship Program [TSA] norities Fellowships in the Sciences, Mathe­ matics and Engineering) [GS] August 1 American Philosophical Society [PS] January 28 National Center for Atmospheric Research (Ad­ August 15 vanced Study Program) [PS] North Atlantic Treaty Organization [TSA] January 30 August 31 * Centro de lnvestigacion del lPN (Solomon Lef­ Australian Institute of Nuclear Science and schetz Research Instructorships) [TSA] Engineering (Research Fellowships) [PS] January 31 Yale University (Josiah Willard Gibbs Instruc­ torships) [PS]

854 AMS Reports and Communications

Recent Appointments L. Bona (1987}, Hermann Flaschka (1988}, John P. Hempel (1988}, Lawrence E. Payne (1987}, and Louis N. Solomon (1989}. Terms expire June 30. Committee members' terms of office on standing committees expire on December 31 of the year Linda Preiss Rothschild (1989}, and Robert given in parentheses following their names B. Warfield, Jr. (1989} have been appointed to unless otherwise specified. ' the Committee on Summer Institutes and Special Symposia by President Irving Kaplansky. Other George D. Mostow has been appointed by Pres­ members of the committee are Albert Baernstein ident Irving Kaplansky to the Centennial Com­ II, chairman (1987}, Eric M. Friedlander (1988}, mittee. Continuing members of the committee are H. Blaine Lawson, Jr. (1987}, John Wermer Felix E. Browder, Harold M. Edwards, Andrew (1988}. Terms expire on February 28. M. Gleason, and Everett Pitcher, chairman. The Committee on Opportunities in Math­ Chung-Chun Yang has been appointed by ematics for Disadvantaged Groups is now a President Irving Kaplansky to the Committee joint committee, AMS-AAAS-MAA Committee on Translation from the Chinese. Continuing on Opportunities in Mathematics for Disadvan­ members of the committee are Sun-Yung Alice taged Groups. Members of the committee are Chang, Tsit-Yuen Lam, chairman, and Tai-Ping Manuel P. Berriozabal (1988}, Sylvia T. Bose­ Liu. man (1989}, William G. Chinn (1987}, James A. Donaldson (1988}, Amassa C. Fauntleroy (1987}, George Lusztig Philip C. Kutzko (1988}, Tepper L. Gill (1987}, Gloria Gilmer, chairman (1988} have been (1988}, and Beth M. Ruskai (1988}, Shirley Malcom (ex officio}, Rogers J. on Research Fellow­ appointed to the Committee Newman (1989}, Argelia Velez-Rodriguez, consul­ ships by President Irving Kaplansky. Continuing tant, Clarence E. Stephens (1989}, and Harold J. the committee are J. William Helton members of Stolberg (1987}. (1987}, Stephen Lichtenbaum (1987}, Kenneth C. Millett, chairman (1987}, and Nancy K. Stanton Ingram Olkin (1988} has been appointed to (1988}. Terms expire June 30. the AMS-IMS-SIAM Committee on Joint Summer Research Conferences in the Mathematical Sci­ (1989}, Frank A. Frederick J. Almgren, Jr. ences by President Paul Meier (IMS). Continuing Raymond (1989} and Richard P. Stanley (1989} members of the committee are Ronald L. Graham by President Irving Kaplan­ have been appointed (1987}, John R. Martin (1987}, Tilla Klotz Milnor sky to the Committee on Steele Prizes. Richard (1987}, and Evelyn Nelson (1987}. Terms expire appointed chairman. W. Beals (1987} has been on June 30. Continuing members of the committee are Jerry

CONTEMPORARY MATHEMATICS terns in two complex variables, while Delanghe and Sommen, Brackx and Pincket, and Lounesto investi­ Plane Ellipticity and Related Problems gate hypercomplex function theory in Rn, that is the class of homogenic functions having values in a Robert P. Gilbert, Editor Clifford algebra. In this collection of papers concepts associated The remaining talks comprising this special meet­ with plane-ellipticity are extended in several ways. ing cannot be categorized as falling into a general For example, the investigations of Begehr and Gilbert, group, but rather explore isolated, albeit important, Begehr and Hsiao, Hile and Snyder treat systems of topics associated w~th ellipticity. elliptic partial differential equations in the plane which resemble in some sense the Cauchy-Riemann Contemporary Mathematics Volume 11, viii + 245 pages (soft cover) equations. Their point of view is to seek general List price $22, institutional member $18, representation formulas and to use these in some individual member $13 cases to solve boundary value problems. Continuing ISBN 0-8218-5012-1; LC 82·11562 with the theme of generalizing the Cauchy-Riemann Publication date: September 1982 equations, Buchanan treats the Bers-Vekua type sys- To order, please specify CONM/11 NA

Shippin~/Handl~ng: 1st book $2, each add'l $1, max. $25; by air, 1st book $5, each add'l $3, max. $100 Prepayme~t IS requ1red. Order from American Mathematical Society, P.O. Box 1571, Annex Station, Providence, Rl 02901-1571, or call toll free 800-556-7774 to charge with Visa or MasterCard.

855 Classified Advertisements

SUGGESTED USES for classified advertising are books or lecture notes for sale, books being sought, positions available, exchange or rental of houses, and typing services. THE RATE IS $.65 per word with a minimum of $30.00. The same ad in 6 consecutive issues is $3.50 per word. Type will be set solid unless centering and spacing are requested. A centered line of any length or the equivalent in white space is $6.00. A word is defined as a group of characters with space at each end. Prepayment is required of individuals but not of institutions. For an additional $10.00 charge, announcements can be placed anonymously. Correspondence will be forwarded. DEADLINES are listed on the inside front cover. U. S. LAWS PROHIBIT discrimination in employment on the basis of color. age, sex, race, religion or national origin. "Positions Available" advertisements from institutions outside the U. S. cannot be published unless they are accompanied by a statement that the institution does not discriminate on these grounds whether or not it is subject to U. S. laws. Details and specific wording may be found following the Classified Advertisements in the January and August issues of the Notices. SITUATIONS WANTED ADVERTISEMENTS from involuntarily unemployed mathematicians are accepted under certain conditions for free publication. Call toll-free 800-556-7774 and speak to Wahlene Siconio for further information. SEND AD AND CHECK TO: Advertising Department, AMS, P. 0. Box 6248, Providence. Rhode Island 02940. Individuals are requested to pay in advance, institutions are not required to do so.

POSITIONS AVAILABLE University of California, Santa Barbara Department of Mathematics THE UNIVERSITY OF ALABAMA DEPARTMENT OF MATHEMATICS Applications are invited for the KY FAN ASSISTANT PROFESSORSHIP. The Ky Fan assistant professorship As part of ongoing development in applied mathemat­ is a special two-year non-renewable position which car­ ics. we have received funding for a new position in applied ries a research stipend. Appointment is effective July 1, mathematics. Salary will be negotiable, up to a maxi­ 1987. Candidates must possess a Ph.D by September mum of $35,000 and rank will depend upon qualifications. 1987. Selection will be based primarily on demonstrated We invite applications from experienced mathematicians research achievement. Teaching experience is desirable. in applied mathematics with excellent records in research Teaching load will consist of four quarter courses per year. and teaching. We are looking for someone who will help To apply send vita and publication list, and arrange to us develop our research group and our graduate pro­ have 3 letters of recommendation sent to: Faculty Search grams in applied mathematics. Applications or inquiries Committee, Department of Mathematics, University of should be sent to: Alan Hopenwasser, Chairman, Depart­ California, Santa Barbara. CA 93106. All applications ment of Mathematics, P. 0. Box 1416, University, AL received by January 9, 1987 will be given thorough con­ 35486. THE UNIVERSITY IS AN EQUAL OPPORTU­ sideration. NITY /AFFIRMATIVE ACTION EMPLOYER UCSB is an equal opportunity/affirmative action em­ ployer. The Department of Mathematics and Statistics at McGill University wishes to sponsor a strong candidate for Department of Mathematics Southern Methodist Uni­ the Natural Sciences and Engineering Research Council of versity Canada (NSERC) 1986-87 University Research Fellowship Applications are invited for a Professor and two tenure­ Competition. These Fellowships are five year research track Assistant Professors beginning September 1987. positions (with a review in the third year). in the nature of Candidates for the senior position should be distinguished Research Assistant Professorships, and carry a teaching scholars providing leadership in one or more areas of load of at most one course throughout the academic year. research in applied mathematics. A successful grant Applicants should have shown some substantial research record and the supervision of doctoral dissertations are ability beyond their Doctoral thesis. They should be desirable. Candidates for the junior positions should have Canadian Citizens or landed immigrants by October 17, an outstanding research record or superior potential. A 1986. commitment to excellence in teaching is expected of all Interested candidates should send their curricula vitae candidates. to: The department has ten active applied mathematicians Professor M. Herschorn, Chairman doing research in areas such as mathematical modeling, Department of Mathematics and Statistics nonlinear wave phenomena. numerical analysis (differential McGill University equations and optimization), and scientific computation. 805 Sherbrooke Street West Recently Lawrence Shampine has been appointed the first Montreal. Quebec, Canada holder of the Betty Clements Chair in Applied Mathemat­ H3A 2K6 ics. They should arrange for at least two letters of reference Applicants should send a vita and three letters of ref­ from competent referees to be sent directly to the same erence (for the senior position. names only) to Richard address. All documentation should reach the department Haberman, Chairman, Department of Mathematics. by September 12, 1986. The department will make Southern Methodist University, Dallas, Texas 75275. or its recommendations to NSERC early in October 1986. call (214) 692-2506. NSERC will announce its decision in March 1987. The university is an Equal Opportunity/Affirmative Action/Title IX employer.

856 POSITIONS AVAILABLE THE CHINESE UIV:I/ERSITY OF HONG KONG invites applications for the t>OSt of Lecturer in Mathematics (two vacancies) tenable from August 1. 1987. Applicants LAWRENCE UNIVERSITY should possess a Ph.D. degree in Mathematics. In ad­ DEPARTMENT OF MATHEMATICS dition to participation in the activities of the Board of Tenure-track position at the assistant professor level Studies in Mathematics. the appointees are expected starting fall 1987. The Department has slight preference to undertake research in their field of specialization for candidates specializing in geometry /topology of proba­ and be responsible for teaching at the undergraduate bility. but will consider especially strong candidates in any and postgraduate level. Annual salary: HK$176,880- area. Lawrence is a liberal arts college with a national rep­ 200.760 by 2 increments BAR 212.700-295.680 by 7 in­ utation. small classes. and excellent students. Teaching crements. (Exchange rate approximately: USSl=HK$7.8. load two courses each ten week term. Salary competitive. £1=HK$11.8). Starting salary will depend on qualifi­ Send resume. transcripts. and three or four supporting cations and experience. Conditions of Service: Benefits letters to Bruce Pourciau. Chair. Department of Mathe­ include long leave with pay, annual :eave. sick leave. su­ matics. Lawrence University, Appleton. WI 54912. These perannuation (University 15%. appointee 5%). education letters should provide specific evidence on the candidate's allowance for children. housing allowance for those whose potential for outstanding undergraduate teaching and con­ annual salary is HK$188.820 or above. and for appointees tinued research. Deadline February 1. Equal Opportunity on overseas terms. passage benefits for themselves and Employer. their dependents as well. Application Procedure: Ap­ plications should be made out in duplicate. giving full particulars. experience and the names and addresses of UNIVERSITY OF CALIFORNIA, IRVINE 3 persons to whom reference may be made, and sent DEPARTMENT OF MATHEMATICS together with copies of certificates/diplomas/testimonials IRVINE, CALIFORNIA 92717 and recent publications to the Personnel Section. The Chi­ Two faculty positions at the level of Assistant Profes­ nese University of Hong Kong. Shatin. N. T .. Hong Kong sor in Applied Mathematics. available beginning academic not later than December 1. 1986. Please quote reference year 1987-88. A Ph. D. degree. publications. and evi­ number 41/509/2/86 and mark 'Recruitment' on cover. dence of active interest in quality teaching are required. Examples of preferred research areas: partial differen­ tial equations. nonlinear phenomena. applied functional DEPARTMENT OF MATHEMATICS & analysis. and numerical analysis. Send applications. a COMPUTER SCIENCE curriculum vitae. and the names of three or more ref­ RUTGERS UNIVERSITY AT NEWARK erences to Professor Martin Schechter. Department of The Department of Mathematics and Computer Science Mathematics. University of California, Irvine. CA 92717. anticipates several tenure track assistant and associate An Affirmative Action/Equal Opportunity Employer. professorships, as well as one one-year visiting research lecturer at the rank of associate or full professor to begin September 1987. Candidates should exhibit strong re­ Department of Mathematics search accomplishments or potential. Salary and teaching VANDERBILT UNIVERSITY load are negotiable. CENTENNIAL PROFESSORSHIP Applicants from all fields are invited. Areas of research in computer-related mathematics interest in the department include number theory, repre­ beginning Fall. 1987 sentation theory and automorphic forms. combinatorics This position will become a named, endowed. chair. and logic, topology, and low dimensional topology and It is intended for a person of distinction whose primary Teichmiiller theory. research involves actual computing. Candidates should send a resume and have three refer- Have curriculum vitae and letters of recommendation ences write to: sent to Jane Gilman. Chair Professor R. R. Goldberg, Chairman Department of Mathematics & Computer Science Mathematics Department Rutgers - The State University Vanderbilt University Newark. New Jersey 07102 Nashville. TN 37235 The closing date for applications is January 15. 1987. Inquiries are welcome. However. late applications will be accepted until the posi­ VANDERBILT UNIVERSITY IS AN EQUAL OPPOR­ tion is filled. Rutgers University is an equal opportunity. TUNITY AFFIRMATIVE ACTION EMPLOYER. affirmative action employer.

