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Notices of the American Mathematical Society

April 1983, Issue 225 Volume 30, Number 3, Pages 249-384 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 the Notices was sent to press. The summer and annual meetings are joint meetings of the Mathematical Association of America and the Ameri­ can Mathematical Society. The meeting dates which fall rather far in the future are subject to change; this 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 second announcements of the meetings will have appeared in earlier issues. 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 tho meet­ ing. Abstracts should be submitted on special forms which are available in many departments of mathematics and from the office of the Society in Providence. 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 ab­ stracts submitted for consideration for presentation at special sessions is usually three weeks earlier than that specified below. For additional information consult the meeting announcement and the list of organizers of special sessions. MEETING ABSTRACT NUMBER DATE PLACE DEADLINE ISSUE 803 April 14-15, 1983 New York, New York EXPIRED April 804 April 29-30, 1983 Salt Lake City, Utah EXPIRED April 805 August 8-12, 1983 Albany, New York MAY 17, 1983 August (87th Summer Meeting) 806 October 28-29, 1983 Fairfield, Connecticut AUGUST 23, 1983 October 807 November 11-12, 1983 San Luis Obispo, California AUGUST 25, 1983 October 808 November 11-12, 1983 Evanston, Illinois AUGUST 29, 1983 October 809 January 25-29, 1984 Louisville, Kentucky NOVEMBER 2, 1983 January (90th Annual Meeting) 1984 April 6-7, 1984 Notre Dame, Indiana January 9-13, 1985 Anaheim, California (91 st Annual Meeting) January 21-25, 1987 San Antonio, Texas (93rd Annual Meeting)

DEADLINES: Advertising (june Issue) April 21,1983 (August Issue) june 8, 1983 News/Special Meetings: Uune Issue) April 5, 1983 (August Issue) May 23, 1983

Other Events Sponsored by the Society April12-13, 1983, AMS-SIAM Symposium on Inverse Problems, New York Statler Hotel, New York, New York. This issue, page 312. May 1983, Symposium on Some Mathematical Questions in Biology, Detroit, Michigan. This issue, page 331. June 5-August 13, 1983, AMS Summer Research Conferences, University of Colorado, Boulder, Colorado. January issue, page 7 4. June 27 -July 8, 1983, AMS-SIAM Summer Seminar on Large-scale Computations in Fluid Mechanics, Scripps Institution of Oceanography, LaJolla, California. February issue, page 199. July 11-29, 1983, AMS Summer Research Institute on Nonlinear Functional Analysis and Applications, University of California, Berkeley. This issue, page 332. August 6-7, 1983, AMS Short Course: Population Biology, Albany, New York. This issue, page 336

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Volume 30, Number 3, April 1983

EDITORIAL COMMITTEE Paul F. Baum, Ralph P. Boas Raymond L. johnson, Mary Ellen Rudin Bertram Walsh, Daniel Zelinsky 251 "Big" News from Archimedes to Friedman, Everett Pitcher (Chairman) C. Smorynski MANAGING EDITOR 257 Newest Ratings of Graduate Programs in Lincoln K. Durst Mathematics, D. C. Rung ASSOCIATE EDITORS 268 Research Briefing Panel on Mathematics: History Hans Samelson, Queries and Background, W. Browder Ronald L. Graham, Special Articles 271 Report of the Research Briefing Panel SUBSCRIPTION ORDERS on Mathematics Subscription for Vol. 30 (1983): 280 National Science Foundation Budget Request $39 list, $20 member. The subscription for FY 1984 price for members is included in the annual dues. Subscriptions and orders 290 Mathematical and Computer Sciences in the FY for AMS publications should be 1984 Federal Budget, W. H. Pell addressed to the American Mathematical 300 Federal Support-Where Do We Stand? Society, P. 0. Box 1571, Annex Station, K. Hoffman Providence, Rl 02901. All orders must be prepaid. 302 Queries 304 News and Announcements ORDERS FOR AMS BOOKS AND INQUIRIES ABOUT SALES, SUBSCRIP­ 306 NSF News & Reports TIONS, AND DUES may be made by 307 Letters to the Editor calling Carol-Ann Blackwood at 308 1983 AMS Elections (Nominations by Petition) 800-556-7774 (toll free in U.S.) between 8:00a.m. and 4:15p.m. eastern time, 310 Future Meetings of the Society Monday through Friday. New York City, April 74-75, 310; Salt Lake City, April 29-30, 323; Symposium on CHANGE OF ADDRESS. To avoid Mathematical Biology, Detroit, May 37, 331; interruption in service please send AMS Summer Research Institute, 332; Albany, address changes four to six weeks in advance. It is essential to include the August 8-72, 333; Synopses, Short Course on member code which appears on the Population Biology, 350; Call for Topics, 352; address label with all correspondence Invited Speakers and Special Sessions, 355 regarding subscriptions. 357 New AMS Publications INFORMATION ABOUT ADVERTISING 359 Miscellaneous in the Notices may be obtained from New Doctorates (Supplement), 359 Wahlene Siconio at 401-272-9500. 360 Special Meetings CORRESPONDENCE, including changes 365 AMS Reports & Communications of address should be sent to American Recent Appointments, 365; Officers of the Mathematical Society, P.O. Box 6248, Society, 1982 and 1983, 366 Providence, Rl 02940. 367 Advertisements Second class postage paid at Providence, Rl, and additional mailing 380 Registration Forms offices. Copyright © 1983 by the Summer List of Applicants, 380, 381 American Mathematical Society. Albany Preregistration and Housing Printed in the United States of America. Reservation Form, 383, 384 JOURNALS PUBLISHED BY THE AMERICAN MATHEMATICAL SOCIETY

TRANSLATION JOURNALS Soviet Mathematics-Doklady is a bimonthly translation journal containing the entire pure mathematics section of the DOKLADY AKADEMII NAUK SSSR. Mathematics of the USSR-Izvestija, a bimonthly journal, is a cover-to-cover translation of IZVESTIY A AKADEMII NAUK SSSR SERIYA MA TEMATICHESKA Y A. Mathematics of the USSR-Sbornik is a monthly journal and is a cover-to-cover translation of MATEMATICHSKii SBORNIK (NOVA Y A SERIY A). Theory of Probability and Mathematical Statistics is the cover-to-cover translation into English of TEORIYA VEROY ATNOSTEi I MA TEMA TICHESKAY A STATISTIKA. Vestnik Leningrad University (Mathematics) is the complete translation into English of the mathematics section of the VESTNIK LENINGRADSKOGO UNIVERSITETA. Transactions of the Moscow Mathematical Society is a translation into English of TRUDY MOSKOVSKOGO MATEMA TICHESKOGO OBSHCHESTV A. Proceedings of the Steklov Institute of Mathematics is a translation of the PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS IN THE ACADEMY OF SCIENCES OF THE USSR.

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OTHER JOURNALS Abstracts of Papers Presented to the American Mathematical Society contains abstracts of invited hour addresses of papers presented in special sessions or in sessions for contributed papers and of papers presented to the Society "by title". Current Mathematical Publications, issued biweekly, contains a subject-classified index of papers and books being published currently in mathematics. Mathematical Reviews is a monthly journal devoted to abstracts and review of the current mathematical literature of the world. Each volume consists of six regular issues plus an index issue. Starting in 1979 the second volume index each year is the author and subject index for the entire year. Abstracts and reviews are grouped under subject headings. Notices of the American Mathematical Society, published eight times a year, contains programs and reports of the meetings of the Society, reports on Society business, communications to the membership, and news items and information of interest to the mathematical community. "Big" News From Archimedes to Friedman by C. Smorynski

"There are some, King Gelon, who think that for all numbers up to X 2 by counting units the number of the sand is infinite in multitude; and remainders. In particular, having traditional and I mean by the sand not only that which names for numbers up to a myriad, Archimedes exists about Syracuse and the rest of Sicily but could immediately obtain names for numbers up also that which is found in every region whether to a myriad myriads. He called these the numbers inhabited or uninhabited. Again there are some of the first order. Using X as a myriad myriads, who, without regarding it as infinite, yet think he concluded the nameability of all numbers up that no number has been named which is great to X 2 (i.e. 1016), and called those between X and enough to exceed its multitude." So saying X 2 the numbers of the second order. Now, it (modulo translation by T. L. Heath), Archimedes is obvious that one could use X 2 as a unit and went on to produce an upper bound on the generate names for numbers up to (X2) 2 = X 4 • number of grains of sand in the universe. This However, Archimedes was a bit more pedestrian might not seem like much of an accomplishment: and only went from a unit Y (e.g. X or X 2) to One has but to pick a lower bound on the size Y . 108 . Thus, numbers of the third order were of a grain of sand and an upper bound on the 2 3 size of the universe. . . . The real difficulty was those between X and X (for X the myriad notational: In sooth, no number had been named myriads). which was great enough to satisfy Archimedes; he Archimedes then noted that, when one has had to invent his own method of generating large carried out this process a myriad myriads of times numbers. (i.e. one has reached xx for X= 108 ), one has The standard Greek system of numeration did obtained a unit P yielding the first period. One not go very far. The 27 numbers 1, 2, ... , 9; can then go on to the first order of the second 10, 20, ... , 90; 100, 200, ... , 900 were denoted by period (i.e. the numbers between P and P·X), the overlined letters of an expanded Greek alphabet, various orders of the third period, and on up to e.g. a = 1, (j = 9. Every number from 1 to the Xth order of the Pth period, i.e. pX = xx•, 999 could be alphabetically represented, e.g. 333 where X = 108 . is r\"j. Thousands, from 1000 to 9000 were This last number cited by Archimedes is represented by pre-subscripting a little stroke, probably the largest number ever named in e.g. ,(3 = 2000. One now has the ability to antiquity. It has remained rather large until represent all numbers from 1 to 9999. The next the present century when it was toppled by number (10,000) was called a myriad and would the googolplex (1010100 ; Archimedes' number is be represented by two myriads (20,000) would 1s.•o3 Sk M; about 1010 ) and also the famous ewes be denoted M." Alternatively, one could write Number (10 101034) and the killion (which I dare Ma and M/J or even a and~. However, Greek not mention). The googolplex was defined for the numeration ended with myriads. In Archimedes' fun of it and Archimedes' number was generated words (translated by Ivor Thomas), "Now we for the sake of example (the actual bound he already have names for the numbers up to a produced for the number of grains of sand being myriad and beyond a myriad we can count in only a tiny 1063); the Skewes Number actually myriads up to a myriad myriads." A myriad appeared in a proof as a bound-it was, until myriads is just 108 . the 1960s I believe, the largest number to ever Suppose one has names for all numbers 1, .. .,X. seriously occur in a mathematical proof. One can then call X the "unit" and give names 1. Dominance. There has long been a fascina­ C. SMORYNSKI received his Ph.D. in 1973 from the tion with large numbers and, consequently, with University of illinois at Chicago Circle. Since then methods of generating such numbers. These he has held visiting positions at nine universities in methods are nothing more than rapidly growing eight cities and five countries on three continents. functions, with more powerful methods being Currently, he is a visiting assistant professor at functions of more rapid growth. The Ohio State University where he teaches his Suppose (X, <,O) is a linearly ordered set with version of the History of Mathematics. His main least element 0 and no greatest element. If g : research interest is the study of self-reference in f, X -+ X are two functions, we say f (eventually) forniallanguages. C. S. dominates g, written g « J, if f(x) > g(x) from

251 some point on. The dominance relation, «, is only he diagonalises: g(n) = f n(n). (Actually, since he a partial ordering of xx. However, it is directed, worked with continuous functions on the reals, he and can have many linear suborderings, some of had to do a little more than this. It is probably which can be used as measures of growth-scales also worth remarking that du Bois-Reymond's use in G. H. Hardy's terminology-for natural classes here of diagonalisation predates Cantor's famous of functions. For functions of sufficiently slow argument.) growth, these measures can be quite meaningful ... Modern logicians prefer to speed this process but I am getting ahead of myself. up a bit. Thus, given a function f satisfying x < In the late 19th century, Paul du Bois­ f(x), the logician will go directly to the function Reymond initiated a study of the dominance g(x) = jX(x), which would only first be obtained relation restricted to continuous functions on, after w + 1 steps in the process mentioned above. say, the nonnegative real numbers. This study Of course, once one realises one can iterate into was furthered and exposited by Godfrey Hardy the transfinite, the logician's impatience really who, in 1910, published a little booklet thereon amounts to no more than cutting down the time entitled Orders of Infinity. Of course, if our it takes to hit a certain level of growth by a factor goal is to generate large numbers, it is the of w and does not really reach higher levels of dominance relation on the set of functions on the growth. Being a logician, however, I prefer the nonnegative integers that most interests us. The logical approach and shall say a fews words about most interesting results concern the restriction of it. the dominance relation to "reasonable" collections The logician's scale of growth of "slowly rapidly of functions. growing" functions is generated by the procedure One nice result I know of is due to Hardy. just described. The logician takes X = w = Let LE (for logarithmico-exponential) be the class {0, 1, ...} and begins with F0(x) = x + 1. He then of functions mapping infinite intervals (r, oo) to defines Fn+l(x) = F';;,(x), the x-fold composition similar intervals (r a nonnegative real number) of Fn applied to x. Thus, generated as follows: F1(x) = 2x, F2(x) = 2x · x (i) LE contains log, exp (i.e. exp(x) =ex), and all constant functions; and (ii) LEis closed under+,·,+-, and composition. (x) larger than 2· } x 2's. Then, LE is linearly ordered by «. F3 is somewhat The arithmetic version of Hardy's Theorem The functions Fn are cofinal in a natural class is also worth noting. Define the class EP of of functions, the primitive recursive functions exponential polynomials as follows: which are just those generated by recursion in one (i) the identity functions and the constant variable. The individual function Fn measures, in functions n, for n a nonnegative integer, are a sense, the maximum growth attainable through exponential polynomials; n applications of recursion. (ii) if/, g E EP, then F + g, f · g, f9 E EP. The difference between the rates of growth of The functions in EP are obviously the restrictions a function Fn and its successor Fn+l is so vast to the nonnegative integers of functions in LE, that we do not really worry about the ordinary whence Hardy's result applies: EP is linearly dominance relation. Instead, for a given function ordered by the dominance relation. There is f, we generate a class 1(!) of functions by closure more: EP is well-ordered by this relation. (Two under various forms of explicit definition, e.g. things are not known, however: (i) it is yet an composition and the replacing of variables by open problem if the ordering is decidable; and (ii) constants. Two functions f and g are then said the order-type of the ordering, though close to to have roughly the same rate of growth if f is of 1(g) and vice versa. being known, is as yet undetermined.) dominated by an element The emphasis is on the word "rough": F1 has 2. Iteration. If the name of the game is BIG roughly the same rate of growth as any linear NUMBERS the strategy is RAPIDLY GROWING function; F2 has that of exponentiation, i.e. 2x or FUNCTIONS, i.e. BIG ELEMENTS of XX under even xx; and F 3 has that of a linear stack of 2's «. So, how do we generate these big elements? or even a linear stack of x's. It should be borne The simplest, most naive approach is to iterate in mind, however, that for such rapid growth as some method of passing from a function f exhibited by F2 , F3 , ... any finer classification of to a larger one g. In Hardy's already cited growth could well be meaningless. exposition of du Bois-Reymond's Infinitiirkalkul, The logician's iteration process is successful he illustrates this as follows: Suppose f is a relative to this notion of rough equivalence. Each function satisfying x < f(x) from some point F n+l is demonstrably not of roughly the same on. Define /I = /, fn+l = f o fn, i.e. fn(x) = rate of growth as Fn. It grows incredibly more tn(x). Then f = h « h « ·· · and we have rapidly-so much so that it cannot be reached successively bigger elements of xx. Moreover, from Fn without the aid of Fn+l itself; such was from a sequence h « h « · · ·, he shows how the significance of the closure conditions put on to find g such that, for all n, fn « g; namely, 1.

252 Of course, if our goal is truly to generate us what we want much more quickly. What plays functions of rapid growth, we will want to go the role of exponentiation relative to our function beyond this sequence. Following Hardy, we iteration? One answer is powerful theories. diagonalise Fw(x) = F,(x). Actually, since we are Any reasonably powerful theory T will be able now playing the roles of logicians, we should find to handle numbers and compute all computable it unthinkable to stop here: For countable ordinals functions. Interestingly enough, however, if T o:, define Fa by is sufficiently sound (i.e. T does not prove too many false arithmetic assertions-the (so-called) Fa(x) = F~(x), ifo: = ,8 + 1, w-consistency of T is sufficient here, though the Fa(x) = Fax(x), if o: = lim o:,, of T is not), T will not be able xEw mere consistency to prove the computability of all functions it can where o:o < o:1 < · · · · compute. One can diagonalise on the T-provably Oops! The definition of Fa, for o: a limit computable functions and obtain a function Fr ordinal, depends on the path o:o < 0:1 < · · · by which dominates every T-provably computable which we choose to reach o:: No matter how function. slowly this sequence grows, Fa will dominate For various theories T, Fr is known: For Primi­ all F 13 's for ,8 < o:; but if we pick a rapidly tive Recursive Arithmetic, a theory embodying growing sequence, Fa could become a much more Hilbert's finitism, Fr is roughly Fw, the prov­ rapidly growing function than we intended: It ably computable functions of Primitive Recursive could become, for example, what we intended Arithmetic being just the primitive recursive ones. Fa+1 to be! In the first problematic case, o: = w, For Peano Arithmetic, the usual formulation of we defined Fw by choosing O:n = n; we could have number theory (even in a mathematical sense: it chosen O:n = Fn(n) (= Fw(n)) or O:n = Fw+l(n). is strong enough to prove all results of analytic

With the former choice we would get something number theory), Fr is roughly F,0 • Many other much bigger than Fw relative to dominance, but examples are known. of roughly the same rate of growth relative to Of course, one can now ask about F 1 for our rough equivalence; with the latter we would T being Zermelo-Fraenkel Set Theory (the usual violate even the rough equivalence. The idea of axiomatic set theory) or some extension by new this example is this: If we try to get beyond all axioms of infinity. And one can now look for Fn's by diagonalising and only allow ourselves progressively more powerful axioms of infinity. access to the rates of growth already at hand­ There are three problems here. First, with respect i.e. the Fn's-then we will get something like to the addition of new axioms, one is required to F w; to get beyond F w, we must use something of have consistent axioms; in fact the axioms must already more rapid growth. yield a theory a bit more than merely consistent. For reasonably small, well-understood ordinals While the study of large cardinals has been an o:, like w and Eo, we have more-or-less canonical active area of research in logic, and while there is choices for the sequences O:n and, relative to our good reason to believe in the consistency of many notion of rough equivalence, it does not matter of the new axioms, there is little evidence for or if we use O:n or o:F13 (n) for some ,8 < o:. For against their additional soundness. larger, less studied ordinals, the situation might The second problem is that, once we have Fr, well be delicate: Two distinct ways of generating we would like to assign some measure to its size, an ordinal o: could readily yield distinct canonical i.e. we would like to say Fr is roughly equivalent choices of the sequence O:n, so widely inequivalent to Fa for some o:. As far as I know, the theory definitions of Fa. It suffices to say that Fa(x) can allowing this has never been fully worked out. be made well defined (up to rough equivalence) Moreover, the determination of the ordinal o: for all "small" o: and this suffices for the results to associate with the theory T has only been we discuss. made for remarkably weak theories. I can repeat But I have begun to digress. . . . Our problem my earlier observation on large ordinals: Proof is not to show that there is a unique (up to rough theorists are hard at work on this, but it is a equivalence) function we want to call Fa for some messy undertaking. want to generate such a particular o:, but that we The third problem is an inherent limitation on rapid growth rate. Thus, function and admire its the theoretical approach. Any decent theory has large ordinals. Proof our problem is to generate axioms we can list. All of its provably computable theorists are hard at work on this, but it is a functions are (theoretically) computable, as is messy undertaking and I do not wish to discuss it their dominating diagonal function. Even the here. So we depart from this topic. iteration process will get us past the computable 3. Theoretical growth. One disadvantage to functions if we simply go on until we have passed using the transfinite iteration of a given function the first noncomputable ordinal. If we decide to to generate ever more rapidly growing functions allow noncomputable functions, we are again faced is that it is such a laborious process; it is with the laborious slowness of the straightforward like generating large numbers by iterating the iteration process and we must look for principles successor function when exponentiation will get allowing bigger jumps in the dominance ordering.

253 There are such: One diagonalises on functions F232 gives a bound. There are, however, other definable in languages (as opposed to provably functions of combinatorial interest of definite computable in theories). However, I am not much transfinite growth rates. In fact, around 1977 interested in noncomputable functions and shall Jeff Paris and Leo Harrington concocted variants not discuss this further. of Ramsey's Theorem, the diagonal functions of which grow roughly as rapidly as FEo. I shall say 4. Rapid growth in real life. The surreal charac­ a few words about, say, Harrington's variant of ter of the preceding discussion is misleading; Ramsey's Theorem. rapid growth is not as divorced from mathemati­ cal reality as it seems. Large numbers and Harrington's Variant. For all n, k there is a p rapidly growing functions exist in mathematical so large that: If q > p and C: [q]k -+ n, there is nature. In calculus, e.g., the difference between a subset Y of q of cardinality at least min(Y) and convergence and divergence can be so slight that greater thank such that C ~ [Y]k is constant. attempts to measure it yield large numbers. Hardy Harrington's Theorem differs from Ramsey's cites the two series Theorem mainly in that, rather than to require the C-homogeneous set Y to have a prescribed 00 1 00 1 cardinality, it requires Y to be relatively large in A: fa n log n(log logn)2' B: fa n log n(log logn) · that its cardinality is no smaller than its least element. Again, there is an underlying function Series A converges to 38.43 ... , but it converges H(n, k) yielding the value of p. The diagonal so slowly that it requires 103·14·1086 terms to give function H(x, x) has roughly the same growth rate two-decimal accuracy; series B diverges, but the as F,0 • partial sums exceed 10 only after a googolplex Recall that the function F,0 has roughly the of terms have appeared. It would be interesting rate of growth that Fr has, where T is the to see the full rates of growth of such moduli of theory Peano Arithmetic. Thus H(x, x), as well convergence and divergence analysed. as the corresponding function .for Paris' original Next to logic the most fruitful source of rapid (uncited) variant of the Ramsey Theorem, has growth is combinatorics, particularly Ramsey the rate of growth of the first "natural" function Theory (at least: until recently-cf. the next to dominate all functions provably computable in section). The two most celebrated results of Peano Arithmetic. Originally, in fact, the rate of Ramsey Theory are Ramsey's Theorem and van growth of H and the Paris function was shown der Waerden's Theorem. To state these, we need to be at least this great by formal considerations a few definitions. For a positive integer n, an n­ about the theory Peano Arithmetic; only more colouring of a set X is just a map C: X -+ n. If recently have there evolved verifications of this X is a set, we let [X]k denote the collection of growth rate that proceed by directly relating the k-element subsets of X. functions in question to the Fa:-hierarchy. While Ramsey's Theorem. For all m, n, k there is a p the verification of rapidity, i.e. the lower bound, is so large that: If X is a set of cardinality at least easily (almost effortlessly to the logician with the p and C: [X]k -+ n is an n-colouring, there is a proper background) established by reference to subset Y of X of cardinality at least m and greater the formal theory, the ordinal analysis is generally thank such that C ~ [Y]k is constant. messy. More briefly put: We call a subset Y of X The work of Paris and Harrington opened homogeneous with respect to the colouring C if the floodgates for many more functions of rapid c ~ [Y]k is constant. Ramsey's Theorem asserts growth. There is something of a recipe for their that, by making X large enough, there will be a concoction: Find a quasi-combinatorial problem homogeneous subset Y of X of any predetermined that is hard to solve (cf. Clote's paper for hints size. The sufficient largeness required of X on this step) and show that the corresponding is a function R(m,n,k). The corresponding function dominates all functions provably total diagonal function R(x, x, x) is known to be roughly in a suitable theory (one generally does this equivalent to F3 , i.e. iterated exponentiation. by showing the function to encode incredibly van der Waerden's Theorem. For all m, n, there many closure properties). Most often, the is a number p so large that for any colouring C: theory in question is Peano Arithmetic and the p--+ n there is an m-term arithmetic progression rate of growth is roughly that of F,0 • There Y ~ p such that C ~ Y is constant. are exceptions: Harvey Friedman has pushed Again, van der Waerden's Theorem determines this growth rate up quite a bit by working a function W(m,n). The "exact" rate of growth with stronger theories, and, among his slower of W, or rather its diagonal W(x, x), is not known. contributions, he, Ken McAloon, and Steve All proofs so far given rely on a double induction, Simpson have even given a variant of Ramsey's whence the recursion defining W(m,n) is a double Theorem which they find rather simple and recursion-yielding F w or a rough equivalent as which has a corresponding function of rate the immediate upper bound. of growth roughly that of Fr0 , where fo is Of course, Fw is only the best known bound the first countable ordinal which cannot be for W(x,x). It may well turn out that F11 or described without reference (if only oblique) to the

254 uncountable. This variant is not simple enough Here, by a finite tree I mean a finite rooted tree for me to state here; besides, it is still one of the i.e. a finite partially ordered set which (i) branche~ "first generation" of functions of rapid growth. only _upwardly, i.e. the predecessors of any element ar~ ~mearly ordered (tree property), and (ii) has a 5. ~he new gene~ation of rapidly growing rmrumum element (rootedness). A homeomorphic functions. The Pans-Harrington work was a embedding is a one-to-one inf-preserving map. remarkable accident. Paris' Variant of Ramsey's Theorem turned out to be independent of Peano The function F(x) which takes one from k to n grows incredibly rapidly. It is, at the time Arit~etic. It f~llowed that its corresponding functwn grew rapidly. Harrington's Variant was of wr~ting, the most rapidly growing computable a cosmetically improved version of Paris' Variant functwn that has been described by a means other and was also independent, etc. The flood of than pointing to an ordinal or a theory. One can, new functions of rapid growth were imitations of course, point to an ordinal and a theory for disguised versions of the proof-theoretic growth F-Friedman has analysed this completely-but described in §3. Now, however, there is a new I had best not do so: Few readers would recognise either. A modest idea of the growth rate and appr~ach yielding much more rapidly growing functwns. proof-theoretic strength of Friedman's finitisation of Kruskal's Theorem can be gleaned from my The new success seems to me as serendipitous relevant recent contribution to the Mathematical as Paris' original success, but it is different in Intelligencer. several important respects: (i) the combinatorial problems were found first, with the theories . In any event, the function F will not long hold located later; (ii) the independent combinatorial Its place of honour. Friedman's Theorem is a direct finitisation of a weak form of Kruskal's theor~ms with corresponding rapidly growing functiOns are much simpler; and (iii) the connexion Theorem; there is the full theorem to contend of the combinatorial problems with ordinals is with-and soine strengthenings. Friedman has more direct. begun the underlying proof-theoretic analyses of To build up the suspense, let me first introduce these results and reports that their ordinals and proof-theoretic strengths dwarf those of the above­ an ordin~lly poorer result. The underlying theorem IS due to R. L. Goodstein and its mentioned finite theorem. When these results are independence of Peano Arithmetic is due to finitised, they will yield functions that dwarf F. Lawrence Kirby and Jeff Paris. Let m n be ~rbitrary positive integers and write m hereditarily References m base n, i.e. write m in base n, write all exponents occurring in the expression in base n, 1. Archimedes, The Sand Reckoner 214?· etc. For example, let m = 71, n = 2: reprinted ~n The Works of Archimedes' (T. L: Heath, editor), Dover Publications, Inc., New 71 = 222+2 + 22 + 2 + 1. York. Given this, alternately apply the following two 2. Peter Clote, Anti-basis theorems and their operations: relation to independence results in Peano arith­ (a) subtract 1; ~etic, in Model Theory and Arithmetic (C. Ber­ hne? K. McAloon and J.-P. Ressayre, editors), (b) replace base n by base n + 1. Sprmger-Verlag, Berlin and New York, 1981. Thus, 71 -+ 70 -+ 333+3 + 33 + 3 ~ (2.0589). 10I4 . 3. Ian Frazier, The killion, The New Yorker, 6 According to Goodstein, one will eventually reach September 1982. 0. It may take a while: Let G(x) denote the 4. R. L. Goodstein, On the restricted ordinal supremum over all m, n ~ x of the number of theorem, Journal of Symbolic Logic, volume 9, steps it takes to reach 0. The function G has 1944, pages 33-41. growth rate roughly at least that of F, • 0 5. R. L. Graham, B. L. Rothschild and J. H. The secret behind Goodstein's Theorem is that Spencer, Ramsey theory, Wiley-Interscience Series the hereditary expression of m in base n minlicks in Discrete Mathematics, a Wiley-Interscience an ordinal notation for an ordinal less than e0 . Publication, John Wiley & Sons, Inc. New York The first operation applied reduces the ordinal· 1980. , ' the second leaves it fixed. Ordinals are also related to well-quasi-orderings, foremost among which are ?·. Godfrey ~· Hardy, Orders of infinity, 2nd editwn, Cambndge University Press Cambridge embeddings of finite trees; hence we see a rich 1924. ' ' source of rapidly growing functions. The simplest example arises from the following theorem which 7. Lawrence Kirby and Jeff Paris Accessible is a finite form of Kruskal's Theorem: ' independence results for Peano arithmetic Bul­ letin of the London Mathematical Society, v'olume . Friedman's Theorem. Let k > 1 be given. There 14, 1982, pages 285-293. ~s a number n such that, for any sequence ~aris, Some independence results for T~, T2_, ... , Tn of finite trees with Ti having car­ 8. Jeff . Peano anthmet~c, Journal of Symbolic Logic, ~znal~ty at most. k + i, there are i < j such that Ti ~s homeomorph~cally embeddable in Tj. volume 43, 1978, pages 725-731.

255 9. Jeff Paris and Leo Harrington, A mathemati­ 11. C. Smoryilski, Some rapidly growing func­ cal incompleteness in Peano arithmetic, in Hand­ tions, Mathematical lntelligencer, volume 2, 1980, book of Mathematical Logic (J. Barwise, editor), pages 149-154. North-Holland Publishing Company, Amsterdam, 12. -, The varieties of arboreal experience, 1977. Mathematical lntelligencer, volume 4, 1982. 10. S. Skewes, On the difference 1r(x) -li(x), Journal of the London Mathematical Society, volume 8, 1933, pages 277-283.

EDITORS' ='

256 Newest Ratings of Graduate Programs in Mathematics by Donald C. Rung

Between September 1982 and January 1983 For quality of graduate faculty (Table A) the the Conference Board of Associated Research scale was Councils published a five-volume assessment 5 . . . distinguished of various graduate programs including mathe­ 4 ... strong matics. Volume 1, published in October 1982, 3 ... good assessed the mathematical and physical sciences. 2 ... adequate The Conference Board includes representatives of the American Council of Learned Societies, 1 ... marginal American Council on Education, Social Science 0 ... not sufficient for doctoral training Research Council, and the National Research X ... don't know well enough to evaluate Council. The scale for rating the effectiveness of doctoral This survey is the fourth such survey since programs (Table B) was 1957 when Hayward Kennison of the University of 3 . . . extremely effective Pennsylvania evaluated graduate programs. The 2 ... reasonably effective second survey was done in 1964 by Allan Cartter, 1 . . . minimally effective American Council on then Vice-President of the 0 . . . not effective Education. A summary of these reports for mathematics graduate programs was published in X ... don't know well enough to evaluate the December 1966 issue of the Notices. The (the nomenclature but not the scale used in the third survey, by Kenneth D. Roose and Charles J. 1970 report for this second parameter was slightly Andersen of the American Council on Education, different). was published in 1970 and the mathematics The 1982 report contains an evaluation of the ratings were summarized in the February 1971 improvement of the program quality (Table C) issue of the Notices. based on the following scale: The 1982 survey is a somewhat more elaborate 2 . . . better than 5 years ago assessment than previous surveys. It rates over 1 . . . little or no change in last 5 years 2,600 programs in 31 fields in the physical and 0 . . . poorer than 5 years ago mathematical sciences, engineering, life sciences, X ... don't know well enough to evaluate social sciences and humanities (about the same The 1970 report had a similar rating separated, breadth as in the 1970 Roose-Andersen survey). however, into two components: improvement It reports sixteen different measures to assess each of faculty and improvement in effectiveness of program as opposed to three measures reported educating research scholars. in the 1970 survey. The measures are grouped into six categories: program size, characteristics To be nominated for inclusion in the 1982 of graduate program, university library, research survey a mathematics program needed either (1) a support, published articles, and survey results. minimum of seven doctoral graduates in the period All but the last are factual measures. The 1976-1978 or at least three doctoral graduates in survey measures, sometimes called reputational 1979, or (2) an average of at least 2.0 in the 1970 measures, are mean ratings of (1) scholarly quality Roose-Andersen assessment of quality of program of program faculty, (2) effectiveness of the program faculty. The 1982 study included assessments in educating research scholars or scientists, (3) for 115 mathematics programs. They accounted improvement in program quality in the last five for 92% of the doctoral degrees in mathematics years), and (4) the evaluator's familiarity with the awarded during the period 1976-1979. Only 102 work of the program faculty. (The reputational programs were assessed in the 1970 survey. survey was conducted in the Spring of 1981.) In Of the 348 faculty members asked to participate this article the first three reputational measures in the reputational part of the recent assessment, are reproduced in Tables A, B and C. These 223 (64%) responded. Each evaluator was asked were the same measures used in the 1970 and · to rate no more than fifty randomly selected 1964 surveys. Furthermore, the ratings scale and programs and was supplied with a list of faculty nomenclature have remained the same for the first members in each program as well as the number two program measures (Tables A and B). of doctorates awarded in the period 1976-1980.