TEXAS A&M UNIVERSITY The Ohio State University Department of Mathematics Applications are invited for tenure-track positions in· Mathematics starting September 1. 1986. ca·ndidates The Department of Mathematics of The Ohio State must have a Ph.D. in Mathematics and salaries are com­ University hopes to fill several positions. both visiting and petitive. We are especially interested in applicants in permanent. effective Autumn Quarter 1987. Candidates algebra. applied mathematics. and non-linear analysis. in all areas of applied and pure mathematics are invited There is a possibility of appointments at Professor. Asso­ to apply. Significant research accomplishments or excep­ ciate Professor. and Assistant Professor ranks. Visiting tional research promise. and evidence of good teaching positions are also expected. Send vita and arrange for ability, will be expected of successful applicants. three reference letters to be sent to Please send credentials and have letters of recommen­ dation sent to Professor Joseph Ferrar. Department of Dr. H. E. Lacey. Head Mathematics. The Ohio State University, 231 W. 18th Department of Mathematics Avenue. Columbus. Ohio 43210. Reviews of resumes will Texas A&M University begin immediately. College Station. TX 77843-3368 The Ohio State University is an Equal Opportu­ Texas A&M is an equal opportunity employer. nityI Affirmative Action Employer.

857 POSITIONS AVAILABLE MATHEMATICAL SCIENCE DEPARTMENT CHAIR DEPARTMENT OF MATHEMATICS & The Mathematical Sciences Department at Worcester COMPUTER SCIENCE Polytechnic Institute seeks a department chair to lead its RUTGERS UNIVERSITY AT NEWARK faculty of 23. Active areas of faculty research include PROFESSOR OF MATHEMATICS classical applied mathematics. statistics-applied probabil­ ity. applications of discrete mathematics. alg~bra. and The Department of Mathematics and Computer Science graph theory. The new department chair will be asked anticipates an opening at the Rank of Professor beginning to guide the department through a period of growth and Fall 1987. Candidates should exhibit strong research development that will see the addition of several new Salary and teaching load are negotiable. accomplishments. faculty positions. a strengthening of the department's re­ Applicants from all fields are invited. Areas of research search and professional activities. and the introduction of interest in the department include number theory. repre­ a graduate program. In addition the person selected must sentation theory and automorphic forms. con1binatorics be committed to maintaining high quality teaching in the logic. topology. and low dimensional topology and and department's project-oriented undergraduate program and Teichmiiller theory. in service courses. Candidates should send a resume and the names of The nation's third oldest engineering college. WPI is three references to: located in Worcester. Massachusetts. It offers degrees Jane Gilman. Chair through the Ph.D. with an undergraduate enrollment of Department of Mathematics & Computer Science 2.500. full time faculty of 200. and a graduate and Rutgers - The State University continuing education enrollment of 1.200. Newark. New Jersey 07102 Nominations and applications should be directed to The closing date for applications is January 15. 1987. D. W. Woods. Mathematical Sciences Department Chair However. late applications will be accepted until the posi­ Search Committee. Department of SS&PS. Worcester tion is filled. Rutgers University is an equal opportunity. Polytechnic Institute. Worcester. MA 01609. affirmative action employer. Worcester Polytechnic Institute is an AA/EOE.

AUBURN UNIVERSITY The Ohio State University The Division of Mathematics at Auburn University CHAIR IN NUMERICAL ANALYSIS AND invites applications and nominations for the position of SCIENTIFIC COMPUTATION Coordinator of Undergraduate Mathematics. whose princi­ The Department of Mathematics has been awarded a pal concern will be freshman and sophomore mathematics Chair in Numerical Analysis and Scientific Computation. service courses. The Coordinator will be responsible for Applications are invited from individuals with outstanding scheduling. teaching assignments. registration of students. credentials in any area of applied mathematics whose oversight of curriculum revisions and textbook selections. research activities make significant use of large scale administration of course and teaching evaluations. and su­ computing. The appointee will hold the academic rank of pervision and evaluation of Graduate Teaching Assistants. Professor of Mathematics and will be expected to provide Ph.D in mathematics and significant undergraduate the academic leadership in the developing field of scientific teaching experience required. Academic rank. eligibility computation at Ohio State. for tenure. and salary commensurate with qualifications. Individuals interested in this position should contact The Division has approximately 70 faculty members and Joseph Ferrar. Chairman 40 Graduate Teaching Assistants (in M.S. and Ph.D. pro­ Department of Mathematics grams) in two departments of mathematics. Enrollment The Ohio State University in about 475 sections of lower division service courses is 231 W. 18th Avenue approximately 13.000 students annually. The Coordinator Columbus. Ohio 43210 will hold a 12 month position and be responsible to the Telephone: 614/422-7173 heads of the two departments. The Ohio State University is an Equal Opportu­ Auburn University. located in Auburn. Alabama. is a nity/ Affirmative Action Employer. state land-grant university enrolling more than 19.000 students. The city of Auburn is a picturesque university community located about 120 miles southwest of Atlanta UCLA DEPARTMENT OF MATHEMATICS in a region of farms and woodlands. TEMPORARY POSITIONS Women and minorities are encouraged to apply. Send Two E. R. Hedrick Assistant Professors. Applicants nominations. or applications including resume and names must show strong promise in research and must have of three references to Robert E. Kribel. Acting Dean. received the Ph.D. during the past three or four years {but College of Sciences and Mathematics. Auburn University. may be of any age): no restrictions as to field; salary AL 36849. $35.800. Three year appointment: research supplement AUBURN UNIVERSITY IS AN EQUAL OPPORTU­ of $3.978 first summer. Teaching load: Four quarter NITY. AFFIRMATIVE ACTION EMPLOYER courses per year. which may include one advanced course in candidate's field. Deadline for applications is January The Ohio State University 1. 1987. Depanment of Mathematics Also. subject to administrative approval. a few adjunct assistant professorships; two year appointment; strong Research Instructorships in Mathematics research and teaching background; no restriction as to Applications are invited for the position of research field. Salary $30.000-$32.000 for academic year. Teaching instructor in mathematics for the academic year 1987- load: Five quarter courses per year. 88. Candidates should hold a Ph.D. (or equivalent) in Also several positions for visitors and lecturers. mathematics and show strong research promise. To apply. write to Yiannis N. Moschovakis. Chair. Please send credentials and have letters of recommen­ Department of Mathematics. University of California. Los dation sent to Professor Joseph Ferrar. Department of Angeles. CA 90024. Mathematics. The Ohio State University. 231 W. 18th UCLA is an equal opportunity /affirmative action em­ Avenue. Columbus. Ohio 43210. The Ohio State Univer­ ployer. sity is an Equal Opportunity/ Affirmative Action Employer.

858 POSITIONS AVAILABLE UCLA DEPARTMENT OF MATHEMATICS REGULAR POSITIONS IN MATHEMATICAL THE UNIVERSITY OF MARYLAND COMPUTER SCIENCE COLLEGE PARK ' One or two positions in mathematical computer sci­ Department of Mathematics ence. Preference Will be g1ven to candidates in analysis New or recent Ph.D.'s with outstanding research po­ of algonthms, codmg theory, computational complexity tential in Applied Mathematics are invited to apply for a and the theory of programming languages. Very strong tenure-track appointment as Assistant Professor in the research and teachmg background required. Positions Mathematics Department with an initial joint appointment m1t1ally budgeted at the assistant professor level. Suffi­ for three years in the Institute for Physical Science and Ciently outstanding candidat~s at higher levels and/or in Technology. The teaching load during the joint appoint­ other f1elds will also be considered. Teaching load: Five quarter courses per year. ment would be one course per semester. Applicants should send a vita with a brief description of current work To apply, write to Yiannis N. Moschovakis. Chair, and arrange for three letters of recommendation to be Department of Mathematics. University of California Los Angeles. CA · sent to Professor N. G. Markley, Chairman. Department 90024. of Mathematics. University of Maryland. College Park. UCLA is an equal opportunity/affirmative action em­ ployer. MD 20742 by February 1. 1987. The University of Maryland is an Equal Opportu­ nity/ Affirmative Action Employer. UCLA DEPARTMENT OF MATHEMATICS REGULAR POSITIONS IN THE UNIVERSITY OF MARYLAND APPLIED/COMPUTATIONAL COLLEGE PARK MATHEMATICS Applications are invited for tenured or tenure-track . Three or four regular positions in applied and computa­ appointments in the Institute for Physical Science and tional mathematiCS. Preference will be given to candidates Technology at the University of Maryland from outstand­ in numerical analysis, mathematical modeling, and scien­ ing research workers in applied mathematics. especially tlflc/engmeenng computmg. Very strong research and in the following fields: asymptotic analysis. computer teachmg background required. Positions initially budgeted graphics. nonlinear dynamics and numerical analysis. The at the ass1stant professor level. Sufficiently outstanding appointments will be joint with another department of candidates at h1gher levels and/or 1n other fields will also the University and carry a one-course teaching load per be considered. Teaching load: Five quarter courses per year. semester. Salaries are negotiable. Applicants should send vita and the names of three. or To apply, write to Yiannis N. Moschovakis. Chair, more, references to Professor J. Yorke, Director. Institute Department of Mathematics. University of California. Los Angeles. CA for Physical Science and Technology, University of Mary­ 90024. land. College Park. MD 20742 by December 1. 1986. All UCLA is an equal opportunity/affirmative action em­ ployer. applications will be treated confidentially. The University of Maryland is an Equal Opportu­ nity/ Affirmative Action Employer. UCLA DEPARTMENT OF MATHEMATICS REGULAR POSITIONS IN PURE Carnegie-Mellon University MATHEMATICS Department of Mathematics Four or five regular positions in pure mathematics. The Department plans to make several tenure-track Spec1f1c f1elds of ~nterest include algebra/number theory, appointments beginning in Fall 1987. Although these ap­ analysiS, d1fferent1al equations. geometry /topology logic pointments are expected to be at the Assistant Professor probability, and statistics. Very strong resear~h and level. we also solicit exceptionally well-qualified applicants teaching background required. Positions initially budgeted for more advanced positions. Areas of particular inter­ at the assistant professor level. Sufficiently outstanding est are numerical analysis. combinatorics/combinatorial candidates at h1gher levels and/or m other fields will also load: Five quarter courses per optimization. and operations research although applicants be considered. Teaching year. with any research interests which strongly intersect those of the Department will be welcomed. Candidates should To apply, write to Yiannis N. Moschovakis, Chair. send a resume to: Appointments Committee. Department Department of Mathematics, University of California. Los Angeles. CA · of Mathematics. Carnegie-Mellon University, Pittsburgh, 90024. PA 15213. Carnegie-Mellon University is an Affirmative UCLA is an equal opportunity /affirmative action em­ ployer. Action/Equal Opportunity Employer.