257 It should be noted that the program faculty each discipline so that those departments which was limited to those who participate significantly had granted ninety percent of the doctoral degrees in the doctoral education program. (In some during the academic years 1976-1977 to 1977- instances, therefore, the faculty si:lie reported in 1978 qualified: for mathematics the cut-off was this study is smaller than that given in the Annual seven or more doctorates in that period; for both AMS Survey reports.) The 1970 assessment survey statistics/biostatistics and computer science, it was based on responses from 240 evaluators. was five. In addition, a qualifying department was required to have granted during 1978-1979 In addition to the raw scores, the 1982 more than one-third of the number of the degrees assessment also reported a normali:!ied score (mean specified in the previous sentence. 50, standard deviation 10) in most of the sixteen Several departments which met the criteria categories. The normali:!ied scores were computed for inclusion in the study did not participate. from exact values of the measures and not from Three programs which met the criteria for the the rounded raw values given in the published mathematics study but were not included are report. Thus programs with the same rounded those at Idaho State University, Lehigh Univer­ raw score may have different normali:!ied scores sity and the University of Northern Colorado. or those with different reported raw scores may Programs which qualified for inclusion in the have been reported as having the same normali:!ied statistics/biostatistics study but did not par­ score, since the latter was also rounded. ticipate are those at Dartmouth College, Univer­ The 1982 assessment also contained evaluations sity of Illinois (Chicago), New York University, of 64 programs in statistics/biostatistics and 58 University of Northern Colorado and the Univer­ programs in computer science. The three reputa­ sity of South Carolina. Programs in computer tional measures reproduced for mathematics are science which were omitted are those at the also given for these programs in corresponding University of Chicago, George Washington Univer­ tables following those for mathematics. [See sity, Harvard University, Northeastern Univer­ Editor's Note, below.] sity, Purdue University, University of Southwest Louisiana and the University of Texas (Health Copies of the complete report, An assessment Science Center, Dallas). Several departments of research-doctorate programs in the United were included in both the mathematics and the States, are available for $10.50 per volume from statistics /biostatistics studies. the National Academy Press, 2101 Constitution In the mathematics study, two departments at Avenue N.W., Washington, DC 20036. Brown University (Mathematics, Applied Mathe­ matics) were grouped together, as were Mathe­ EDITOR'S NOTE. In addition to the tables matics and Biomathematics at UCLA, but both on the three following pages giving the ratings Mathematics and Applied Mathematics at the on mathematics programs discussed above by University of Maryland (College Park) were in­ Professor Rung, we reproduce corresponding cluded separately. tables for Statistics/Biostatistics and Computer Tables A, B, and C for Statistics/Biostatistics Science. (pages 262, 263, 264) and for Computer Science The criteria for inclusion differ somewhat for (pages 265, 266, 267) follow those for Mathematics the latter two, since the cut-off level was set for (pages 259, 260, 261).

258 Graduate Programs in Mathematics Table A. Ranked by Scholarly Quality of Program Faculty

Princeton University 4.9 Pittsburgh, University of 2.5 California, University of (Berkeley) 4.9 SUNY at Albany 2.4 Massachusetts Institute of Technology 4.9 Florida State University 2.4 Chicago, University of 4.8 Oregon State University 2.4 Harvard University 4.8 Temple University 2.4 Stanford University 4.6 California, University of (Riverside) 2.3 New York University 4.5 Claremont Graduate School 2.3 Yale University 4.5 Syracuse University 2.3 Columbia University 4.4 Virginia Polytechnic Institute & State Wisconsin, University of (Madison) 4.2 University 2.3 Brown University 4.1 California, University of (Davis) 2.3 Michigan, University of 4.1 Florida, University of 2.3 Cornell University 4.0 Georgia, University of 2.3 California, University of (Los Angeles) 4.0 Kansas, University of 2.3 illinois, University of 4.0 Iowa, University of 2.2 Minnesota, University of 3.9 Delaware, University of 2.2 California Institute of Technology 3.8 Georgia Institute of Technology 2.2 Brandeis University 3.8 Wayne State University 2.2 SUNY at Stony Brook 3.7 Iowa State University 2.1 Pennsylvania, University of 3.7 New Mexico, University of 2.1 Rutgers University 3.6 North Carolina State University 2.1 Washington, University of 3.6 Polytechnic Institute of New York 2.1 CUNY Graduate School 3.5 Tennessee, University of 2.1 Indiana University 3.5 Oklahoma, University of 2.0 Maryland, University of (Applied Math.) 3.5 Vanderbilt University 2.0 Maryland, University of (Mathematics) 3.5 Connecticut, University of 2.0 Northwestern University 3.5 Auburn University 1.9 Purdue University 3.4 Cincinnati, University of 1.9 California, University of (San Diego) 3.4 Colorado State University 1.9 Johns Hopkins University 3.4 Houston, University of 1.9 Rice University 3.4 Wesleyan University 1.9 Texas, University of (Austin) 3.3 Nebraska, University of 1.8 Utah, University of 3.2 Kent State University 1.8 Washington University 3.1 South Carolina, University of 1.8 illinois, University of (Chicago) 3.0 Missouri, University of (Columbia) 1.7 Carnegie-Mellon University 3.0 SUNY at Binghamton 1.7 North Carolina, University of 3.0 Boston University 1.7 Ohio State University 3.0 Clemson University 1.7 Pennsylvania State University 3.0 Oklahoma State University 1.7 Virginia, University of 3.0 Texas Tech University 1.7 Oregon, University of 2.9 Texas, University of (Arlington) 1.7 Colorado, University of 2.9 Wisconsin, University of (Milwaukee) 1.6 Duke University 2.8 Clarkson College of Technology 1.5 Kentucky, University of 2.8 Southern illinois University 1.5 Southern California, University of 2.8 Emory University 1.5 Notre Dame, University of 2.7 South Florida, University of 1.5 Rensselaer Polytechnic Institute 2.7 Bowling Green State University 1.3 Rochester, University of 2.7 Denver, University of 1.3 SUNY at Buffalo 2.7 Ohio University 1.3 California, University of (Santa Barbara) 2.7 Montana, University of 1.2 Louisiana State University 2.7 Stevens Institute of Technology 1.2 Massachusetts University of (Amherst) 2.7 Missouri, University of (Rolla) 1.1 Michigan State University 2.7 Western Michigan University 1.1 Tulane University 2.7 Alabama, University of 1.1 Arizona, University of 2.6 illinois Institute of Technology 1.1 Case Western Reserve University 2.6 Adelphi University 0.9 Dartmouth College 2.6 Saint Louis University 0.7 Northeastern University 2.5 New Mexico State University NA

259 Graduate Programs in Mathematics Table B. Ranked by Effectiveness in Educating Research Scholars

Princet.on University 2.8 California, University of (Riverside) 1.6 Califor,lia, University of (Berkeley) 2.7 Duke University 1.5 Harvard University 2.7 Florida State University 1.5 Massachusetts Institute of Technology 2. 7 Iowa, University of 1.5 Chicago, University of 2.7 Northeastern University 1.4 Stanford University 2.6 Oregon State University 1.5 New York University 2.6 California, University of (Davis) 1.4 Yale University 2.5 Claremont Graduate School 1.4 Wisconsin, University of (Madison) 2.4 Syracuse University 1.4 Michigan, University of 2.4 Virginia Polytechnic Institute & State Brown University 2.4 University 1.4 Cornell University 2.3 Florida, University of 1.4 California, University of (Los Angeles) 2.3 Tennessee, University of 1.4 Columbia University 2.3 Georgia, University of 1.3 illinois, University of 2.3 Kansas, University of 1.3 Minnesota, University of 2.2 Delaware, University of 1.3 California Institute of Technology 2.2 Georgia Institute of Technology 1.3 Brandeis University 2.1 Temple University 1.3 SUNY at Stony Brook 2.0 New Mexico, University of 1.3 Pennsylvania, University of 2.1 Wayne State University 1.2 Rutgers University 2.0 Iowa State University 1.2 Washington, University of 2.0 North Carolina State University 1.2 Purdue University 2.1 Polytechnic Institute of New York 1.2 Rice University 2.. 0 Oklahoma, University of 1.2 Indiana University 2.0 Vanderbilt University 1.2 Maryland, University of (Mathematics) 2.0 Cincinnati, University of 1.2 California, University of (San Diego) 2.0 Houston, University of 1.2 Northwestern University 2.0 Wesleyan University 1.1 Carnegie-Mellon University 1.9 Clemson University 1.1 CUNY Graduate School 1.9 Auburn University 1.1 Maryland, University of (Applied Math.) 1.9 Kent State University 1.1 Texas, University of (Austin) 1.8 Connecticut, University of 1.0 Washington University 1.8 Nebraska, University of 1.0 Ohio State University 1.8 SUNY at Binghamton 1.1 Pennsylvania State University 1.8 Oklahoma State University 1.0 Vrrginia, University of 1.8 Wisconsin, University of (Milwaukee) 1.0 Kentucky, University of 1.8 Colorado State University 1.0 Oregon, University of 1.8 Texas Tech University 1.0 Colorado, University of 1.8 South Carolina, University of 0.9 Johns Hopkins University 1.7 Missouri, University of (Columbia) 0.9 Utah, University of 1.7 Texas, University of (Arlington) 0.9 North Carolina, University of 1. 7 Emory University 0.9 Notre Dame, University of 1.7 South Florida, University of 0.9 Rochester, University of 1.7 Clarkson College of Technology 0.9 Tulane University 1. 7 Stevens Institute of Technology 0.8 illinois, University of (Chicago) 1.7 Missouri, University of (Rolla) 0.9 Rensselaer Polytechnic Institute 1. 7 Boston University 0.8 SUNY at Buffalo 1. 7 Southern illinois University 0.8 California, University of (Santa Barbara) 1.7 Denver, University of 0.7 Massachusetts, University of (Amherst) 1.7 Adelphi University 0.7 Michigan State University 1. 7 Bowling Green State University 0.6 Pittsburgh, University of 1. 7 Montana, University of 0.6 Southern California, University of 1.6 Ohio University 0.6 Louisiana State University 1.6 Western Michigan University 0.6 Arizona, University of 1.6 illinois Institute of Technology 0.5 Case Western Reserve University 1.5 Saint Louis University 0.5 Dartmouth College 1.6 Alabama, University of 0.4 SUNY at Albany 1.5 New Mexico State University NA

260 Graduate Programs in Mathematics Table C. Ranked by Improvement in Quality in Last Five Years

Utah, University of 1.7 Purdue University 1.1 Texas, University of (Austin) 1.6 Case Western Reserve University 1.1 Kentucky, University of 1.6 Rice University 1.1 lllinois, University of (Chicago) 1.5 Oregon, University of 1.1 Delaware, University of 1.5 Rensselaer Polytechnic Institute 1.1 California, University of (San Diego) 1.4 Nebraska, University of 1.1 Texas, University of (Arlington) 1.4 Montana, University of 1.1 Rutgers University 1.4 Cornell University 1.1 Pennsylvania State University 1.4 lllinois, University of 1.1 Northeastern University 1.4 Auburn University 1.1 Virginia Polytechnic Institute & State SUNY at Binghamton 1.1 University 1.4 Clarkson College of Technology 1.1 Texas Tech University 1.4 Western Michigan University 1.1 South Carolina, University of 1.4 California Institute of Technology 1.1 Maryland, University of (Mathematics) 1.3 Southern California, University of 1.1 Tennessee, University of 1.3 SUNY at Buffalo 1.1 California, University of (Los Angeles) 1.3 Princeton University 1.1 SUNY at Stony Brook 1.3 Columbia University 1.1 Washington, University of 1.3 Wisconsin, University of (Madison) 1.1 Georgia, University of 1.3 Washington University 1.1 Massachusetts, University of (Amherst) 1.3 Rochester, University of 1.1 Arizona, University of 1.3 Louisiana State University 1.1 Georgia Institute of Technology 1.3 Tulane University 1.1 North Carolina State University 1.3 California, University of (Davis) 1.1 Cincinnati, University of 1.3 Ohio University 1.1 Brown University 1.3 Alabama, University of 1.1 Clemson University 1.3 Stanford University 1.0 South Florida, University of 1.3 Carnegie-Mellon University 1.0 North Carolina, University of 1.3 California, University of (Santa Barbara) 1.0 Colorado, University of 1.2 California, University of (Riverside) 1.0 Wayne State University 1.2 Chicago, University of 1.0 Colorado State University 1.2 Harvard University 1.0 Minnesota, University of 1.2 Notre Dame, University of 1.0 Oregon State University 1.2 Wisconsin, University of (Milwaukee) 1.0 Houston, University of 1.2 Claremont Graduate School 1.0 Ohio State University 1.2 New Mexico, University of 1.0 Duke University 1.2 Missouri, University of (Columbia) 1.0 Temple University 1.2 Pennsylvania, University of 0.9 Maryland, University of (Applied Math.) 1.2 Dartmouth College 0.9 Pittsburgh, University of 1.2 Kansas, University of 0.9 Oklahoma, University of 1.2 Iowa State University 1.0 Connecticut, University of 1.2 CUNY Graduate School 0.9 Kent State University 1.2 Missouri, University of (Rolla) 0.9 Boston University 1.2 Denver, University of 0.9 Indiana University 1.2 New York University 0.9 Northwestern University 1.2 Syracuse University 0.9 Michigan State University 1.2 Florida State University 0.8 SUNY at Albany 1.2 Brandeis University 0.8 Florida, University of 1.2 Virginia, University of 0.8 California, University of (Berkeley) 1.2 Johns Hopkins University 0.8 Massachusetts Institute of Technology 1.1 Emory University 0.6 Michigan, University of 1.2 Polytechnic Institute of New York NA Iowa, University of 1.1 Bowling Green State University NA Vanderbilt University 1.1 Stevens Institute of Technology NA Wesleyan University 1.1 lllinois Institute of Technology NA Oklahoma State University 1.2 Adelphi University NA Southern lllinois University 1.2 Saint Louis University NA Yale University 1.1 New Mexico State University NA

261 Graduate Programs in Statistics/Biostatistics Table A. Ranked by Scholarly Quality of Program Faculty

California, University of (Berkeley) Southern Methodist University Statistics 4.9 Statistics 2.6 Stanford University, Statistics 4.9 Ohio State University Chicago, University of, Statistics 4.7 Statistics & Biostatistics 2.5 Wisconsin, University of (Madison) Pennsylvania, University of Statistics 4.3 Statistics 2.6 North Carolina, University of California, University of (Riverside) Statistics 4.0 Statistics 2.4 Cornell University, Statistics 4.0 Texas, University of (Houston) Harvard University, Statistics 4.0 Biomathematics {Anderson Hospital} 2.4 Iowa State University, Statistics 4.0 Florida, University of, Statistics 2.4 Princeton University, Statistics 4.0 North Carolina State University Columbia University Biomathematics 2.4 Mathematical Statistics 3.8 Missouri, University of (Columbia) Florida State University Statistics 2.3 Statistics 3.8 SUNY at Stony Brook Purdue University, Statistics 3.9 Applied Mathematics & Statistics 2.2 Washington, University of Connecticut, University of Biomathematics & Biostatistics 3.8 Statistics 2.1 Minnesota, University of Minnesota, University of Statistics 3.7 Biometry 2.2 North Carolina, University of New Mexico, University of Biostatistics 3.7 Mathematics and Statistics 2.1 Yale University, Statistics 3.7 Yale University California, University of (Los Angeles) Epidemiology & Public Health 2.0 Public Health & Mathematics 3.7 South Florida, University of Carnegie-Mellon University Mathematics 2.0 Statistics 3.5 Kansas State University illinois, University of (Urbana) Statistics 1.8 Mathematics 3.4 Pittsburgh, University of Colorado State University Biostatistics {Public Health) 1.9 Statistics 3.2 SUNY at Buffalo, Statistics 1.9 Rutgers University, Statistics 3.2 Wyoming, University of Michigan State University Statistics 1.9 Statistics & Probability 3.1 Delaware, University of Michigan, University of Applied Sciences 1.8 Statistics 3.1 Georgia, University of North Carolina State University Statistics & Computer Science 1.7 Statistics 3.1 Indiana University, Mathematics 1.7 Pittsburgh, University of Oklahoma State University Mathematics and Statistics 3.1 Statistics 1.7 Texas A&M University Temple University, Statistics 1.8 Statistics 3.1 Case Western Reserve University Iowa, University of, Statistics 3.0 Biometry 1.6 Rochester, University of Maryland, University of (College Park) Statistics 3.0 Mathematics 1.6 Virginia Polytechnic Institute Missouri, University of (Rolla) Statistics & Statistical Laboratory 2.9 Statistics 1.5 Johns Hopkins University Boston University, Mathematics 1.4 Biostatistics 2.8 Virginia Commonwealth University Oregon State University Biostatistics {Medical College) 1.1 Statistics 2.8 _A_l!lerican University Pennsylvania State University Mathematics, Statistics f!3 Computer Statistics 2.7 Science 0.9 George Washington University Michigan, University of Statistics 2.6 Biostatistics {Public Health) NA Kentucky, University of Statistics 2.6

262 Graduate Programs in Statistics/Biostatistics Table B. Ranked by Effectiveness in Educating Research Scholars

Stanford University, Statistics 2.8 Florida, University of, Statistics 1.4 California, University of (Berkeley) Kentucky, University of Statistics 2.6 Statistics 1.4 Wisconsin, University of (Madison) North Carolina State University Statistics 2.4 Biomathematics 1.4 Chicago, University of, Statistics· 2.3 Yale University Iowa State University, Statistics 2.3 Epidemiology & Public Health 1.4 Cornell University, Statistics 2.2 California, University of (Riverside) Florida State University Statistics 1.3 Statistics 2.2 George Washington University North Carolina, University of Statistics 1.3 Biostatistics 2.2 Missouri, University of (Columbia) North Carolina, University of Statistics 1.4 Statistics 2.2 Oklahoma State University Washington, University of Statistics 1.3 Biomathematics & Biostatistics 2.2 New Mexico, University of Minnesota, University of Mathematics and Statistics 1.3 Statistics 2.1 Pittsburgh, University of Purdue University, Statistics 2.1 Mathematics and Statistics 1.3 California, University of (Los Angeles) Texas, University of (Houston) Public Health & Mathematics 2.0 Biomathematics (Anderson Hospital) 1.3 Carnegie-Mellon University Minnesota, University of Statistics 2.0 Biometry 1.2 Columbia University Kansas State University Mathematical Statistics 2.0 Statistics 1.2 North Carolina State University Connecticut, University of Statistics 1.9 Statistics 1.1 Yale University, Statistics 2.0 SUNY at Buffalo, Statistics 1.1 Colorado State University SUNY at Stony Brook Statistics 1.9 Applied Mathematics & Statistics 1.1 Harvard University, Statistics 1.9 Temple University, Statistics 1.1 Princeton University, Statistics 1.9 Wyoming, University of Iowa, University of, Statistics 1.8 Statistics 1.1 Michigan, University of Georgia, University of Statistics 1.8 Statistics & Computer Science 1.0 Texas A&M University Pittsburgh, University of Statistics 1.8 Biostatistics {Public Health} 1.0 Michigan State University Boston University, Mathematics 0.8 Statistics & Probability 1.8 Case Western Reserve University Rutgers University, Statistics 1.8 Biometry 0.9 Virginia Polytechnic Institute Indiana University, Mathematics 0.8 Statistics & Statistical Laboratory 1.8 Virginia Commonwealth University Johns Hopkins University Biostatistics {Medical College) 0.8 Biostatistics 1.7 Delaware, University of Oregon State University Applied Sciences 0.7 Statistics 1.7 Maryland, University of (College Park) Rochester, University of Mathematics 0.7 Statistics 1.7 Missouri, University of (Rolla) Pennsylvania State University Statistics 0.7 Statistics 1.6 South Florida, University of Southern Methodist University Mathematics 0.6 Statistics 1.6 American University illinois, University of (Urbana) Mathematics, Statistics & Computer Mathematics 1.6 Science 0.4 Ohio State University Michigan, University of Statistics & Biostatistics 1.6 Biostatistics (Public Health} NA Pennsylvania, University of Statistics 1.4

263 Graduate Programs in Statistics/Biostatistics Table C. Ranked by Improvement in Quality in Last Five Years

Pittsburgh, University of Kansas State University Mathematics and Statistics 1.8 Statistics 1.0 Washington, University of Kentucky, University of Biomathematics & Biostatistics 1.8 Statistics 1.0 Carnegie-Mellon University Oregon State University Statistics 1.6 Statistics 1.0 California, University of (Riverside) Boston University, Mathematics 1.0 Statistics 1.6 Columbia University SUNY at Stony Brook Mathematical Statistics 1.0 Applied Mathematics & Statistics 1.5 Cornell University, Statistics 1.0 Texas A&M University Florida State University Statistics 1.4 Statistics 1.0 California, University of (Berkeley) Michigan, University of Statistics 1.3 Statistics 1.0 South Florida, University of Southern Methodist University Mathematics 1.3 Statistics 1.0 Delaware, University of Temple University, Statistics 1.0 Applied Sciences 1.3 Wyoming, University of Texas, University of (Houston) Statistics 1.0 Biomathematics (Anderson Hospital) 1.2 Minnesota, University of Chicago, University of, Statistics 1.2 Biometry 1.0 Purdue University, Statistics 1.2 Missouri, University of (Rolla) Virginia Polytechnic Institute Statistics 1.0 Statistics & Statistical Laboratory 1.2 Pittsburgh, University of Stanford University, Statistics 1.2 Biostatistics {Public Health} 1.0 Pennsylvania, University of Yale University, Statistics 1.0 Statistics 1.2 American University Wisconsin, University of (Madison) Mathematics, Statistics & Computer Statistics 1.2 Science 0.9 Florida, University of, Statistics 1.1 New Mexico, University of lliinois, University of (Urbana) Mathematics and Statistics 0.9 Mathematics 1.2 North Carolina State University Iowa State University, Statistics 1.2 Statistics 0.9 California, University of (Los Angeles) Oklahoma State University Public Health & Mathematics 1.1 Statistics 0.9 Colorado State University Johns Hopkins University Statistics 1.1 Biostatistics 0.9 Georgia, University of Michigan State University Statistics & Computer Science 1.1 Statistics & Probability 0.9 Maryland, University of (College Park) Missouri, University of (Columbia) Mathematics 1.1 Statistics 0.9 North Carolina, University of Rutgers University, Statistics 0.9 Biostatistics 1.1 North Carolina, University of Pennsylvania State University Statistics 0.9 Statistics 1.1 Princeton University, Statistics 0.9 Iowa, University of, Statistics 1.1 Virginia Commonwealth University Minnesota, University of Biostatistics {Medical College) 0.8 Statistics 1.1 North Carolina State University Ohio State University Biomathematics 0.7 Statistics & Biostatistics 1.1 Case Western Reserve University Yale University Biometry 0.6 Epidemiology & Public Health 1.1 Indiana University, Mathematics 0.6 Rochester, University of George Washington University Statistics 1.1 Statistics 0.4 Connecticut, University of SUNY at Buffalo, Statistics 0.1 Statistics 1.0 Michigan, University of Harvard University, Statistics 1.0 Biostatistics {Public Health} NA

264 Graduate Programs in Computer Science Table A. Ranked by Scholarly Quality of Program Faculty

Stanford University, Computer Science 5.0 Ohio State University Massachusetts Institute of Technology Computer & Information Science 2.4 Electrical Engineering & Computer Rice University, Mathematical Sciences 2.4 Science 4.9 Rutgers University, Computer Science 2.4 Carnegie-Mellon University Syracuse University, Computer Sciences 2.4 Computer Science 4.8 Arizona, University of California, University of (Berkeley) Computer Sciences 2.4 Electrical Engineering & Computer Indiana University, Computer Science 2.3 Sciences 4.5 SUNY at Buffalo, Computer Science 2.3 Cornell University, Computer Science 4.3 California, University of (Santa Barbara) California, University of (Los Angeles) Electrical & Computer Engineering 2.1 Computer Science 3.8 Pennsylvania State University illinois, University of (Urbana) Computer Sciences 2.1 Computer Science 3.8 Kansas, University of Yale University, Computer Science 3.5 Computer Science 1.9 Washington, University of Pittsburgh, University of Computer Sciences 3.4 Computer Science 1.9 Southern California, University of Vanderbilt University Computer Science 3.2 Computer Science 1.8 Texas, University of (Austin) Iowa State University Computer Sciences 3.2 Computer Science 1.7 Wisconsin, University of (Madison) Virginia, University of Computer Sciences 3.2 Applied Mathematics (lj Computer Maryland, University of (College Park) Science 1.7 Computer Science 3.1 Connecticut, University of Princeton University Electrical Engineering & Computer Electrical Engineering & Computer Science 1.7 Science 3.0 Iowa, University of, Computer Science 1.7 Brown University, Computer Science 2.9 Southern Methodist University Massachusetts, University of Computer Science & Engineering 1.6 Computer & Information Sciences 2.8 Washington State University New York University, Computer Science 2.8 Computer Science 1.5 Utah, University of, Computer Science 2.8 Michigan State University Georgia Institute of Technology Computer Science 1.5 Information & Computer Science 2.7 Washington University Minnesota, University of Computer Science 1.4 Computer Science 2.7 Case Western Reserve University North Carolina, University of Computer Engineering, Computing & Computer Science 2.7 Information Science 1.3 Pennsylvania, University of Missouri, University of (Rolla) Computer & Information Science 2.7 Computer Science 1.2 Rochester, University of Polytechnic Institute of New York Computer Science 2.7 Electrical Engineering & Computer SUNY at Stony Brook Science 1.2 Computer Science 2.7 Stevens Institute of Technology California, University of (San Diego) Electrical Engineering & Computer Electrical Engineering & Computer Science 1.2 Science 2.6 Texas A&M University California Institute of Technology Industrial Engineering 1.1 Computer Science 2.5 Kansas State University Columbia University, Computer Science 2.5 Computer Science 0.9 California, University of (Irvine) Oklahoma, University of Information & Computer Science 2.4 Electrical Engineering & Computer Duke University, Computer Science 2.4 Science 0.8 Northwestern University Michigan, University of Electrical Engineering & Computer Computer & Communication Sciences NA Sciences 2.4

265 Graduate Programs in Computer Science Table B. Ranked by Effectiveness in Educating Researeh Scholars

Stanford University, Computer Science 2.8 California, University of (Irvine) Massachusetts Institute of Technology Information & Computer Science 1.4 Electrical Engineering & Computer Rutgers University, Computer Science 1.4 Science 2.8 Indiana University, Computer Science 1.3 Carnegie-Mellon University Pennsylvania State University Computer Science 2.7 Computer Sciences 1.3 California, University of (Berkeley) California, University of (San Diego) Electrical Engineering & Computer Electrical Engineering & Computer Sciences 2.6 Science 1.2 Cornell University, Computer Science 2.5 Columbia University, Computer Science 1.2 illinois, University of (Urbana) Iowa State University Computer Science 2.3 Computer Science 1.2 California, University of (Los Angeles) Pittsburgh, University of Computer Science 2.2 Computer Science 1.2 Yale University, Computer Science 2.1 Virginia, University of Texas, University of (Austin) Applied Mathematics & Computer Computer Sciences 2.1 Science 1.2 Washington, University of California, University of (Santa Barbara) Computer Sciences 2.0 Electrical & Computer Engineering 1.2 Maryland, University of (College Park) Iowa, University of, Computer Science 1.1 Computer Science 1.9 Kansas, University of Wisconsin, University of (Madison) Computer Science 1.1 Computer Sciences 1.9 Vanderbilt University Princeton University Computer Science 1.1 Electrical Engineering & Computer Connecticut, University of Science 1.9 Electrical Engineering & Computer Utah, University of, Computer Science 1.9 Science 1.0 SUNY at Stony Brook Washington University Computer Science 1.8 Computer Science 1.0 Southern California, University of Southern Methodist University Computer Science 1.8 Computer Science & Engineering 1.0 North Carolina, University of Washington State University Computer Science 1.7 Computer Science 1.0 Pennsylvania, University of Michigan State University Computer & Information Science 1.8 Computer Science 0.9 Rochester, University of Polytechnic Institute of New York Computer Science 1.7 Electrical Engineering & Computer Brown University, Computer Science 1.7 Science 0.9 New York University, Computer Science 1.7 Stevens Institute of Technology Massachusetts, University of Electrical Engineering & Computer Computer & Information Sciences 1.7 Science 0.9 Minnesota, University of Missouri, University of (Rolla) Computer Science 1.6 Computer Science 0.8 Georgia Institute of Technology Case Western Reserve University Information & Computer Science 1.6 Computer Engineering, Computing & Ohio State University Information Science 0.8 Computer & Information Science 1.6 Texas A&M University Rice University, Mathematical Sciences 1.6 Industrial Engineering 0.7 California Institute of Technology Kansas State University Computer Science 1.5 Computer Science 0.6 Duke University, Computer Science 1.5 Oklahoma, University of Northwestern University Electrical Engineering & Computer Electrical Engineering & Computer Science 0.3 Sciences 1.5 Michigan, University of SUNY at Buffalo, Computer Science 1.4 Computer & Communication Sciences NA Arizona, University of Computer Sciences 1.4 Syracuse University, Computer Sciences 1.4

266 Graduate Programs in Computer Science Table C. Ranked by Improvement in Quality in Last Five Years

Georgia Institute of Technology California, University of (San Diego) Information & Computer Science 1.8 Electrical Engineering & Computer Rochester, University of Science 1.1 Computer Science 1.7 Cornell University, Computer Science 1.1 Washington, University of North Carolina, University of Computer Sciences 1.7 Computer Science 1.1 Columbia University, Computer Science 1.6 Ohio State University Indiana University, Computer Science 1.6 Computer & Information Science 1.1 Vanderbilt University Pennsylvania, University of Computer Science 1.5 Computer & Information Science 1.1 Brown University, Computer Science 1.5 Yale University, Computer Science 1.1 Rutgers University, Computer Science 1.4 Stanford University, Computer Science 1.1 Arizona, University of illinois, University of (Urbana) Computer Sciences 1.3 Computer Science 1.0 California, University of (Berkeley) New York University, Computer Science 1.0 Electrical Engineering & Computer Stevens Institute of Technology Sciences 1.3 Electrical Engineering & Computer California, University of (Los Angeles) Science 1.0 Computer Science 1.3 Utah, University of, Computer Science 1.0 Southern California, University of Washington University Computer Science 1.3 Computer Science 1.0 Duke University, Computer Science 1.3 Iowa, University of, Computer Science 1.0 Wisconsin, University of (Madison) Texas A&M University Computer Sciences 1.3 Industrial Engineering 1.0 Massachusetts, University of SUNY at Buffalo, Computer Science 0.9 Computer & Information Sciences 1.3 California, University of (Irvine) Texas, University of (Austin) Information & Computer Science 0.9 Computer Sciences 1.3 Michigan State University Virginia, University of Computer Science 0.9 Applied Mathematics & Computer Kansas State University Science 1.3 Computer Science 0.9 Connecticut, University of Pittsburgh, University of Electrical Engineering & Computer Computer Science 0.9 Science 1.2 California Institute of Technology Maryland, University of (College Park) Computer Science 0.8 Computer Science 1.2 Kansas, University of Minnesota, University of Computer Science 0.8 Computer Science 1.2 Southern Methodist University SUNY at Stony Brook Computer Science & Engineering 0.8 Computer Science 1.2 Polytechnic Institute of New York Washington State University Electrical Engineering & Computer Computer Science 1.2 Science 0.7 Iowa State University Princeton University Computer Science 1.2 Electrical Engineering & Computer California, University of (Santa Barbara) Science 0.7 Electrical & Computer Engineering 1.1 Case Western Reserve University Carnegie-Mellon University Computer Engineering, Computing & Computer Science 1.1 Information Science 0.4 Massachusetts Institute of Technology Pennsylvania State University Electrical Engineering & Computer Computer Sciences 0.4 Science 1.1 Michigan, University of Missouri, University of (Rolla) Computer & Communication Sciences NA Computer Science 1.1 Oklahoma, University of Northwestern University Electrical Engineering & Computer Electrical Engineering & Computer Science NA Sciences 1.1 Rice University, Mathematical Sciences 1.1 Syracuse University, Computer Sciences 1.1

267 Research Briefing Panel on Mathematics: History and Background by William Browder

In most respects the report speaks for itself, public, both scientific and general, as well as but I would like to give a little history and industry. background for this effort, as well as some of my During the last fifteen years, while there were general conclusions. certainly efforts to make the case for mathematics We are engaged in an effort to delineate the by various individuals and by the staffs of the case for mathematics in both an intellectual and NSF Mathematical Sciences Section and the other a political way. Both halves of this effort are agencies, there was no clear forceful voice of necessary, on the one hand to show the importance the mathematical community speaking to the of the subject and its health, and on the other to real needs, and the serious support problems show its needs and the problems of its support, of mathematical research. Mathematicians felt and to demonstrate how, and on what scale, to isolated from, and unappreciated by, the general attack these problems. scientific community. Many mathematicians themselves doubted the importance or relevancy Such an effort must be made in a strong and of their work, and the long lead time from coherent way, by the most forceful and effective theoretical innovation to practical application means available. The effort must be maintained seemed to them infinite. over a long period of time to be effective in It seems clear now that mathematicians and the long run, and different initiatives should the mathematical community suffered from a be coordinated to avoid fragmentation of effort, severe case of modesty, or inadequate aspiration. the cancellation effects of diverse or conflicting Many of the new linkages of mathematics and directions, or inept approaches. It should be its applications have shown clearly even to directed into all channels available, toward the mathematicians, that mathematics, as well as Congress and Administration, and toward the being a profound subject in its own right, is of EDITOR'S NOTE: This article is the first of central significance in the development of science several in this issue of the Notices devoted to and technology, both in its uses and in the mathematics and government and, in particular, to philosophical frame of reference it provides. Our government support of research in the mathematical efforts to communicate with scientists in other sciences. In the first article WILLIAM BROWDER areas, and with science adininistrators on the of Princeton, who is Chairman of the Office of serious support problems in our field have usually Mathematical Sciences of the National Academy of been met with sympathy, understanding, and very Sciences and chairman-designate of the Advisory active help. Subcommittee for Mathematics of the National We are not isolated, intellectually or institu­ Science Foundation, describes the events leading to tionally, unless we ourselves choose to be. the preparation and presentation of the Report of the Briefing Panel on Mathematics. If support of mathematics has suffered in the The second article contains the text of the past, it may be partly due to a lack of vigor in Briefing Panel Report itself. presenting our case, and a lack of confidence in Next is this year's version of the annual report · the true value of our own subject. analyzing the budget request of the National Science On the other hand, the new direction in NSF Foundation and especially that of the Mathematical support must be attributed not only to new and Sciences Section of the Foundation. influential voices raised in the discussion, such The fourth article presents an account of the as that of Edward E. David, Jr., chairman of Federal support for the mathematical sciences in the Committee on Resources in the Mathematical the Research and Development effort proposed Sciences, but also to a new interest in, and in the Administration's budget request submitted receptivity to, the message in the Administration, to Congress January 31; it was prepared by most notably in the White House and its science WILLIAM H. PELL for the Conference Board of advisor George Keyworth, and in the NSF and its the Mathematical Sciences. director Edward A. Knapp. Finally KENNETH M. HOFFMAN of the Mas­ This also reflects the stated cominitment of sachusetts Institute of Technology, who is chairman the current adininistration to improvement of the of the Society's Committee on Science Policy, support of basic research, a cominitment which we presents a survey of the entire situation from the look forward to seeing manifested in the research point of view of a close observer of the problems involved in Federal support for mathematics. budgets of the other federal agencies, including DOD.

268 The ongm of my own involvement in these one of them being mathematics. I was asked questions dates to the time of my becoming to chair the Mathematics Panel, and with some Chairman of the Office of Mathematical Sciences trepidation, I agreed and drove off to Ohio to of the National Academy of Sciences-National deliver my daughter to college. Research Council. My predecessor, G. D. Mostow, My misgivings were threefold: had just given a clear demonstration of the First, if we were to single out certain promising possibilities of this position and the opportunities areas of research, and the result was, by for communication with other sciences that it administrative fiat, to divert large support into represented in the role that it played in the those directions at the expense of momentarily creation of the Postdoctoral Fellowship program less fashionable areas, the end result might be a in the NSF. The idea emerged with strong support serious distortion of the support of mathematical from AMPS (Assembly of Mathematical and research and a disservice to scientific progress. Physical Sciences-NAB/NRC), and, finding favor Secondly, the time scale for assembling a panel, in the National Science Board, was implemented meeting, agreeing, and preparing a document was in the Mathematical Sciences Section of NSF. extremely short. We were to have a written On this occassion, the need for this program in document at the office of COSEPUP by October mathematics, and the paucity of other resources 11, 1982, to be presented at the COSEPUP meeting in this area, could be, and was, demonstrated October 16, to be presented at the White House to groups of scientists in diverse areas, struck (to Keyworth) on October 26. a responsive chord, and received their strong Thirdly, the term mathematics might be support. construed too narrowly. It was therefore This was a time when the disastrous financial important that the panel membership reflect picture in mathematical research was becoming the breadth and diversity of the mathematical understood widely in the mathematical com­ sciences. munity, and mechanisms of an indirect nature, However, I felt that this was too important such as the NSF Institutes, were being proposed an opportunity to miss, possibly a unique one. to rectify it. The presentation of a case for the support of With Mostow's experience of the sympathy mathematics on this level of government had not and interest of other scientists in the welfare of occurred before, and the very act of doing it, even mathematics, an idea of a direct approach to if imperfect, or flawed in its recommendations, the serious problems of support of mathematical might have a positive benefit. research was suggested to me by J. J. Kohn: The day after Labor Day, September 7, we Set up a high-level committee composed of both began assembling a panel, the panel met in New mathematicians and scientists to . consider the York on September 25 and 26, J. K. Goldhaber problems of the mathematical research enterprise and I wrote the first draft of the report September and report on it in such a way as to have maximal 28 and 29, numerous comments and suggestions impact on the decision-making process. poured in from panel members and others during This Committee on Resources in the Mathe­ the following week, and the final draft was matical Science (CRMS), set up under AMPS, delivered Sunday morning, October 10, to the managed to attract a truly distinguished group Academy, where it was typed in the afternoon and of scientists and mathematicians, an indication of delivered on Monday, October 11, to COSEPUP the value placed on mathematics as a discipline for distribution. in the scientific community. The report, due The panel was by no means unanimous on later this year, should have a significant impact. all questions, but in the course of the meeting The act of running this study and the activities a consensus emerged (see the account by Hirsh of its chairman E. E. David and its executive Cohen in the SIAMNEWS, January 1983, page director Kenneth Hoffman, may already have had 16). There was general agreement on the dangers some impact, as well as the activities of Hoffman of distorting the flow of research by arbitrarily as chairman of the invigorated Science Policy diverting resources heavily into currently hot Committee of the AMS. areas, with some reservations on this point in This was where matters stood to my knowledge, the case of scientific computing. The general when on September 1, 1982 (the morning after feeling was that unlike some areas of science in my return from a trip to Europe), I received which great funding efforts might be necessary to a call from the executive director of COSEPUP solve important scientific problems (i.e., for par­ (Committee on Science, Engineering and Public ticle accelerators, large astronomical instruments, Policy-NAB/NRC), Allan Hoffman (no relation oceanographic vessels), in mathematics there were to Ken). Dr. Keyworth, the Science Advisor to no scientific reasons to justify major diversions the President, had asked the Academy, through of funds into special directions. The vast range the mechanism of COSEPUP, to prepare briefings of mathematical research was appreciated, and for the Office of Science and Technology Policy the necessity of its support, both in diversity of (OSTP) of the White House on areas of special subject and geography was endorsed by all, as well opportunities in research in seven areas of science, as the high priority which should be placed on

269 the encouragement of young people, who would 'vote with their feet' on the most important Research Briefing Panel on Mathematics directions to pursue. The fact that important The members of the Research Briefing Panel research is done by mathematicians spread over on Mathematics were WILLIAM BROWDER a tremendous variety of institutions all over the (chairman), Princeton University, HERMAN country was clear to all, as was the importance of CHERNOFF, Massachusetts Institute of Tech­ support and encouragement for these productive nology, HIRSH COHEN, IBM Watson Re­ people wherever they are found. search Center, LOUIS N. HOWARD, Florida The panel document was presented to Dr. State University, ARTHUR M. JAFFE, Harvard Keyworth on October 26, at his office, by G. D. University, PETER D. LAX, Courant Institute, Mostow, I. M. Singer, and myself, in the presence New York University, JOEL L. LEBOWITZ, of Allan Hoffman of COSEPUP, Frank Press, Rutgers University, New Brunswick, G. D. President of the NAS, and of Douglas Pewitt and MOSTOW, Yale University, MICHAEL RABIN, other members of Keyworth's staff. Our interview Harvard University, I. M. SINGER, University of lasted about an hour and proceeded in a very California, Berkeley, and SHING-TUNG YAU, cordial atmosphere. Institute for Advanced Study. The same group with the addition of J. K. The members of the Panel were assisted in Goldhaber and J. Glimm, chairman of the their tasks by the following guests: DONALD NSF Advisory Subcommittee for Mathematics, AUSTIN, U.S. Department of Energy, STUART presented the briefing document to Dr. Knapp, BRODSKY, U.S. Department of Defense, JAG­ director of NSF, at an hour-long meeting on DISH CHANDRA, U.S. Army Research Office, December 17, in the presence of a number of NSF DAVID Fox, Mathematical and Information staff members. (Our original group had previously Sciences, Bolling Air Force Base, KENNETH presented it to Knapp, informally, on October 26, HOFFMAN, Massachusetts Institute of Technol­ when he was still assistant director for MPS). ogy, E. F. INFANTE, National Science Founda­ At both these interviews, our presentation tion; and by the following members of the NAS was received with great interest, and we came staff: J. K. GOLDHABER, Executive Secretary, away with the message that there was a serious Office of Mathematical Sciences, and ALLAN R. commitment by the Administration to improving HOFFMAN, Executive Director, Committee on support of basic research and an interest in and Science, Engineering and Public Policy. appreciation of the high quality of mathematical ·research in the U.S., as well as concern for its of Mathematical Sciences of NAS/NRC and its welfare and development. executive director J. K. Goldhaber. The existence The importance of the ongoing effort of CRMS of this office in the Academy made possible the (the David Committee) and its preliminary work quick reaction to the OSTP request, gave a context to our efforts is worth emphasizing again. Its final in which to found the CRMS committee, provided report should be an important next step in our a channel of communication to other sciences, efforts. and of paramount importance, provided a locus The help and advice of E. F. Infante, Director where the health of the mathematical sciences as a of the Division of Mathematical and Computer discipline was the primary object of concern. The Sciences of the NSF, in our work was of great continued existence of this Office in its present importance. Our work utilized presentations form is now in question, due to financial stringency prepared over the years by Infante, and others in the Academy; and the long term effect of its in the NSF staff, in particular William G. Rosen, elimination on the mathematical enterprise is a Head of the Mathematical Sciences Section, who cause for worry. have been fighting the battle for many years. Finally the active presence in Washington of a Maureen Virkham who hand carried a 'voice of mathematics,' in the person of Kenneth manuscript to Washington, and Judy Marshall Hoffman as chairman of the Science Policy Com­ who typed draft after draft, revision after revi­ mittee of AMS has played an important role. The sion, at short notice and at outrageous times, constant contact between government agencies deserve special thanks. and the mathematical community through this Let me again emphasize the crucial role mechanism is of great significance. played throughout this history by the Office

270 Report of the Research Briefing Panel on Mathematics

Preface. In response to an invitation from OSTP or to photocopy important documents. At a to present a research briefing on the current state time when the development of research in mathe­ of mathematical research in the United States, matics is making unparalleled progress, when the a panel was convened on September 25 and 26, influence of mathematics is pervasive in the other 1982. In addition, opinions were solicited from sciences, when American mathematicians lead the the chairmen of the top 27 research departments world in most areas, research support for graduate of mathematics in the United States. students and young Ph.D.'s, the lifeblood of the This document is an account of the deliberations enterprise, is insufficient to ensure the quality of of the panel. Our charge was to identify special future generations. opportunities in the mathematical sciences. We In Section 1, we describe some of the exciting could identify many promising areas ripe for contributions and important developments in development, but it is our belief that the most mathematical research and the highly promising dramatic mathematical tendency in recent years opportunities that arise from the recent opening is the drawing together of mathematics (often the of new bridges between mathematics and other most abstract science) with other sciences and sciences. In Section 2, we document the decay the interplay between them. Some results have of the infrastructure and support of mathematical already been achieved; with encouragement, the research, and indicate possible actions to prevent interplay can be made increasingly fruitful. further decay of the mechanisms that have enabled U.S. mathematics to flourish. In Section 3, we A healthy mathematical enterprise must have recommend steps to be taken to rehabilitate the effective means for nurturing new developments. ailing infrastructure and to provide the flexibility (1) support of gifted young These means involve needed for the exploitation of new opportunities. investigators who will choose from among the new directions, and (2) flexibility for scientific leaders allowing them to develop and expose recent Section 1: Some Recent Developments breakthroughs. Unfortunately, severe underfund­ in Mathematical Research ing has limited the mathematical community's will be in two parts, the first capacity to respond in these ways. For this Our exposition of the pervasive influence of reason, in addition to spelling out the current giving examples mathematics in other sciences and the second state of mathematical research, we have tried recounting some of the recent significant advances to specify the additional resources that will be in theoretical mathematics. Some speculations needed to expolit new opportunities. on future possibilities will be interspersed. (For Introduction. There is a striking contrast between a somewhat fuller review, we refer the reader the importance of mathematical sciences in the to the chapter "On some recent developments United States and the perception of them. On in mathematics" in Outlook for Science and the one hand, mathematics and its applications Technology, the Next Five Years, NAS, W. play an ever increasing role in science, technology, Freeman & Co., San Francisco, 1982, pages 467- business, and everyday life in this country, and 510.) on the other, mathematical research is almost Influence and Applications of Mathematics. The completely unknown to and poorly understood by research area of mathematics most used today in the general public, and even the scientific public. technology is numerical analysis and mathemati­ The reputation and achievements of the cal modeling. In the area of industrial design, for American mathematical community make the example, a given process must be described and United States first in the world in mathematics. understood in a mathematical way, and the details Yet, at the same time support for mathemati­ of the mathematical description will interact with cal research erodes at a steady rate, and the the design process. Analysis and design become institutional infrastructure that supports the en­ mathematically interdependent. For example, the terprise exhibits symptoms of decay. In a year in design of the fuel efficient, weak shock transonic which two out of three of the quadrennial Field's airfoil, currently flying on the Boeing 767, would medalists (the mathematical equivalent of the not have been possible without the mathematical Nobel Prize) are American mathematicians, we work of Garabedian, Cole, and Jameson. On find highly ranked departments of mathematics the speculative side, a project to mathematically lacking in enough research support to send their model the human circulatory system now under­ most productive people to professional meetings way might eventually have important medical

271 consequences. Among them would be the pos­ Other new and important mathematical inputs sibility of indirect ways to measure the heart in into physics have been the introduction of abstract a situation where the direct measurement of the probability into statistical mechanics and material heart itself is impractical. Computer-aided design science with the notion of Gibbs state and the (CAD) is now used in the design of artificial heart input of the theory of dynamical systems and valves, using a mathematical model of the left ergodic theory into the study of turbulence. All side of the heart. these phenomena illustrate the drawing together Mathematical design of efficient compression of abstract and applied mathematics and their and turbine blades is a reality today, while fruitful interaction. the design of efficient combustion chambers is a As a result of the rise of the computer, subject of intense research. the theory of computation has become an area In national defense, the replacement of ex­ of mathematical research. Solidly based on perimentation by numerical modeling, made pos­ methods and fields in the mainstream of modern sible by advances in computers and dramatic mathematics, such as probability, combinatorics, improvements in mathematical algorithms, has algebraic geometry, and number theory, it creates resulted in great savings in the cost and im­ important tools for the practicing computer provement in the quality of design. This has scientist. The main themes in this new area been particularly significant in weapons-related are the study of algorithms and of programming. research and development, where experimentation Efficient algorithms often have important prac­ is costly, dangerous, and physically impossible in tical importance. Notable examples are the Fast the early stages of a project. Fourier Transform with its application to signal the recently developed randomized In economics, mathematics is playing an ever processing and algorithms in number theory and finite fields with increasing role as witnessed by three recent Nobel their application to error correcting codes and prizes in mathematical economics. cryptography. In oil prospecting, mathematical results are Developments in coding and cryptography used in a fundamental way in the separation of provide dramatic examples of unexpected ap­ primary signals from multiple reflections. The plications of 'pure' mathematics to applied areas. modern theory of inverse scattering is becoming a Number theoretical work of A. Weil in 1948 was basic tool in this area. Mathematical modeling is applied to coding theory some years ago. Last year important in the study of efficient secondary oil a group of Soviet mathematicians showed how to recovery. use the latest work of Deligne, Rapoport, Ihara, In electrical engineering, the mathematical work and Langlands, in the most abstract areas of al­ of Wiener has proved fundamental in several gebraic geometry, in the design of error-correcting areas, and mathematical control theory plays an codes of a theoretical efficiency heretofore deemed important role. impossible. In medicine, great advances in diagnostic In the field of robotics, the development techniques (tomography-the CAT scanner; and of automatic industrial processes depends on NMR) are strongly related to mathematical successful mathematization or modeling of the research. In the latter, methods from singular processes involved. In many industrial areas, integrals, complex function theory and Hilbert progress is in its infancy, and some of the most space were used. Statistics and statistical methods simple tasks seem the least likely to yield to are crucial in epidemiology, drug testing, and automation. It is extremely difficult to design a many other areas, and mathematical modeling robot arm with sensors that will enable it to avoid is an important tool in the development of new obstructions while picking up a target object, one drugs. The list could be extended indefinitely of the most routine of human abilities. The with examples drawn from biology, chemistry, parameters of this problem can be interpreted as neuroscience, and other sciences. a problem in algebraic geometry, and progress There are many recent examples where math­ here may have some effect on the solution of other ematical research, driven by the inner dynamic practical problems. of the subject without reference to practical prob­ There are current proposals for initiatives in lems, has been found to be of great significance large-scale scientific computing that would have in other areas. An outstanding illustration of sizable components of mathematical research and this has been in the development of Gauge Field important applications to applied mathematics. Theory in physics. Nobel Prizewinner, C. N. Yang In pure mathematics, Thurston (one of this year's wrote, "I found it amazing that gauge fields are Field's Medalists) has made a surprising use of the exactly connections on fibre bundles, which the computer as an experimental tool in his work on matheJ!laticians developed without reference to the topology of 3-dimensions, although the solution physical world." Algebraic geometry produced all of the famous four-color map problem a few years self-dual solutions for the Yang-Mills equations. ago required the computer essentially in the proof. But the physical theory led also to important Recent advances in computer technology and consequences in topology, as we will relate later. software are having a deep influence on the

272 nature of work subjected to statistical analysis, • Physicists have introduced gauge theories in on the methods of analysis, and on theoretical 4-dimensions (space-time) as a unifying principle questions in statistics. The computer and in field theory. The study of Yang-Mills equations space technologies provide vast amounts of high of motion in this context led S. Donaldson to a dimensional multivariate data which standard remarkable description of certain 4-dimensional classical methods no longer fit, because the spaces. A little earlier, M. Freedman, using purely underlying assumptions of normality and linearity topological methods, had produced a powerful are no longer satisfied. Methods justified comprehensive theory of 4-dimensional manifolds. under those assumptions would lead to serious These results of Donaldson and Freedman have errors. Novel methods of pattern recognition and combined to give the following result in the robust regression and new methods of graphical topology of 4-dimensional space. In all other representation that allow the comprehension of dimensions there is essentially one mode of doing these data are being developed. The interactive calculus in a Euclidean space (Rn has a unique kinematic displays of Friedman and Tukey are differential structure for n # 4), but an entirely examples. different situation exists in dimension 4 ~there are New interactive statistical packages with at least two different structures on R ). This properly built-in diagnostics will permit naive qualitative difference between dimension 4 and users to observe phenomena formerly accessible other dimensions is a startling development for only to trained and ingenious statisticians. topology, and it may also be the reflection of some Statistics has aided computer science in that deeply significant physical principles. simulations and experimental designs have been • The unifying role of group symmetry in used to find good hardware configurations and geometry, so penetratingly expounded by Felix software designs. Klein in his 1872 Erlanger Program, has led to a century of progress. A worthy successor to the Progress in Theoretical Mathematics. By Erlanger Program seems to be Langlands' program "theoretical mathematics" we mean research to use infinite dimensional representations of Lie motivated by the inner dynamic of the subject groups to illuminate number theory. rather than by the needs of research in other That the possible number fields of degree sciences. It is remarkable how often much of this n are restricted in nature by the irreducible research, seemingly irrelevant, turns out to have infinite dimensional representations of GL(n) was important practical impact. Who, for example, the visionary conjecture of R. P. Langlands. in the 1920s and 1930s would have predicted that His far-reaching conjectures present tantalizing the most abstract work in mathematical logic­ problems whose solution will lead us to a better recursive functions and "Turing machines" - understanding of representation theory, number would provide the philosophical framework for theory and algebraic geometry. Impressive von Neumann's introduction of the stored pro­ progress has already been made, but very much gram computer, in which the instructions to the more lies ahead. machine can be manipulated and modified by the Closely related to the Langlands program is the machine itself? The development led eventually remarkable and mysterious connection between to a multibillion dollar industry. counting points in finite spaces and computing the Progress has been spectacular in recent years topological invariants of continuous spaces. First and we cite some notable examples: propounded in the Weil conjecture, the connection -The work of Deligne, proving the famous "Weil is being made more accessible by the Goreski­ conjectures" in number theory. MacPherson-Deligne homology theory. The whole -The classification of finite simple groups, as the thrust of these developments is to force the end of a 20-year effort. next generation of mathematicans to embrace -The work ofYau on the Calabi conjecture, with heretofore widely separated areas of mathematics. important applications to algebraic geometry. The expected unifications are awesome. -The work of Thurston, showing how to employ • In analysis, the old problem of the regularity methods of (mostly non-Euclidean) geometry properties of the Cauchy integral (for Lipschitz curves) was recently solved by the work of to attack problems in the topology of 3- Calderon, Coifman, Mcintosh, and Meyer. Crucial dimensions. to the solution of this problem were the techniques -The work of Khachian on polynomial algo­ of Hardy spaces developed within the last decade, rithms in linear programming. as well as recent methods for dealing with singular -The discovery of solitons and strange attrac­ integrals with "rough" coefficients. It seems very tors. likely that these ideas will be applicable to a -The work of Connes on operator algebras. host of important problems in partial differential Instead of continuing this list at length, we give equations, as is indicated by their role in recent in greater detail short accounts of some recent advances in the solution of "Kato's conjecture" dramatic examples, starting with a startling (dealing with square roots of Laplacians) and advance in pure mathematics resulting from solutions of parabolic equations with minimal interaction with physics. smoothness assumptions.

273 Section 2: The Mathematical has clear potential relation to the goals of a Research Enterprise mission-oriented agency. e) The number of top-ranking graduate stu­ In analyzing the state of mathematical research dents seems to be declining, and many of them and its needs, we must keep in mind special are from abroad. features that distinguish mathematics from the f) The strength of some of the leading depart­ other sciences. Among those features are these: ments of mathematics is being undermined by the 1. Mathematics is the most labor intensive of lack of federal funding for research, a lack that all sciences. Little equipment is involved, except the universities cannot replace, especially in states for computers, which are heavily used in statistics whose economies are suffering. and areas of applied mathematics and are an Yet the small scale of the mathematical experimental tool for a few pure mathematicians. enterprise would make it rather inexpensive to 2. The vast majority of research mathe­ alleviate many of the serious shortcomings in the health maticians are employed in universities as teachers. support configuration, and to ensure the Industry and national laboratories support only a and vitality of American mathematical research handful. into the next century. research by 3. Very few federal agencies support research As the support of mathematical federal agencies has eroded over the last decade, in mathematics. NSF supports 60 percent of all extent, taken up the research in mathematics and almost 100 percent the universities have, to some support, as for example with postdoctoral research of pure mathematics; most of the rest is supported instructorships. The support has, however, by DOD and DOE. This contrasts strongly with become more and more difficult for the financially other disciplines. troubled universities to continue. 4. The magnitude of the total research support of mathematics by the federal government in Support in the Mathematical Sciences. Currently, comparison with other fields is miniscule, less the United States ranks first in the quality of than $60 million annually. research in the mathematical sciences. But the 5. Mathematics is "small science." Though vitality conceals a variety of problems that, if collaboration among two or three researchers left unsolved, will inevitably lead to a substan­ is not uncommon, large projects with many tial deterioration in the nation's mathematical researchers devoted to specific goals are relatively sciences' research enterprise. The same can be rare. Mathematics thrives on the interaction of said substantially of other sciences, but it is our independent viewpoints and different approaches. purpose here to document the special strains in 6. The health of the mathematical enterprise mathematics that are reaching the crisis stage. in the United States hinges on the strength and The consequences of the financial stringency are vitality of the departments in the leading research falling most heavily on the young mathematicians, universities. graduate students, and recent recipients of the research support is available to The following special factors have strongly Ph.D. Little students, though many teaching assis­ influenced the pattern of decay in the support of graduate tantships are available, particularly at the large mathematical research that we perceive. state schools. This means that mathematics a) It is now generally accepted that the impact graduate students, unlike those in other sciences, of inflation is much greater in labor-intensive seldom have the opportunity to work full time enterprises than in the general economy. The in research. The lack of postdoctoral research impact of declining resources and inflation has, appointments in mathematics creates a similar therefore, been most severe in mathematics. problem for the young Ph.D.'s who, in addition, b) The universities as a whole are subject to are finding it increasingly difficult to obtain re­ the same effect, so that the resources available search grants. At a time of real opportunities, to them as the main supporters of mathematical in the drawing together and mutually fruitful in­ research have dwindled proportionately. teraction of mathematics and applications, the c) There has been no organization of math­ young innovators who will be needed to exploit the ematicians expressing their descipline's support opportunities are not receiving the nourishment requirements for research. In sciences needing they need for full development. large instruments or projects to achieve their The chairman of a prestigious mathematics scientific goals, organized support has evolved department writes (in a letter to the Panel): and served effectively. But in the small-scale "Mathematical research has been flourishing in individualistic atmosphere of mathematics, no the past decade, but the institutional structure of mechanism has evolved for calling attention to mathematical research is in trouble. Recruitment the alarming decline in funding. of young talent for the future looks to be in d) Inflexibility is inevitable when few funding even more serious trouble. The level of research agencies support mathematical research; inves­ support has been very low in terms of the tigators not supported by NSF, for example, often percentage of active research people supported, have no place else to turn, unless their research and recent cuts in support have produced signs of

274 a serious deterioration of morale especially among problem areas, and for each area estimate the younger mathematicians." dollar cost of alleviating the problem. Another chairman writes: "We are some one TABLE 1 hundred in number, we are invariably ranked among the top twelve departments in the country, Percentage of Research in we continue to recruit good graduate students, Universities Not Sponsored and I claim with confidence that of the one November 1978-0ctober 1979 hundred at least ninety are seriously engaged in research and scholarship. Yet, after two severe Engineering 16 years, we are down from one-half to about one­ Environmental Sciences 16 Life Sciences 13 third of the faculty on NSF grants. Moreover, Mathematical/Comouter Sciences 59 we have sustained these severe losses without any Physical sciences 22 sense of the prevalent quality of work having (Source: Activities of science and engineering faculty in declined at all; on the contrary, several colleagues universities and 4-year colleges 1978-1979; Surveys of science have lost grants in the very year when they have resources series NSF final report, NSF 81-323, page 2.) done their best work. . . . At the same time, universities are increasingly affected by lack of TABLE 2 money. Here, for example, loss of NSF grants has reduced departmental income from overhead Estimated Federal Obligations just when the university, which in any case had for Researchl Performed at always counted on strong departments like ours Universities and Colleges, FY 1981 to earn much of its research support outside, is quite unable to raise the level of state support. Engineering $ 350,208 Also, of course, we find little endowment money Environmental Sciences 332,063 coming in earmarked for mathematics." Life Sciences 2,088,893 Mathematics 55,906 Yet another writes: "Many young mathe­ Physical Sciences 511,638 maticians are discouraged at their prospects for a successful career in mathematics because of (1 "Research" does not include "R&D Plant" which is defined as follows: "R&D plant (R&D facilities and decreased research funds, poor salaries and the fixed equipment, such as reactors, wind tunnels, and radio shortage of openings in universities. Several I telescopes) includes acquisition of, construction of, major know are actively looking for jobs in other fields repairs to, or alterations in structures, works, equipment, where they can expect much better treatment facilities, or land, for use in R&D activities at federal or non-federal installations. Excluded from the R&D plant economically, and even the top departments are category are expendable equipment and office furniture finding it increasingly difficult to attract qualified and equipment. Obligations for foreign R&D plant are graduate students." limited to federal funds used in support of foreign research What is the research support picture in the and development.") Ibid, page 2. (Source: Federal funds for research and development, Fiscal mathematical sciences? One can glean an Years 1979, 1980, 1981, NSF 80-318, page 117.) indication from Tables 1 and 2. Lest the disparity in funding exhibited in Postdoctoral Positions in the Mathematical Table 2 be totally attributed to differences in Sciences. Table 3 (based upon data in Academic the numbers of professionals in the various areas, science: Graduate enrollment and support for we note that in January 1980 the numbers of 1980, NSF 81-330, Table A-30) gives dramatic full-time scientists and engineers at doctorate­ evidence of the disparity in the numbers of granting institutions in the various areas were as postdoctorates in various sciences and in engineer­ follows: ing. Engineering 20,511 TABLE 3 Life Sciences 93,309 Physical Sciences 16,845 Number of ?ostdoctorates in All Environmental Sciences 5,891 Graduate Institutions: 1980 Mathematical Sciences 9,146* Engineering (*Data from Academic sciences: Scientists and engineers, 981 NSF 81-307, Table B-5. These figures include both Environmental Sciences 3ll research and non-research scientists and engineers.) Life Sciences 11,715 Mathematics 143 In light of these data, it is not surprising that Physical Sciences 4,261 departmental chairmen speak of discouragement and of deterioration of the morale of young The excellence of science in the United States mathematicians. If it is in the national interest today derives from postdoctoral opportunities in to maintain a healthy and vigorous mathematical the past. Clearly, if the current postdoctoral sciences research enterprise, then it is imperative pattern in the mathematical sciences persists, we that conditions contributing to this deterioration will jeopardize the quality of the mathematical be altered. We discuss the more important sciences at our leading universities in the years

275 to come. To be sure, the pool of outstanding be used in offering it: fellowships in NSF, DOE, candidates for tenure positions at the leading DOD; institutional support at major centers and three or four universities will be large enough. research institutes; enhancement of grants by However, one can expect a serious drop in the providing support for postdoctoral positions. quality of candidates at the next five ranking universities, and an even more serious drop at the Research Grants in the Mathematical Sciences. next ten and twenty. Table 4 indicates many categories where fund­ ing for the mathematical sciences is markedly There is a clear need to provide a significant avail­ number of outstanding recent Ph.D.'s in the insufficient. The amount of dollar support mathematical sciences with the opportunity to able for graduate students in the mathematical with a devote full-time to research in association with sciences is inadequate. We illustrate this a major scientific figure of their own choosing. very particular example. A member of this Panel The most creative future mathematicians in the recently directed two Ph.D. dissertation students, United States will emerge from this group, and it one in the mathematical sciences, the other in must be nurtured. We are not advocating a large another science. The non-mathematics student move away from teaching, the traditional mode was federally supported, and his sole task was of mathematical support. Typically, all graduate to devote his time to study and research. The students will do some teaching while preparing mathematics student, on the other hand, was for their doctorates. However, we are suggesting a university supported and, in addition to study small shift to allow promising young investigators and research, had to devote his time to grading a few years after their degrees to develop their papers, teaching, registering students, and holding research talents. office hours. The difference in treatment did not The Panel estimates that there should be an go unnoticed by these students nor, assuredly, did additional 120 postdoctoral appointments each others fail to notice it. Mathematics students year, each appointment being for two years. At need the opportunity for a year or two of uninter­ a cost of $25,000 per appointment per year, this rupted research during their graduate study to amounts to an increment of $6 million per year. fully develop their research abilities, as students It is also essential, in the view of the Panel, in most other fields of science do. that there be flexibility in the nature of the Along with decreased research support there is postdoctoral support. A variety of modes should a rapid increase in the demand for mathematics

TABLE 4 - Budgetary Categories of NSF Awards: FY 1981

Mathematical Sciences MPS Directoratel

Percentage Percentage Budgetary Constraints Amount of Total Amount of Total

Personnel: Senior Personnel $11,710 41.6 $ 23,253 10.2 Postdoctoral Associates 1,699 6.0 17,068 7.5 Graduate Students 1,189 4.2 26,611 ll.7 Other Personnel Costs2 2,538 9.0 29,027 12.7 Subtotal: Wages, Salaries, and Fringe Benefits 17,136 60.8 95,959 42.1

Permanent Equipment 50 . l 38, 550 16.9 Other Direct Costs3 2,022 7.2 41,141 18.0 Indirect Costs 8,972 31.9 52,370 23.0

Total: All Budgetary $28,180 100.0 $228,020 100.0 Categories

~Excludes mathematical sciences section, MCS. Includes undergraduate students, secretarial-clerical, other 3 professionals, technicians, and fringe benefits. Includes domestic and foreign travel, materials and supplies, publication costs, consultant costs, and computer costs.