University of VIrginia UNIVERSITY OF CRETE, GREECE Department of Applied Mathematics DEPARTMENT OF MATHEMATICS Tenure-track assistant professorships and senior posi­ Permanent positions at all levels and areas of Mathe­ will be filled within the next two tions for January, 1987. and later. Preferred areas are matiCS are available and are invited from mathemati­ rartial differential equations. scientific computing, and years. Expressions of interest optimization. Ph.D. required with strong research and Cians holdmg the Ph.D. degree (Greek citizenship required teaching ability. Applicants for senior positions must have by law). Write to Prof. Susanna Papadopoulou. Chair­ established record of research and grant support. Send person. Mathematics Department. University of Crete vita and names of three references to James M. Ortega, lraklion. Greece, including vita and representative research of position will be announced Chairman, Applied Mathematics. Thornton Hall, Univer­ work. The content and level sity of Virginia. Charlottesville, VA 22903. An EO/AA m the near future and will depend upon the availability of needs of the Department. Employer. qualified applicants and the

859 POSITIONS AVAILABLE The Department of Mathematics at Boston University anticipates an opening for an Assistant Professor in Fall 1987. Preference given to applicants in Applied Mathe­ THE GEORGIA INSTITUTE OF TECHNOLOGY matics. Dynamical Systems. Statistics and related fields. Women and minorities are encouraged to apply. Send The School of Ma_thematics expects to have available vita and three letters of reference to: Search Commit­ some visiting and tenure-track positions beginning in tee. Department of Mathematics. Boston University, 111 the Fall of 1987. Priority will be given to applicants Cummington Street. Boston. MA 02215. in statistics. scientific computing (parallel and vector computing. robotics) and differential equations. Excellent accomplishments or potential in research is required. Send Senior level appointment in Mathematical Statistics resume toW. F. Ames. Director. School of Mathematics. anticipated for Fall1987. Record of distinguished achieve­ Georgia Institute of Technology. Atlanta. Georgia 30332. ments in research. commitment to excellence in teaching Georgia Tech. a unit of the University System of Georgia. required. Women. minorities esp. encouraged to apply. is an Equal Opportunity/Affirmative Action Employer. Send nominations and applications to: Search Committee, Department of Mathematics. 111 Cummington Street. Boston University, Boston. MA 02215. Applied Mathematics Illinois Institute of Technology Two tenure-track faculty positions will be available in the Department of Mathematics Department of Mathematics beginning August 1987. University of Alberta Applicants should have a Ph.D. in applied mathematics. Edmonton. Alberta. Canada statistics. numerical analysis. engineering science or a Applications are invited for one tenure-track position at closely related area. and should show evidence of strong the Assistant Professor level starting July 1. 1987. Re­ research potential and teaching ability. quirements are a Ph.D. and proven ability or demonstrated Candidates are especially sought who are interested in potential for research and teaching. Current salary range is solving scientific and/or engineering problems using $31.612-$45.340 (Canadian) per annum. depending upon mathematical tools. U.S. citizenship or resident status is qualifications. Send vitae and arrange for three letters required. The closing date is January 31. 1987. of reference to be sent to: Professor H. I. Freedman. To apply, send a vita and names of three references to Acting Chairman. Department of Mathematics. University Chairman. Department of Mathematics. Illinois Institute of Alberta. Edmonton, Canada, T6G 2G1. The University of Technology, Chicago, IL 60616. of Alberta is an equal opportunity employer. but in accor­ An Equal Opportunity/Affirmative Action Employer. dance with Canadian Immigration requirements, priority will be given to Canadian citizens and permanent residents of Canada. Closing date for applications is November 30. DEPARTMENT HEAD 1986. Please refer to File AMD1 when responding to this advertisement. MATHEMATICS AND COMPUTER SCIENCE

SOUTH DAKOTA SCHOOL OF MINES AND SOUTHERN ILLINOIS UNIVERSITY AT CARBON­ TECHNOLOGY DALE. DEPARTMENT OF MATHEMATICS, CARBON­ Nominations and applications are invited for the posi­ DALE, IL 62901. Applications are invited from qualified tion of Head of the Mathematics and Computer Science candidates for a tenure track position in Discrete Math­ Department. Candidates should have an appropriate ematics at the rank of Assistant or Associate Professor earned doctorate. established leadership and administra­ beginning on August 16, 1987. Ph.D. in mathematics tive potential. and a strong interest in developing an with a strong background in discrete mathematics re­ innovative undergraduate program in Mathematics and quired. Candidates must have demonstrated excellence Computer Science as well as graduate and research pro­ in research or potential for such. Evidence of teaching grams in Computer Science. The anticipated appointment effectiveness is required. Substantial record of published date is August 1987. research required for appointment at the Associate level. Salary will be commensurate with qualifications. Ap­ The closing date is December 15. 1986 or until positions plications with a resume. letter of transmittal indicating are filled. Send letter of application, resume and three philosophy and objectives in education and research. and letters of recommendation to: Discrete Math Position: references should be submitted not later than November cfo Ronald Kirk, Chair: Department of Mathematics: 15. 1986 to: Southern Illinois University; Carbondale, IL 62901. SIU-C Chairman. Search Committee is an Equal Opportunity/Affirmative Action Employer. Dr. G. L. Scofield Vice President University of California, Santa Barbara South Dakota School of Mines and Technology Department of Mathematics 501 E. St. Joe Rapid City, SD 57701 Applications are invited for two tenure track appoint­ ments at the assistant professor level, effective July 1, SDSM&T is an Equal Opportunity/Affirmative Action 1987. Candidates in the area of applied discrete mathe­ Employer. matics are especially sought, but junior candidates in all areas of the mathematical sciences will be given serious Associate or Assistant Professor position in Probability consideration. Outstanding research and teaching accom­ is anticipated for Fall 1987. Demonstrated excellence in plishments and potential will be the primary criteria for research and a strong commitment to teaching at the selection. Ph.D. required by the time of appointment. graduate and undergraduate level required. Candidates Applicants should send vitae and publication list. and with established research records as well as new Ph.Ds arrange to have three letters of recommendation sent to: are encouraged to apply. Women and minorities are Alex Rosenberg, Chairman. Department of Mathematics. especially encouraged to apply. Send vita and three letters South Hall 6607, University of California, Santa Barbara. of reference to: Professor Murad Taqqu, Probability CA 93106 by January 10, 1987. Position. Department of Mathematics. 111 Cummington UCSB is an equal opportunity/affirmative action em­ Street. Boston University. Boston. MA 02215. ployer.

860 POSITIONS AVAILABLE RESEARCH PROGRAMMER Software Options, Incorporated MILLS COLLEGE Software Options. Inc.. is a small Cambridge, Mas­ Department of Mathematics and Computer sachusetts company doing research, prototyping and de­ Science velopment in the areas of computer languages. experi­ Oakland, California 94613 mental compiler design, programming environments. large s~stem development and maintenance, symbolic computa­ Mills College is seeking outstanding candidates for a tion. and programming methodology. We have permanent tenure-track position as Assistant. Associate or Full Pro­ positions at intermediate and senior levels for persons fessor of Mathematics commencing Fall. 1987. Successful with a strong mathematics background (in any area) and candidates must have demonstrated superior teaching and a love for programming. Our firm includes M.S.'s and research abilities and a commitment to become involved Ph.D.'s in both Computer Science and Mathematics. and in a highly innovative and energetic department. Rank offers industry-competitive salary and benefits in an inti­ and salary will depend on experience and qualifications. mate. academic work environment. Send resume. copies The _i~itial ~ontract will be for three years. subject to final of transcripts. and a list of references to the attention of ad~m1strat1ve approval. Mills College is an Affirmative Chip Morris. Software Options. 22 Hilliard St .. Cambridge. Action/Equal Opportunity Employer. MA 02138. Send vita and direct three letters of reference to: Professor Diane McEntyre Mathematics Search Committee UNIVERSITY OF WISCONSIN-MADISON Mills College Department of Mathematics Oakland. CA 94613 The Department of Mathematics at the University of Wisconsin-Madison solicits applications for tenure-track or tenure positions to begin fall1987. We are interested in MATHEMATICS: tenure track, Ph.D. in a mathemati­ candidates of established excellence as researchers as well cal science. assistant professor, start September 1987. as recent recipients of the PhD who exhibit outstanding Stro_n~ com_mi~ment. t~ u_ndergraduate teaching, willing to potential. All candidates should have a strong commit­ part1~1pate m mte~d1sc1plmary general education program. ment to good teaching. Consideration of established contmued professional development. Interviews begin by candidates will begin November 24. 1986. Consideration February 1987. Send application. resume, three reference of recent PhD recipients will begin December 31. Ap­ letters. complete transcripts of college work to Prof. Wal­ plications will continue to be accepted until all positions ter Walker. Eckerd College, St. Petersburg Florida 33733 EOE. . . are filled: however. only applications received by these dates along with all supporting materials are assured full consideration. Department of Mathematics Application forms are available from the Tenure Track University of Kansas Screening Committee. Department of Mathematics. 223 Van Vleck Hall. 480 Lincoln Drive. Madison. WI 53706. Anticipate some instructorships beginning fall semester The University of Wisconsin-Madison is an AA/EOE. 1987. which are normally renewable for second and third year. Salary to be determined. Research interests should be in areas closely related to those of current staff. DREXEL UNIVERSITY Ph.D. or dissertation accepted with only formalities to Head, Department of Mathematics be completed. Send detailed resume and dissertation and Computer Science abstract: arrange for three letters of reference to be sent Candidates must have a strong research record and a directly to C. J. Himmelberg, Chairman. Department of demonstrated commitment to applied mathematics. prob­ Mathematics. University of Kansas. Lawrence. KS 66045- ability and statistics, and computer science. Send vita and 2142. Deadline date: December 1. 1986. then monthly names of references to Dr. Jet Wimp, Search Commit­ until August 1987. 1. tee. Department of Mathematics and Computer Science. The University of Kansas is an Affirmative Action/ Drexel University, Philadelphia. Pennsylvania 19104. Equal Opportunity Employer. Applications from women and minorities are actively solicited. University of California, Riverside Drexel University is an AA/EOE. Applications are invited for a tenure-track or tenure position in Computer Science beginning Fall 1987. Candi­ Department of Mathematics dates must have demonstrated excellence in research and University of Kansas te~ching. . Research ~pecialties in all areas of Computer Applications are invited for tenure-track and temporary Sc1ence w1ll be cons1dered but we are particularly inter­ positions at all levels. commencing August 16. 1987 or ested in research areas in Computer Systems or Computer as negotiated. Field is unrestricted but preference will Methodology and Applications. The position is open as · be given to algebra and numerical analysis, and to areas to the level of appointment. meshing well with the department's needs. Require Ph.D Applicants should send a curriculum vitae and see that or Ph.D. dissertation accepted with only formalities to be at least three letters of recommendation are sent to: completed. · Professor Theodore J. Barth, Chair Application. detailed resume with description of re­ Computer Science Search Committee search. and three recommendation letters shoud be sent Department of Mathematics and Computer Science to C. J. Himmelberg. Chairman. Department of Mathe­ University of California matics. University of Kansas. Lawrence. KS 66045-2142. Riverside. CA 92521 Deadlines: December 1. 1986 for first consideration. University of California. Riverside. is an Affirmative then monthly until August 1. 1987. Action/Equal Opportunity Employer. The University of Kansas is an AA/EOE.