276 courses in universities. Those who enter and results that lie at the boundary of two, or more, remain in the mathematical sciences must neces­ scientific disciplines. Those mechanisms that do sarily teach more and devote less time to research. exist are in disrepair. Support here is especially This is now characteristic of the mathematical important because of the new opportunities sciences but not of most of the other sciences. made possible by the recent liaisons between Because of this dearth of support and less mathematics and other sciences. than optimal conditions for the acquisition of There has been substantial discussion in the knowledge in mathematics, many talented stu­ mathematical sciences community concerning dents select other areas for study. As a conse­ means of supporting the infrastructure of the quence, the quality and excellence of the graduate research enterprise. In Table 5 we list some of student body in the mathematical sciences is these means and the amount of funds that the diminishing, and, as with the postdoctorates, Panel feels will be required per year to support the implications for the future excellence of the them. discipline in the United States are ominous. Table 6 is a summary of the per annum dollar Implicit in Tables 1 and 4 is that a substantial amounts discussed above (not listed in priority amount of support for research in the mathemati­ order). cal sciences must be contributed by universities in the form of reduced teaching loads, secretarial TABLE 5 services, travel costs, and other aids. This ($ Mill ion) support, as we have noted, is crumbling. summer schools, special It is the view of the Panel that an additional 300 years, mini-institutes $ 5.3 graduate students should be supported per year. Mid-level fellowships 3.6 At a cost of $20,000 per student (including indirect Travel grants, senior costs) this amounts to an increment of $6 million research associate programs 4.'5 per year. In addition, the amount allocated Computer time and equipment 2.0 in federal mathematical sciences research grants for secretaries, travel, and publication should be Total Sl5.4 increased by $3,000 per individual investigator. amounts to an increment of $5.4 million per This TABLE 6 year. ($Million) Percentage of Active Research People Supported. We saw in Table 1 that, in fiscal year 1979, postdoctoral positions $ 6.0 59 percent of the mathematical sciences research Graduate students 6.0 in grants 5.4 done in universities was unsponsored. Since Operating expenses in number of grants 10.0 even further: Increase then, the situation has deteriorated Infrastructure (total) 15.4 approximately 200 active researchers doing high­ quality work have lost support during the past two Total S42.8 years. Moreover, the research of many excellent new Ph.D.'s in the mathematical sciences-more Section 3: Recommendations than 86 percent-goes unsupported. We are not capitalizing on the investment made in the The mathematical sciences lie at the core of development of mathematical scientists. science, technology, and the national defense. An To stem the decay and to put a measure of excellent mathematical sciences enterprise does vitality into the mathematical sciences research not automatically produce excellence in science enterprise requires the allocation of sufficient and technology and strength in the national funds to support an additional 500 researchers. defense, but one cannot have quality and strength At a cost of $20,000 per researcher, including in the latter without quality and excellence in the indirect costs, this amounts to $10 million per mathematical sciences. year. The cost to the federal government of ensur­ State of the Infrastructure. The stresses addressed ing excellence in the mathematical sciences is thus far pertain primarily to those that affect the relatively very small, the leverage of the dollars individual researcher or gradute student. Severe invested very large, the ratio of benefit to cost problems also exist in the area of communication enormous. and interaction between researchers. There is In times of economic uncertainty or stress, a paucity of mechanisms in the mathematical the implementation even of programs with high sciences for maintaning the vitality of researchers benefit-to-cost ratios is sometimes delayed. When working at a distance from elite centers, for this occurs, one must also look at the "disbenefit" generating young people's interest in promising associated with delay. The current exciting and important new subfields, and for informing opportunities will not be fully exploited in the and educating the research community about new United States; the erosion of excellence in ideas and results-particularly those ideas and the mathematical sciences will accelerate; the

277 disbenefit to our nation of not doing something • If there is no increase, we recommend no now is too large to allow the erosion to continue. changes. The mathematical community, after The Panel makes the following recommenda­ deliberation, has recently reallocated the resources tions: available to it. It shifted monies into an alternative 1. The federal dollar allocation for research in mode of research support and cut down on the the mathematical sciences should be increased over number of individual research grants. It will the next three years by approximately 80 percent­ take time to absorb these changes. We must i.e., there should be a total increment over the emphasize again, however, that no increase in next three years of$42.8 million (in 1982 dollars). support will spell disaster for all but a few of the 2. The increments should be allocated in propor­ top mathematical research centers. tions deemed appropriate to the NSF, the DOD •If there is a 10 percent increase ($5.7 million), agencies, and the DOE. we recommend that the increment be disbursed 3. The managers of mathematical science fund­ approximately as in Table A. ing programs should have sufficent flexibility and •If there is a 20 percent increase ($11.5 million), freedom to choose the areas of research to be we recommend that the increment be disbursed supported and the mechanisms for support. approximately as in Table B. The Panel recognizes that it has no formal •If there is a 50 percent increase ($28 million), status as representative of the mathematical we recommend that the increment be disbursed community. No single group does. Nevertheless, approximately as in Table C. because it realizes that increased support may be •If there is an 80 percent increase ($42.8 slow in coming, despite the emergency, the Panel million), we recommend that the increment be tried to give priority to its recommendations as disbursed approximately as in Table D. follows.

T~BLE ~- 10 PERCENT INCRE~SE ($5.7 Million)

~llocation Use ($ Million) $1.5 Postdoc~oral Positions (in a variety of modes) 1.2 Increastng the Operating Expense ~llocations in Grants 0.5 Mini-Institutes 0.5 Senior Research ~ssociate and Visiting Positions in Grants 0.5 Mid-level Fellowships 0.5 Graduate Student Support 0.5 Equipment and Computer Time 0.25 Increasing the Number of Grants 0.25 Travel Grants, Special Years, etc.

T~BLE B - 20 PERCENT INCREASE ($11.5 Million)

Allocation Use ($ Million) $2.0 Increasing the Operating Expense ~llocations in Grants 2.0 Postdoctoral Positions (in a variety of modes) 1. 75 Increasing the Number of Grants l.O Senior Research ~ssociate and Visiting Positions in Grants 1.0 Graduate Student Support 0.75 Mini-Institutes 0.75 Mid-level Fellowships 0.75 Research Institute 0.75 Equipment and Computer Time 0.75 Travel Grants, Special Years, etc.

278 TABLE C - 50 PERCENT INCREASE ($28 Million) Allocation Use ($ Million) $4.8 Increasing the Number of Grants 4.6 Postdoctoral Positions (in a variety of modes) 4.3 Increasing the Operating Expense Allocations in Grants 3.5 senior Research Associate and Visiting Positions in Grants 3.2 Graduate Student Support 2.2 Mini-Institutes 1.7 Mid-level Fellowships 1.4 Equipment and Computer Time 1.4 Research Institute 0.9 Travel Grants, Special Years, etc.

TABLE D - 80 PERCENT INCREASE ($42 Million) Allocation Use ($ Million) $ 6.0 PoStdoctoral Positions 6.0 Graduate Students 5.4 Operating Expenses in Grants 10.0 Increasing the Number of Grants 5.3 summer Schools, Special Years, Mini-Institutes 3.6 Mid-level Fellowships 4.5 Travel Grants, Senior Research Associate Programs 2.0 Computer Time and Equipment

279 National Science Foundation Budget Request for Fiscal 1984

The Administration's budget request for the to the mathematical sciences (Table II), and fiscal year 1984 was sent to Congress in January the effects of inflation on the NSF budget 1983. This report examines the budget requests (Table Ill). for the National Science Foundation. An • Second is the text of the material sent to accompanying article prepared for the Conference Congress in support of the budget request Board of the Mathematical Sciences by William H. for the Mathematical Sciences Section; this Pell traces a larger picture, outlining the budget describes some of the work supported by requests for support of mathematical activities by the Section, as well as other significant a variety of agencies of the Federal government, mathematical work; it is designed to explain including NSF. to members of Congress why support of As in recent years, this report concentrates mathematical research is important. primarily on NSF support for research in the • Third is a selection of material, extracted mathematical sciences and on the fate of support from the budget submission, devoted to the for science education within the Foundation. Foundation's efforts in support of education in These reports have appeared annually for more science and engineering. than a decade-the most recent one was published Tables I and II report the big news this in the April 1982 issue of the Notices, pages 238 year, that the Administration is asking for more to 244. money for the mathematical and physical sciences, This year there are three components to the 21.6% more for Fiscal 1984 than the predicted report: expenditures for Fiscal 1983. The increase in • First is the customary set of tables depicting the request for mathematics is the same as that the part of the entire NSF budget request for computer research, physics, chemistry, and allocated to support of the mathematical materials research, combined (see Table II). This sciences (Table I), the fraction of the budget increase for MPS is larger than the corresponding request for the Division of the Mathematical increase in the budget request for the entire and Physical Sciences (MPS) which is devoted Foundation, which comes to 17.8% (see Table I).

TABLE I. NATIONAL SCIENCE FOUNDATION BUDGET (Millions of Dollars)

1980 1981 1982 1983 1984 Actual Change Actual Change Actual Change Plan Change Request

(!)Mathematical Scienees Researeh Support $ 25.0 13.2% $ 28.3 7.8% $ 30.5 13.8% $ 34.7 21.6% $ 42.2 (2)0ther Research Support (Note A) 817.7 6.8% 873.7 0.23% 875.7 8.9% 953.9 19.2% 1136.6 (3)Education, Information, Foreign Currency Program (Note B) 95.4 -16% 80.6 -63% 26.9 30% 38.6 24% 47.7 (4)Program Development and Management ("Overhead") (Note C) 58.2 1.7% 59.2 6.8% 63.2 10.4% 69.8 -5.4% 66.0

(5)Totals $996.3 4.6% $1041.8 -4.1% $999.1 9.8% $1097.0 17.8% $1292.5 (6)(1) as% of (1) and (2) 2.97% 3.14% 3.37% 3.51% 3.58% (7)(1) as % of (5) 2.51% 2.72% 3.05% 3.16% 3.26%

NOTE A. Scientific research and facilities (excluding mathematics and science information). National and special research programs, and national research centers. Support for mathematics has been excluded, cf. items (1) and (3). NOTE B. The programs in this group are ones in which there is some support for projects in every field, including mathematics. The foreign currency program involves both cooperative scientific research and the dissemination and translation of foreign scientific publications. Foreign currencies in excess of the normal requirements of the U.S. are used. NOTE C. This heading covers the administrative expenses of operating the Foundation; the funds involved are not considered to constitute direct support for individual projects.

280 TABLE II. DMSION OF MATHEMATICAL AND PHYSICAL SCIENCES (Millions of Dollars)

Section 1980 Actual 1981 Actual 1982 Actual 1983 Plan 1984 Request Mathematical Sciences $ 25.0 (11.01 %) $ 28.3 (11.03%) $ 30.5 (11.18%) $ 34.7 (11.58%) $ 42.2 (11.58%) Computer Research 18.5 ( 8.15%) 22.3 ( 8.69%) 25.7 ( 9.42%) 29.1 ( 9.71%) 34.7 ( 9.53%) Physics 63.4 (27.93%) 72.1 (28.11%) 75.3 (27.60%) 88.9 (29.66%) 107.6 (29.54%) Chemistry 51.4 (22.64%) 57.6 (22.46%) 61.4 (22.51%) 66.2 (22.09%) 80.2 (27.34%) Materials Research 68.7 (30.26%) 76.2 (29.71%) 79.9 (29.29%) 80.8 (29.96%) 99.6 (27.34%) Total $227.0 $256.5 $272.8 $299.7 $364.3 Table TI updates an annual report in this series (1) (2) (3) (4) (5) (6) and relates the same old story-no matter what Fiscal Budget Increase Amount Increase Fraction Request Requested Spent Obtained Realized else may change, Mathematical Sciences gets the Year same share of the MPS fiscal pie, somewhere 1971 $12.9 between 11% and 12% of the Division budget. 1972 $15.9 23% 13.7 6.2% 27% 1973 14.4 5.1% 14.1 2.9% 57% Tables illa and illb give this year's version of 1974 14.4 2.1% 14.5 2.8% 133% the ten-year report on what inflation has done 1975 17.1 18% 16.4 13% 72% to the NSF budget: in 'constant dollars' the 1976 18.0 9.8% 17.3 5.5% 56% Foundation's total expenditures dropped about 1977 20.9 21% 20.1 16% 73% 20% between 1975 and 1982. In the same period, 1978 21.8 8.5% 21.4 6.5% 76% support for mathematics remained more or less 1979 23.1 7.9% 22.8 6.5% 82% constant. 1980 25.5 12% 25.0 9.6% 80% Readers who have studied the articles in this 1981 29.3 17% 28.3 13% 76% series in the Notices, and others interested in 1982 34.4 22% 30.5 7.8% 35% 1983 33.9 11% 34.7* 14%* 127%* such matters, realize-of course-that much can 1984 42.2 22%* happen and usually does happen between the submission to Congress of an Administration's Figures published each year in this series reveal budget request in the winter and the Congressional that Congressional appropriations do not account action in the summer or fall which results in the entirely for what happens, since the so-called appropriation of funds needed to carry out the 'current plans' (based on appropriations and activities in question. on decisions made within the Foundation) do The following list shows what has happened not always serve as accurate predictors of the to budget requests for the Mathematical Sciences actual expenditures reported later for the year Section (in millions of dollars) for more than a in question. Asterisks in the list above mark decade. Column (2) gives the budget request quantities based on the 1983 'Plan' rather than for each year. Column (4) shows what the 'Actuals' (which will not be known until the close actual expenditure turned out to be for that year. of the 1983 fiscal year). These figures indicate Column (3) gives the percentage increase for the that the amount finally spent on the mathematical current year's request over the previous year's sciences during a given fiscal year is often some actuals. Column (5) gives the percentage increase 70% or 80% of the original budget request for that of the current year's actuals over the previous year. In two cases, it has actually been higher year's actuals, and column (6) gives the ratio of than the original budget request, and in some column (5) to column (3) (as a percentage) and thus cases it has been well below the average. LKD measures that fraction of the increase originally requested of Congress which was actually spent by [Tables ITia and ITib are spread across the two the Foundation during the budget year in question. pages following this one.]

281 and the essential tools for the needs of technology .. The following report was prepared by the staff of The detached spartan elegance and inner drive the Mathematical Sciences Section of NSF and was toward abstractness of the mathematical sciences submitted to Congress as a part of the material form the key to their usefulness and applicability. which accompanied the Administration's Budget They are, with astronomy, the oldest of the Request for the Fiscal Year 1984. sciences. A staggering number and a bewildering variety Mathematical Sciences Subaetivity of applications of mathematics result from its intimate connections with the other sciences and technology. Some of these connections are (Millions of dollars) predictable, others are not. The predictable Activity Actual Plan Request connections make use of statistics, mathematical FY1982 FY1983 FY1984 modeling, and computational mathematics. That Classical Analysis $3.2 $3.3 $4.1 overbooking on airlines leads only to relatively Modern Analysis 3.3 3.4 4.2 rare inconveniences is not a matter of blind luck, Geometric Analysis 2.9 3.1 3.9 but of careful statistical analysis. Similarly, the Topology & Foundations 4.0 4.2 5.2 design of efficient aircraft requires sophisticated Algebra & Number Theory 5.0 5.3 6.6 Applied Mathematics 3.8 4.0 5.3 mathematical modeling by partial differential Statistics & Probability 3.4 3.6 4.7 equations of the flow of air over wings and Special Projects 4.9 7.7 8.3 the computation of their approximate solutions on high speed digital computers. Unpredictable Total $30.5 $34.7 $42.2 applications of mathematics are equally striking. The discovery and classification or a class of Objectives and Description important solutions to the set of equations The objectives of the NSF's Mathematical proposed by C. N. Yang and R. Mills to describe Sciences program are to promote and foster weak and strong interactions in particle physics significant American research in the mathematical depend on mathematical tools developed in sciences and to ensure the continuing vitality topology and algebraic geometry, among the most and long-range health of the discipline. Each abstract parts of the mathematical sciences. The subdiscipline of the mathematical sciences, from very abstract mathematical concept of fractional those with a sharp disciplinary focus to those dimension, a space with a dimension that is not that reach out to other disciplines, is supported a whole number, is being applied to the study of to encourage interplay among the subdisciplines turbulence. and a healthy balance among them. Various Unlike research in scientific disciplines with an support mechanisms are used to foster the creation empirical background, mathematical research is of new mathematical methods and techniques focused on a variety of themes and problems. and to promote their use in research directed Some of these are motivated by applications, at improving our understanding of physical, others by the goal of developing unified abstract biological, and social phenomena. theories. The mathematical sciences attack no Among the scientific disciplines, the mathe­ central agreed-upon problem at any given time. matical sciences occupy a unique position. They Research proceeds on seemingly unrelated fronts: are to a great extent driven by their own abstract however, the concepts, methods, and techniques internal goals, but provide the language of science that result unify and illuminate broad areas of

TABLE ITia. TEN-YEAR COMPILATION OF THE NSF BUDGET (Millions of Dollars) Current dollars are converted to 1967 dollars using the wholesale/producer index

1975 1976 1977 1978 1979 1980 Actual Actual Actual Actual Actual Actual (1) Mathematical Sciences Research Support $ 16.4 $ 17.3 $ 20.1 $ 21.4 $ 22.8 $ 25.0 1967 dollars 9.4 9.4 10.4 10.0 9.7 9.9 (2) Other Research Support 551.9 592.1 642.9 702.8 761.0 817.7 1967 dollars 915.7 929.9 991.1 926.8 922.7 904.2 (3) Education, Information, Foreign Currency Program 83.0 72.8 83.3 84.3 88.4 95.4 1967 dollars 47.5 99.7 42.9 99.2 97.5 95.5 (4) Program Development and Management ("Overhead") 37.9 42.2 45.5 48.7 54.7 58.2 1967 dollars 21.7 29.0 29.4 22.6 29.2 21.7

(5) Totals $693.2 $724.4 $791.8 $857.2 $926.9 $996.3 1967 dollars 996.5 995.5 407.8 998.6 993.0 970.6

282 mathematics and science, despite their diverse those that model natural phenomena and, origins. consequently, often display nonlinearities or During the recent past, a striking feature discontinuities in their definition. The theories of research in all the sciences has been the of special functions, several complex variables, convergence of investigations within different and harmonic analysis form an interdepen­ disciplines on problems that transcend individual dent network of research which provides a sciences. Phenomena of phase transitions, for theoretical base for classical analysis. example, are studied by physicists, chemists, • Modern Analysis consists of that part of materials scientists, and mathematicians. This mathematical analysis that has been developed development in research is particularly evident in the 20th century by abstracting and refining in the mathematical sciences, where researchers the essential concepts from classical analysis from various subfields attack and solve problems and introducing new algebraic, geometric, from other areas of mathematics and science. and topological methods. A major goal is The recent findings in linear programming by to provide unified approaches that explain the geometer Steven Smale, described in the phenomena and results from classical analysis recent achievements section, is illustrative of this and have more general applicability. important trend. • Geometric Analysis encompasses the study of Historically, the core of the Foundation's geometric objects such as curves and surfaces, program of research support in the mathematical as well as the use of geometric models in the sciences has been the Scientific Research Project analysis of non-linear phenomena such as those Support (SRPS) award. Recently there has been an that arise in ordinary and partial differential emphasis on an extended program of investment equations, in functions of several complex in special projects designed to develop and variables, and in variational problems. Many maintain the infrastructure of the mathematical of these mathematical phenomena are of research community. These special projects, interest because of their relation to theoretical which include two research institutes, postdoctoral problems in physics and other sciences. and visitor support, postdoctoral fellowships, • Topology and Foundations: Topology studies and research conferences and workshops, are in a formal and abstract manner those important in furthering research interactions properties of geometric objects which persist among mathematical subfields and mathematical after deformation by stretching, shrinking, scientists. Special efforts continue to be made and twisting without tearing or cutting. in postdoctoral and graduate student support in Foundations research involves the study of order to maintain the vitality brought by young the logical structure and basic concepts that people to mathematical research. undergird mathematics. Numerous and varied A brief description of research areas supported problems have topological formulations, for under the Mathematical Sciences subactivity example, the stability of orbits. Foundations follows: has intimate connections with the theory and • Classical Analysis: Research in classical application of computing. analysis is centered on properties of solutions • Algebra and Number Theory: Algebraic struc­ of differential equations and the development tures such as groups, rings, and fields form of mathematical techniques to estimate these the basis for all mathematical structures by solutions. Equations of primary concern are describing the ways in which elements of a

TABLE ITib. TEN-YEAR COMPILATION {Continued} (Millions of Dollars) Current dollars are converted to 1967 dollars using the wholesale/producer index

1981 1982 1983 1984 Increase Increase Actual Actual Plan Request 1975-1982 1975-1984

(1) Mathematical Sciences Research Support $ 28.3 $ 30.5 $ 34.7 $ 42.2 86% 157% 1967 dollars 9.6 10.2 8.5% (2) Other Research Support 873.7 875.7 953.9 1136.6 58.6% 106% 1967 dollars 295.9 291.6 -7.6% (3) Education, Information, Foreign Currency Program 80.6 29.7 38.6 47.7 -64% -43% 1967 dollars 27.2 9.9 -79% (4) Program Development and Management ("Overhead") 59.2 63.2 69.8 66.0 67% 74% 1967 dollars 20.0 21.0 -1.8%

(5) Totals $1041.84 $999.1 $1097.0 $1292.5 44% 86% 1967 dollars 952.1 992.7 -16%

283 set can combine to form new elements. Com­ The NSF role in support of academic basic binatorial structures and the resulting discrete research in the mathematical sciences is crucial. analysis can be used for the mathematical In FY 1982, the Foundation provided 38 percent description of situations where no obvious of the support in applied mathematics, 44 percent analytic or algebraic structure exists. Num­ in statistics and probability, and 97 percent in ber theory is the study of the structure and the other mathematical sciences disciplines, for an properties of number systems. All have exten­ overall total of 67 percent. sive and unique applications because of the NSF coordinates its support of research in the basic nature of their formulations. mathematical sciences with its counterpart federal • Applied Mathematics centers on the use of agencies through the Interagency Committee for mathematics for the modeling, analysis, and Extramural Mathematics Programs. This group explanation of real world phenomena. This meets regularly to share information on policies broad field includes work in such diverse areas of support and to discuss areas of emphasis as continuum mechanics, mathematical biol­ and of unusual scientific opportunity. Extensive ogy, systems theory, control theory, optimiza­ day-to-day contact between program officers at tion, statistical mechanics, numerical analysis, the various federal agencies is maintained by and numerical partial differential equations. telephone and personal visits. Conferences and jointly by the • Statistics and Probability: Statistics is regarded workshops are sometimes supported as both a branch of mathematics and a field Foundation and one or more of the other agencies. also supports postdoctorals at the of science. Problems in statistics arise as The Foundation Research Center at the University scientists collect, organize, and analyze data Mathematics of Wisconsin, which has principal support from to discover underlying relationships among the Army Research Office, so NSF-ARO contact is variables. Ideas and methodology developed continual and close. to deal with these problems develop lives of their own as objects of theoretical study, so Major Programmatic Topics that mathematical statistics is also a subarea • Many phenomena, such as the distribution of of mathematics. Probability theory is the heat in a nuclear reactor or weather conditions basis of statistical reasoning. It is concerned in the atmosphere, are accurately described with the study of phenomena that are random by solutions of partial differential equations. or are modeled as random owing to incomplete The behavior of these solutions in any region understanding. Probability models are used is strongly dependent on their properties at in a wide variety of scientific endeavors. the boundary of the region. Can solutions • Special Projects foster the increasingly fruitful be reconstructed from knowledge of these research interactions among different mathe­ boundary features in a coherent manner? matical subfields and their areas of applica­ To what extent does the geometry of the tion. Research conferences and workshops boundary affect this process? encourage these interactions directly. The • Transformations of Hilbert space, and infinite­ Mathematical Sciences Postdoctoral Research dimensional extension of the usual Euclidean Fellowships and grants to the new re­ plane, form von Neumann algebras, a fertile search institutes and other research centers ground for physicists seeking mathematical provide research opportunities for junior inves­ expressions for quantum theory. Alain tigators. Increasing requirements for special­ Connes, a 1982 Fields medalist, obtained new purpose computer equipment for mathemati­ information about mathematical physics by cal sciences research are addressed through a several striking extensions of geometric and limited number of awards for equipment. analytic structures into the setting of C*­ algebras, variants of von Neumann algebras NSF Role which reflect more of the geometry of the NSF is the only Federal agency with respon­ mathematical situation. Why do certain sibility for support across the entire spectrum geometric structures make sense in these of the mathematical sciences. Research in the algebras while others seemingly do not? mathematical sciences is also funded by the • Most of the commonly used statistical methods Department of Defense (DOD), the Department of were developed when computation was slow Energy (DOE), the National Institutes of Health and expensive. During the recent past there (NIH), and the National Aeronautics and Space has been a surge of development in new statis­ Administration (NASA). For the past decade, tical methods which take advantage of high the research agencies of the DOD have focused speed computation and make the analysis free their support on applied mathematics and statis­ of distributional assumptions. For example, tics. Further, Foundation support of applied methods using only the sample data have been mathematics and statistics generally involves re­ suggested for assigning a reliable measure of search on a broader number of topics than the accuracy to a statistical estimate, without more project-oriented research sponsored by the concern for the mathematical complexity of mission agencies. the estimator or the non-bell-shapedness of the

284 data. How can one provide solid theoretical of probability for linear programming problems. foundations to these methods? How can these He identified a collection of "atypical" problems, methods be implemented in an efficient way? which have low probability, and proved that the • Recent work on 4-dimensional manifolds has remaining "typical" problems are solved by the demonstrated the existence of non-standard simplex algorithm extremely quickly. Previously, smooth structures on 4-dimensional space­ theorists had guessed that the time to reach a time. Is this an isolated anomaly or is there solution of a good problem grows in proportion to a systematic way to detect and classify such some positive power, like the square, of the num­ pathologies? What significance have they for ber of variables involved. Smale's proof rigorously geometric structures on space-time? establishes a much greater efficiency: the time • Novel computer architectures create pos­ grows more slowly than any fractional power (like sibilities for non-sequential numerical com­ the square root) of the number of variables. putations. What particular mathematical Positive Mass Conjecture. In general relativity, structures and formulations of problems can the gravitational effect of mass is formulated as be devised to take advantage of distributed, curvature of space-time near the location of the parallel, or sequential architectures? Con­ mass. Recently, two mathematicians, Shing-Tung versely, what architectures are optimal for the Yau (Institute for Advanced Study) and Richard numerical implementation of specific mathe­ Schoen (University of California, Berkeley), have matical structures? used ideas from differential geometry to show • Using concepts of distance based on divisibility that these gravitational effects are not confined of numbers by prime numbers, researchers in to points near the location of the mass, but can number theory developed a technique known in principle also be detected at the edge of the as p-adic analysis which has proven its value universe. This assertion, known as the "positive in many situations. The technique is now mass conjecture," was subsequently reproved seeing expanded use and is undergoing rapid using different ideas by the physicist E. Witten development. How can special functions (Princeton). Yau and Schoen's proof rests on a of traditional analysis be realized in p-adic geometric study of the difficulties of constructing analysis? What geometric connections exist? surfaces with positive scalar curvature, which has Can the results of p-adic analysis be carried led to further results in differential geometry. For back to provide information on the questions his work in this area, Yau was honored in 1982 of traditional analysis? with the award of the Fields Medal, the "Nobel • A fundamental technique in the theory of Prize" of mathematics. phase-transitions is the renormalization group Kinky 4-Spaee. In the summer of 1982, a method, developed by a number of physicists sequence of results by several young researchers (including the 1982 Nobel Laureate, Kenneth in four-dimensional topology and mathematical Wilson). Recently, this technique has been physics culminated in the surprising observation applied within mathematics, leading to the that fundamentally distinct kinds of differential establishment of certain universal patterns in calculus are possible in four-dimensional space. the bifurcation of simple model systems of A curve has a kink at a point where it bends chaotic behavior. Can similar patterns be abruptly and has no tangent, even though it has verified rigorously for realistic models of fluid continuously varying tangents at nearby points. flow and thus be related to understanding Kinks are similarly defined for higher dimensional phenomena of turbulence? On the other hand, manifolds, although failure to have well-defined is it possible to place these renormalization tangents can happen in a greater variety of ways schemes on a firm mathematical foundation? in higher dimensions. An underlying differentiable Significant Recent Achievements structure is needed to make precise the notion of Efficiency of the Simplex Algorithm. Compli­ tangency. Spaces such as spheres and Euclidean cated practical problems concerning optimal al­ spaces have a natural differentiable structure. location of resources, known as linear program­ However, it has long been known that complicated ming problems, are usually solved with computers twisting and turning of the structures given to using the simplex algorithm, and algorithm based certain spaces can lead to structures other than on a combinatorial search devised by George the natural ones and hence to incompatible Dantzig in the 1950s. The method is extremely notions of which curves or functions are smooth efficient in practice, although in principle it is and which have kinks. The first such exotic possible to formulate problems which the simplex differentiable structure was discovered in 1956 algorithm is very slow to solve. The apparent on the seven-dimensional sphere by John W. contradiction between theoretical mathematical Milnor at Princeton. Exotic structures were analysis and practical observation has been subsequently discovered and classified on many resolved recently by Steven Smale (University of spaces using algebraic invariants. For Euclidean California, Berkeley). Drawing on concepts from spaces (which are simpler than spheres) these his earlier work in differential topology, Smale invariants are all zero, so that exotic structures studied a natural geometrically-defined measure were not expected to exist on any Euclidean

285 space. Indeed, the impossibility of such structures late passage of the FY 1983 Appropriations Act was known with the possible exception of four­ and are located in the Special Projects program dimensional Euclidean space. It came as a element. considerable surprise, therefore, when an exotic The small increase of $0.1 million is in response differentiable structure was discovered on four­ to strongly felt needs for increased support dimensional Euclidean space. The principal actors of graduate students and young postdoctoral in the discovery of exotic 4-space were Michael H. investigators in the mathematical sciences. The Freedman, a topologist (University of California, New Obligational Authority (NOA) for the Special San Diego) and Simon Donaldson, a graduate Projects program element is increased by $0.4 student in physics (Oxford), building on earlier million above the FY 1983 Budget Request of work of Karen Uhlenbeck (University of illinois, $6.6 million to provide enhanced support of young Chicago) and Clifford Taubes (Harvard). The investigators through the Mathematical Sciences discovery poses fundamental new questions for Postdoctoral Research Fellowship program. A topologists, including a re-examination of the balancing decrease is distributed over the other algebraic methods they have used so successfully programs of the subactivity; even within these in the past, but which can yield no information decreases, emphasis is placed on support of about this new phenomenon. graduate students above the originally planned Theory of Algorithms and Card ShufHing. How levels. many times does a deck of cards have to be shuffled until it is close to random, in the FY 1984 Budget Highlights sense that any possible arrangement has an and Explanation of Increases, Decreases, approximately equal chance to occur? This and Continuing Emphases classical combinatorial question is related to the design of algorithms for generating random Obligations permutations. Probabilists are interested in FY1982 Actual ...... $30,488,529 these algorithms, since they are heavily used FY1982 Carryover ...... $ 683,437 in investigating the stochastic properties of FY1983 New Obligational Authority ....$34,023,000 FY1983 Current Plan ...... $34,706,437 various statistical procedures through computer FY1984 Request ...... $42,175,000 simulations. Persi Diaconis and a graduate Difference FY1984/FY1983 (NOA) ..... $ 8,152,000 student at Stanford University recently solved this problem by employing an abstract mathematical Obligations. The FY 1984 Budget Request tool known as noncommutative Fourier analysis. of $42.2 million is $8.2 million or 24.0 percent This tool is now being used by statisticians to above the FY 1983 New Obligational Authority. solve a variety of other problems. This represents a very significant increase in the Other useful card shuffling problems involve research support for the mathematical sciences. "perfect shuffles," which cut the deck in half and During FY 1981, the Foundation initiated a set interlace perfectly. There are two ways to obtain a of projects in the mathematical sciences designed perfect shuffle. The out shuffle leaves the original to develop and maintain the infrastructure of top card on top, whereas the in shuffle leaves it the mathematical research community which had second from top. Mathematicians study perfect seriously eroded as a result of a long period of shuffles because combinatorial questions involving very restricted budgets. Included in these projects these shuffles are central to various scientific were the creation of the Mathematical Sciences problems. For example, the shuffle exchange Research Institute in Berkeley and the Institute net in modern parallel processors shuffles data for Mathematics and its Applications at the in exactly the same way that the cards move and is the basis for the fastest known ways of University of Minnesota and enhanced support for postdoctorals and visitors at the Institute multiplying, sorting, doing matrix transpositions, and performing various other operations. An for Advanced Study, the Mathematics Research important related problem of how many different Center at the University of Wisconsin, and the arrangements of 2n cards can be achieved by Courant Institute of Mathematical Sciences of combining the in shuffle and the out shuffle was New York University. These projects, together also recently solved by Diaconis and two scientists with the support of research conferences and at Bell Laboratories. In July 1982, Diaconis was workshops and a significantly enhanced program awarded a MacArthur Prize Fellowship. of postdoctoral fellowships, were particularly designed to stimulate research interactions and Changes Between FY 1983 Budget Request to provide the means for support of young and FY 1983 Current Plan researchers, whose vitality is essential to the The FY 1983 Current Plan is for $34.7 million, continued well-being of the mathematical sciences. $0.8 million above the FY 1983 Budget Request In spite of restricted budgets, these activities were of $33.9 million. This represents an increase of expanded in FY 1982 and FY 1983 and support 2.3 percent and includes an appropriated increase for computational equipment for research in the of $0.1 million and an FY 1982 carryover of mathematical sciences was added in FY 1982. All $0.7 million. The carryover funds result from these projects resulted in a decrease in the number

286 of senior investigators that could be supported on Change, FY 1983 to FY 1984 SRPS grants. $Millions Percent The FY 1984 Budget Request provides the opportunity to continue the needed expansion of Classical Analysis $0.78 23.5% Modern Analysis 0.72 21.0 some of these activities, with special emphasis Geometric Analysis 0.73 23.4 on programs in support of graduate students, Topology & Foundations 0.96 22.9 young investigators, and research visitors at major Algebra & Number Theory 1.27 23.8 centers, and to respond to important scientific Applied Mathematics 1.25 30.9 opportunities in computational mathematics and Statistics & Probability 1.14 32.0 Special Projects 1.30 18.5 scientific computing: • In the Special Projects program, the num­ Total Mathematical Sciences $8.15 24.0% ber of new 24-month Mathematical Sciences Postdoctoral Research Fellowships awarded annually is increased from twenty to thirty; The following account is from the Budget report this requires a $0.6 million increase in the sent to Congress by the National Science Founda­ cost of this fellowship program, to a total of tion. It is the first sign of a renewed interest $1.8 million. Other postdoctoral and visitor in education, which until recently constituted a support provided by the Special Projects pro­ significant component of the Foundation's efforts. gram is increased by approximately 15 percent to $1.27 million. Support for computational Science and Engineering equipment in the Special Projects program is increased by $0.1 million to $1.1 million. Education (SEE) • The operation of the mathematical sciences The FY 1984 Budget Request for the Science research institutes in Berkeley and Minnesota and Engineering Education activity is $39.0 is maintained at a constant effort level of million. The request includes $19.0 million $2.9 million. The Berkeley institute will for the NSF Graduate Research Fellowships concentrate on low-dimensional topology and subactivity and $20.0 million for the Precollege on the connections among operator algebras, Teacher Improvement in Science and Mathematics foliations, and index theory. The Minne­ subactivity. sota institute will concentrate on continuum physics and nonlinear partial differential equa­ The FY 1983 Current Plan for the SEE activity tions. totals $30.0 million. The Graduate Research Fellowships subactivity will be funded at $15.0 • An increase of $3.7 million in SRPS awards million and the new programs under the Precollege emphasizes the provision of support through Teacher Improvement in Science and Mathematics research grants to established senior inves­ subactivity will be funded at $15.0 million. In tigators for postdoctoral research associates, FY 1983 NSF will introduce two Presidential graduate research assistants, and senior and midlevel research visitors. This increase initiatives in science education addressing the is distributed throughout all subdisciplinary critical problem of secondary school instruction in programs. It is planned that $0.75 million will science and mathematics: be devoted to the support of twenty additional • Presidential Awards for Teaching Excellence postdoctoral research associates, $1.60 million in Science and Mathematics to recognize to that of eighty graduate students, and $1.35 outstanding secondary school teachers of million to the support of midlevel and senior science and mathematics who can serve research visitors. as models for their colleagues and provide • Significantly increased support is provided for increased status and rewards for demonstrated research activities in computational mathe­ professionalism; and matics and on mathematical aspects of large­ • Presidential Secondary School Science and scale scientific computing. This new activity, Mathematics Teaching Improvement program considered an area of high scientific oppor­ which will support workshops and training tunity, is budgeted at $1.3 million, distributed activities to improve the knowledge of science in all the mathematical sciences programs and mathematics for teachers of these subjects with particular emphasis in the Applied Math­ in grades 6 through 12. ematics, Statistics and Probability, and Al­ The request for FY 1984 reflects an increase gebra and Number Theory programs. of $9.0 million over the FY 1983 Current Plan. • An additional increase of $1.9 million in Four million of the increase will be used to SRPS awards, from $27.0 million to $28.9 raise the stipend and cost-of-education allowance million, provides modest grant enrichment for of approximately 1,390 graduate fellows. The such direct costs as travel, publications and stipend will be increased $1,200 from $6,900 to supplies. $8,100 while the cost-of-education allowance will Percentage increases for the eight program ele­ be increased by $900 from $4,000 to $4,900. ments of FY 1984 over FY 1983 New Obligational The remaining $5.0 million will be used Authority are: to expand the Presidential Secondary School

287 Science and Mathematics Teaching Improvement teachers of mathematics and science in grades program. 6 through 12. There is national recognition of Improving the Educational Base the fact that the quality of teaching in these subjects has deteriorated in the last decade or so The educational base for producing the Na­ at the same time that graduation requirements tion's futur.e research scientists, engineers, and and admission requirements to college have been technicians consists of a continuum extending reduced. from primary and secondary schools through un­ The Presidential Awards for Teaching Ex­ dergraduate colleges and universities to graduate cellence in Science and Mathematics program and professional schools. As students move identifies outstanding secondary school teachers through these institutions, they make a series of of the subjects who can serve as models for their choices leading to adult careers in science and colleagues. The FY 1984 Budget Request main­ engineering. These choices are influenced very tains this program at the FY 1983 level, providing much by the quality of the faculty and teaching support for 100 Awards-half in science, half in which students encounter and, at later stages, mathematics-to teachers who will be selected by the opportunities they have to gain firsthand nationally from each state and other jurisdictions knowledge of the research process. such as Puerto Rico and the District of Columbia. Support for the Presidential Secondary School EDITOR'S NOTE. For many years Science Science and Mathematics Teaching Improvement Education has been an important concern of the program is increased in FY 1984 over the FY National Science Foundation. (LOWELL PAIGE, 1983 Current Plan. This program, initiated in FY in the mid-1970s, was Head of the then Office 1983, supports workshops and training activities of Science Education at NSF.) During the latter to improve the subject matter knowledge of half of the decade of the seventies, the annual science and mathematics for teachers in grades 6 actual expenditures in the office were usually in through 12. It is estimated that by FY 1984, it the range of $70 to $80 million dollars (1976 was an will have reached some 11,500 teachers. exception: only $62.5 million were spent then). In In April 1982, the National Science Board 1982, however, there was a drastic reduction in this established a Commission to examine the current program and only $20.9 million were actually spent. state of education at the precollege level in The major component of the 1982 activity was the mathematics, science and technology. It was Graduate Research Fellowship program which was charged to develop an agenda which defined supported at the level ($15 million) required to the roles of all sectors of society involved with continue the fellowships contracted in prior years. or having an interest in the Nation's public In 1983 the other (relatively small) commitments school system. Twenty persons of various were scheduled to be discontinued and the "plan" expertise, who are highly respected in their fields, called for $15 million for the Fellowships and, in were asked to serve as Commissioners. They addition, $15 million was added for a program represent the business and industrial communities, in Precollege Teacher Improvement in Science & higher education, precollege education and local Mathematics. The Budget Requests for FY1984 governing groups. call for increases in the Fellowship program to $20 Another critical part of the link in the educa­ million and in the Precollege Teacher program to tional base which provides the Nation's future en­ $19 million. Some of the Administration's ideas gineers and scientists is the small, predominantly for ways to spend these funds are presented in the undergraduate college. A significant proportion accompanying text. The total "planned" for 1983 of those awarded the Ph.D. in science, mathe­ is $30.0 million and the request for 1984 is $39.0 matics and engineering receive their baccalaureate million. at these institutions. For the period between 1967-1976, for example, more than 9,500 Ph.D. The Foundation's major responsibilities lie in recipients in chemistry had their baccalaureates graduate education and training in research, from 198 predominantly undergraduate clleges. which it supports through fellowships, stipends in In FY 1982, 20 percent of the NSF Graduate research awards for graduate research assistants, Research Fellowships went to graduates of such and by monitoring and evaluating personnel colleges. Since these colleges serve as such an trends and the status of science and engineering important pipeline for entry into the system of education. In primary and secondary education graduate education, and finally, into the overall NSF can play an important role by helping scientific and technical enterprise of the countrJ, to identify critical needs and by encouraging it is important that faculty in these institutions scientific and professional societies, academic receive support and encouragement for maintain­ institutions, state and local governments, private ing their research capability. industry, and other federal agencies to meet these In prior years, the NSF operated two re­ challenges. search activities especially oriented towards this In FY 1983 two new Presidential initiatives were audience: the Two- and Four-Year College Re­ instituted, aimed at improving the knowledge and search Instrumentation Program, aimed at in­ science and mathematics instructional skills for stitutional physical resource improvement, and a

288 small college faculty research opportunities effort Finally, the FY 1984 Budget Request will that provided a way for motivated faculty at continue support of the Research Improvement these small institutions to participate in research in Minority Institutions (RIMI) program that was under the aegis of NSF investigation at larger established in FY 1982 in response to Executive institutions. Order 12320, to help predominantly minority The FY 1984 Budget Request builds on and colleges and universities with graduate science or refocuses these previous efforts by proposing a engineering programs to improve their research new program, Undergraduate College Research capabilities. Four awards were made in FY 1982 Support, to be supported at the level of $3.0 and an additional eight awards are anticipated in million. It will be coordinated through the FY 1983. Support in FY 1984 will be level with STIA Activity, but managed and budgeted for in that provided in the FY 1983 Current Plan. all of the Foundation's major research activities One project funded in FY 1982 at Howard to insure that the work which it supports is University strengthens the semiconductor research adequately coordinated with other NSF-supported program at the only historically minority institu­ research and facilities at the Nation's leading tion in the U.S. that offers a doctoral degree in research institutions. electrical engineering. Another project at Jackson The program will emphasize independent re­ State University uses its chemical physics research search by faculty at these predominantly under­ laboratory to study the structure and dynamics of graduate colleges, using the physical resources molecular ions and other unique small molecular of major university, industrial or government re­ systems. Researchers at the University of Puerto search centers, and will encourage the inclusion Rico are using a high resolution scanning electron of advanced undergraduate students as part of microscope with an energy dispersive spectrometer the research team. Funds for essential permanent to study polycrystalline semiconductors, selective equipment will also be provided. This program surfaces and other materials. A project at At­ will thus help improve the quality of the research lanta University supports studies which form the environment at small institutions which serve as initial core of an institutional research program in major sources of future scientists, engineers and applied mathematics and computational physics. technicians. Support of similar projects at predominantly minority institutions will be provided in FY 1984.

289 Mathematical and Computer Sciences in the FY 1984 Federal Budget by William H. Pell

Distribution of Federal Support range experimental research. NSF, DOD, DOE, and NASA all have substantial software support. The financial support of research in the DARPA is the prime support for the study of mathematical and computer science by the federal intelligent systems. NSF, NASA and the DOD government is concentrated largely in the National agencies are supporting fundamental research on Science Foundation (NSF), the Department of computer systems designs. NSF and DARPA (for Defense (DOD)-Office of Naval Research (ONR), its own research) are the principal federal sources Army Research Office (ARO), Air Force Office of funds for computer research equipment. of Scientific Research (AFOSR), and the Defense It should be noted that a certain difficulty al­ Advanced Research Projects Agency (DARPA)­ ways arises in any survey associated with research the Department of Energy (DOE), the National support of the sciences which includes the mathe­ Institutes of Health (NIH), and the National matical and computer sciences. Today virtually Aeronautics and Space Administration (NASA). every scientific and engineering endeavor which is All other federal sources provide miniscule basic beyond its most primitive stage has a mathemati­ research support, but do provide some funds for cal, statistical, and computing component. One very applied work in consonance with the basic finds, then, almost a continuum of activities in mission of such agencies. the sciences ranging from the purely mathemati­ NSF is the only federal agency with respon­ cal to those in which mathematics and/or the sibility for the support of research across the entire computer play a role, vital or not so vital, but the spectrum of the mathematical sciences; all other research objective lies completely within another agencies support research which is in larger or science. The boundaries between the mathemati­ smaller measure related to their various missions. cal sciences, computer science, and other scientific As a consequence, the NSF provides virtually all EDITOR'S NOTE. This article was prepared I federal support for core mathematics, but also by WILLIAM H. PELL, former Head of the gives substantial support to the applied math­ Mathematical Sciences Section of the National ematical sciences, to statistics and probability, Science Foundation, on behalf of the Conference and to computer science. The mission-oriented Board of the Mathematical Sciences, for the agencies spread their support widely over the American Association for the Advancement of applied mathematical sciences, statistics, and, especially computer science. Emphasis at any Science's annual report Intersociety Preliminary Analyses of R&D in the FY 1984 Budget. given agency varies in time and in accord with changing mission needs. Readers of the Notices may appreciate some remarks on the apparent differences in the way The pattern of support of computer science figures for the current year are presented in is quite different from that of the mathe­ this report and in the previous one. Dr. Pell's matical sciences because of the non-academic numbers for 1983 are those of the so-called world. Presumably an exigent need for computer "Obligational Authority for 1983"; in the annual hardware and software also play a role. Inasmuch reports published in the Notices for many years, as questions of national security arise in connec­ the current year's figures are those of the current tion with communication of the results of research "Plan," i.e. for the amount presently expected and development in (certain areas of) computer to be spent in the current fiscal year. (That science, it is not easy to arrive at accurate these numbers may differ somewhat is indicated figures on the federal financial commitment to the in the table headed "Obligations" in the part science. of the previous report under the subheading NSF provides most of the support for theoretical ''Mathematical Sciences Section.") In the FY computer science. The DOD agencies, DOE, 1984 Budget Request, the Administration has and NSF (especially for non-mission oriented) departed from its (and its predecessors') previous provide the bulk of support for numerical analysis. usage by describing its Requests as "Estimates" DARPA is coordinating with NSF and ONR an rather than "Requests," a practice which Dr. ambitious program involving research facilities Pell has chosen to follow. at selected universities for undertaking long-

290 TABLE l. National Science Foundation Budget Authority ($ Millions)

Actual Current Estimate %Increase FY 1982 _f'Y __l~El_~_f_'f_:!__cJEl_~ __ f'I.___l~~!/83

Research & Related Activities $ 97l.li $1064.1 $1250.7 17.5 Science & Engineering Education ?0.9 30.0 39.0 30.0 Special Foreign Currency Program 3.1 3.1 2.6 -12.9 ------Total s 995.6 $1097.2 $1292.3 17.8 and engineering fields of endeavor are therefore phenomenon must be noted. In what constitutes poorly demarcated. Compounding the difficulty a complete about-face, the administration is is the fact that the administrative structure and recommending substantially increased funding for disciplinary interests of, and terminology used by, science education (30 percent at NSF). the various federal agencies are each distinct. In The NSF budget request contains three view of all this, it seems clear that it is not Presidential Initiatives aimed at recognizing and feasible to do other than to accept each agency's encouraging outstanding young scientists and en­ own judgement as to the primary (or essential) gineers to pursue the academic research careers nature of the R&D which it supports. for which they have high potential, and at significantly raising the quality of teaching in Overview science and mathematics in our secondary schools. The overall FY 1984 budget for R&D is $45.8 billion, up from $38.9 billion (estimated) in FY I. National Science Foundation 1983. Of this total, $31.8 billion is slated for The overall level of the :\'SF budget request for military R&D, and only $14.0 billion for non­ FY 1984 and its context is indicated briefly in the defense R&D, which is about the same level as last summary tabulation in Table 1. year. Increases in selected areas of the physical and mathematical sciences and engineering are The most notable feature of the FY 1984 balanced by the cutting of support in other budget is the surprisingly large increase in R&D sciences. funding, both in absolute and in percentage terms ($195.1 million; 17.8 percent). It will be seen in what follows that the treatment Also surprising, accorded the mathematical and computer sciences and perhaps signalling a change in philosophy as in the agencies considered is not at all uniform in important in the long run as large increases in funds for research, the FY 1984 budget. Specifically, NSF is slated for is the $9.0 million (30 percent) rather handsome increases in the mathematical for manpower training. and computer sciences, the physical sciences, and The principal locus of :-.!SF activity in R&D engineering. The DOD agencies are being held in the mathematical and computer sciences is at very nearly a constant level, although in some in the Division of Mathematical and Computer areas there is mild growth, especially in computer Sciences within the Directorate for Mathematical science. The remaining agencies discussed will and Physical Sciences. The budget for the also be held more or less at a level of constant Directorate is in Table 2. effort, though again there will be areas of mild A. Mathematical Sciences. The objectives of growth. the NSF Mathematical Sciences Program are Although it does not have to do with research to promote significant American research in (at least not immediately or directly), one the mathematical sciences and to ensure the

TABLE 2. NSF--Directorate of Mathematical and Physical Science£ Budget Authority by Division ($ Millions)

Actual Current Estimate % Increase Activity ______!!__]: 2..._8_!1 __ FY 1 9Jl_~_B 1 98 4______IT__l~? y 8 3 Mathematical Sciences $ 30.5 $ 34.0 s 42.2 24.1 Computer Research 25.7 2q,l 34.7 19.2 Physics 75.3 89.1 107.6 20.8 Chemistry 61.4 66.4 80.2 20.7 Materials Research 79.9 81.0 99.6 23.0 ------~ ------Total $ 272.8 s 2q9.6 $ 364.3 21.6

291 TABLE 3. NSF--Mathematical Sciences Budget Authority by Program Element ($ Thousands)

ll.ctual Current Estimate % Increase Program Element FY 1982 FY 1983 FY 1984 FY 1984/83

Classical ;a.na1ysis $ 3,165 $ 3,320 $ 4,100 23.5 Modern l\na1ysis 3,258 3,430 4,150 21.8 Geometric ll.nalysis 2,q27 3,120 3, 8 50 23.4 Topology and Foundations 3,980 4,190 5,150 22.9 ll.lgebra and Number Theory 5,048 5,330 6,600 23.8 1\pplied Mathematics 3,768 4,050 5,300 30.9 Statistics and ~robability 3,432 3,560 4, 700 32.0 Special Projects 4,911 7,706 8, 325 8.0

Total $30,489 $34,706 $42,175 21.5 continuing vitality and long range health of the • Special Projects activity will be expanded discipline. A breakdown of the Mathematical as follows: the number of new 24-month Sciences budget by program element is given in Mathematical Sciences Postdoctoral Research Table 3. Fellowships will be increased from 20 and Perhaps the feature of the FY 1984 budget 30 (cost $0.2 million); other postdoctoral most deserving of comment is the surprisingly and visitor support increased by about $0.17 large increase in all of the disciplinary program million; support for computational equipment elements, and, therefore, in the Section's total increased to $1.1 million ($0.1 million increase). budget. The past few years have seen the in­ • Operation of the mathematical sciences re­ stallation of two institutes of the mathematical search institutes at Berkeley and the Univer­ sciences, increased support of postdoctoral study, sity of Minnesota will be maintained at a computational equipment for research, and an constant level of effort. Cost: $2.9 million. expansion of a program of conferences by the • Significantly increased support will be Section. These initiatives, coupled with unprece­ provided for research activities in computa­ dented inflation, have resulted in some erosion of tional mathematics and on mathematical support of senior investigators. The essentially aspects of large scale scientific computing. across-the-board program increases of the FY This is a new activity, considered to be an area 1984 budget will allow a reversal of this trend, as of high scientific opportunity. It is budgeted well as a continuation of the other activities just at $1.3 million distributed throughout the mentioned. Specifically: Section's programs, but with special emphasis • $3.7 million increase in personnel support on in the applied mathematics, statistics and grants: $0.75 million for twenty additional probability, and algebra and number theory postdoctoral research associates, $1.60 million programs. for eighty graduate students, and $1.35 million for midlevel and senior research visitors. B. Computer Research. The objectives of the • $1.9 million increase for enrichment of re­ Foundation's Computer Research Program are to search grants for such direct costs as travel, generate new fundamental knowledge about the publications, and supplies. structure and design of computer systems, both

TABLE 4. NSF--Computer Research Budget Authority by Program Element ($ Thousands)

Actual Current Estimate % Increase Program Element ____F_Y 1982 FY 1983 FY 1984 FY 1984/~

Theoretical Computer Science $ 3,225 s 3,200 $ 3,700 15.6 Software systems Science 3,150 3,100 3,700 19.4 Software Engineering 3,133 3,000 3,500 16.7 Intelligent Systems 3,348 3,000 3,500 16.7 Computer Systems Design 3,078 3,100 3,600 16.1 Coordinated Experimental Research 8,553 11,175 13,725 22.8 Special Projects 1,2Fi0 1,245 1,450 16.5 Computer Research Equipment -0- 1,300 1,500 15.4

Total $25,747 $29,120 $34,675 19.1

292 TABLE 5. NSF--Electrical, Computer, and Systems Engineering Budget Authority by Program Element ($ Thousands)

Actual Current Estimate % Increase Program Element FY 1982 FY 1983 FY 1984 FY 1984/83

Automation, Bioengineering/ Sensing Systems $ 4,758 $ 4,939 $6,350 28.6 Electrical/Optical Communications 4,478 5,141 6,350 23.5 Computer Engineering 2,607 2,500 3,700 48.0 Quantum Electronics, Waves/Beams 4,095 4,534 5,400 19.1 Solid State/Microstructures Engineering 4,793 5,556 7,200 29.6 Systems Theory/Operations Research 4,405 2,000 5,400 16.6 Science/Technology to ~id the Handicapped 647 2,000 2,300 15.0

Total $25,784 $29,300 $36,700 25.3 hardware and software, synthesize accumulated of a large variety of man-made systems. These knowledge into coherent theories which point out systems range from integrated circuits with mini­ new directions for exploration and testing, and mum feature size of a few billionths of a meter to train sophisticated research personnel essential global communications networks. to the continuing development of both computer The Electrical, Computer, and Systems En­ research and the computer industry. A breakdown gineering (ECSE) subactivity supports basic and of the budget of the Compuuter Research Section applied research on electrical materials and by program element is given in Table 4. devices, sensors and imaging systems, very large The budget request for FY 1984 is $5.6 million, scale integrated circuits, computer engineering, or 19.1 percent, above the FY 1983 current plan. integrated optics and opto-electronics, informa­ This provides substantial real growth in all of the tion theory, signal processing and communica­ Computer Research programs, but especially in tion systems, machine intelligence and robotics, experimental computer research. Highlights of automation, and control and system design the FY 1984 budget are: methodologies. The budget for this subactivity is • Support for coordinated experimental research in Table 5. facilities will be increased by $3.4 million, or The principal contributions to R&D in the 35.3 percent, to a total of $13.0 million. mathematical and computer sciences reside This will permit the addition of 4 to 5 in the Computer Engineering and Systems new facilities, bringing the total to 18 or Theory/Operations Research program elements. 19. Support of CSNET (computer research In particular, network), the other activity in the Coordinated • Computer research in the Computer Engineer­ Experimental Research program element, will ing Program, directed at broadening the un­ be reduced by $0.6 million to $0.7 million (46.1 derstanding of the design and construction of percent), reflecting the shift of that project computers, is coordinated with and comple­ toward self-supporting status. ments work supported by the Computer Re­ • The research equipment program will be search Section. In FY 1984 novel computer increased by $0.2 million, or 15.4 percent, architectures and interfaces for intelligent to a total of $1.5 million. This is a new robotic systems will be emphasized. The program element established in FY 1983. program increase of 48 percent is much the • Research project support will be increased by largest in the Division. $2.8 million, or 16.9 percent, to a total of • The Systems Theory and Operations Research $19.5 million. This will allow support of new Program supports development of mathemati­ opportunities in experimental and theoretical cal and computational methods for analysis, computer research projects. modelling, simulation, optimization, and con­ trol of systems and processes. C. Electrical, Computer, and Systems Engineer­ • Increased support for equipment to inves­ ing. R&D in the mathematical and computer tigators is planned, and for research initiation sciences figure to a significant extent in the grants to new faculty. Electrical, Computer, and Systems Engineering subactivity of the NSF Directorate for Engineer­ D. Information Seienee and Technology. A limited ing. The objective of that subactivity is to amount of R&D in the mathematical and com­ gain new knowledge of fundamental engineering puter sciences may be found in the programs of the principles applicable to the analysis and design Information Science and Technology Division of

293 TABLE 6. NSF--Information Science and Technology Budget Authority by Program Element ($ Thousands)

Actual Current Estimate % Increase Program Element FY 1982 FY 1983 FY 198 4 FY 1984/83

Information Science $ 4,028 $ 4,000 $ 4, 000 0.0 Information Technology 370 597 1,000 67.5 Information Impact 800 800 1,100 37.5 ---- Total $ 5,198 $ 5,397 $ 6,100 13.5 the Directorate for Biological, Behavioral and So­ a constant level to constitute the heart of the cial Sciences. Work within this subactivity draws Division's effort. upon and is coordinated with that supported by A, B, and c above. The objective of this subactivity is to discover IT. Department of Defense and formulate the principles governing the genera­ tion, transmission, and use of information. DOD support of R&D in the mathematical and computer sciences is accomplished in two ways: The NSF is the primary federal supporter of basic research in information science and 1) by grants and contracts with academic, and non-profit organizations (primarily technology. In contrast, the DOD concentrates industrial, operations on requirements for communications, command, the first of these), and 2) through control and intelligence systems, administrative conducted in-house and through contracts, in their and office systems, and on human factors in own laboratories or other installations. Such R& D work is concentrated primarily in the following weapons systems design; the NIH (National Library of Medicine) focuses its research on departmental entities: improving medical information systems. The 1) Army Research Office (ARO) budget for the subactivity is given in Table 6. 2) Office of Naval Research (ONR) The substantial increases in the Information 3) Air Force Office of Scientific Research Technology and Information Impact Programs (AFOSR) respond to research opportunities relating to electronic components and computing machinery, 4) Defense Advanced Research Projects Agency as well as to increasing international research (DARPA) efforts in information technology. The basic The first three are reasonably similar in their research portion of the program continues at method of operation. DARPA differs in that it is TABLE 7. ARO--Mathematical and Computer Sciences ($ Thousands)

Actual Current Estimate % Increase Subactivity FY 1982 FY 1983 FY 1984 FY 1984/83

Nonlinear Analysis* $ 1,900 $ 1,670 $ 1,940 16.2 Computational Methods and Mathematical Software* 1,800 1,700 1,900 ll.8 Statistical Methods and Operations Research* 1,300 1,200 1, 400 16.7 Systems Theory, Control, and Modelling 900 900 1,200 33.3 Conferences and Special Projects** 100 160 160 0.0 Computer Science 1,000 1,000 1, 000 0.0

Total $ 7,000 $ 6,630 $ 7, 600 14.6

* The FY 1983 and FY 1984 numbers for Special Projects include $80,000 for Special Graduate Fellowships. ** The support of the Mathematics Research Center, University of Wisconsin, is imbedded in the first three subactivities listed. These amounts are FY 1982, $2,121,000; FY 1983, $2,200,000; FY 1984, $2,200,000.

294 TABLE B. AFOSR--Mathematica1 and Information Sciences ($ Thousands)

Actual Current Estimate % Increase Topic FY 1982 FY 1983 FY 1984 FY 1984/83

Control Theory s 853 $ 1,051 $ 1,298 23.5 Computer Science 3,3461 4,3702 5,1803 18.5 (1,800) (1,970) (1,980) (0. 5) Computational Mathematics 1,397 1,900 1,975 3.9 Physical Mathematics 1,033 867 1,000 13.3 Probability and Statistics 1,798 1,860 3,3204 78.5 (1,820) (-2 .1) Systems Science 1,659 1,355 1,400 3.3 Information Electronics 580 605 637 5.3

Total $10,6661 $12,0082 $14,8103,4 23.3 (9, 120) (9 ,608) (10, 110) (5. 2)

~Includes Systems Automation Initiative of $1,546 thousand. Includes Systems Automation Initiative of $1,200 thousand 3and Image Understanding Initiative of $1,200 thousand. Includes Systems Automation Initiative of $1,200 thousand 4and Image Understanding Initiative of $2,000 thousand. Includes Reliability for Real Systems Initiative of $1,500 thousand. not attached to any one department, and is not The Air Force also supports a substantial so strongly focused on basic research. amount of R&D intramurally at several Air Force installations. In FY 1984 support is Army Research Offiee. In the tabulation in planned as follows: Eglin Air Force Base, $525,000; Table 7, all of the listed programs are in the Rome Air Development Center, $437,000; Wright­ Mathematical Sciences Division except Computer Patterson Air Force Base, $713,000; and Kirkland Science, which is a program of the Electronics Air Force Base, $84,000. This work is highly Division. mission-directed, with about a 50-50 split between A certain amount of R&D is carried on in computer science and mathematically-oriented Army laboratories such as Aberdeen, Picatinny, research. and Watervliet. This is estimated to amount to about $1 million in each of FY 1982 and FY 1983, Office of Naval Research. Recent reorganization and is not included in Table 7. of the Office of Naval Research has resulted in the former Division of Mathematical and Information Air Foree Office of Seientifie Research. The titles Sciences being split up into a Division of Mathe­ of the first six of the seven listed topics in Table matical Sciences and a Division of Information 8 carry their commonly associated meanings; the Sciences, with the Fluid Dynamics Program being seventh topic, Information Electronics, is actually folded into the Division of Mechanics. concerned with mathematics related to electronic communications. A. Information Sciences Division. The Informa­ The footnotes indicate special funds dedicated tion Sciences Division is concerned with research to providing research emphasis, or, as the Air in such topics as: computer architecture, robotics Force puts it, "initiative" in certain years. and advanced automation, artificial intelligence, Numbers in parentheses are base program funds, software engineering, and user-machine interfaces. i.e., program funds without the Initiative money. The budget for this Division is in Table 9.