861 POSITIONS AVAILABLE Mathematics library. Request list. Bert Ross. Univer­ sity of New Haven. West Haven, CT 06516. The University of Wyoming Head-Department of Mathematics Proceedings of the Nineteenth The Department of Mathematics at the University of Nordic Congress of Mathematicians, Wyoming invites applications for the position of Depart­ University of Iceland, August 13-17,1984 ment Head. The University is the sole four-year institution Detailed advertisement with list of contents in the of higher education in the state of Wyoming with an en­ January 1986 Notices. p. 181. Send order and pay­ rollment of 10.000 students. The mathematics program ment ($20) to: Icelandic Mathematical Society, Science offers degrees in mathematics, applied mathematics. and Institute. Dunhaga 3. IS-107 Reykjavik. Iceland. several joint-degree options at the bachelor. master and doctoral levels. The department has a growing major research component in applied mathematics including a A special textbook by Gilbert Strang petroleum research institute with funding from major in­ INTRODUCTION TO dustrial supporters. Other active research areas in the APPLIED MATHEMATICS department include numerical analysis, partial differen­ tial equations. functional analysis. optimization theory, (750 pages. $36) dynamical systems, rigidity theory, and combinatorics. Candidates should have a strong research record com­ Please order directly from Wellesley-Cambridge Press, patible with department interests and a commitment to Box 157. Wellesley, MA 02181 617-235-9537 excellence in instruction. Applicants should submit a current curriculum vita and the names of at least three suitable references to Announcement Myron P. Ballen, Chair The Books: "The Open Mapping and Closed Graph Search Committee Theorems in Topological Vector Spaces" and "Introduc­ Mathematics Department tion to Topological Groups" by T. Husain are now available University of Wyoming from: The Book-Store. McMaster University, Hamilton, Laramie. WY 82070 Ontario. Canada, L8S 4Kl. Applications will be considered through January 31. 1987. The University of Wyoming is an equal opportunity Technical book specialists employer. New - Old - Out of print Want lists welcome POSITIONS WANTED Top prices paid for your books. MATHEMATICS PROFESSOR. Ph.D. 1969. Long BOOK SCIENTIFIC years of teaching at American universities and abroad. 18 E. 16th Street Research in geometry. Resume upon request. Available New York. NY 10003 immediately. Write to: Notices. October Classifieds. (212) 206-1310 American Mathematical Society, P. 0. Box 6248. Provi­ Toll-free 1-800-621-1220 dence. Rl 02940. Men - Sat. 10am-6pm EST FOR SALE Foundations of Semiological Theory of Numbers H. A. Pogorzelski and W. J. Ryan Volume 1 {1982). General Semiology, 597 pp., $29.95 Volume 2 (1985), Semio. Functions. 695 pp .. $34.95 (Eight Volumes Projected) UMO Press. Univ. of Maine. Orono. ME 04469 (Post free within US if check included)

Ordered Fields and Real Algebraic Geometry D. W. Dubois and T. Recio, Editors

"The book's most significant contribution is its breadth. It encompasses the entire spectrum of present research in ordered fields and real algebraic geometry. The field is very active. and this is the first attempt at such a comprehensive coverage." - W Vogel (Halle) Periodica Mathematica Hungarica. V. 15(2) 1984

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862 UNIVERSITY OF PETROLEUM & MINERALS DHAHRAN- SAUDI ARABIA

MATHEMATICAL SCIENCES DEPARTMENT

Needs

Faculty members for teaching graduate and undergraduate courses. Candi­ dates should have a Ph.D. in Mathematics, teaching experience, and research interests preferably in Applied Mathematics and/or Numerical Analysis.

The University offers attractive salary and benefits which are tax-free.

Send resume with supporting documents to:

University of Petroleum & Minerals Houston Office, Department 519 5718 Westheimer, Suite 1550 Houston, Texas 77057

Chaotic dynamics are a vital area of applied mathematics. the impli· cations of which are increasingly being pursued in a range of biolog· ical and physical sciences. This volume sets out the basic applied mathematical and numerical methods of chaotic dynamics and illus­ trates the wide range of phenomena, inside and outside the labora· tory, that can be treated as chaotic _processes.

Contributors to the volume are K. Aihara. J. Belair, M. Conrad, W. J. Firth, L. Glass, P. Grassberger, A.V. Holden, M. Kot, H. A. Lauwerier, G. Matsumoto, A. Mees, M. A. Muhamad, P. E. Rapp, 0. E. Rossler, W. M. Schaffer, A. Shrier, C. Sparrow, K. Tomita, and A. Wolf.

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863 THE UNIVERSITY OF NEWCASTLE New South Wales, Australia LECTURER DEPARTMENT OF MATHEMATICS, STATISTICS & COMPUTER SCIENCE (Position No. A107/86) The University of Newcastle invites applications for the position of Lecturer in Mathematics within the Department of Mathematics, Statistics and Computer Science. The successful candidate must have a Ph.D degree or equivalent in the fields of Pure or Applied Mathematics. The new lecturer will be expected to participate in teaching of undergraduate courses in Mathematics, supervision of graduate students, and the research activities of the Department. The particular field of Mathematics is open, but Statistics and Computer Science are excluded. At present, the Department of Mathematics, Statistics and Computer Science has 24 full-time academic staff members, including 17 in Mathematics, 2 in Statistics and 5 in Computer Science. Further information can be requested from the Head of the Department by writing or by phoning (049) 68 5657. The successful applicant will be expected to commence duties in early 1987. Commencing salary will be in the range $A27,859- $A36,600 per annum depending upon qualifications and experience. Applications close 15th October, 1986. General Information: Conditions of employment, including method of application and other particulars, may be obtained from the Staff Office, the University of Newcastle, N.S.W. 2308 Australia. Applications (in duplicate) should be addressed to this Unit. Equality of employment opportunity is University policy.

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MATH TEXT THE UNIVERSITY OF VIRGINIA Version 1.85 Applications are invited for: The easiest, most natural (1) Tenure track or senior positions starting technical word processor available. Sept. 1, 1987. Preference will be given to Fully what you see is what you get. applicants in algebraic geometry and POE but all strong applications will be considered. < sin 2nf } d~(~) (2) The Whyburn Research Instructorship, a two I { e~+ln(5)+12 0 year appointment with reduced teaching load

starting Sept. 1 , 1987. Dead I i ne for receipt Confif~urable for IBM compatibles, of applications including three letters of the Z-100, Tandy 2000, and Tl-PC. reference: Feb. 1, 1987. Many supported printers. Applications should be mailed to: Full edition $199 David C. Brydges (Usable Uemo/Mini edition $9) Department of Mathematics Spelling Checker $29 Mathematics·Astronomy Building, Cabell Drive University of Virginia H l J F F s o r t. \>\1 FJ r e Charlottesville, Virginia 22903 410 S. Fraser Street The University of Virginia is an Equal Oppor­ State College, PA 16801 tunity/Affirmative Action Employer. V!SA/MC WELCOME (814)238 0885

864 UNIVERSITY OF ZURICH At the Swiss Federal Institute of Technology in Zurich there will be between now and 1988 three openings for faculty positions in mathe­ matics. Duties of the new professors include teaching at DEPARTMENT of MATHEMATICS undergraduate and graduate levels for students of There is a vacancy from 15.1 0.1988 for a mathematics as well as for engineering students Professor of Numerical and the supervision of graduate students and Analysis Ph.D. candidates in their thesis work. to complement the existing fields of teach­ Applicants should have a higher degree in mathe­ ing and research. It is hoped, if possible, to matics and a record of successful teaching and appoint an expert in both the theoretical research. Preferred research interests are in the and the practical, computer oriented numeri­ following fields: cal analysis of partial differential equations. numerical mathematics, scientific computation Applications, with curriculum vitae and algorithmics, combinatories list of publications, (no algebra, geometry publications, please) should reach analysis, number theory the Dekanat der Philosophischen partial differential equations Fakultat II, Universitat Zurich, Ramistrasse 71, 8006 Zurich, Switzerland, before 31.1 0. Applications with curriculum vitae and list of pub­ 1986. lications should be submitted before October 31, 1986, to the President of the ETH Zurich, Prof. Applications from overseas may be accepted H. Ursprung, ETH-Zentrum CH-8092 Zurich. until the end of December 1986.

41C••.C••.C•·

865 THE UNIVERSITY OF FLORIDA DEPARTMENT OF MATHEMATICS The Department of Mathematics is beginning a five year program to fill over :lO new tenure track faculty positions with mathematicians of exceptional caliber. At least one third of these positions will be senior positions. The Department invites applications for six tenure track positions beginning in Fall, 1987, at least two of which will be filled with senior appointments. Senior candidates chosen by the department will be given a role in the selection of this year's candidates for junior positions. The Department is especially interested this year in attracting candidates in the fields of Differential Geometry (includin~ Dynamical Systems); Mathematical Algorithms and Complexity Theory; Analysis and Probability (particularly Complex Variables and Stochastic Differential Equations). Candidates should have strong research potential, an interest in teaching and at least two years of postdoctoral experience. Rank and salary will be commensurate with experience and achievements. Candidates should forward a resume (including a list of publications) and should arrange for at least three letters of reference to be sent to: Gerard G. Emch, Chairman Department of Mathematics 201 Walker Hall University of Florida Gainesville, Florida 32611 All applications for the academic year 1987-1988 must be complete by January 15, 1987. The University of Florida is an equal opportunity employer.

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TENURED POSITION ll CARNEGIE MELLON UNIVERSITY l HEAD DEPARTMENT OF MATHEMATICS l of California, Berkeley University d f l Department of Mathematics l Nominations' . . and . .applications are JnvJte or 1 Berkeley, CA 94720 ~ the position of Professor and Head of the De- ~ should We invite applications for one or more I partment of Mathematics. Candidates mathemat- positions effective July 1, 1987, at tenure ~ have an international reputation for 1 h' level (Associate or full Professor), subject ) ical research and be able to provide leaders 1p to budgetary approval, in the areas of I to a diverse and growing department of applied algebra, analysis, applied mathematics, and pure mathematics with strong programs on foundations, or geometry and topology. both graduate and undergraduate levels. Demonstrated leadership in research is 1 Nominations and applications should be expected of applicants_ Send by November ~ addressed to: 15, 1986 a curriculum vitae, list of publi­ ~ Professor Victor J. Mizel cations, a few selected reprints or pre­ ~ Head of Search Comittee prints, and the names of three references, l Department of Mathematics to Marc A. Rieffel, Vice Chair for Faculty \ Carnegie Mellon University Affairs, at the above address_ The Univer­ 1 Pittsburgh, PA 15213 sity of California is an Equal Opportunity, l (412) 268-2562/2545_ Affirmative Action Employer. l Carnegie Mellon University is an equal \ opportunity /affirmative action employer.

866 Personal AT LAST: Mathematical Typesetting TEX Capability for PC Users! Inc ... the producers of PC'JEX offer a complete line of software, hardware and fonts for journal quality output. T_EX is a state-of-the-art typesetting program developed by Professor Donald E. Knuth at Stanford University. T_EX is being supported as a standard language for mathematical type­ setting by the American Mathematical Society. The AMS has developed a special package of mathematical typesetting tools for T_EX, called AMS-'!EX, which greatly simplifies the setting of complex mathematical formulas. TEX inputs a standard ASCII computer file, and generates output which can be directed to print on a variety of devices, from dot matrix printers to laser printers to phototypesetters. This entire ad was typeset using PCT_EX and printed on the Corona LP300 Laser Printer. At the recent AMS Conference in New Orleans, we asked mathematicians to suggest com­ plex formulas which we then typeset using T_EX. The following was submitted by Bernard Harris, Department of Statistics, University of Wisconsin.

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868 Reidel ~rn~ athematics Announcing a new Journal K-Theory An Interdisciplinary Journal for the Development, Applica­ tion, and Influence of K-Theory in the Mathematical Sciences K-Theory is a relatively new research area which overlaps many others and has taken on an interdisciplinary role in mathematics. Areas it touches upon include linear algebra, ring theory, number theory, algebraic geometry, algebraic topology, differential geometry, functional analysis, operator algebras, and dynamical systems. It is also finding applica­ tions in applied subjects such as systems control, econometrics, and quantum physics. The journal K-Theory will provide a forum for the presentation, discussion, and critical evaluation of significant, as well as outstanding, mathematical advances in which K­ Theory plays a part. It will be open to all mathematical disciplines and it is expected that this will bring together work having close conceptual and methodological relationships which, hitherto, have been scattered in the literature. K-Theory will also publish survey articles describing the role and influence of K-Theory in mathematics. Editorial board A. Bak, Managing Editor Fakultat fur Mathematik, Universitat Bielefeld, F.R.G. S. Bloch (University of Chicago) A. Connes (College de France) R. K. Dennis (Cornell Universit)l) E. Effros (University of California at Los Angeles) E. Friedlander (Northwestern Universit)/) W.-C. Hsiang (Princeton UniversittJ M. Karoubi (Universite de Paris) G. Kasparov (Soviet Academy of Sciences) D. Quillen (Oxford Universit)/) A. Suslin (Leningrad Universit)/) . B. Williams (University of Notre Dame)

Submissions of original papers in the field are welcomed. Enquiries regarding sub­ mission of papers, information for authors, subscriptions and sample copies should be directed to the publisher. D. Reidel Publishing Company a member of the Kluwer Academic Publishers Group P.O. Box 17, 3300 AA, Dordrecht, the Netherlands 190 Old Derby Street, Hingham MA 02043 U.S.A. Falcon House, Queen Square, Lancaster LA 1, 1 RN, U.K.

869 New titles from Birkhiuser Boston Your sourcefor major mathematical developments

PROGRESS IN MATHEMATICS PROGRESS IN PROBABILITY AND STATISTICS

Manifolds of Nonpositive Curvature Ergodic Theory of Random Transfonnations Werner Ballman, Mikhael Gromov, Yuri Kifer and Viktor Schroeder llardmver/291 pages/S.)1.00II'I'S. \i>L 10 Hardcover/277 pagestsroo/PM. \bl. 61 ISBN: 0-8n>-.~.m-- ISBN: 0-8176-3181-X Dependence in Probability and Statistics A Ia Recherche de Ia 'fupologie Perdue Ernst Eberlein and Murad 'lllqqu I Du cote de chez Roblin Hardcover/.f96 pages/S W SO/I'PS. \bill II Le rote de Casson ISBN: !)-817C>-3:12:i-S Lucien Guillou and Alexis Marin Har-3.~:11-6 Paris 1984-85 Catherine Goldstein, ed. Hardcover/252 pages/S36.0011'M. Vol. 63 Other new titles in mathematics . . . ISBN: 0-817(>-.~.~47-2 Lectures on Complex Approximation Malcev-Admissible Algebras Dieter Gaier Hyo Chul Myung Hardcover/216 pages/512. ()()/ISBN: ()-817(>-)117 -X Hardcover/353 pages/SSS.OOII'M. Vol. 64 ISBN: 0-8176-334S-6 linear Differential Equations and Group Theory from Riemann to Poincare Functional Calculus of Pseudo-Differential Jeremy Gray Boundary Problems Hardcover/490 pages/SSt. SO/ISBN: 0-8n>-3318-9 Gerd Grubb Hardcover/512 pages/S49.00/I'M,Vol. 65 ASource Book in Matroid Theory ISBN: 0-8176-3349-9 Joseph Kung, ed. Hardcover/512 pages/S6S.!lO(tent.)IISBN: ()-817(>-:1173-9

SCIENTISTS OF OUR TIMES The Ancient Tradition of Geometric Problems Wilbur Knorr Discrete Thoughts Hardcover/411 pages/569.00/ISBN: 0-81"(>-.~118-8 ~ys on Mathematics, Science, and Philosophy Mark Kac, Gian-Carlo Rota and Jacob T. Schwartz New Directions in the Philosophy of Hardcover/276 pages/$65.00/ISBN: 0-8176-3285-9 Mathematics Thomas Tymoczko Science, Computers, and People llardcover/320 pages/SS~()()/ISBN: ()-81"(>-3163-1 From the Tree of Mathematics by STANISlAW UIAM Geometric Theory of Foliations Gian-Carlo Rota and Mark Reynolds, eds. Cesar Camacho and A. Lins Neto Hardcover/264 pages/SWOOIISBN: 0-8176-327(>-X llardcover/216pages/S12.00/ISBN: 0-81-6-:11.~')-9

Available at your local scientitic hkstore or order directly from the publisher: Birkhiiuser Boston, Inc. D Distribution Center, P.O. Box 2485, Secaucus, NJ 070l).f c/o Springer-Verlag

870 INSTITUTE FOR MATHEMATICS AND ITS APPLICATIONS UNIVERSITY OF MINNESOTA announces a program on APPLIED COMBINATORICS September 14, 1987 to June 25, 1988 Organizing Committee: Victor Klee (Chairman). Daniel Kleitman, Dijen Ray-Chaudhuri, and Dennis Stanton For the scientific program see the article in the news and announcements section of the October 1986 issue of the AMS Notices. A four-week program on ROBOTICS is scheduled to precede the APPLIED COMBINATORICS program, and a ten-week program on SIGNAL PROCESSING will follow in the Summer of 1988. VISITING MEMBERSHIPS AVAILABLE

POSTDOCTORAL MEMBERSHIPS are available. All requirements for a doctorate should be completed before the membership begins. The following materials must be submitted: • Personal statement of scientific interests, research plans, and reasons for wishing to participate in the Applied Combinatorics program. Applicants who wish to begin with the Robotics program and/or remain for the Signal Processing program should include these topics in their statements. (The personal statement is an essential part of the application.) • Curriculum vitae and a list of publications. • Three letters of recommendation, to be sent directly to the IMA. All material should arrive by January 15, 1987. SENIOR MEMBERSHIPS for periods of four weeks to a year are also available. All member­ ships include logistic support. Preference for financial support will be given to persons with partial support from sabbatical leaves, fellowships, or other stipends. Applications for all memberships are invited from mathematicians, computer scientists, statisticians, and operations researchers, and from those in other scientific areas whose work makes significant use of combinatorial methods. All correspondence should be sent to VISITING MEMBERSHIPS COMMITTEE INSTITUTE FOR MATHEMATICS AND ITS APPLICATIONS UNIVERSITY OF MINNESOTA 514 VINCENT HALL 206 CHURCH ST. S.E. MINNEAPOLIS, MN 55455-0436 (612) 624-6066 The University of Minnesota is an equal opportunity educator and employer, and specifically invites and encourages applications from women and minorities.

IMA PARTICIPATING INSTITUTIONS: INDIANA UNIVERSITY, IOWA STATE UNIVERSITY, MICHI­ GAN STATE UNIVERSITY, NORTHERN ILLINOIS UNIVERSITY, NORTHWESTERN UNIVERSITY, OHIO STATE UNIVERSITY, PURDUE UNIVERSITY, UNIVERSITY OF CHICAGO, UNIVF.:RSITY OF CINCINNATI, UNIVERSITY OF ILLINOIS (CHICAGO), UNIVERSITY OF ILLINOIS (URBANA), UNIVER­ SITY OF IOWA, UNIVERSITY OF MICHIGAN, UNIVERSITY OF MINNESOTA, WAYNE STATE UNIVERSITY.

871 PURE AND APPLIED MATHEMATICS Founded by Paul A. Smith and Samuel Eilenberg Edited by Hyman Bass, Armand Borel, JDrgen K. Moser, and Shing-Tung Yau

Recent volumes FUNDAMENTALS OF THE THEORY OF OPERATOR ALGEBRAS, VOL. II: ADVANCED THEORY Richard V. Kadison and John A. Ringrose 1986. 688 pp. $79.50 casebound ISBN 0-12-393302-1 $39.50 paperback ISBN 0-12-393303-X

NONSTANDARD METHODS IN STOCHASTIC ANALYSIS AND MATHEMATICAL PHYSICS Sergio Albeverio, Jens Erik Fenstad, Raphael HCilegh-Krohn, and Tom Lindstl'(llm 1986. 554 pp. $59.50 casebound ISBN 0-12-048860-4 $34.50 paperback ISBN 0-12-048861-2

COMPLEX COBORDISM AND STABLE HOMOTOPY GROUPS OF SPHERES Douglas C. Ravenel 1986. 440 pp. $90.00 casebound ISBN 0-12-583430-6 $45.00 paperback ISBN 0-12-583431-4

A THEORY OF SETS, 2nd edition Anthony P. Morse 1986. 216 pp. $59.00 casebound ISBN 0-12-507952-4

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AN INTRODUCTION TO NONSTANDARD REAL ANALYSIS Albert E. Hurd and Peter A. Loeb 1985. 244 pp. $35.00 casebound ISBN 0-12-362440-1

Reprinted in paperback NUMBER THEORY Z. I. Borevich and I. A. Shafarevich 1966. 448 pp. $34.95 paperback ISBN 0-12-117851-X

INTRODUCTION TO LIE GROUPS AND LIE ALGEBRAS Arthur A. Sagle and Ralph E. Walde 1973. 371 pp. $39.95 paperback ISBN 0-12-614551-2