TABLE 9. ONR--Information Sciences Division ($ Thousands)

Actual Current Estimate % Increase Programs FY 1982 FY 1983 FY 1984 FY 1984/83

Computer Architecture $ 510 $ 563 $ 547 -2.6 Theory and Techniques 3,504 3,538 3,432 -3.0

Total $ 4,019 $ 4,101 $ 3,979 -3.0

295 'rABLE 10. ONR--Mathematical Sciences Divis ion ($ Thousands)

Actual Current Estimate % Increase Group or ~P~r~o~g~r~a~m~------~F~Y~l~9~8~2~F~Y~l~9~8~3~F-=Y~1~9~8~4~~F~Y~l~9~8~4~/~8~3 Mathematics Applied Mathematics $ 1, 92 3 $ 2,873 $ 2, 725 -5.2 Numerical Analysis 1, 685 2,813 3,300 17.3 Operations Research (Math.) 1,540 1,609 1,534 -4.7 Statistics and Probability Statistics 3, 931 3,826 3,400 -11.1 Operations Research (Stoch.) 640 1,054 1,005 -4.6 Systems Science 838 525 531 1.1 Total $10,557 $12,700 $12,495 -1.6 At present there is a particular focus in this FY 1982 the Agency's Information Processing Division on: 1) revolutionary software, 2) the Technology Office supported research at about construction of intelligent robots, and 3) artificial the level of $75 million, about one-half of which intelligence. was in information processing technology; support for FY 1984 is expected to be at about the same B. Mathematical Sciences Division. The Math­ level. DARPA does not support research in the ematical Sciences Division is composed of two mathematical sciences. groups: 1) a Mathematics Group and 2) a Statis­ tics and Probability Group. The budget is in Table 10. m. Department of Energy Normally, about 30-40 percent of the ONR Last year's plan for reorganization Program's funds are devoted to "special focus" of the federal energy establishment, the programs. These are analogous to the AFOSR scenario of which "initiatives." had the Department of Commerce becoming the parent cabinet-level organization for most of DOE These figures do not contain monies which ONR activities, with the R&D functions of DOE being handles for R&D work at the Naval Research incorporated into a new Energy Research and Laboratory (NRL), Naval Air Systems Command, Technology Administration (ERTA) seems to have Naval Sea Systems Command, and the Naval been put to one side, if not dropped. Electronic Systems Command. The order of In FY 1984 DOE funds for R&D will drop, magnitude of funds for each of these is about primarily in the area of energy $1 million though NRL funds programs. R&D will be somewhat funds for basic science larger. will, however, increase from $283 million in FY 1983 to $350 million. Most Defense Advanced Research Projects Agency. of the increase will go to the physics facilities at This agency seems to be more directly sensitive the Brookhaven and Lawrence laboratories, with to the research and development needs of the an overall increase for mathematics and computer DOD than the research agencies of the individual science of about 6 percent. departments, and, indeed, appears to be the Activities supporting R&D in the mathemati­ central motive office for long-range research cal and computer sciences at DOE are lodged and development within DOD. DARPA's role primarily in a program of Applied Mathematical is analogous to that of the corporate research Sciences within a division of Engineering, Mathe­ division of private sector organizations. Because matical, and Geosciences. This is in turn a sub­ of this, DARPA is able to initiate investigations activity of a program of Basic Energy Sciences, in new directions or involving new concepts which is the principal program in the DOE applicable to military departments without regard category of Supporting Research and Technical to role or mission limitations. A key element of Analysis. the DARPA mission is its ability to effect the The objective of the Applied Mathematical transition from research effort to development Sciences (AMS) research program is to advance and application. DARPA accomplishes this the understanding of the fundamental concepts by the execution of research programs through of mathematics, statistics, and computer science appropriate Service agents for performance by in­ underlying the complex mathematical models of house R&D laboratories or by contracts awarded the key physical processes in energy systems. to university or industrial organizations. The AMS program funds basic research at DARPA does not have well-defined programs many of the national laboratories, universities in either the mathematical or computer sciences and private research institutions in three major or in basic, as opposed to applied, research. In categories: analytical and numerical methods,

296 TABLE 11. DOE--Applied Mathematical Sciences Budget Authority by Component ($ Thousands)

Actual Current Estimate % Increase Component FY 1982 FY 1983 FY 1984 FY 1984/83

Analytical and Numerical Methods s 6,000 $ 6,800 $ 7,000 2.9 Information ~nalysis Techniques 1,900 2,000 2,000 0.0 ~dvanced Computer Concepts 3,200 4,350 4,970 14.3 Special Projects 500 700 700 0.0 ---- Total $11,600 $13,850 $14,670 5.9 information analysis techniques, and advanced amount of R&D, much of it basic research, in computing concepts. The budget for the AMS all disciplines connected with its basic mission. program is in Table 11. This is carried out in-house in NASA centers The FY 1984 budget request is $820 thousand and laboratories and extramurally through grants (5.9 percent) larger than that for FY 1983, and contracts. Support for the mathematical about three-quarters of which will go into the sciences is relatively small, while that for com­ computer science rather than the mathematical puter science is considerably larger. In fact, NASA aspects of the program. The increase will does not support research in the mathematical permit continued emphasis on the support of the sciences per se but only those portions which have algorithmic, language and architectural aspects of direct relevance to the NASA mission, e.g. control high performance parallel computing systems for theory, numerical work in ordinary and partial computation. In FY 1984-1985 a major new thrust differential equations, theoretical and numerical in large-scale scientific computing is planned, with fluid mechanics. emphasis on both numerical and non-numerical NASA has undergone considerable administra­ algorithms for large-scale parallel architectures tive unheaval lately, and the Office of University based on several experimental computer systems Affairs, which formerly provided figures on re­ under study. Some additional support for analysis search support, has been abolished. No predeter­ of the mathematics of non-linear systems will be mined amount is budgeted for support in a given possible. Support for computational mathematics area, so NASA has used the device of projecting and statistics will be held at the FY 1983 level. support for the forthcoming year to be the same as that for the current year. This is not, of course, an IV. National Aeronautics illuminating procedure for detecting trends. The and Space Administration breakdown for the university grants and contract The National Aeronautics and Space Ad­ program in mathematics and computer science is Table 12. ministration (NASA) supports a very considerable in

TABLE 12. NASA--Mathematics and Computer Science ($ Thousands)

Actual Current Estimate % Increase Classification FY 1982 FY 1983 FY 1984 FY 1984/83 Mathematics (Basic) $ 813 $ 853 $ 853 0.0 (~pplied) 124 150 150 0.0 Computer Science (Basic) 1,006 1,000 1,000 0.0 (Applied) 1,272 1,350 1,350 0.0 Mathematics/Computer Science (N.E.C!') (Basic) 677 650 650 0.0 Mathematics/Computer Science (N.E.C:*) (Applied) 91 100 100 0.0 ----- Total s 3,983 s 4,103 $ 4,103 0.0

* N.E.C. means Not Elsewhere Classified, and is used for interdisciplinary projects and others not listed by discipline name. This is a formal classification specified by OMB Circular A-46.

297 TABLE 13. NIH--Mathematical Sciences Budget Authority by Type of Research ($ Thousands)

~ctual Current Estimate % Increase Typ~ of Research ______--~1982 X!_l,~__!!l__FY 1984 FY 1984/83

Mathematics (Basic) $ 5,409 $ 6,126 $ 6,342 3.5 Mathematics (Applied) ---~02 ___ 5,0l_i 5,183 3.4 Total $ 9,916 $11,140 $11,525 3.5

In the past it has been the case that NASA has of so complex a nature that it is difficult to conducted intramural research in the mathemati­ obtain data on the support of a specific discipline. cal and computer sciences (mostly) at some of its There are no specific programs of support for facilities. In FY 1978 this was estimated to be mathematics, mathematical sciences, or computer about $800 thousand. In the time available it has science. What one finds instead is that for each not been possible to obtain an estimate of this for grant or contract an estimate is made of the FY 1984. fraction of the research done thereon which is attributable to mathematics, chemistry, etc. The V. National Institutes of Health total of such fractions converted to dollars is reported as the support for the discipline. See The research efforts (mathematical or otherwise) Table 13. supported by the National Institutes of Health It should be noted that only mathematics is (NIH) fall into two distinct and unrelated catego­ reported by NIH. It may be further pointed out ries, or programs. The first is the "extramural" that research in the mathematical sciences is not research which is that resulting from a very large very purposefully supported by NIH (see above on grants and contracts program conducted by NIH, how it is reported), but is a secondary by-product while the second is ''intramural" research which is or concomitant of other research. produced by NIH employees. The grants and con­ tracts program is somewhat similar to grants and contracts programs at other federal agencies such VI. National Bureau of Standards as NSF or DOD, but differs in important respects. Some 80-90 percent of the NIH budget for any The National Bureau of Standards (NBS) in given year is dispensed through this extramural the Department of Commerce has within its program. National Engineering Laboratory a Center for The "in-house" or intramural research program Applied Mathematics. The Center, which in the mathematical and computer sciences seems comprises Divisions of Mathematical Analysis, to be little known to the scientific community Statistical Engineering, Operations Research, outside of its medical and biological sectors and Scientific Computing, and Computer Services, is perhaps not really familiar to them. A has as its primary goal to ensure that the most discussion of this program cannot be pursued in appropriate mathematical, computational, and detail here, but the following remark may be statistical procedures are available for use in the illuminating. The in-house researchers are found NBS technical programs. The operations of the in the biometrics branches of the various Institutes Center are entirely in-house, and in part these (eleven), in theoretical research groups, in the operations involve R&D in the mathematical and Division of Computer Research and Technology computer sciences. The figures in Table 14 are (the central resource), and in the Computer from the FY 1984 budget submission to Congress Division. It is difficult to estimate the funding for the Center. required to support this research, but there is no doubt that it is not inconsiderable vis a vis the A portion of the support of the Center comes budgets of Agencies discussed above. from contracts with or transfers of funds from other federal departments or agencies. Currently The NIH is so organized and the principal focus of the extramural research which it supports is the extent of this is about $1 million annually.

TABLE 14. NBS--Center for Applied Mathematics ($ Thousands)

~ctual ~urrent Estimate % Increase Subactivity FY 1982 FY 1983 FY 1984 FY 1984/83 Mathematical Sciences $ 4,730 $ 5,013 $ 5,503 9.8

298 One other NBS activity, tinder the heading which is operated in association with the National of Central Technical Support, deserves brief men­ Research Council, was $2,190,000 in FY 1981. The tion, namely, the Research Associates Program. current plan budget of the program for FY 1983 The objective of this program is to provide is $2,224,000, and its budget authority request for scientists and engineers from industry, academia, FY 1984 is $2,412,000. The Postdoctoral Research and other federal agencies who demonstrate out­ Associateships component of the program typi­ standing promise and ability and opportunity cally supports around twenty-five such associates to participate in NBS research programs in per year for the entire NBS. Normally, two or the various branches of physical, engineering, three of these will be associated with the Center and mathematical sciences. The overall mag­ for Applied Mathematics. nitude of the NBS Research Associates Program,

299 Federal Support-Where Do We Stand? by Kenneth Hoffman

This is a brief report from the AMS Com­ will exact one more toll this year in the form of mittee on Science Policy, written to accompany reduced grants for established investigators. It is publication of the FY84 budgets of the various these last two issues which we wish to focus on. federal agencies which support research in the Let's talk first about DOD. Basically what is mathematical sciences and the report of the going on is that, as NSF support of fundamental Mathematics Briefing Panel (the Browder Panel). research in the mathematical sciences is being The Panel was established in the fall of 1982 by pushed up, such support in the three militalf' COSEPUP (the Committee on Science, Engineer­ services is going down. This is something which 1s ing and Public Policy of the National Academy of of deep concern to us, because DOD carries half of Sciences), to brief the President's Science Advisor, the federal support for applied mathematics and George A. Keyworth. Our main purpose here statistics. In a bit more detail, here is what is is to point out two major concerns of the mathe­ happening: The fundamental research money at matical sciences community as we move into the AFOSR, ARO and ONR (the 6.1 money, as they spring of 1983. As individuals you no doubt call it) is being diverted into special 'initiatives' will have other concerns. We encourage you to or 'thrusts' of brief duration, motivated by earlier express them, especially to us: AMS Committee impacts on technologies. Thus, while the overall on Science Policy, Room 2-280, MIT, Cambridge, budgets of the mathematical offices of these Massachusetts 02139. agencies are growing, their 'core' programs--:­ Spring makes a useful metaphor for where we the ones which provide stable funding for bas1c now stand relative to research support: It is still research-are shrinking. A careful reading of Bill winter, but a thaw is coming. Those who are Pell's preceding summary of FY84 federal support actively working on funding issues on your behalf will show this to be the case. The situation we see are confident that our research support picture is not unique to the mathematical sciences. It is will improve dramatically over the next few years, even more extreme in other sciences. The matter is even though we have another difficult year to face. obviously troublesome, not only because of short­ What does this mean in plainer English? First term erosion-the extension of our winter-but that, through hard work over the last couple because there is involved a fundamental policy of years by some of our colleagues and the question for the federal government: Through central science policy/ science funding people in which agencies will basic research be supported Washington, we have turned a corner. The fact over the long haul? We will comment more on this that 'mathematics' funding and mathematical in future articles. For now, suffice it to say that science departments have major problems and the issue is most serious, we are working as hard that something substantial must be done about as we can to try to help turn the situation around, them is now ingrained in science funding policy. and-although things at DOD may get worse The principal manifestations of this are the before they get better-we remain confident that proposed 24% increase for the Mathematical the efforts will succeed. Sciences Section in the FY84 National Science offset Foundation budget just submitted to Congress The DOD losses should be more than and the accompanying statements, which suggest by NSF gains, starting next year. Since this that this may be only the first of several planned issue of the Notices contains the complete text of injections of resources. the FY84 budget of NSF's Mathematical Sciences can see not only the size of the Why is it then still 'winter'? Not because we Section, one increase in funds but also the planned still have Congressional approval to be concerned proposed funds. The document speaks for itself, about. There is a genuine problem there, but use of the things. we think that (as things are shaping up) it will but we would like to call attention to a few prove surmountable. It is still winter because In reading the budget, it is important to bear in (i) we won't feel the effects of the sizable NSF mind that increase for a year or more; (ii) meanwhile, • only certain of our needs are addressed by the the support of fundamental research by the planned use of additional resources; we feel Department of Defense (DOD) continues to erode that those addressed are of high priority and the modifications in the pattern of use of • the priorities were set after much reflection funds at NSF, started a couple of years ago, by the NSF's Advisory Subcommittee for the

300 Mathematical Sciences, a diverse group of our KENNETH M. HOFFMAN (Massachusetts Institute colleagues of Technology) is Chairman of the Society's Com­ • what you see is only the first step in mittee on Science Policy. Other members of restructuring and stabilizing our funding the Committee are Hyman Bass (Columbia Univer­ patterns for the future. sity), ~elix E. Browder (University of Chicago), Of the several needs not yet addressed perhaps Fredenck W. Gehring (University of Michigan), none is as serious or as much on people's minds Andrew M. Gleason (Harvard University), Joseph J. Kohn (Princeton University), William J. LeVeque as the need to stop, and even reverse, the trend (Executive Director, AMS), Cathleen S. Morawetz of reduction of the number of established (non­ (New York University, Courant Institute), George junior) people supported by NSF grants. We D. Mostow (Yale University), Julia B. Robinson support whole-heartedly the heavy emphasis on (University of California, Berkeley), Hugo Rossi young people in the existing and planned use of (University of Utah), I. M. Singer (University of additional funding and we are fully aware that we California, Berkeley), James D. Stasheff (University will probably be forever stuck with some form of a of North Carolina, Chapel Hill) and Guido L. Weiss two-class system relative to federal support, since (Washington University). least two and one-half times as many there are at 1. That, beginning in FY84, there be no further productive research people in the mathematical planned reduction in the number of grants for sciences as are now supported. But we seem established investigators (there is always a small perilous point with regard to to have reached a amount of unplanned reduction by attrition, of who is supported and who is not: Thoughtful course). people in the mathematical community cannot grant tell the difference at the cut-off line between 2. That in FY84 part of the existing go to the quality of the established people who are money (a part which would probably anyway) be combined with part of supported and those who are not; the criteria for young people new resources earmarked in the very young mathematicians to get grants have the anticipated postdoctoral support to form a special become difficult for all but a privileged few to budget for the specific purpose of increasing the meet; confidence in the peer review system is fund for These weakening; morale is going down at distinguished number of grants for young investigators. The research centers, which are suffering under a large should be adequate grants, not skimpy ones. as the number of other pressures in addition to the fund should be phased out after a few years, a stable "lopping off of heads" syndrome. pool of general resources increases and structure takes shape. The Committee on Science Policy is convinced in­ that the nature of these issues is quite well 3. That, from FY85 onward, a planned crease of several percent per year in the number understood by the MSS Advisory Subcommittee investigators supported be imple­ and the administration of the Foundation. It is of (established) also convinced that the problem exists in other mented. fields of science as well (even though our situation The purpose of these suggested policies is may be more extreme), that plans are in the to make clear to the mathematical sciences works to ease the strain, and that the timing community that a destructive trend will be and scope of plans to reverse the downward trend reversed, as we go about reconstructing a support in the number of established people supported pattern which addresses many needs. The pace reflect a number of very real political constraints of the reversal will seem too slow to many; the under which those who set and implement policy wisdom of it will seem questionable to others. We must work. In connection with the last point, encourage members of the research community we urge you to read in the February Notices to communicate their thoughts to us and to see the remarks made at the Denver AMS Meeting our Committee as one channel for getting major by Edward A. Knapp, Director of the National concerns across to policy makers. Our experience Science Foundation, as well as the editorial in the has been that the Advisory Subcommittee and February 18 issue of Science magazine, written by administration at NSF, as at other agencies, are George Keyworth. open and receptive to our inputs. They are also quite plain-spoken) about some of the We say these things as a preamble to some open (and specific recommendations we are transmitting to practical limitations we face, so we of necessity communicate some of them to you. But this is the NSF Advisory Subcommittee for the Mathe­ both matical Sciences, recommendations for policies to the way proposals are developed which are complement those which have been evolving in effective and realistic. the light of prospects for significant increases:

301 Queries Edited by Hans Samelson 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. REPLIES from readers will be edited, when appropriate, into a composite answer and published in a subsequent column. All answers received will ultimately be forwarded to the questioner. QUERIES and RESPONSES should be typewritten if at all possible and sent to Professor Hans Samelson, American Mathematical Society, P. 0. Box 6248, Providence, Rhode Island 02940. Queries (when and where for the 1st time) formulated the following two conjectures? What are the 271. Jean Paul Van Bendegem (Seminarie Voor latest results on them? Both conjectures look Logica en Wijsbegeerte van de Wetenschappen plausible. (1) e is very transcendental, i.e., for B-9000 Ghent, Belgium). I only know of two ar­ any term t obtained from 1 by +, ·, j, exp we ticles by Alexander Yessenin-Volpin [A. S. Esenin­ have t = 1 iff Vn: t(n) = 1, where t is the result Vol'pin] on finitistic analysis: Le programme ultra­ of the substitution e t - ,_. n t - in t (a t b intuitionniste des fondements des mathematiques, means a b). (2) {ln(p)IP is prime} are algebraically Infinitistic Methods (Proceedings of the Sym­ independent. The strongest result I saw toward posium on Foundations of Mathematics, Warsaw, (2) is: the ln(p) are linearly independent over the September 2-9, 1959), Pergamon Press, Oxford, algebraic numbers. These conjectures are related to the Skolem problem on exponential functions; 1961; PWN, Warsaw, 1961; The ultra-intuitionistic if (1) and (2), then there is a decision method for criticism and the antitraditional program for foun­ inequalities between (1, exp )-terms as in (1). dations of mathematics, Intuitionism and Proof +, ·, /, Theory (Kino, Myhill and Vesley, editors), North­ 276. J. R. Higgins (Department of Science, Holland, Amsterdam, 1970. Cambridgeshire College of Arts and Technology, My problem is two-fold: (i) is any other work Cambridge, England). I am trying to assess ofYessenin-Volpin known; (ii) does anybody know the current status, history and development of what Yessenin-Volpin is doing at the present the "nearest neighbor" problem: Let M denote of moment (or even simply where he is!) Even a a subset of Rn and S a discrete subset M. the distance d from any point of S to negative answer would be helpful. Consider its nearest neighbors in S. From among all such 272. Francis D. Lonergan (8 Spring Street, subsets S with a given density in M find those Webster, Massachusetts 01570). Is it known for which maximise the value of d averaged over all which integer values of l, m, n, t the "generalized" points of S. I am interested in tracking down binary polyhedral groups, (l, m, nCtl), given by the complete proofs for the solution in the plane, and group presentations generators: R, S, T, relations: would like to know if complete solutions exist R 1 = sm = rn = RSTt are or are not residually for n points in a plane square, a plane circle, finite? More generally, has anything been on the surface of a sphere, a torus. I would appreciate references to any aspect of this published or is much known about these groups? greatly problem, at any level. 273. David Mesehelotf (Moshav Hemed 50295, 277. Albert A. Mullin (506 Seaborn Drive, Israel). (1) Has the simple probability function Huntsville, Alabama 35806). There is considerable P(x) = (1/rr) (x +sin x), 0 ::;; x ::;; 1r, been studied mathematical information available on distribu­ or applied anywhere? References will be greatly tion functions for the prime numbers. Far appreciated. (2) This function is monotonically less is known about distribution functions for increasing, and is clearly invertible. Is there products of two primes whose difference is d 2:: 1; an analytic expression for the inverse? Good so-called d-bi-composites. I would appreciate approximations of the inverse near the upper end information on tight asymptotic estimates (e.g., of the range are hard to get since P'(1r) = 0. What exact orders of magnitude) for the following dis­ is a good analytic way around this obstacle? tribution functions: (1) f3d(n) = card{m::;; n: m is d-bi-composite}; (2) f3e(n) = card{m :S; n: m 274. Elvin Lee (1502 S. 17th Street, Fargo, North is d-bi-composite for all even d}; and (3) (30(n) = Dakota 58103). What is the state of the conjecture card { m ::;; n : m is d-bi-composite for all odd that every prime number divides at least one d}. Note. Put (3(n) = I:d>l f3d(n). Then (3(n) ~ amicable number? Is it true that the number of nloglogn/logn. - distinct prime divisors of an amicable number is unbounded (assuming there are infinitely many Problem List amicable numbers)? Does anybody have computer files of amicable numbers? Ergodic Theory 275. R. Gurevich (Maurice Thorez 33, Apt. The following is a list of open questions and 205, Leningrad 194223, USSR). Has anybody conjectures which were introduced at a problem

302 session directed by A. Katok (notes by R. Spatzier) then there exists a finite r-invariant probability at a special session on Ergodic Theory. The special on the circle equivalent to Lebesgue measure and session was organized by N. Martin and held at T is a one-sided Bernoulli shift. What is the the AMS meeting in Toronto, Canada, August 23- density function of this measure? 26, 1982. 4. (A. del Junco). In order to find the value of 1. (M. Brin). It has recently been proven the Ornstein isomorphism between two Bernoulli (independently by K. Burns and W. Ballman­ shifts with the same entropy at a point, one M. Brin) that if M is a compact manifold of must know all terms in the sequence which nonpositive curvature and if there exists a geodesic constitutes the point. Sinai, on the other hand, in M, the universal cover of M, that does not obtained a map intertwining the shifts which bound a flat half plane then the geodesic flow of only requires knowledge of the "past" times in M is ergodic. Examples of manifolds which do a sequence. Meshalkin's isomorphism between not satisfy the second hypothesis are: (a) products B(1/2, 1/8,1/8,1/8, 1/8) and B(1/4, 1/4,1/4, 1/4) is of manifolds of nonpositive curvature, (b) locally of the Sinai type. Question. If two Bernoulli shifts symmetric spaces of rank ~ 2, (c) manifolds that have the same entropy is there a "one-sided" are "glued" from these. Assuming that the factor isomorphism ¢, i.e., such that 4> is determined manifolds in (a) have ergodic geodesic flows then by the "past" and ¢-1 by the "future"? Quite (1) the geodesic flows on these examples are not generally, the relation between Ornstein's and ergodic, (2) the ergodic components are compact Sinai's theorem is not well understood. smooth manifolds, and (3) any ergodic component 5. (N. Markley). Let Z2 act on {0, l}z• by intersects a maximal flat subspace only a finite the rule given by the matrix n~). Is there number of times. Question. Do properties similar a unique invariant measure p, for this action to (1), (2), and (3) with suitable modification, hold whose measure theoretic entropy is equal to the for compact manifolds M of nonpositive curvature topological entropy? Give a closed formula (for all of whose geodesics in M bound a flat half example, in terms of the eigenvalues of the matrix plane? Classify the manifolds M of nonpositive as in the one-dimensional case) for the topological curvature all of whose geodesics in M bound a entropy. Numerical computations show that h ~ flat half-plane. Is every such manifold locally of 0.41. Compute the measure of a block without type (a) or (b)? taking a thermodynamic limit. 2. (D. Rudolph). Does there exist a smooth 6. (A. Katok). (a) del Junco has shown that Riemannian metric on each compact manifold so there exists an interval exchange T with minimal that the geodesic flow is ergodic, or even Bernoulli? self-joining. On the other hand, generically an Does there exist a geodesic flow on some manifold interval exchange Tis rigid (i.e., for some sequence which is K but not Bernoulli? Do there exist kn, Tkn ---+ id; Katok). Conjecture: All nontrivial zero entropy, weak mixing but not mixing, etc., self-joinings of T appear from transformations geodesic flows on manifolds? The answers may commutmg with T, so that there are uncountably depend on the smoothness of the metric. For many transformations commuting with T. (b) example, there exists a C 1 metric on 8 2 whose Generically the only spectral multiplicities of geodesic flow is ergodic. In all known smooth T X T for an interval exchange T are 2 and examples the geodesic flow is Bernoulli whenever 4 (Katok). Conjecture: Generically TXT has it is ergodic. homogeneous spectrum of multiplicity 2. Does transformations 3. (N. Martin). Let this hold true generically for all on a measure space with respect to one of the n z-a· m 1-o·z standard topologies? Are there examples of t(z) = II ~ II __J transformations with homogeneous spectrum of k=l z-bj i=1 1-ajz higher multiplicity? for z, ai, bi complex, Iail < 1; lbil < 1 and r denote the restrictions of f to the unit circle. If ~ 1-lail ~ 1-lbkl L.---- L....--->1 j=11 + lail k=11 + lbkl

303 News and Announcements

Mina Rees Awarded with applications of research to social problems NAS Public. Welfare Medal including the delivery of health care. She is a member of the visiting committees of two Mina Rees, President Emerita of the Graduate universities and works with several foundations School and University Center of the City Univer­ concerned with improving the effectiveness of the sity of New York, will be awarded the National educational establishment. Academy of Sciences Public Welfare Medal at spe­ She published her account of the wartime effort cial ceremonies in Washington on April 25, 1983. in 1980: The mathematical sciences and World The award is to be made for her contributions to War II, American Mathematical Monthly, volume the scientific enterprise, especially in mathematics 87 (1980), pages 607 to 621. and computer sciences, from wartime through the transition from war to peace and continuing Martin Kruskal Awarded today. Heineman Prize in Mathematical Physics The Public Welfare Medal is one of the most prestigious honors which the Academy can bestow; The American Physical Society (APS) presented the first recipients were George W. Goethals (who the Dannie Heineman Prize in Mathematical built the Panama Canal) and William Crawford Physics for 1983 to Martin D. Kruskal of Princeton Gorgas (who helped eradicate yellow fever in the University. Canal Zone). Other recipients have been J. Edgar Hoover, John D. Rockefeller, Jr., David E. The Heineman Prize, which includes a cash Lillienthal, Alan T. Waterman, John Gardner, award of $5000, is endowed by the Heineman Leona Baumgartner and Walter Sullivan. Foundation for Research, Education, Charitable and Scientific Purposes, Inc., and is administered Mina Rees earned her Ph. D. from the University jointly by the APS and the American Institute of Chicago in 1931, with a thesis written under the of Physics. The prize recognizes outstanding supervision of A. A. Albert. She was a member of publication in the field of mathematical physics. the faculty of Hunter College from 1926 to 1961. Kruskal was cited for "his important contributions From 1943 to 1946 she was Assistant to the to general relativity, plasma physics and especially Chief of the Applied Mathematics Panel of the to the theory of some special nonlinear differential Office of Scientific Research and Development. equations useful in many areas of physics." For this work she was honored by both the British government (King's Medal for Service, Kruskal's understanding and ability in mathe­ 1948) and the U.S. government (President's matics have contributed to knowledge in many Certificate of Merit, 1958). Mter the war she was areas of physics. His contributions have stemmed invited by the Navy to establish the mathematics from interests that include general relativity, research program in the newly created Office controlled fusion and the behavior of plasmas, of Naval Research; she served as Head of the magnetohydrodynamics, asymptotic phenomena, Mathematics Branch (1946 to 1949), Director of logic, and minimal surfaces. the Mathematical Sciences Division (1949 to 1952) After obtaining his Ph.D. in mathematics from and Deputy Scientific Director (1952 to 1953). New York University in 1952, Kruskal joined She was Dean of the Faculty of Hunter College the staff of Princeton University as a research from 1953 to 1961. At the Graduate Center, she scientist. He began his career at Princeton as part was Dean of Graduate Studies (1961 to 1968), of Project Matterhorn; he became associate head Provost (1968 to 1969) and President (1969 to of the theoretical division of the Plasma Physics 1972). Laboratory in 1956 and senior research associate In 1962, She was the first recipient of the of the laboratory in 1959. In 1961 he became a Mathematical Association of America's Award for professor of astrophysical sciences, and since 1968 Distinquished Service to Mathematics. She was he has simultaneously been serving as chairman a member of the National Science Board (1964 of the program in applied mathematics. to 1970) and, since 1973, a member of the Board -Physics Today of Directors of the Society for Industrial and Applied Mathematics's Institute for Mathematics AMS Postdoctoral Research Fellowship and Society. She was also active in the American Association for the Advancement of Science: Vice The Society has awarded one Postdoctoral President and Chairman of Section A (1954), Research Fellowship for 1983-1984. The recipient President (1971) and Chairman of the Board is Russell David Lyons, who received his Ph.D. (1972). from the University of Michigan in early 1983. He Since her retirement in 1972, Dr. Rees has been plans to begin his fellowship year at the Universite active as a member of several boards concerned de Paris-Sud.

304 The AMS Research Fellowship fund was estab­ The format of these Postdoctoral Fellowships lished in 1973 in response to the need for funds has been changed significantly from those of for postdoctoral research. The fellowships are former years to provide increased flexibility for awarded on the basis of mathematical merit to the recipients. The stipend of $52,800 provides recent recipients of Ph.D.'s (regardless of their support for two nine-month academic years and age) who are citizens or permanent residents of three two-month summers, with the awardee's a country in North America. The awards are having two options for receiving the academic­ intended to support research fellows for a period year support: as full-time support for any eighteen of one year, and at present carry a stipend of academic-year months in a three-year period, $24,000 each, with an expense allowance of $1,000. in intervals not shorter than three consecutive The competition was under the supervision of the months (the Research Fellowship option), or as Society's Committee on Postdoctoral Fellowships, a combination of full-time and half-time support consisting of Richard M. Karp, Daniel J. Kleit­ over a period of three academic years, usually one man, Susan Montgomery, Stewart B. Priddy, academic year full-time and two academic years Paul H. Rabinowitz (chairman), Irving Reiner, half-time (the Research Instructorship option). and Donald E. Sarason. The recipients are listed below with current There were many strong candidates among the affiliation in parentheses, followed by the name of 48 applicants. The fellowship is financed by the the fellowship institution: generous contributions of supporters of mathe­ Kenneth S. Alexander (Mathematical Sciences matical research, supplemented with Society funds Research Institute, Berkeley), University of appropriated according to a matching formula. California, Berkeley; Christopher R. Ander­ The continuation of the AMS Research Fellowship son (University of California, Berkeley), Stan­ program depends on contributions the Society ford University; Linda June Davis (Rutgers receives. Every member of the Society is urged University), Stanford University; David L. to contribute to the Fund. Contributions are, of Donoho (Harvard University), Princeton Univer­ course, tax deductible. Checks should be made sity; Charles L. Epstein (Courant Institute, New payable to the American Mathematical Society, York University), Stanford University; James clearly marked "AMS Research Fellowship Fund" Lee Hafner (California Institute of Technology), and sent to the American Mathematical Society, University of California, San Diego; John L. Harer Post Office Box 1571, Annex Station, Providence, (Columbia University), University of Maryland; Rhode Island 02901. Neil Immerman (Tufts University), Yale Univer­ sity; John William Lott (University of Califor­ NSF Announces Mathematical Sciences nia, Berkeley), Harvard University; Russell D. Postdoctoral Research Fellowships Lyons (University of Michigan), University of Il­ linois, Urbana-Champaign; David E. Marker (Yale Twenty-one recent recipients of doctoral de­ University), University of California, Berkeley; grees in the mathematical sciences have been Stephen A. Mitchell (Massachusetts Institute of offered fellowship awards designed to contribute Technology), Princeton University; Tobias B. to the future vitality of the Nation's scientific Orloff (Princeton University), Massachusetts In­ effort. The awards will be made under the stitute of Technology; Tudor S. Ratiu (University National Science Foundation's (NSF) Mathemati­ of Michigan), University of California, Berkeley; cal Sciences Postdoctoral Research Fellowship Jonathan D. Rogawski (Yale University), In­ program, now in its fifth year. They will per­ stitute for Advanced Study and Harvard Univer­ mit recipients to choose research environments sity; Stephen William Semmes (Mittag-Leffier that will have maximal benefit to their scientific Institute), Yale University; James A. Sethian development. (University of California, Berkeley), University The awards are made to United States citizens of California, Berkeley and Courant Institute, or nationals. Selections are made on the New York University; Joseph H. Silverman basis of ability of the applicant and the likely (Massachustts Institute of Technology), Harvard improvement on his or her future in science. University; Clifford H. Taubes (Harvard Univer­ A panel of mathematical scientists, chosen by sity), University of California, Berkeley; Samuel the American Mathematical Society, the Institute A. Weinberger (Princeton University), Princeton for Mathematical Statistics, and the Society for University; Deane Yang (Harvard University), New Industrial and Applied Mathematics, evaluated York University. 107 applications; final selections were made by NSF.