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872 An Informal Introduction to Kurt Godel: Collected Works Theoretical Fluid Mechanics Volume 1: Publications 1929-1936 Editor-in-chief: SOLOMON FEFERMAN, JAMES LIGHTHILL, Provost, University College, Stanford University London The initial volume of a comprehensive edition of of the W?rld's leading authorities on theoretical Of~:e Gi:idel's works, this book makes available-for the !IUJd mec.hamcs here of!ers an unusually accessible first time in a single source-all his publications mtroduchon to the subject, blending practical and from 1929 to 1936. The volume begins with an over­ theoretical aspects in a useful and different view of Gi:idel's life and work and features facing approach. English translations for all German originals, exten­ (IMA Monograph Series) sive explanatory and historical notes, and a com­ 1986 300 pp.; 103 illus. 853631-3 $45.00 plete bibliography. $35.00 Rubik's Cubic Compendium 1986 496 pp.: 6 illus. .503964-5 ERNO RUBIK, University ofBudapest; TAMAs VARGA Undergraduate Algebra: Professor of Education, Budapest; GERZSON KERI: A First Course Computer & Automation Institute, Budapest; C.W. NORMAN, Royal Holloway and Bedford New GYO~GY MARX, Eotvos University, Budapest; and of London TAMAS VEKERDY, Psychologist, Budapest College, University second-year mathematics students, this Co-written by the cube's inventor, this book serves as Designed for systematic approach to a comprehensive guide to the Rubik's cube. It opens book offers a modern, highly students with the theory of up a wealth of fascinating mathematics and offers a thoroughly familiarize and particularly with the vast number of new ideas and possibilities to those rings, fields, vector spaces, matrix manipulation. The method is who have solved the cube as well as to those who techniques of and remain puzzled. both strong in its presentation of linear algebra relevant to computer science. (Recreations in Mathematics) 400 pp.; 63 illus. 853249-0 $39.95 1986 200 pp.; 142 illus., 61 color 853202-4 $14.95 1986 Finite Projective Spaces Diophantine Inequalities of Three Dimensions Royal Holloway College R.C. BAKER, J .W.P. HIRSCHFELD, University of Sussex University of London ' This self-contained and highly detailed study consid­ new series of London The first in the prestigious ers projective spaces of three dimensions over a finite Monographs, this book Mathematical Society field, covering both topics which are analogues of of the theory of nonlinear presents an overview geometry over the complex numbers and topics that It concentrates on the Diophantine approximation. arise out of the modern theory of incidence structures. important pr?gre~s made during the last ten years by .such dtstmgUJshed contributors as Vinogradov, (Oxford Mathematical Monographs) Hetlbronn, and Schmidt. 1986 370 pp.; 21 illus. 853536-8 $65.00 (London Mathematical Society Monographs) Groups 1986 288 pp. 853545-7 $65.00 Loop ANDREW PRESSLEY, King's College, London, and GRAEME SEGAL, University of Oxford Discrete Mathematics Drawing together the many branches of mathematics NORMAN L. BIGGS, University of London from which current theory developed, this book gives This book provides a well-structured introduction to a complete and self-contained account of what is ?isc:ete m<;tthematics that covers a wide range of top­ known about loop groups from a geometrical and ICS, mcludmg graph theory, cornbinatorics, number analytical point of view and discusses applications theory, coding theory, combinatorial optimization, to simple particle physics. and abstract algebra. (Oxford Mathematical Monographs) 1986 480 pp.; 125 illus. 853252-0 $29.95 1986 350 pp. 853535-X $69.00 Local Class Field Theory KENKICHIIWASAWA, Princeton University This readable introduction to local class field theory, a theo~ of algebr~ic extensions, covers such topics ~ as abehan extenttons. Almost self-contained the book is accessible to any reader with a basic back­ ground in algebra and topological groups. ------~~~------­ (Oxford Mathematical Monographs) OXFORD UNIVERSITY PRESS 160 pp. 504030-9 $39.95 1986 200 MADISON AVENUE, NEW YORK, N.Y. 10016

873 ANNALS OF QUEUEING SYSTEMS OPERATIONS RESEARCH THEORY AND APPLICATIONS

Editor in chief: Editor-in-Chief: Peter L. Hammer, RUTCOR- Rutgers Center for Operations N.U. Prabhu, 338 Upson Hall, Cornell University, Ithaca, NY Research, Hill Center for the Mathematical Sciences, Rutgers 14853, U.S.A. University, Busch Campus, New Brunswick, NJ 08903 Associate Editors: Editorial Board: U.N. Bhat, Dallas, TX, USA; A.A. Borovkov, Novosibirsk, J. Abadie, Paris; E. Balas, Pittsburg, PA; E.M.L. Beale, USSR; O.J. Boxma, Amsterdam, The Netherlands; P. Bremaud, Milton Keynes; R.E. Burkard, Graz; D. de Werra, Lausanne; Paris, France; J.A. Buzacott, Waterloo, Canada; K.M. Chandy, P.C. Fishburn, Murray Hill, NJ; M. Grigoriadis, Austin, TX, USA; B.T. Doshi, Holmdel, NJ, USA; P. Franken, New Brunswick; P. Hansen, Mons; M. Iri, Tokyo; P. Kall, Berlin, GDR; C.M. Harris, Charlottesville, VA, USA; J. Hunter, ZUrich; B. Korte, Bonn; W.F. Lucas, Claremont, CA; Auckland, New Zealand; N.K. Jaiswal, New Delhi, India; H. Miiller-Merbach, Kaiserslautern; G.L. Nemhauser, Ithaca, T. Rolski, Wroclow, Poland; M. Rubinovitch, Haifa, Iarael; NY; A. Prekopa, Budapest; A.H.G. Rinnooy Kan, Rotterdam; R. Schassberger, Berlin, W.-Germany; R.F. Serfow, Atlanta, F.S. Roberts, New Brunswick, NJ; S.M. Robinson, Madison, GA, USA; S. Stidham, Jr., Raleigh, NC, USA; WI; B. Rosen, Minneapolis, MN; M. Segal, Holmdel, NJ; J.G.C. Templeton, Toronto, Canada; J. Walrand, Berkeley, CA, B. Simeone, Rome; R. Wets, Lexington, KY. USA; W. Whitt, Holmdel, NJ, USA. The Annals of Operations Research will publish a series The journal will publish papers on the theory and of volumes dedicated to the presentation of the current applications of queueing systems. It will cover a broad level and of the main trends of the development of spectrum of topics of current research on queueing specific areas of the field. systems, crossing existing boundaries of academic Each volume will consist of original manuscripts, survey disciplines. Particularly welcome are papers on computer articles, selected and tested computer programs, etc. performance, communication systems, flexible The Annals of Operations Research hope to play an manufacturing systems and transportation. Summaries of active role in the publication of well-refereed Conference computational work and reports of case studies will be Proceedings or selected papers thereof and in publishing encouraged. There will be review articles and a section volumes of contributed papers in well defined areas of on short communications. OR from highly theoretical to the algorithmic and also to The Journal will be exacting and scholarly in its the very applied. standards. Every effort will be made to promote Every volume will have one or more Guest Editors who innovation, vitality and communication between will be personnally responsible for the collection of disciplines. papers to appear in that volume, for the refereeing Our objective is to make QUESTA the definitive Journal process and for the time schedule. on queueing systems, deserving of the loyalty of all researchers in theory and applications of these systems. All papers will be subject to peer refereeing and there will be no page charge. Forthcoming papers:

Available as journal edition and as separate hardbound J .A. Buzacott and David D. Yao: On queueing network models of flexible maPufacturing systems volumes. Bharat T. Doshi: Queueing systems with vacations - a survey Publication schedule: 1984: vol. 1 & 2. 1985: vol. 3 & 4. Edward G. Coffman, Jr. and Micha Hoffri: Queueing analysis of 1986: vol. 5, 6 and 7. secondary storage devices U. Narayan Bhat and S. Subba Rao: Statistical analysis of ISSN 0254-5330 queueing systems Edward H. Lipper and Bhaskar Sengupta: Assembly-like queues Separate volumes, hardbound with finite capacity: bounds, asymptotics and approximations queues vol. 1: Archetti & Maffioli, Stochastics and Optimization. 1984. U. Narayan Bhat: Finite capacity assembly-like birth and vol. 2: Thompson & Thrall, Normative Analysis in Policy Bhanu G. Bhaskaran: Almost sure convergence of systems Decisions, Public and Private. 1985. death processes with applications to M/M/s queueing vol. 3: Stecke & Suri, Flexible Manufacturing Systems: Operations Research Models and Applications. 1985. Volume I, 4 issues, 1986, 400 pages, ISSN 0256-0130, for vol. 4-5: Monma, Algorithms and Software for Optimization. Sfr. 304.00/US$ 112.50 incl. postage. Special price postage. 1985. ORSA/TIMS members Sfr. 85.00/US$ 42.50 incl. vol. 6: Osleeb & Ratik, Locational Decisions: Methodology and Applications. 1986. vol. 7: Blazewicz, Cellary, Slowinski, and Weglarz, Scheduling under Resource Constraints - Deterministic Models. 1986. vol. 8: Albin & Harris, Statistical and Computational Problems in Probability Modeling. 1987.

Price per volume including postage: Sfr. 411.--/US$ 152.50 Reduced price for ORSA/TIMS members: Sfr. 85.--/US$ 42.50 Please request

Journal subcription, paperbound vol. S-7, 1986, Sfr. 377.--/US$ 139.50 per vol. incl. postage free sample copies. vol. 1-4, 1984-1985, Sfr. 411,--/US$ 152.50 per vol. incl. postage.

Please request a free sample copy.

k J.C. BALTZER AG, SCIE:\TIFIC Pl'BLISHI:\G COMPA:\\ '-\'H \\ rttstl•inplatz 10, CH-4058 Basel, S\\ itzrrland

874 AMS Books in BIOMATHEMATICS

0 Some mathematical questions in biology: DNA SEQUENCE ANALYSIS Robert M. Miura, Editor This book is based on the 18th Annual Symposium on Some Mathematical Questions in Biology, held in conjunction with the annual AAAS meeting. The papers within, presented by speakers knowledgeable in both biology and mathematics. discuss developments in DNA sequence analysis and emphasize the need for rigorous. efficient computational tools such as biologically relevant definitions of sequence similarity and string matching algorithms. ISBN 0-8218-1167-3, LC 80-646696, ISSN 0075-8523 136 pages, Softcover. April 1986 Individual member $17: Institutional member $22. List price $28 To order, please specify LLSCI/17NA o Some mathematical questions in biology: MUSCLE PHYSIOLOGY Robert M. Miura, Editor This volume contains six papers presented at the 17th Annual Symposium on Some Mathe­ matical Questions in Biology held in conjunction with the annual AAAS meeting. The papers deal with overlapping areas of muscle physiology: cross-bridge dynamics as well as distinctions between striated and cardiac muscles and the control of muscular contractions by action po­ tentials. Focusing on both experimental techniques and theoretical underpinnings, the authors present the recent technological advances that provide an improved database for obtaining a better understanding of the biochemical mechanics and developing better mathematical models. ISBN 0-8218-1166-5. LC 85-28613. ISSN 0075-8523 248 pages. Softcover, April 1986 Individual member $21: Institutional member $28. List price $35 To order. please specify LLSCI/16NA o Some mathematical questions in biology: NEUROBIOLOGY Robert M. Miura, Editor This is an excellent collection of articles on some of the more interesting and timely problems of cellular neurobiology. Some of the topics include: the analysis of models for excitable membranes, neuronal plasticity, and diffusion in the brain cell microenvironment. The volume is based on lectures presented at the 15th Annual Meeting on Mathematical Biology held in January, 1982. ISBN 0-8218-1165-7. LC 82-18418, ISSN 0075-8523 132 pages. Softcover, 1982 · Individual member $12: Institutional member $16, List price $20 To order, please specify LLSCI/15NA

To order Volumes 1-14 in this series, call 800-556-7774

Shipping/Handling: 1st book $2, each PREPAYMENT REQUIRED. Order from: additional $1, maximum $25; by air, 1st book American Mathematical Society $5, each additional $3, maximum $100. PO Box 1571 Annex Station Providence, Rl 02901-1571 or call 800-556-7774 to use VISA or MasterCard.

875 Reviews in K-theory Bruce Magurn, Editor

''This is a very important set of reviews. chosen with an eye toward completeness and comprehensibility. It is a vital research tool for a wide range of mathematicians: algebraists. number theorists. topologists. and algebraic geometers. among others. Professor Magurn has rendered a vital service to the mathematical community by compiling this excellent and timely collection of reviews in the relatively fast devel- oping subject matter of K -theory." - Irving Reiner Unlv. of Illinois, Urbana-Champaign Reviews in K-theory is a collection of about 3.000 reviews of papers and books from Mathematical Reviews on this exciting and relatively new research area. together with a listing of about 1.200 papers in the subject which were published before 1985 but not yet reviewed. The reviews are organized by subject. according to the new classification scheme for K-theory developed by Professor Magurn and his collaborators. This new classification system was introduced in 1986 into Mathematical Reviews as a secondary classification only. Because K-theory is strongly cross-disciplinary. it has been difficult in the past to connect papers with close conceptual relationships because the classification scheme of MR was not well suited to the purpose. With the appearance of this important book and the adoption of the new scheme. these difficulties have been relieved. Under each chapter heading in the book is a description of other chapters and sections where related papers are reviewed; each section has a list of papers elsewhere which have secondary classifications in that section. Many reviews are followed by a "See Also" list of reviews cited in or relevant to the paper at hand. A second list of papers citing the current one. is also .Provided. and Zentralblatt review numbers are given when available. Research mathematicians and fairly advanced students will find the book accessible. and through it will become acquainted with the most current research related to K-theory. Less sophisticated readers will find the many reviews of introductory texts especially useful. Reviews in K-theory 194o-B4 Edited by Bruce Magurn 1985. 811 pages (LC 85-7481; ISBN 0-8218-0088-4} List price $240. Institutional member $192. Individual member $144 To order. please specify REVK/NA

Also available from the AMS:

Reviews In Number Theory 1973-83 Edited by Richard K. Guy 1984. Set of six volumes. 3573 pages (LC 84-11159; ISBN 0-8218-0218-6} May also be ordered individually. List price $675. Institutional member $540. Individual member $405 To order. please specify REVNUM/83NA

Reviews In Ring Theory 194o-1979 Edited by Lance W. Small 1981. Two volumes. 1114 pages (LC 81-10770; ISBN 0-8218-0215-1} List price $250. Institutional member $200. Individual member $150 To order. please specify REVRING/NA

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876 PREREGISTRATION/HOUSING FORM, SAN ANTONIO, TEXAS January 21-24, 1987

MUST BE RECEIVED IN PROVIDENCE NO LATER THAN NOVEMBER 15, 1986 Please complete this form and return it with your payment to Mathematics Meetings Housing Bureau P.O. Box 6887, Providence, Rhode Island 02940- Telephone: 401-272-9500, Ext. 239 DEADLINES: Prere istration: November 15, 1986. Changes Cancellations: Before January 12, 1987, make all changes to or cancellations of hotel reservations with the Mathematics Meetings Housing Bureau in Providence; after that date, changes or cancellations should be made directly with the hotel assigned. Refunds: 50% of preregistration fee(s) for the AMS Short Course or the Joint Meetings will be refunded in Providence on or before January 16, 1987. After this date, there will be no refunds. REGISTRATION FEES Preregistration by mail by 11/15 At Meeting JOINT MATHEMATICS MEETINGS Member of AMS, ASL, MAA, NCTM $59 $77 Nonmember $90 $117 * Student, Unemployed, or Emeritus $16 $21 AMS SHORT COURSE Member/Nonmember $35 $45 * Student or Unemployed $10 $15 EMPLOYMENT REGISTER- Employer fee $75 $100 -Applicant fee $15 $20 - Posting fee for job descriptions for noninterviewing employers $10 $15 (N. B.: A separate form appears in this issue for preregistration for MAA Minicourses.) *All full-time students currently working toward a degree or diploma qualify for the student registration fees, regard­ less of income. The unemployed status refers to any person currently unemployed, actively seeking employment, and who is not a student. It is not intended to include persons who have voluntarily resigned from their latest position. The emeritus status refers to any person who has been a member of the AMS or MAA for twenty years or more and is retired on account of age from his or her latest position.

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877 PREREGISTRA TION/HOU!l ING FORM, SAN ANTONIO, TEXAS HOUSING SECTION: JANUARY 1987 Please rank hotels in order of preference by writing 1, 2, 3, in the spaces at left on form, and by circling the requested room type and rate. If the rate requested is no longer available, you will be assigned a room at another hotel at the next available rate. If not all hotels are ranked, and all rooms have been filled at the ranked hotels, the assignment will be made at an unranked hotel with the next available rate. Rates listed below are subject to 11% hotel/motel occupancy tax. GUARANTEE REQUIREMENTS: $50 by check, VISA, or MasterCard credit cards. No other credit cards will be accepted. PLEASE SUPPLY THIS INFORMATION ON THE REVERSE together with mailing address for confirmation of room reservation,

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San Antonio Marriott (1) $60 $60 $60 $66 $66 $70 $70 $125 -' $300 (Headquarters Hotel)

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DEADLINE: Preregistration deadline for Minicourses: November 15, 1986 (Send form to address above.) CANCELLATIONS: 50% of Minicourse fees will be refunded in Washington for cancellations received on or before January 12, 1987. No refunds will be made after this date. NOTE: All Minicourse registration fees paid to MAA by professional mathematicians are tax deductible. IMPORTANT: Please be sure to send a separate Joint Meetings preregistration form and fee to the Mathematics Meetings Housing Bureau, PO Box 6887, Providence, RI 02940. Preregistration for the Joint Meetings is a requirement in order to participate in the MAA Minicourses.

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No. MINICOURSES: Participants are limited to two Minicourses each.

A microcomputer linear algebra course using LIN-KIT (Howard Anton) 2 Introduction to computer graphics (Joan P. Wyzkoski, Fairfield University)

3 The teaching of applied mathematics (Gilbert Strang, Massachusetts Institute of Technology) 4 Interesting applications of elementary mathematics (JoAnne s. Growney, Bloomsburg University) 5 Discrete mathematics using difference equations (James T. Sandefur, Jr., Georgetown University)

6 Using microcomputer software in teaching calculus (David A. Smith, Benedict College, and David P. Kraines, Duke University)

NOTE: There is no Minicourse i/7.

8 Computer simulation of discrete systems (Zaven A. Karian, Denison University)

9 Recurrence relations (Margaret Barry Cozz~ns, Northeastern University)

10 Integrating history into undergraduate mathematics courses (Judith V. Grabiner, Pitzer College)

11 Teaching mathematical modeling (Maurice D. Weir, Naval Postgraduate School, and Frank R. Giordano, US Military Academy) 12 True BASIC in freshman calculus (James F. Hurley, University of Connecticut)

13 For all practical purposes (Solomon A. Garfunkel, COMAP, Inc.)

14 Applications of discrete mathematics (Fred S. Roberts, Rutgers University)

15 Constructing placement examinations (John W. Kenelly, Clemson University)

First choice: [ Second choice: [

879 MATHEMATICAL SCIENCES COMBINATORIAL EMPLOYMENT REGISTER METHODS IN Instructions for Applicant's Fonn on facing page The form. Applicants' forms submitted for the TOPOLOGY AND Employment Register will be photographically repro­ ALGEBRAIC GEOMETRY duced in the December 1986 issue of Employment Information in the Mathematical Sciences. Resumes of John R. Harper and Richard those attending will be posted at the meeting. Mandelbaum, Editors The forms must be carefully typed using a fresh black ribbon. The best results are obtained with a This collection marks the recent carbon-coated polyethylene film ribbon, but satisfac­ made of resurgence of interest in combinatorial tory results may be obtained using a ribbon is exer­ methods, resulting from their deep and nylon or other woven fabric if suitable care make diverse applications both in topology cised. It is important that the keys be clean and and algebraic geometry. Nearly thirty a sharp, clear impression. Do not erase-it causes Use mathematicians met at the University of smudges which reproduce when photographed. fluid if Rochester in 1982 to survey several of a correcting typewriter or correction tape or only. the areas where combinatorial methods necessary. Submit the original typed version accept­ are proving especially fruitful: topology Copies will not reproduce properly and are not and combinatorial group theory, knot able. Hand lettered fonns will be returned. theory, 3-manifolds, homotopy theory Applicants' forms must be received by the Society and infinite dimensional topology, and by November 15, 1986 in order to appear in the four manifolds and algebraic surfaces. special issue of ElMS, and must be accompanied This material is accessible to advanced by the Preregistration/Housing Form printed in this Meeting graduate students with a general course issue, if attending the meeting. See the in algebraic topology along with some Announeement in this issue for information. Forms be work in combinatorial group theory received past the deadline or not completed will and geometric topology, as well as to returned. established mathematicians with interests The summary strip. Information provided here will in these areas. For both student and be used to prepare a printed list of applicants for dis­ professional mathematicians, the book tribution to employers. Please supply all information provides practical suggestions for research requested, and confine your characters to the boxes directions still to be explored, as well provided. Use the codes below. Circled letters identify as the aesthetic pleasures of seeing the corresponding items on the form and the strip. interplay between algebra and topology @ Specialties which is characteristic of this field. AL = Algebra AN = Analysis BI = Biomathematics BS = Biostatistics In several areas the book contains the CB = Combinatorics CM = Communication first general exposition to be published CN = Control CS = Computer Science on the subject at hand. In topology, for CT = Circuits DE = Differential Equations example, the editors have included M. EC = Economics ED = Mathematical Education Cohen, W. Metzler and K. Sauerman's FA = Functional Analysis FI = Financial Mathematics article on "Collapses of K x I and group FL = Fluid Mechanics GE = Geometry LO = Logic and Metzler's "On the HM = History of Math presentations" ME = Mechanics and related MB = Mathematical Biology Andrews-Curtis-Conjecture MP = Mathematical Physics Montesino MO = Modelling problems." In addition, J. M. MS =Management Science NA = Numerical Analysis has provided summary articles on both NT = Number Theory OR = Operations Research 3- and 4-manifolds. PR = Probability SA = Systems Analysis ST = Statistics TO = Topology Career Objectives 1980 Mathematics Subject Classifications: @ 14Jxx, 20Fxx, 55Pxx, 55Sxx, 57lvlxx, and AR = Academic Research AT = Academic Teaching others. NR = Nonacademic R&D NC = Nonacad. Consulting ISBN 0-8218-5039-3, LC 85-11244 NS = Nonacademic Supervision JSSN 0271-4132 376 pages (softcover), October 1985 @CD Duties List price $32, Institutional member $26, T =Teaching U = Undergraduate $19 Individual member G =Graduate R =Research specify CON!vi/44NA To order, please C = Consulting A = Administration S = Supervision IND = Industry GOV = Government DP = Data Processing Location Shipping/Handling: 1st book $2, each add'! $1, $25 max. By air, 1st book $5, each add'} $3, $100 max. E =East S = South Prepayment required. Order from AMS, P.O. Box C =Central M =Mountain or 1571, Annex Station, Providence, RI 02901-1571, W =West 0 = Outside U.S. I = Indifferent call 800-556-7774 to use VISA or MasterCard. U.S. Citizenship Status C = U.S. Citizen P = Permanent Resident T = Temporarily in U.S. N = Non-U.S. Citizen

880 MATHEMATICAL SCIENCES EMPLOYMENT REGISTER APPLICANT FORM JANUARY 21-23, 1987 SAN ANTONIO, TEXAS

APPLICANT: Name______Mailing address (include zip code)------

@Specialties'------­ @ Career objectives and accomplishments ACADEMIC: D Research, D Teaching NON-ACADEMIC: D Research and Development, D Consulting, D Supervision Near-term career goals______

Significant achievements or projects, including role ______

Honorsandoffires'------­ Other (e.g., paper to be presented at THIS meeting)------

Selected titles of papers, reports, books, patent"------

@Degree Year Institution ------@No. of abstracts, internal report.______------@No. of papers accepteu______------@No. of books and patents.______EMPLOYMENT IDSTORY: Present Previous Previous @Employer ------­ Position @Duties Years ______to. ______to ______to. ______DESffiED POSITION: ______Q) Duties______Q)Available mo. __ fyr. __ Locatiol'------­ Salary____ _ @References (Name and Institution)

@Citizenship------­ @AVAJLABLE FOR INTERVIEWS: (Interviews for Session 4 scheduled on the· basis of employer's request only.) Session 1 D Session 2 D , Session 3 D Session 4 D Thurs. AM 9:30-11:45 Thurs. PM 1:15-5:00 Fri. AM 9:30-11:45 Fri. PM 1:15-5:00 I do not plan to attend the Winter Meeting D

Family Name Firat Name Mailing Addreu SUMMARY STRIP I I I I I I I I I I I I I, I I I I I I I I 1, lr-TITI-rl lr-TI--r-1 -r--1lr-rl--r-1 -,-,1lr-TI--r-1 -,-,11""'1--r-1 ,--,1 I, Addreu (eont'd.) Addreu (cont'd.) State & Zip Code @ Specialties I I I I I I I I I I I I I I I, I I I I I I I I I I I I I I I I, I I I I I I I I I, [I] [I] [I] [I] CIJ, @ Career objectives @M~\~::t@ Yr. @ Institution @ @ ® @Most recent employer [I] [I] [I] [I] CIJ. ITDJ. CIJ. I I I I I I I I I I I, CIJ, CIJ. CIJ. I I I I I I I I I I I. @ Preeent duties (!) Desired duties 0 !,~~?;r~le @ @ Seuions I I I I I I I I I I I, ITDJ, ITDJ, CIJ/CIJ. D. D D D D

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00 EMPLOYMENT INFORMATION IN THE MATHEMATICAL SCIENCES

The American Mathematical Society, the Mathematical Association of America, and the Society for Industrial and Applied Mathematics publish Employment Information in the Mathematical Sciences six times a year. The following resolution was passed on October 25, 1974, by the Council of the American Mathematical Society: "The Council of the AMS adopts principles that all positions in the mathematical sciences shall insofar as practicable be advertised, and that the standard place for the advertisements to appear is the publication Employment Informa­ tion." A similar resolution was subsequently approved by the Board of Governors of the Mathematical Association of America. The November, January, March, May, and August issues contain listings of open positions, information for which has been provided by heads of mathematics departments of colleges and universities in the United States, Canada, and overseas. In addition, these issues contain descriptions of open positions within government, industrial, and other nonacademic areas. The December issue contains resumes of job applicants who will be participating in the Employment Register at the January Annual Meeting. Subscription rates include first-class delivery in North America and airmail delivery elsewhere. The 1987 subscription starts with the November 1986 issue and ends with the August 1987 issue. All subscribers receive all six issues regardless of when the order is received.