305 National Science Foundation News f3 Reports

Scandinavia and The Netherlands Positions Open in NSF Cooperative Science Program Applicants for the following positions should The U. B.-Scandinavia and The Netherlands submit resumes to NSF, Personnel Administration Cooperative Science program offers support for Branch, Room 212, 1800 G Street, N.W., cooperative research projects, joint workshops Washington, DC 20550. Attn: E. Paul Broglio; and seminars, and long-term visits. There are no 202-357-7841. NSF is an Equal Opportunity deadlines for proposals; processing lead time is six Employer. months. Please request program guidelines before NSF's Division of Mathematical and Computer applying. Sciences is seeking qualified applicants for posi­ Further information and quidelines may be tions which periodically become available. The obtained from the U. 8.-Scandinavia and The positions are filled on a one- to two-year rotational Netherlands Program, Division of International basis and are excepted from the competitive civil Programs; 202-357-7554. -NSF Bulletin service. The salaries range from $41,277 to $57,500 per annum (Gs-14/15 equivalent). Applicants United States-Eastern Europe should have a Ph.D. or equivalent experience in an appropriate field, plus six to eight years Cooperative Research of successful scientific research experience. A NSF's Eastern Europe Cooperative Science broad general knowledge of the field and some program seeks to foster and support scientific administrative experience are also required. and technological cooperation between the U.S. NSF's Division of Mathematical and Computer and Bulgaria, Hungary, and Romania. Sciences is seeking qualified applicants for rotator The program offers financial support for three positions which periodically become available in computer research. The positions will be filled typ~s of activities: cooperative research projects, semmars and workshops, and scientific visits for on a one- or two-year rotational basis and are planning purposes. Proposals must be submitted excepted from the competitive civil service. The to NSF by American institutions, and to the ap­ salaries range from $34,930 to $57,500 per annum propriate foreign agency by foreign institutions. (Gs-13 to 15 equivalent). One anticipated opening Proposals should be prepared according to stan­ in academic year 1983-1984 will be in the field dard NSF guidelines, but must also contain a of Intelligent Systems. Applicants should have a Ph.D. or equivalent experience and training in ~ection on "International Cooperation," which describes the cooperative aspects of the work plan an appropriate field, plus six to eight years of and the mutual benefits to be obtained. Proposals successful scientific research experience. A broad for joint research should be submitted at least general knowledge of computer research and some administrative experience are also required. nine m~nths before the requested starting date; for semmars twelve months, and for short-term -NSF Bulletin visits, six months. For further information, write or phone Deborah Wince, Division of International Programs; 202- 357-9516. -NSF Bulletin

306 Letters to the Editor

Gurevich To the list of refuseniks which was printed Policy on Letters to the Editor in the Notices, November 1982, pages 626-628, Letters submitted for publication in the Notices are reviewed by the Editorial Committee, whose task is should be added the name of Reuben Gurevich to determine which ones are suitable for publication. [Gurevil'.], with whom I and a number of other The publication schedule requires from two to four mathematicians have been corresponding for the months between receipt of the letter in Providence and last two years. Gurevich graduated from the publication of the earliest issue of the Notices in which Leningrad University and is not married; I do not it could appear. know his age. He was first refused permission Publication decisions are ultimately made by majority vote of the Editorial Committee, with ample provision to emigrate in February 1980, though he has for prior discussion by committee members, by mail or not had anything to do with state secrets. His at meetings. Because of this discussion period, some mathematical interests are in logic and its relations letters may require as much as seven months before a to topology and analysis. (See, for example, final decision is made. Abstracts of Papers Presented to the American The committee reserves the right to edit letters. Mathematical Society 3 (August 1982), page 387, The Notices does not ordinarily publish complaints about reviews of books or articles, although rebuttals #82T-03-402, or Israel Journal of Mathematics, and correspondence concerning reviews in the Bulletin of Volume 42 (1982), Number 4.) the American Mathematical Society will be considered for I would urge readers of the Notices to write publication. to Gurevich, send him your reprints, preprints, Letters should be mailed to the Editor of the etc. His address is Maurice Thorez 33, Apt. 205, Notices, American Mathematical Society, Post Office Box 6248, Providence, Rhode Island 02940, and will be Leningrad 194223, USSR. His phone number is acknowledged on receipt. 5521363. C. Ward Henson University of illinois, World Directory Urbana of Mathematicians 1982 Letters to the Editor The seventh edition of this directory, which is I find that, in recent issues, the Letters to the based on material supplied by the National Com­ Editor section of the Notices has become small, mittees for Mathematics, is published by the Bureau conservative and less interesting. Is this the result of the World Directory of Mathematicians of the of lack of imagination on the part of members or International Mathematical Union. Some of the national committees failed to offer new entries; in does it indicate that the editors have established these cases the entries from the sixth edition have censorship? been used. The seventh edition contains a list of Please publish this question as a letter. It important mathematical organizations, an alphabeti· certainly will not be space consuming. cal Iist of mathematicians with addresses, and a Paul Dedecker geographical list. The AMS is distributing the Wissenschaftskolleg zu directory. Berlin World Directory of Mathematicians 1982 Approximately 550 pages (soft cover) NOTE: The Editors have instituted no changes in List price $23, institutional member $21, policy with respect to letters. The procedure con­ individual member $21 Publication date: November 1982 tinues to be that the entire Editorial Committee To order, please specify WRLDI R/7N judges each proposed letter on its own merits.

307 1983 AMS Elections Nominations by Petition

Vice-President or Member-at-Large Rules and Procedures Two positions of vice-president and member of the Council ex officio for a term of two years are Use separate copies of the form for (a) to be filled in the election of 1983. The Council candidates for vice-president, (b) candidates intends to nominate four candidates, whose names for member-at-large, (c) candidates for the may be expected to appear in the June issue of Nominating Committee. the Notices, which is scheduled to be mailed by 1. To be considered, petitions must be ad­ the printer on May 13. Nominations by petition dressed to Everett Pitcher, Secretary, P.O. Box as described in the box are acceptable. 6248, Providence, Rhode Island 02940, and Five positions of member-at-large of the Council must arrive by July 9, 1983. for a term of three years are to be filled in the same election. The Council intends to nominate 2. The name of the candidate must be given seven candidates, whose names may be expected as it appears in the Combined Membership List. to appear in the June Notices. Nominations by If the name does not appear in the list, as petition in the manner described in the box are in the case of a new member or by error, it acceptable. The Council has stated its intent to must be as it appears in the mailing lists, for have at least ten candidates and will bring the example on the mailing label of the Notices. number up to ten if the nominations by petition If the name does not identify the candidate do not do so. uniquely, append the member code, which may Petitions are presented to the Council, which, be obtained from the candidate's mailing label according to Section 2 of Article Vll of the or the Providence office. bylaws, makes the nominations. The Council of 23 January 1979 stated the intent of the Council 3. The petition for a single candidate may of nominating all persons on whose behalf there consist of several sheets each bearing the were valid petitions. statement of the petition, including the name Prior to presentation to the Council, petitions in of the position, and signatures. The name of aid of a candidate for the position of vice-president the candidate must be exactly the same on all or of member-at-large of the Council must have sheets. at least 50 valid signatures and must conform 4. On the facing page is a sample form for to several rules and operational considerations, petitions. Copies may be obtained from the which are described in the box. Secretary; however, petitioners may make and use photocopies or reasonable facsimiles. The Nominating Committee for 1984 5. A signature is valid when it is clearly that The Council has approved the continuation of of the member whose name and address are the procedure of filling places on the Nominating given in the left-hand column. Committee by election. There will be four con­ 6. The signature may be in the style chosen tinuing members of the Nominating Committee, by the signer. However, the printed name and namely address will be checked against the Combined Linda Keen Membership List and the mailing lists. No 0. Carruth McGehee attempt will be made to match variants of Wolfgang M. Schmidt names with the form of name in the CML. A James D. Stasheff name neither in the CML nor on the mailing There will be four places filled by election in a lists is not that of a member. (Example: The preferential ballot. The President will name six name Everett Pitcher is that of a member. The candidates for these four places. The names may name E. Pitcher appears not to be. Note that be expected to appear in the June issue of the Notices. Nominations by petition, in the manner the mailing label of the Notices can be peeled described in the box, will be accepted. Should the off and affixed to the petition as a convenient final number of candidates be less than eight, the way of presenting the printed name correctly.) President will bring it up to eight. 7. When a petition meeting these various The name of a candidate for member of the requirements appears, the Secretary will ask Nominating Committee may be placed on the the candidate whether he is willing to have his ballot by petition. The candidate's assent and name on the ballot. Petitioners can facilitate petitions bearing at least 100 valid signatures are the procedure by accompanying the petitions required for a name to be placed on the ballot. with a signed statement from the candidate In addition, several other rules and operational giving his consent. considerations which are described in the box should be followed.

308 NOMINATION PETITION FOR 1983 ELECTION The undersigned members of the American Mathematical Society propose the name of

as a candidate for the position of (check one):

D Vice-President D Member-at-Large of the Council D Member of the Nominating Committee

of the American Mathematical Society for a term beginning January 1, 1984.

Name and Address (printed or typed, or Notices mailing label)

Signature

Signature

Signature

Signature

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309 New York City, April14-15, 1983, New York Statler Program for the 803rd Meeting

The eight hundred and third meeting of the Megibben, John D. O'Neill, K. M. Rangaswamy, J. D. American Mathematical Society will be held at the Reid, Charles Vinsonhaler, Elbert Walker, and W. J. New York Statler Hotel, Seventh Avenue and 33rd Wickless. Street, New York City, on Thursday and Friday, Transcendence theory and Diophantine prob­ April 14 and 15, 1983. The New York Statler is lems, W. DALE BROWNAWELL. The speakers will located directly across from Madison Square Garden be Enrico Bombieri, Richard T. Bumby, Gregory V. and Pennsylvania Station. Chudnovsky, Thomas W. Cusick, Howard Kleiman, J. The meeting will be preceded by a symposium C. Lagarias, Richard C. Mason, Julia Mueller, Gerald on Inverse Problems to be held on Tuesday Myerson, Melvyn B. Nathanson, A. M. Odlyzko, and Wednesday, April 12 and 13. The topic Charles F. Osgood, Eric Reyssat, M. L. Robinson, was selected by the AMS-SIAM Committee on Joseph H. Silverman, Cameron L. Stewart, Wolfgang Applied Mathematics, whose members are Roger M. Schmidt, Rob Tubbs, and Jeffrey D. Vaaler. Brockett, John Dennis, Norman Lebovitz, Alan Combinatorial group theory, ANTHONY M. Newell (chairman), George C. Papanicolaou, and GAGLIONE, United States Naval Academy. The Robert S. Warming. The members of the Organizing speakers are Michael Anshel, S. Bachmuth, Joan S. Committee are Robert Burridge, Courant Institute Birman, Frank B. Cannonito, Leo P. Comerford, of Mathematical Sciences, New York University; Jr., Joan Dyer, Benjamin Fine, Chander Kanta Joseph B. Keller, Stanford University; Robert B. Gupta, Narain Gupta, Robert M. Guralnick, Richard Marr, Brookhaven National Laboratory; David W. M. Hain, Ronald Hirshon, L.-C. Kappe, John P. McLaughlin (chairman), University of Arizona; and Labute, R. C. Lyndon, S. K. Mahdavianary, Dennis C. Ray Smith, University of Wyoming. Support is Spellman, Ruth Rebekka Struik, F. C. Y. Tang, anticipated under a grant from the National Science Carol L. Tretko:lf, and K. Weston. Foundation. Sensitivity of functionals, VADIM KOMKOV, Invited Addresses West Virginia University. The speakers will be Edward J. Haug, Vadim Komkov, Jerzy Kyparisis, By invitation of the Committee to Select Hour M. Zuhair Nashed, Pauli Pedersen, Herschel Rabitz Speakers for Eastern Sectional Meetings, there will and W. M. Wonham. be four invited one-hour addresses. The speakers, Singularities of algebraic and analytic varieties, their affiliations, titles of their talks, and scheduled HENRY LAUFER, SUNY, Center at Stony Brook. times ofpresentation, are as follows: The speakers are Piotr Blass, Ragnar-0. Buchweitz, W. DALE BROWNAWELL, Pennsylvania State Robert Ephraim, Henry Laufer, Anatoly Libgober, University, University Park, Contemporary trends David R. Morrison, Donal B. O'Shea, Richard in transcendence theory, 1:30 p.m. Thursday. Randell, Michael Schlessinger, Susan Szczepanski, RICHARDS. HAMILTON, Mathematical Sciences Hiroaki Terao, Philip Wagreich, Jonathan M. Wahl, Research Institute, Berkeley, and Cornell University, and Stephen Yau. Three-manifolds with positive Ricci curvature, 1:30 The mathematical legacy of Hans Rademacher, p.m. Friday. JOSEPH LEHNER, Institute for Advanced Study, OLIVER A. MCBRYAN, Courant Institute of and MARVIN I. KNOPP, Temple University. The Mathematical Sciences, New York University, Com­ speakers are Raymond G. Ayoub, Bruce C. Berndt, putational methods for discontinuous solutions, D. M. Bressoud, Harvey Cohn, Ronald J. Evans, 11:00 a.m. Friday. E. Grosswald, James Lee Hafner, Marvin I. Knopp, WOLFGANG ZILLER, University of Pennsyl­ Joseph Lehner, Thomas A. Metzger, Morris Newman, vania, On the existence of closed geodesics on L. Alayne Parson, N. Purzitsky, M. Ram Murty, V. Riemannian manifolds, 11:00 a.m. Thursday. Kumar Murty, Mark Sheingorn, Harold M. Stark, All of these talks will be presented in the Georgian Kenneth B. Stolarsky, Robert Alan Styer, and Albert Room. L. Whiteman. Computational methods and fluid mechanics, Special Sessions OLIVER A. MCBRYAN. The speakers will be By invitation of the same committee, there will James Bramble, P. R. Garabedian, Harold Grad, be nine special sessions of selected twenty-minute Anthony Jamison, Malvin H. Kalos, Keith Miller, papers. The topics of these special sessions, the Steven A. Orszag, and Norman J. Zabusky. names of the organizers, and the lists of speakers are Value distribution theory and its applications, as follows: CHUNG-CHUN YANG, United States Naval Re­ Abelian groups, KHALID BENABDALLAH, search Laboratories. The speakers will be Chi-tai University of Montreal. The speakers will be Donna Chuang, A. Edrei, Fred Gross, Simon Hellerstein, Beers, Khalid Benabdallah, Jutta Hausen, Paul Hill, Yong-Xing Ku, J. K. Langley, Peter A. Lappan, Len­ John Irwin, Samir Khabbaz, George Kolettis, Charles nox S. 0. Liverpool, Seiki Mori, Kiyoshi Niino, S. M.

310 Shah, Daniel F. Shea, Leonard M. Smiley, Guo-dong of reservations by the hotel was March 14, and Song, Charles S. Stanton, Sh. Strelitz, Nobushige requests received after that date will be honored Toda, Chia-Chi Tung, and Niro Yanagihara. on a space available basis. The rates shown below Variational problems in Riemannian geometry, are subject to 8.25 percent sales and $2 per night WOLFGANG ZILLER. The speakers will be Paul occupancy taxes. A rollaway cot for an extra person Baird, Victor Bangert, Melvyn S. Berger, Dennis can be added to double or twin sleeping rooms only. DeTurck, Doris Fischer-Colbrie, Michael E. Gage, Note that there is no charge for a maximum of two Carolyn S. Gordon, Alfred Gray, David Hoffman, children under the age of 17 years when occupying Anatoly Katok, Tilla Klotz Milnor, Wei-Ming Ni, a room with parents, provided a rollaway cot is not L. M. Sibner, David A. Singer, Karen K. Uhlenbeck, required. If the reservation is to be guaranteed and S. Walter Wei, and Frank Williams. the room held for arrival after 6:00p.m., an American Express charge card number (or check to cover the Contributed Papers first night's accommodations) must accompany the There will also be sessions for contributed ten­ reservation. No other charge cards are accepted minute papers on Thursday afternoon, Friday for guaranteed reservations, although they will be morning and Friday afternoon. Late papers cannot accepted for payment of the bill for rooms occupied. be accepted. The address to which reservations must be sent is Audio-Visual shown below. For your convenience, a reservation form can be found on page 248 in the February issue All session rooms will be equipped with an of the overhead projector and screen. Equipment for the Notices. preparation of transparencies will be available at New York Statler Hotel the meeting registration desk all day Thursday and Room Reservations Office 7th Avenue and 33rd Street Friday morning. Blackboards will not be available. New York, New York 10001 Council Meeting Telephone: 212-736-5000 The Council of the Society will meet at 5:00 p.m. Single $64, 70, 78, 88, 98 on Thursday, April 14, in the Gold Ballroom on the Double/Twin $81, 87, 95, 105, 115 third level of the ballroom floor at the New York Rollaway $16 extra per person Statler. The Statler offers a variety of restaurants, including Registration the Deli-Express off the lower lobby where moderately priced food and beverages are served cafeteria-style The registration desk will be located in the to either eat in or take out. Cornell/Dartmouth Rooms of the Ivy Suite on the ballroom floor of the New York Statler Hotel, and Travel will be open from 8:30 a.m. to 4:00 p.m. Tuesday The New York Statler is conveniently located through Thursday, and from 8:30 a.m. to 2:00 p.m. across Seventh Avenue from Pennsylvania Station, on Friday. Registration fees for the symposium and where Amtrak trains arrive from Washington, D.C., meeting are: Philadelphia, Boston, and Montreal, in addition to Symposium Only Long Island Railroad commuter trains. Seventh Nonmember $30 Avenue is reached from the main concourse by stairs Member AMS/SIAM $20 and escalator. Amtrak trains from upper New York Student/Unemployed $10 State, Toronto, and Chicago, as well as Conrail Meeting Only commuter trains, come into Grand Central Terminal which is located at 42nd Street between Park and Nonmember $16 Vanderbilt Avenues. Member AMS/SIAM $10 All of the major airlines and many commuter Student/Unemployed $ 5 airlines serve the New York metropolitan airports. Symposium and Meeting From LaGuardia Airport, located 8 miles northeast Nonmember $46 of New York City, Carey Bus provides ground Member AMS/SIAM $30 transportation every 20 to 25 minutes between 6:50 Student/Unemployed $15 a.m. and 12:30 a.m. to 42nd Street and Park Avenue Book Exhibits and Sale at a cost of $4 per person. From John F. Kennedy An exhibit of assorted mathematics books offered International Airport, located 15 miles southeast by various publishers, and a sale of recent books of New York City, Carey Bus operates every 20 published by the American Mathematical Society, to 30 minutes between 5:20 a.m. and midnight, will be located in the Cornell/Dartmouth Rooms dropping passengers off at the East Side Airlines during the above registration hours Tuesday through Terminal at 38th Street and First Avenue; the Friday. fare is approximately $5 per person. Passengers arriving at Newark International Airport, 16 miles Accommodations southwest of New York City, can take the New A block of rooms is being held at the New York Jersey Transit Express bus, which operates every 15 Statler Hotel for use by participants attending· the to 30 minutes from 5:00 a.m. to midnight; it costs symposium and meeting. The deadline for receipt approximately $4 per person and drops passengers

311 off at the Port Authority Bus Terminal. Mini-Bus Parking Service also operates from Newark Airport to some of The Statler Hotel offers in-once valet parking for the midtown Manhattan hotels every one-half hour guests, the current rate for which is $15 per day. Monday through Friday. The fare ranges up to $10, There are several parking garages in the vicinity of and further information can be obtained by calling the hotel, the closest one being on 31st Street (across 212-586-8280. from Madison Square Garden). The Port Authority Bus Terminal is located at 41st Street and Eighth Avenue; it serves all buses coming into New York City, including the New Jersey Transit Express buses from Newark Airport.

Program for the AMS-SIAM Symposium on Inverse Problems

All sessions will be held in the Georgian Room, Ballroom Floor, New York Statler Hotel

Tuesday, April 12

First Session. Geophysical Inverse Problems Chaired by ROBERT BURRIDGE 9:00- 9:45 An inverse problem of electromagnetism arising in geophysics. ROBERT PARKER, University of California, San Diego 10:00-10:45 The joint inversion of seismic wave forms for lateral variations in earth structure and earthquake source parameters. JOHN WOODHOUSE, Harvard University 11:00-11:45 Inverse scattering for stratified media. DAVID STICKLER, Courant Institute of Mathematical Sciences, New York University

Second Session. Computed Tomography and Inverse Problems in Medicine Chaired by ROBERT MARR 1:30- 2:10 Radon's problems-old and new. A.M. CORMACK, Tufts University 2:15- 2:55 Inversion of the x-ray transform. KENNAN T. SMITH, Oregon State University 3:00- 3:40 Computed tomography from ultrasound scattered by biological tissues. JAMES F. GREENLEAF, Mayo Foundation 3:50- 4:30 The limited angle problem in tomography and some related mathematical questions. F. ALBERTO GRUNBAUM, University of California, Berkeley 4:35- 5:15 The need to choose different models for each of x-ray, NMR, and emission CT. LAWRENCE A. SHEPP, Bell Laboratories, Murray Hill

Wednesday, April13

Third Session. Developments in Mathematical Inverse Theory Chaired by JOSEPH B. KELLER

9:00- 9:45 Review of the inverse spectral problem in R 3 . ROGER G. NEWTON, Indiana University, Bloomington 9:50-10:35 Isospectral periodic potentials on Rn. JAMES V. RALSTON, University of California, Los Angeles 10:45-11:30 Some aspects of inverse problems in several-dimensional wave propagation. W. SYMES, Michigan State University 11:35-12:20 A discussion of the determination of conductivity based on boundary measurements. MICHAEL S. VOGELIUS, University of Maryland, College Park

Fourth Session. Methods of Maximum Information Entropy Chaired by C. RAY SMITH 1:45- 1:55 Introduction. C. RAY SMITH 2:00- 2:45 Maximum entropy, minimum cross-entropy, and logical inference. JOHN SHORE, U. S. Naval Research Laboratory 3:00- 3:45 Prior information and ambiguity in generalized inverse problems. EDWIN T. JAYNES, Washington University 4:00- 4:45 The entropy of an image. JOHN SKILLING, University of Cambridge, England

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

Abeles, F., 181 Gerstenhaber, M., 88 •Mahdavianary, S. K., 52 •Silverman, J. H., 117 Anacker, S. E., 176 •Gordon, C. S., 35 •Mason, R. C., 20 *Singer, D. A., 125 •Anshel, M., 57 *Grad, H., 45 •McBryan, 0. A., 144 *Smiley, L. M., 152 Arkin, J., 86 •Gray, A., 34 * Megibben, C., 170 *Song, G.-D., 100 •Ayoub, R. G., 137 •Gross, F., 48 •Metzger, T. A., 138 •Spellman, D., 14 •Bachmuth, S., 15 •Grosswald, E., 31 *Miller, K., 147 • Stanton, C. S., 5 •Baird, P., 70 •Gupta, C. K., 53 •Milnor, T. K., 72 *Stark, H., 69 •Bangert, V., 127 *Gupta, N., 111 •Mori, S., 3 •Stewart, C. L., 18 Bazley, N. W., 81 * Guralnick, R. M., 13 •Morrison, D. R., 107 • Stolarsky, K. B., 66 * Strelitz, S., •Beers, D., 24 •Hafner, J. L., 30 •Mueller, J., 59 4 * Struik, R. R., 113 •Benabdallah, K., 21 •Rain, R. M., 12 •Murty, M. R., 140 •Styer, R. A., 63 •Berger, M. S., 36 •Hamilton, R. S., 145 •Murty, V. K., 139 *Szczepanski, S., 156 •Berndt, B. C., 28 •Haug, E. J., 141 •Myerson, G., 17 Szpunar-Lojasiewicz, W., •Birman, J. S., 159 •Hausen, J., 120 * Nashed, M. Z., 39 82 •Blass, P., 108 * Hellerstein, S., 101 *Nathanson, M. B., 116 •Tang, F. C. Y., 55 •Bombieri, E., 62 Herda, H., 84 •Newman, M., 64 Temkin, B., 85 •Bramble, J., 148 •Hill, P., 25 •Ni, W.-M., 174 •Terao, H., 7 •Bressoud, D. M., 32 •Hirshon, R., 110 •Niino, K., 102 Thrash, J., 79 eBrownawell, W. D., 42 •Hoffman, D., 71 * Odlyzko, A. M., 58 * Toda, N., 47 •Buchweitz, R.-0., 154 Huang, B. C., 180 *O'Neill, J. D., 169 Trench, W. F., 132 •Bumby, R. T., 115 •Irwin, J., 23 • Orszag, S. A., 44 •Tretkoff, C. L., 54 •Cannonito, F. B., 158 *Jamison, A., 146 •Osgood, C. F., 163 Tsao, A., 188 Chow, Y., 177 Jerri, A. J., 78 *O'Shea, D. B., 10 •Tubbs, R., 165 •Chuang, C.-T., 49 Jung, R., 187 •Parson, L.A., 68 *Tung, C.-C., 151 •Chudnovsky, G. V., 166 •Kalos, M. H., 46 •Pedersen, P., 40 * Uhlenbeck, K. K., 73 Chung, I. Y., 91 •Kappe, L.-C., 56 •Purzitsky, N., 27 * Vaaler, J. D., 61 •Cohn, H., 136 •Katok, A., 126 Puttaswamy, T. K., 129 Verma, R. U., 185 •Comerford, L. P., Jr., •Khabbaz, S., 22 •Rabitz, H., 142 Verma, S., 133 161 •Kleiman, H., 16 •Randell, R., 8 • Vinsonhaler, C., 121 Courter, R. C., 92 •Knopp, M. I., 65 •Rangaswamy, K. M., 119 •Wagreich, P., 105 *Cusick, T. W., 114 Kohlmayr, G. F., 178 Rao, K. M., 74 *Wahl, J. M., 157 *Walker, 168 D'Archangelo, J. M., 80 •Kolettis, G., 167 Rao, V. V., 183 E., Waterman, D., 184 Deretsky, Z., 182 Raykhshteyn, B. Z., 90 •Komkov, V., 37 Watson, B., 95 •DeTurck, D., 172 Kruelle, M. F., 131 •Reid, J.D., 122 •Wei, S. W., 171 DiPaola, R. A., 179 •Ku, Y.-X., 51 •Reyssat, E., 164 *Weston, K., 112 •Dyer, J., 109 •Kyparisis, J., 38 •Robinson, M. L., 60 *Whiteman, A. L., 135 •Edrei, A., 1 •Labute, J.P., 11 Roxin, E. 0., 83 * Wickless, W. J., 123 Eggermont, P., 77 •Lagarias, J. C., 118 Salzer, H. E., 76 •Williams, F., 33 •Ephraim, R., 9 •Langley, J. K., 99 Schiffman, J., 89 Wolper, J. S., 93 Erdelyi, I., 186 •Lappan, P. A., 103 * Schlessinger, M., 104 * Wonham, W. M., 143 •Evans, R. J., 29 *Laufer, H., 6 Schmeelk, J., 128 Woods, D., 87 Fabrikant, V.I., 134 •Lehner, J., 26 •Schmidt, W. M., 19 * Yanagihara, N., 150 •Fine, B., 160 Lerner, B. T., 97 *Shah, S. M., 153 Yang, J. S., 96 •Fischer-Colbrie, D., 173 •Libgober, A., 106 •Shea, D. F., 50 * Yau, S., 155 *Gage, M. E., 124 •Liverpool, L. S. 0., 2 •Sheingorn, M., 67 * Zabusky, N. J., 149 Galambos, J., 75 Lonergan, F. D., 94 * Sibner, L. M., 175 Zahid, M. 1., 98 *Garabedian, P. R., 43 •Lyndon, R. C., 162 Siddiqi, J. A., 189 •Ziller, W., 41

313 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 April 1983 issue of Abstracts of papers presented to the American Mathematical 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.

Thursday, 8:20 a.m. Special Session on Value Distribution Theory and its Applications, I Albany Room 8:20- 8:40 (1) Zeros of the partial sums of Mittag-Leifter's expansion. A. EDREI*, Syracuse University, E. B. SAFF, University of South Florida, Tampa, and R. S. VARGA, Kent State Univesity (803-30-98) 8:50- 9:10 (2) Exceptional sets for entire functions. LENNOX S. 0. LIVERPOOL* and KENNETH B. YUGUDA, University of Jos, Nigeria (803-30-42) 9:20- 9:40 (3) Remarks on holomorphic mappings. SEIKI MORI, Yamagata University, Japan (803-32-58) (Introduced by Chung-Chun Yang) 9:50-10:10 (4) Asymptotic properties of entire transcendental solutions of algebraic differential equations. SH. STRELITZ, University of Haifa, Israel (803-30-59) 10:20-10:40 (5) Value distribution and integral means. Preliminary report. CHARLES S. STANTON, University of North Carolina, Chapel Hill (803-30-97)

Thursday, 8:20a.m. Special Session on Singularities of Algebraic and Analytic Varieties, I Village Room 8:20- 8:40 (6) Generators for the adJunction algebra. Preliminary report. HENRY LAUFER, State University of New York, Stony Brook (803-32-118) 8:50- 9:10 (7) The bifurcation set and logarithmic vector fields. HIROAKI TERAO, University of Wisconsin, Madison (803-14-26) 9:20- 9:40 (8) The monodromy group of quasihomogeneous singularities. RICHARD RANDELL, University of Iowa (803-14-133) 9:50-10:10 (9) A generalized discriminant criterion for equisingularity. ROBERT EPHRAIM, George Mason University (803-32-138) 10:20-10:40 (10) Vanishing folds and limits of tangent spaces. Preliminary report. DONAL B. O'SHEA, Mount Holyoke College (803-14-145)

Thursday, 8:20 a.m. Special Session on Combinatorial Group Theory, I Hudson/Sutton Suite 8:20- 8:40 (11) The determination of the Lie algebra associated to the lower central series of a group. JOHN P. LABUTE, Harvard University {803-20-27) 8:50- 9:10 (12) Non abelian intersections and the lower central series of link groups. RICHARD M. RAIN, University of Washington (803-20-16) 9:20- 9:40 (13) Generation of the lower central series. ROBERT M. GURALNICK, University of Southern California (803-20-32) 9:50-10:10 (14) The structure of the Abelian groups Fn/(Fn,F'). DENNIS SPELLMAN, Widner University (803-20-02) 10:20-10:40 (15) Automorphism groups of 2-generator metabelian groups. Preliminary report. S. BACHMUTH* and H. MOCHIZUKI, University of California, Santa Barbara, and G. BAUMSLAG, City College, City University of New York, and J. DYER, Herbert H. Lehman College, City University of New York (803-20-15)

Thursday, 8:20a.m. Special Session on Transcendence Theory and Diophantine Problems, I Georgian Room 8:20- 8:40 {16) On the diophantine equationf(x,y) = 0. (II). HOWARD KLEIMAN, Queensborough Community College, City University of New York (803-10-52) 8:50- 9:10 (17) A measure for polynomials in several variables. GERALD MYERSON, University of British Columbia (803-10-73)

314 9:20- 9:40 (18) On the greatest square-free factor of a Lucas or Lehmer number. CAMERON L. STEWART, University of Waterloo (803-10-93) 9:50-10:10 (19) Homogeneous diophantine equations. WOLFGANG M. SCHMIDT, University of Colorado, Boulder (803-10-161) 10:20-10:40 (20) Equations over function fields. RICHARD C. MASON, St. John's College, Cambridge, England (803-10-182) (Introduced by W. Dale Brownawell)

Thursday, 8:20a.m.

Special Session on Abelian Groups, I Town Room 8:20- 8:40 (21) Strong centers of isomorphism in primary abelian groups. KHALID BEN ABDALLAH, Universite de Montreal (803-20-86) 8:50- 9:10 (22) The optimality theorem for finite p-groups. SAMIR KHABBAZ, Lehigh University (803-20-130) (Introduced by Khalid Benabdallah) 9:20- 9:40 (23) Abelian p-groups containing proper pw+n_projective subgroups. DOYLE CUTLER, University of California, Davis, JOHN IRWIN*, Wayne State University, and TOM SNABB, University of Michigan, Dearborn (803-20-174) (Introduced by Togo Nishiura) 9:50-10:10 (24) A listing of finite simply presented valuated p-groups. DONNA BEERS*, Wellesley College, and ROGER HUNTER and ELBERT WALKER, New Mexico State University, Las Cruces (803-20-45) 10:20-10:40 (25) On the classification of abelian p-groups. PAUL HILL, Auburn University, Auburn (803-20-131)

Thursday, 8:40 a.m.

Special Session on The Mathematical Legacy of Hans Rademacher, I Empire Suite B 8:40- 9:00 (26) Reminiscences of Rademacher. JOSEPH LEHNER, Institute for Advanced Study (803-10-10) 9:10- 9:30 (27) Invariant convex subdomains of the unit disc for fuchsian groups. Preliminary report. N. PURZITSKY, York University (803-99-185) (Introduced by Marvin Knopp) 9:40-10:00 (28) Sums involving the greatest integer function and Riemann-Stieltjes integration. BRUCE C. BERNDT*, University of illinois, Urbana-Champaign, and ULRICH DIETER, Technische Universitat Graz, Austria (803-10-18) 10:10-10:30 (29) The octic and bioctic character of certain quadratic units. Preliminary report. RONALD J. EVANS, University of California, San Diego (803-10-23) 10:40-11:00 (30) Explicit estimates in the theory of cusp forms. JAMES LEE HAFNER, California Institute of Technology (803-10-24) 11:10-11:30 (31) Dedekind and related sums. Preliminary report. E. GROSSWALD, Temple University (803-10-08) 11:40-12:00 (32) Some applications of the Burge connection between partitions and binary words. Preliminary report. G. E. ANDREWS and D. M. BRESSOUD*, Pennsylvania State University, University Park (803-10-36)

Thursday, 8:50a.m.

Special Session on Variational Problems in Riemannian Geometry, I Pennsylvania Room 8:50- 9:10 (33) Bounded deformations in Riemannian manifolds. JERROLD SIEGEL, University of Missouri, St. Louis, and FRANK WILLIAMS*, New Mexico State University, Las Cruces (803-58-17) 9:20- 9:40 (34) Riemannian manifolds which resemble harmonic spaces up to high order. Preliminary report. ALFRED GRAY*, University of Maryland, College Park, and 0LDRICH KOWALSKI, Charles University, Czechoslovakia (803-53-11) 9:50-10:10 (35) Isospectral deformations of compact solvmanifolds. Preliminary report. CAROLYN S. GORDON*, Lehigh University, and EDWARD N. WILSON, Washington University (803-53-107) 10:20-10:40 (36) A new variational principle for Einstein metrics on Kiihler manifolds. Preliminary report. MELVYN S. BERGER, University of Massachusetts, Amherst (803-53-14)

Thursday, 8:50a.m.