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887 Fair Allocation

H. Peyton Young, Editor (Proceedings of Symposia in Applied Mathematics. AMS Short Course Lecture Notes. Volume 33)

This collection of six papers provides a valuable source of material on the real-world problem of allocating objects among competing claimants. The examples given show how mathematics. particularly the axiomatic method. can be applied to give insight into complex social problems. Originally presented as an AMS Short Course. these papers could serve as a suitable text for courses touching on game theory. decision sciences. economics. or quantitative political science. Most of the material is accessible to the mathematically mature undergraduate with a background in advanced calculus and algebra. Each article surveys the recent literature and includes statements and sketches of proofs. as well as unsolved problems which should excite stCJdent curiosity. The articles analyze the question of fair allocation via six examples: the apportionment of political representation. the measurement of income inequality. the allocation of joint costs. the levying of taxes. the design of voting laws. and the framing of auction procedures. In each of these examples fairness has a somewhat different significance. but common axiomatic threads reveal broad underlying principles. Each of the topics is concerned with norms of comparative equity for evaluating allocations or with standards of procedures for effecting,them: it is this focus on normative properties which suggests that a mathematical analysis is appropriate. Though game theory provides a useful tool in many of these allocation problems. the emphasis here is on standards rather than strategy and equity rather than rationality. an approach which aims to more accurately mirror real-world social problems. Contents M. l. Balinski and H. P. Young. The apportionment of representation James E. Foster. Inequality measurement H. Peyton Young. Cost allocation H. Peyton Young. The allocation of debts and taxes Herve Moulin. Fairness and strategy in voting Robert J. Weber. Auctions and competitive bidding

1980 Mathematics Subject Classification: 90-XX ISBN 0094, LC 85-18542 ISSN 0160-7634 xiv + 170 pages, December 1985 Hardcover: Individual member 116, Institutional member S21. List price S26 Softcover: Individual member 113, Institutional member $17, List price S21 Shipping and handling charges must be added To order, please specify PSAPM/33NA (hardcover). PSAPMS/33NA (softcover)

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888 M.C. Escher: Art and Science The Numerical Solution of Proceedings of the Interdisciplinary Congress on Volterra Equations M.C. Escher, Rome, Italy, March 26-28, 1986 By H. Brunner and Edited by H.S.M. Coxeter, M. Emmer, R. Penrose and P.J. van der Houwen M.L. Teuber CWI Monographs, 3 A unique book with over 200 black and white Illustrations This monograph presents the theory and modern numerical and 16 pages of colour illustrations. analysis of Volterra integral and integra-differential equations, including equations with Contents: Escher and Symmetry. Escher, Mathematics and weakly singular kernels. Visual Perception. Escher and Geometry. Escher, Cinema 1986. xvi + 588 pages. Price US $60 00/Dfl 150.00. and Computer Graphics. Escher and the Physical World . ISBN 0-444-70073-0 Escher and Art. Escher and the Humanities. Colour Section. 1986. About 400 pages. Price: US $50.00/Dfl 140.00. Topology, Theory and Applications ISBN 0-444-70011-0 Edited by A. Csaszar Colloquia Mathematica Societatis Janos Bolyai, 41 Analysis on Real and The topics examined in this volume cover many aspects of general topology (various types of topological space, Complex Manifolds generalized topological structures, classes of mappings, etc.) with special emphasis on the application of methods of By R. Narasimhan modern set theory. North-Holland Mathematical Library, 35 1985. 728 pages. Price US $116.00/Dfl 290.00. (formerly part of the series Advanced Studies in ISBN 0-444-87757-6 Pure Mathematics) Distributed in the East European Countries, China, Cuba, This book is meant for people interested in analysis, who Mongolia, North Korea and Vietnam by Kultura, Budapest. have little analytical background. It is supposed that the reader is well acquainted with linear and multilinear algebra as well as set topology. Theory of Algorithms Contents: Chapters: 1. Differentiable Functions Rn 2. Manifolds. 3. Linear Elliptic Differential Operators. Edited by L. Lovasz and E. Szemeredi 2nd Rev. Ed. 1973. 1st reprint. 1985. Colloquia Mathematica Societatis Janos Bolyai, 44 xiv + 246 pages. These papers embrace many topics of theoretical computer Price: US $55.00/Dfl 150.00 science, including algorithmic problems ISBN 0-444-87776-2 on lattices, parallel computing, maze searching , NP hard and NP-complete problems and some new approaches to characterize computational complexity. Large Scale Eigenvalue 1985. 430 pages. Price US $72.00/Dfl 180.00. Problems ISBN 0-444-87760-6 Distributed in the East European Countries, China, Cuba, Proceedings of the IBM Europe Institute Workshop, Mongolia, North Korea and Vietnam by Kultura, Budapest. Oberlech, Austria, July 8-12, 1985 Edited by J. Cullum and R.A. Willoughby North-Holland Mathematics Studies, 127 The papers presented in this volume fall into four principal North-Holland categories: In the U.S.A. and Canada: 1. Novel algorithms for solving large eigenvalue problems. Elsevier Science Publishing Co. Inc. , 2. The use of novel computer architectures, vector and P.O. Box 1663, Grand Central Station, parallel 3. Computationally-relevant theoretical analyses. New York, NY, 10017, U.S.A. 4. Science and engineering problems where large scale In all other countries: eigenelement computations have provided new insight. Elsevier Science Publishers, 1986. viii+ 330 pages. Book Order Department, Price: US $56.00/Dfl. 140.00. P 0. Box 211, 1000 AE Amsterdam, ISBN Q-444-70074-9 The Netherlands.

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Field Arithmetic Partial Differential Relations M.D. Fried and M. Jarden M. Gromov Contents: Introduction. Infinite Galois Theory and Contents: A Survey of Basic Problems and Results. Profinite Groups. Algebraic Function Fields of One Methods to Prove the h-Principle. Isometric C- Variable. The Riemann Hypothesis for Function immersions. Fields. Plane Curves. The Cebotarev Density 1986/approx. 372 pp./1 illus./Cloth $70.00 Theorem. Ultraproducts. Decision Procedures. Ergebnisse der Mathematik und ihrer Grenzegebiete, Algebraically Closed Fields. Elements of Algebraic 3 Folge, Vol. 9 Geometry. Pseudo Algebraically Closed Fields. ISBN: 0-387-12177-3 Hilbertian Fields. The Classical Hilbertian Fields. Nonstandard Structures. Nonstandard Approach to Hilbert's Irreducibility Theorem. Profinite Groups and Ergodic Theory Hilbertian Fields. The Haar Measure. Effective Field R. Maiie Theory and Algebraic Geometry. The Elementary 1986/approx. 300 pp./32 illus./Cloth $65.60 (tent.) Theory of e-free PAC Fields. Examples and Ergebnisse der Mathematik und ihrer Grenzegebiete, Applications. Projective Groups and Frattini Covers. 3 Folge, Vol. 8 Perfect PAC Fields of Bounded Corank. ISBN: 0-387-15278-4 Undecidability. Frobenius Fields. On w-free PAC Fields. Galois Stratification. Galois Stratification over Of related interest Finite Fields. Open Problems. References. Index. Arithmetic Geometry 1986/approx. 472 pp./Cioth $94.50 Ergebnisse der Mathematik und ihrer Grenzgebiete, Edited by G. Cornell and J.H. Silverman 3 Folge, Vol. II This book is the result of a conference on arithmetic ISBN: 0-387-16640-8 geometry, held July 30-August 10, 1984 at the University of Connecticut at Storrs, the purpose of Einstein Manifolds which was to provide a coherent overview of this field. A.L. Besse This subject has enjoyed a resurgence of popularity due in part to Faltings' proof of Mordell's Conjecture. Contents: Introduction. Basic Material. Basic Included are extended versions of almost all the Material: Kahler Manifolds. Relativity. Riemannian international lectures and, in addition, an English Functionals and Variational Principles. Ricci translation of Faltings' paper. Arithmetic Geometry Curvature as a Partial Differential Equation. Einstein will be of great use to students wishing to enter this Manifolds and Topology. Homogeneous Riemannian field, as well as those already working in it. Manif(Jlds. Compact Homogeneous Kahler Manifolds. Riemannian Submersions. Holonomy Groups. Kahler- 1986/353 pp./5 illus./Cloth $34.00 Einstein Metrics and the Calabi Conjecture. The ISBN: 0-387-96311-1 Moduli Space of Einstein Structures. Self-Duality. Quatemion-Kahler Manifolds. A Report on the Non- Herman Weyl 1885-1985 Compact Case. Generalizations of the Einstein Cemenarv Lectures Delivered by C.N. Yang, R. Condition. Appendix: Sobolev Spaces and Elliptic Penrose, and A. Borel at the ETH, Zurich i: LU Operators. Bibliography. Notation Index. Subject Edited by K. Chandrasekharan 0 0 N v Index. 0\ 1986/approx. 136 pp./Cioth $30.50 (tent.) 0'\ Z5 N 1986/approx. 0\ V'l 512 pp./22 illus./Cioth $89.00 ISBN: 0-387-16843-5 c'l 0 Ergebnisse der Mathematik und ihrer Grenzgebiete, 0 ...J 3 Folge. Vol. 10 0 <( C2 0 u ~~ ISBN: 0-387-15279-2 u z i= c: liJ order. or tor more information, write to: V'l <( "CC V'l ~ Springer-Verlag New York, Inc. :::!: .. LU ·;; ..c Attn: G. Kiely S320 -V'l :X: ....0 .. 175 Fifth Avenue :::!: 1- 0.. .. <( <( ~ New York. NY 10010 ~ ~ ~ :::!: ~c' (Opt" v @lj ,<~G~ ..c: N •.fcON'VENTlON \ \ ... z 10 ~ Springer-Verlag I I 1• YEAJt!j , '+- <( 0 >< i: \:--\·'~i !'"-/ u (lj. Q New York Berlin Heidelberg I .-., "\~.\... --;! ~ ~ i London Paris li>k yo .. 1!-E!iY u"' C2 ... ·;:; LU :J 0 :::!: 0 z <( a.: J'