Special Session on Sensitivity of Functionals, I Belmont Room 8:50- 9:10 (37) Uniqueness for gradient methods in engineering optimization. Preliminary report. VADIM KOMKOV* and CAUL TON L. IRWIN, West Virginia University, Morgantown (803-49-173) 9:20- 9:40 (38) Convexity and concavity properties of the optimal value function in nonlinear programming. ANTHONY V. FIACCO and JERZY KYPARISIS*, George Washington University (803-49- 77) (Introduced by Vadim Komkov)

315 9:50-10:10 (39) Perturbation analysis and correctness functionals for singular problems. M. ZUHAIR NASHED, University of Delaware (803-65-140) 10:20-10:40 (40) Sensitivity analysis for non-selfadjoint problems. PAULI PEDERSEN, Technical University of Denmark (803-49-122) (Introduced by Vadim Komkov)

Thursday, 11:00 a.m. Invited Address Georgian Room 11:00-12:00 (41) On the existence of closed geodesics on Riemannian manifolds. WOLFGANG ZILLER, University of Pennsylvania (803-53-22)

Thursday, 1:30 p.m. Invited Address Georgian Room 1:30- 2:30 (42) Contemporary trends in transcendence theory. W. DALE BROWNAWELL, Pennsylvania State University, University Park (803-10-180)

Thursday, 1:30 p.m. Special Session on Computational Methods and Fluid Mechanics, I Belmont Room 1:30- 2:10 (43) Nonparametric solution of the Euler equations for steady flow. P. R. GARABEDIAN, New York University (803-76-78) 2:20- 3:00 (44) Small-scale structure of the Taylor-Green vortex. STEVEN A. ORSZAG, Massachusetts Institute of Technology (803-76-114) (Introduced by Oliver McBryan) 3:10- 3:50 (45) Efficient numerical algorithms for QDE and plasma equilibrium and transport. HAROLD GRAD, Courant Institute of Mathematical Sciences, New York University (803-99-188) 4:00- 4:40 (46) Monte Carlo calculations of quantum fluids. MALVIN H. KALOS, Courant Institute of Mathematical Sciences, New York University (803-82-177) (Introduced by Oliver McBryan)

Thursday, 2:45p.m. Special Session on Value Distribution Theocy and its Applications, II Albany Room 2:45- 3:05 (47) On meromorphic solutions of some differential equations. Preliminary report. NOBUSHIGE TODA, Nagoya Institute of Technology, Japan (803-30-43) 3:15- 3:35 (48) On fixed points of composite entire functions. Preliminary report. FRED GRoss*, University of Maryland, Catonsville, and U.S. Naval Research Laboratory, and CHARLES F. OSGOOD, U.S. Naval Research Laboratory (803-30-30) 3:45- 4:05 (49) Generalizations of the notion of normal families of meromorphic functions. Preliminary report. CHI-T AI CHUANG, Peking University, People's Republic of China (803-30-62) (Introduced by Chung-Chun Yang) 4:15- 4:35 (50) On the growth of meromorphic functions of order less than one. DANIEL F. SHEA, University of Wisconsin, Madison (803-30-108) 4:45- 5:05 (51) On tests of normality of meromorphic functions. YONG-XING Ku, Nan-chun Normal College, People's Republic of China (803-99-190)

Thursday, 2:45p.m. Special Session on Combinatorial Group Theory, II Hudson/Sutton Suite 2:45- 3:05 (52) Groups with many subgroups 2-subnormal. S. K. MAHDAVIANARY, State University of New York, Binghamton (803-20-40) 3:15- 3:35 (53) Certain subgroups of free groups. Preliminary report. CHANDER KANTA GUPTA, University of Manitoba (803-20-48) (Introduced by N. D. Gupta) 3:45- 4:05 (54) Groups generated by two infinite permutations of the integers. Preliminary report. CAROL L. TRETKOFF, Brooklyn College, City University of New York (803-20-55) 4:15- 4:35 (55) On conjugacy separability ofl-relator groups with torsion. R. B. J. T. ALLENBY, University of Leeds, England, and F. C. Y. TANG*, University of Waterloo (803-20-50) 4:45- 5:05 (56) Engel elements and margins in center-by-metabelian groups. Preliminary report. L.-C. KAPPE* and W. P. KAPPE, State University of New York, Binghamton (803-20-41) 5:15- 5:35 (57) Complexity of problems for infinite groups. MICHAEL ANSHEL, City College of New York (803-99-189)

Thursday, 2:45 p.m. Special Session on Transcendence Theory and Diophantine Problems, II Georgian Room 2:45- 3:05 (58) Knapsack-type public key cryptosystems and the lattice reduction algorithm. A. M. ODLYZKO, Bell Laboratories, Murray Hill (803-10-162)

316 3:15- 3:35 (59) On effective approximation of certain algebraic numbers by algebraic numbers. Preliminary report. JULIA MUELLER, Fordham University (803-10-69) 3:45- 4:05 (60) Effective irrationality measures for certain algebraic numbers. M. L. ROBINSON, U.S. Naval Academy (803-10-136) 4:15- 4:35 (61) On Siegel's lemma. ENRICO BOMBIERI and JEFFREY D. VAALER*, Institute for Advanced Study (803-10-87) 4:45- 5:05 (62) Applications of a new form of Siegel's Lemma. ENRICO BOMBIERI, Institute for Advanced Study (803-10-181) (Introduced by W. Dale Brownawell)

Thursday, 2:45p.m. Sper.ial Session on The Mathematical Legacy of Hans Rademacher, II Empire Suite B 2:45- 3:05 (63) Eisenstein series over number fields when h > 1. Preliminary report. ROBERT ALAN STYER, Temple University (803-10-81) 3:15- 3:35 (64) Rademacher's work on the Ramanuian congruences. MORRIS NEWMAN, University of California, Santa Barbara (803-10-29) 3:45- 4:05 (65) Rademacher's work on J(r) and Eichler cohomology. MARVIN I. KNOPP, Temple University (803-10-20) 4:15- 4:35 (66) Open problems about partitions. KENNETH B. STOLARSKY, University of illinois, Urbana­ Champaign (803-10-12) 4:45- 5:05 (67) Simple closed geodesics on arithmetic Riemann surfaces. Preliminary report. JOSEPH LEHNER, Institute for Advanced Study, and MARK SHEINGORN*, Bernard M. Baruch College, City University of New York (803-10-70) 5:15- 5:35 (68) Bounding the norm of the theta operator. Preliminary report. L. ALAYNE PARSON*, Ohio State University, Columbus, and MARK SHEINGORN, Bernard M. Baruch College, City University of New York (803-10-91) 5:45- 6:05 (69) Fourier coefficients of modular forms. HAROLD STARK, University of California, San Diego (803-99-191)

Thursday, 2:45a.m. Special Session on Variational Problems in Riemannian Geometry, II Pennsylvania Room 2:45- 3:05 (70) Harmonic maps into spheres and certain Hamiltonian systems. Preliminary report. PAUL BAIRD, Massachusetts Institute of Technology (803-53-76) (Introduced by Wolfgang Ziller) 3:15- 3:35 (71) A representation theorem for minimal surfaces in S 3 . DAVID HOFFMAN*, University of Massachusetts, Amherst, and ROBERT OSSERMAN, Stanford University (803-53-166) 3:45- 4:05 (72) Intrinsic curvature of the induced metric on harmonically immersed surfaces. Preliminary report. TILLA KLOTZ MILNOR, Rutgers University, New Brunswick (803-53-95) 4:15- 4:35 (73) Conservation laws for harmonic maps. Preliminary report. KAREN K. UHLENBECK, Harvard University (803-58-66)

Thursday, 2:45a.m. Session on Applied and Applicable Analysis Village Room 2:45- 2:55 (74) Characterization of the convergence in energy for Markov processes. Z. R. POP-STOJANOVIC and K. MURALI RAO*, University of Florida (803-60-125) 3:00- 3:10 (75) Estimates on multivariate extreme value distributions in terms of their marginals. JANOS GALAMBOS, Temple University (803-62-89) 3:15- 3:25 (76) A generalization of Layman-Lotockii: exponential interpolation. HERBERT E. SALZER, Brooklyn, New York (803-65-01) 3:30- 3:40 (77) Tomographic reconstruction on logarithmic polar grids. PAUL EGGERMONT, University of Delaware (803-65-171) (Introduced by M. Z. Nashed) 3:45- 3:55 (78) A transform (spectral)-iterative method for nonlinear problems. ABDUL J. JERRI, Clarkson College of Technology (803-65-159) 4:00- 4:10 (79) The A-transformation. Preliminary report. J. THRASH, University of Southern Mississippi (803-65-112) 4:15- 4:25 (80) Acoustical high-frequency scattering by elastic cylindrical shells. JAMES M. D'ARCHANGELO*, U.S. Naval Academy, and J. W. DICKEY and D. A. NIXON, David Taylor Naval Ship R&D Center (803-44-172) 4:30- 4:40 (81) Bifurcation of periodic solutions for wave equations with scalar nonlinearities. NORMAN W. BAZLEY*, University of Cologne, Federal Republic of Germany, and RICHARD J. WEINACHT, University of Delaware (803-47-09)

317 4:45- 4:55 (82) Time optimal control problem for systems described by autonomous one dimensional inclusions. WANDA SZPUNAR-I:,OJASIEWICZ, University of Rhode Island (803-49-74) (Introduced by Emilio 0. Roxin) 5:00- 5:10 (83) On order relations induced by optimal control problems. Preliminary report. BARBARA KASKOSZ and EMILIO 0. ROXIN*, University of Rhode Island (803-49-64)

Thursday, 2:45p.m. Session on Number Theory and Algebraic Structures Town Room 2:45- 2:55 (84) The series of prime square reciprocals. HANS HERDA, University of Massachusetts, Boston (803-10-68) 3:00- 3:10 (85) The problem of Frobenius for three variables. Preliminary report. BHARTI TEMKIN, Berkeley Heights, New Jersey (803-10-90) (Introduced by A. Odlyzko) 3:15- 3:25 (86) An extension of the Fibonacci numbers-revisited. JOSEPH ARKIN, Spring Valley, New York (803-10-149) 3:30- 3:40 (87) Construction of even type B amicable pairs. Preliminary report. CAL GUTHRIE, Central State University, Oklahoma, and DALE WOODS*, Central State University, Oklahoma, and Northeast Missouri State University (803-10-83) 3:45- 3:55 (88) Twisted simplicial complexes and Tic-Tac-Toe algebras. MURRAY GERSTENHABER*, University of Pennsylvania, and S. D. SCHACK, State University of New York, Buffalo (803-16-128) 4:00- 4:10 (89) A survey of examples and classes of finite solvable groups. Preliminary report. JAY SCHIFFMAN, Kean College of New Jersey (803-20-109) (Introduced by Fran Abeles) 4:15- 4:25 (90) Some applications of linear operators to cubic forms. Preliminary report. BORIS Z. RAYKHSHTEYN, Susquehanna University (803-15-153) 4:30- 4:40 (91) Derivation modules of group rings and algebraic integers of cyclotomic fields. I. Y. CHUNG, University of Cincinnati, Cincinnati (803-15-152) (Introduced by David Styer) 4:45- 4:55 (92) The maximum dimension of nilpotent subspaces of Kn satisfying the identity Sd. R. C. COURTER, New York, New York (803-15-103)

Thursday, 2:45p.m. Session on Geometry and Topology Headquarters Room 2:45- 2:55 (93) Some intersection properties of the fibres of Springer's resolution. JAMES S. WOLPER, University of California, San Diego (803-14-121) 3:00- 3:10 (94) Four $-dimensional manifolds with the same infinite fundamental group. Preliminary report. FRANCIS D. LONERGAN, Webster, Massachusetts (803-55-100) 3:15- 3:25 (95) Fine tuning U(n)-structures. BILL WATSON, St. John's University, Jamaica, New York (803-53-54) 3:30- 3:40 (96) On a class of groups more general than SIN. ROBERT W. BAGLEY, University of Miami, T. S. Wu, Case Western Reserve University, and J. S. YANG*, University of South Carolina, Columbia (803-22-101) 3:45- 3:55 (97) Fuzzy semitopological semigroups. Preliminary report. BAO TING LERNER, U.S. Naval Academy (803-00-148) 4:00- 4:10 (98) One-point extensions of locally para-H-closed spaces. R. W. HEATH, University of Pittsburgh, Pittsburgh, and M. I. ZAHID*, University of Pittsburgh, Bradford (803-54-53)

Friday, 8:20 a.m. Special Session on Value Distribution Theory and its Applications, III Albany Room 8:20- 8:40 (99) Analogues of Picard sets for meromorphic functions and their derivatives. J. K. LANGLEY, University of lllinois, Urbana-Champaign (803-30-92) (Introduced by Chung-Chun Yang) 8:50- 9:10 (100) On pseudo-primality of the combination of meromorphic functions satisfying linear differential equations. GUO-DONG SONG*, Cornell University, and CHUNG-CHUN YANG, U.S. Naval Research Laboratory (803-30-31) 9:20- 9:40 (101) Reality of the zeros of the second derivative of a class of meromorphic functions. SIMON HELLERSTEIN*, University of Wisconsin, Madison, LI-CHIEN SHEN, California Institute of Technology, and JACK WILLIAMSON, University of Hawaii, Honolulu (803-30-176) 9:50-10:10 (102) Deficiency sum of holomorphic curves of lower order less than one. Preliminary report. KrYOSHI NIINO, Kanazawa University, Japan (803-30-56) 10:20-10:40 (103) A necessary condition for the Bloch ordering. Preliminary report. PETER A. LAPPAN, Michigan State University (803-30-07)

318 Friday, 8:20a.m. Special Session on Singularities of Algebraic and Analytic Varieties, II Village Room 8:20- 8:40 {104) Hilbert scheme of space curues. MICHAEL SCHLESSINGER, University of North Carolina, Chapel Hill {803-14-146) 8:50- 9:10 {105) Growth functions of abelian groups. PHILIP WAGREICH, University of lllinois, Chicago {803-13-21) 9:20- 9:40 {106) Irregular cyclic multiple planes. Preliminary report. ANATOLY LIBGOBER, University of lllinois, Chicago {803-14-165) 9:50-10:10 {107) Gorenstein threefold singularities with small resolutions. Preliminary report. DAVID R. MORRISON, Princeton University {803-14-163) 10:20-10:40 {108) Singularity theory and a new class of factorial rings. Preliminary report. PIOTR BLASS, Clark University {803-14-134)

Friday, 8:20a.m. Special Session on Combinatorial Group Theory, III Hudson/Sutton Suite 8:20- 8:40 {109) On non-automorphism groups. JOAN DYER, Herbert H. Lehman College, City University of New York {803-20-28) 8:50- 9:10 {110) Some J replacement results. Preliminary report. RONALD HIRSHON, Polytechnic Institute of New York {803-20-25) 9:20- 9:40 {111) Extensions of groups by tree automorphisms. Preliminary report. NARAIN GUPTA*, University of Manitoba, and SAID SIDKI, University of Brasilia, Brazil {803-20-49) 9:50-10:10 {112) Elementary properties of Steinberg groups. Preliminary report. M. NEWMAN, Australian National University, and K. WESTON*, University of Wisconsin, Parkside {803-20-35) 10:20-10:40 {113) Using graphs to construct groups with specified properties. Preliminary report. RUTH REBEKKA STRUIK, University of Colorado, Boulder {803-20-51)

Friday, 8:20a.m. Special Session on Transcendence Theory and Diophantine Problems, III Georgian Room 8:20- 8:40 {114) Regulators in certain totally real fields. Preliminary report. THOMAS W. CUSICK, State University of New York, Buffalo {803-10-106) 8:50- 9:10 {115) Isolation and continued fractions. Preliminary report. RICHARD T. BUMBY, Rutgers University, New Brunswick {803-10-88) 9:20- 9:40 {116) On the exact order of subsets of additi'lle bases. MELVYN B. NATHANSON, Rutgers University, Newark {803-10-160) 9:50-10:10 {117) The S-unit equation O'ller function fields. Preliminary report. JOSEPH H. SILVERMAN, Massachusetts Institute of Technology (803-10-82) {Introduced by W. Dale Brownawell) 10:20-10:40 {118) An "isolation" result for two-dimensional best simultaneous diophantine approximation denominators. Preliminary report. J. C. LAGARIAS, Bell Laboratories, Murray Hill {803-10-111)

Friday, 8:20 a.m. Special Session on Abelian Groups, II Town Room 8:20- 8:40 {119) Homological properties of torsion-free separable abelian groups as modules O'ller their endomorphism rings. K. M. RANGASWAMY, University of Nevada, Las Vegas {803-20-67) 8:50- 9:10 {120) Abelian groups whose fully in'llariant subgroups form a chain. JUTTA HAUSEN, University of Houston, Houston {803-20-60) 9:20- 9:40 {121) Pure subgroups of finite rank completely decomposable groups. ill. DAVE ARNOLD, New Mexico State University, Las Cruces, and CHARLES VINSONHALER*, University of Connecticut, Storrs {803-20-61) 9:50-10:10 {122) On finitely E-generated abelian groups. J.D. REID, Wesleyan University {803-20-157) 10:20-10:40 {123) Projecti'IJe classes of torsion free groups. Preliminary report. C. I. VINSONHALER and W. J. WICKLESS*, University of Connecticut, Storrs {803-20-33)

Friday, 8:50 a.m. Special Session on Variational Problems in Riemannian Theory, III Pennsylvania Room 8:50- 9:10 {124) An isoperimetric inequality with applications to curue shortening. MICHAEL E. GAGE, University of Pennsylvania {803-53-116)

319 9:20- 9:40 (125) On the total squared curvature of curves. JOEL LANGER and DAVID A. SINGER*, Case Western Reserve University (803-58-72) 9:50-10:10 (126) Exponential growth of closed geodesics on certain manifolds of non-positive curvature. ANATOLY KATOK, University of Maryland, College Park (803-99-184) 10:20-10:40 (127) Homology generated by iterated closed geodesics. VICTOR BANGERT*, Universitii.t Freiburg, Federal Republic of Germany, and WILHELM KLINGENBERG, Universitii.t Bonn, Federal Republic of Germany (803-53-144) (Introduced by Wolfgang Ziller)

Friday, 9:00a.m. Session on Integral and Dift'erential Equations Headquarters Room 9:00- 9:10 (128) Operational techniques for solving partial differential equations. Preliminary report. JOHN SCHMEELK, Virginia Commonwealth University (803-99-187) 9:15- 9:25 (129) Asymptotic behavior of solutions of a certain nth order differential equation. T. K. PUTTASWAMY, Ball State University (803-34-126) 9:30- 9:40 (131) Liouville-constructible solutions for a hypergeometric equation. Preliminary report. MARK F. KRUELLE, Wright State University (803-34-104) 9:45- 9:55 (132) Asymptotics of differential systems with deviating arguments. WILLIAM F. TRENCH, Drexel University (803-34-105) 10:00-10:20 (133) On Laplace transforms via derivatives and their weighted jumps. Preliminary report. SADANAND VERMA, University of Nevada, Las Vegas (803-44-124) (Introduced by L. J. Simonoff) 10:15-10:25 (134) A new type of integral operators and their applications in potential theory. V.I. FABRIKANT*, T. S. SANKAR, and G. D. XISTRIS, Concordia University (803-31-110) (Introduced by Efim A. Galperin)

Friday, 9:10 a.m.

Special Session on The Mathematical Legacy of Hans Rademacher, III Empire Suite B 9:10- 9:30 (135) Rademacher's proof of the von Staudt-Clausen theorem. Preliminary report. ALBERT L. WHITEMAN, University of Southern California (803-10-04) 9:40-10:00 (136) Identities on the unit diagonal for Hilbert modular functions. Preliminary report. HARVEY COHN, City College of New York (803-12-06) 10:10-10:30 (137) Rademacher's contributions to additive and multiplicative number theory in algebraic number fields. Preliminary report. RAYMOND G. AYOUB, Pennsylvania State University, University Park (803-12-155) 10:40-11:00 (138) Taylor coefficient of automorphic forms. THOMAS A. METZGER, University of Pittsburgh, Pittsburgh (803-30-65) 11:10-11:30 (139) Prime divisors of Fourier coefficients of modular forms. M. RAM MURTY, McGill University, and V. KUMAR MURTY*, Institute for Advanced Study (803-10-39) 11:40-12:00 (140) Bounds for eigenvalues of Heeke operators. M. RAM MURTY, McGill University (803-10-38)

Friday, 9:20 a.m.

Special Session on Sensitivity of Functionals, II Belmont Room 9:20- 9:40 (141) Shape optimization of doubly-connected elastic bars in torsion. EDWARD J. HAUG* and JEAN W. Hou, University of Iowa (803-49-117) (Introduced by Vadim Komkov) 9:50-10:10 (142) Sensitivity methods for mathematical modelling. HERSCHEL RABITZ, Princeton University (803-68-80) (Introduced by Vadim Komkov) 10:20-10:40 (143) Sensitivity and the internal model principle for control systems. W. M. WONHAM, University of Toronto (803-93-178) (Introduced by Vadim Komkov)

Friday, 11:00 a.m.

Invited Address Georgian Room 11:00-12:00 (144) Computational methods for discontinuous solutions. OLIVER A. MCBRYAN, Courant Institute of Mathematical Sciences, New York Uiriversity (803-65-170)

Friday, 1:30 p.m.

Invited Address Georgian Room 1:30- 2:30 (145) Three-manifolds with positive Ricci curvature. RICHARD S. HAMILTON, MSRI, Berkeley, and Cornell University (803-53-175)

320 Friday, 1:30 p.m. Special Session on Computational Methods and Fluid Mechanics, II Belmont Room 1:30- 2:10 (146) Solutions of the Euler equations for transonic flow around aircraft. ANTHONY JAMISON, Princeton University (803-99-183) (Introduced by Oliver McBryan) 2:20- 3:00 (147) Node control in the Moving Finite Element method. KEITH MILLER, University of California, Berkeley (803-65-132) 3:10- 3:50 (148) An efficient numerical procedure for the computation of steady state harmonic currents in fiat plates. Preliminary report. JAMES BRAMBLE*, Cornell University, and JOSEPH PASCIAK, Brookhaven National Laboratory (803-65-96) 4:00- 4:40 (149) V-states: Steady state solutions of the Euler equations, their stability and nonlinear evolution. NORMAN J. ZABUSKY, University of Pittsburgh, Pittsburgh (803-76-79) (Introduced by Oliver McBryan)

Friday, 2:45 p.m. Special Session on Value Distribution Theory and its Applications, IV Albany Room 2:45- 3:05 {150) Meromorphic solutions of some functional equations. NIRO YANAGIHARA, Chiba University {803-30-57) (Introduced by Chung-Chun Yang) 3:15- 3:35 {151) On analytic covering spaces. Preliminary report. CHIA-CHI TUNG, Mankato State University {803-32-123) 3:45- 4:05 (152) Analogues for holomorphic curves of R. Nevanlinna's "five-values" theorem. LEONARD M. SMILEY, University of Alaska, Anchorage (803-30-37) 4:15- 4:35 (153) Entire solutions of linear differential equations and bounds for growth numbers. S. M. SHAH, University of Kentucky (803-30-120)

Friday, 2:45 p.m. Special Session on Singularities of Algebraic and Analytic Varieties, III Village Room 2:45- 3:05 {154) Deformations of cones over flag varieties. RAGNAR-0. BUCHWEITZ, Brandeis University (803-14-147) {Introduced by Henry B. Laufer) 3:15- 3:35 (155) Singularities of algebraic and analytic varieties. STEPHEN YAU, University of illinois, Chicago (803-32-85) {Introduced by Henry B. Laufer) 3:45- 4:05 (156) Cobordism and isotopy of algebraic knots. SUSAN SZCZEPANSKI, Lehigh University {803-14-63) 4:15- 4:35 {157) Quasi-homogeneous complete intersections in dimension 2. Preliminary report. JONATHAN M. WAHL, University of North Carolina, Chapel Hill (803-14-164)

Friday, 2:45 p.m. Special Session on Combinatorial Group Theory, IV Hudson/Sutton Suite 2:45- 3:05 {158) Algorithms for solvable groups. Preliminary report. FRANK B. CANNONITO, University of California, Irvine (803-20-44) 3:15- 3:35 {159) An algorithm for simple curves on surfaces. JOAN S. BIRMAN*, Columbia University, and CAROLINE SERIES, University of Warwick, United Kingdom {803-57-46) 3:45- 4:05 {160) Representations of Riemann surface groups over number fields. II. Preliminary report. BENJAMIN FINE, Fairfield University (803-20-158) 4:15- 4:35 (161) Quadratic equations over free products of free groups with cyclic amalgamation. Preliminary report. LEO P. COMERFORD, JR.*, University of Wisconsin, Parkside, and CHARLES C. EDMUNDS, Mount Saint Vincent University (803-20-47) 4:45- 5:05 (162) Infinite Eulerian tessellations. J. L. BRENNER, Palo Alto, California, and R. C. LYNDON*, University of Michigan, Ann Arbor (803-05-03)

Friday, 2:45p.m. Special Session on Transcendence Theory and Diophantine Problems, IV Georgian Room 2:45- 3:05 (163) Approximation of functions. Preliminary report. CHARLES F. OSGOOD, U.S. Naval Research Laboratory (803-10-135) 3:15- 3:35 (164) Gel'fond's criterion and measure of algebraic independence. Preliminary report. ERIC REYSSAT, Harvard University {803-10-137) {Introduced by W. Dale Brownawell) 3:45- 4:05 {165) Lower bounds on linear forms and measures of algebraic independence. Preliminary report. ROB TUBBS, Pitzer College (803-10-168) 4:15- 4:35 (166) Diophantine approximations to values of functions. Preliminary report. GREGORY V. CHUDNOVSKY, Columbia University (803-10-169)

321 Friday, 2:45 p.m. Special Session on Abelian Groups, III Town Room 2:45- 3:05 (167) p-maps. GEORGE KOLETTIS, University of Notre Dame (803-20-156) 3:15- 3:35 (168) Applications of matm representations to direct sum decompositions of finite valuated p-rrroups. DONNA BEERS, Wellesley College, and ROGER HUNTER and ELBERT WALKER*, New Mexico State University, Las Cruces (803-20-139) 3:45- 4:05 (169) Direct products and sums of isomorphic abelian rrroups. Preliminary report. JOHN D. O'NEILL, University of Detroit (803-20-102) 4:15- 4:35 (170) w1-separable p-rrroups. Preliminary report. CHARLES MEGIBBEN, Vanderbilt University (803-20-75) Friday, 2:45p.m. Special Session on Variational Problems in Riemannian Geometry, IV Pennsylvania Room 2:45- 3:05 (171) Generalized idea of Synge. S. WALTER WEI, Michigan State University (803-53-141) 3:15- 3:35 (172) Elastic deformations with prescribed principal strains. DENNIS DETURCK*, University of Pennsylvania, and DEANE YANG, Harvard University (803-35-113) 3:45- 4:05 (173) The structure of minimal surfaces of finite index. DORIS FISCHER-COLBRIE, Columbia University (803-53-143) 4:15- 4:35 (17 4) Conformal metrics with prescribed curvatures. Preliminary report. CARLOS E. KENIG and WEI-MING NI*, University of Minnesota, Minneapolis (803-35-84) 4:45- 5:05 (175) Removable isolated singularities of coupled Yang-Mills fields in R 3 . L. M. SIBNER*, Polytechnic Institute of New York, and R. J. SIBNER, Brooklyn College, City University of New York (803-35-142)

Friday, 2:45p.m. Session on Logie and Combinatoric:& Headquarters Room 2:45- 2:55 (176) Covers of trees. Preliminary report. STEVEN E. ANACKER, Central Michigan University (803-05-127) 3:00- 3:10 (177) An operator approach to enumeration of matchings and forests of a rrraph. Preliminary report. C. J. LIU, Marquette University, and YUTZE CHOW*, University of Wisconsin, Milwaukee (803-05-99) (Introduced by Bennett Chow) 3:15- 3:25 (178) A refutation of the axiom of infinity (an exercise in functional analysis). GERHARD F. KOHLMAYR, Glastonbury, Connecticut (803-03-115) 3:30- 3:40 (179) A very nonextensional formula of arithmetic with undecidable fixed points in many theories. ROBERT A. DIPAOLA, Queens College and the Graduate Center, City University of New York (803-99-186) 3:45- 3:55 (180) Search for the CSPBIB designs by a new method of the generalized difference set. B. C. HUANG* and B. H. HUANG, People's Republic of China, and T. P. LIN, California State University, Northridge (803-05-05) (Introduced by Kwang-Nan Chow) 4:00- 4:10 (181) Power in decisions between multiple alternatives. Preliminary report. FRANCINE ABELES, Kean College of New Jersey (803-90-71) 4:15- 4:25 (182) On the symmetry of the Smith normal form for (v, k, A.) designs. ZAKHAR DERETSKY, University of California, Santa Barbara (803-05-34) (Introduced by Morris Newman)

Friday, 2:45p.m. Session on Fourier Series and Spaces of Functions Empire Suite B 2:45- 2:55 (183) Fourier coefficients on non-analytic wave forms. V. V. RAO, University of Regina (803-30-94) 3:00- 3:10 (184) Convergence of Fourier series under change of variable. Preliminary report. DANIEL WATERMAN, Syracuse University (803-42-151) 3:15- 3:25 (185) The essential spectra of a D-commuting system. Preliminary report. RAM U. VERMA, Universidad de Oriente, Venezuela (803-47-13) 3:30- 3:40 (186) Spectral duality theorems for closed operators. WANG SHENGWANG, Nanjing University, People's Republic of China, and I. ERDELYI*, Temple University (803-47-150) 3:45- 3:55 (187) The structure of Chebyshev spaces. Preliminary report. ROBERT JUNG, Wesleyan University (803-41-167) 4:00- 4:10 (188) On a conjecture of Zalcman. Preliminary report. ANNA TSAO, U.S. Naval Academy (803-30-129) 4:15- 4:25 (189) On the equivalence of classes of infinitely differentiable functions. Preliminary report. JAMIL A. SIDDIQI, Universite Laval (803-46-154)

W. Wistar Comfort Middletown, Connecticut Associate Secretary

322 Salt Lake City, April 29-30, 1983, University of Utah Program for the 804th meeting

The eight hundred and fourth meeting of the Fredric D. Ancel, Steven Bleiler, James W. Cannon, American Mathematical Society will be held at the Tim Cochran, Donald Coram, Mark Culler, William University of Utah in Salt Lake City on Friday and T. Eaton, Allan L. Edmonds, Ronald Fintushel, Saturday, April 29 and 30, 1983. This meeting Patrick M. Gilmer, Robert E. Gompf, Cameron MeA. will be held in conjunction with a meeting of the Gordon, Joel Haas, Michael Handel, John Hempel, Mathematical Association of America (MAA). The Terry Lawson, Youn Lee, Charles Livingston, Ken­ sessions will be held in the James C. Fletcher Building neth Millett, Luis Montejano, Ulrich Oertel, Daniel (JFB), John A. Widstoe Building (JWB), and Physics Ruberman, T. B. Rushing, Nobuyuki Sato, Martin G. South Building (PSB). Scharlemann, David J. Schorow, Laurence R. Taylor, Bruce Trace, John J. Walsh, and David G. Wright. Invited Addresses Contributed Papers By invitation of the Committee to Select Hour Speakers for Far Western Sectional Meetings, there There will also be sessions for contributed ten­ will be two invited one-hour addresses. The speakers minute papers. and the titles of their talks are: MAAProgram WILLIAM ARVESON, University of California, Berkeley, Quasitriangular operator algebras. The MAA program will include two invited LANCE W. SMALL, University of California, speakers. RICHARD D. ANDERSON of Louisiana San Diego, Finitely generated algebras satisfying a State University, Baton Rouge, will deliver an polynomial identity. address entitled, Some recent results in infinite dimensional topology. JEAN J. PEDERSEN of Special Sessions the University of Santa Clara will deliver an address By invitation of the same committee, there will be entitled, Approximating n-gons by folding paper: five special sessions of selected twenty-minute papers. An interplay of analysis, geometry and number The topics of these special sessions, the names of the theory. organizers, and lists of speakers are as follows: There will be a session for student papers Computer-aided geometric design, PETER AL­ organized by L. J. KRATZ of Idaho State University. FELD and ROBERT BARNHILL, University of A panel discussion on Industrial Opportunities Utah. The speakers will be Peter Alfeld, Paul R. and Applications will be moderated by R. L. Arner, R. E. Barnhill, Gerald Farin, Gary Herron, CARPENTER of Thiokol Corporation. Thomas Jensen, Peter Kochevar, Gregory M. Nielson, Immediately following the AMS session on Chip Petersen, Kestutis Salhaushas, Sarah E. Stead, Computer-aided geometric design, there will be a Tracy Whelan, and Andrew J. Worsey. one-hour computer-aided geometric design interactive Theory and applications of cocycles for an graphics workshop. irrational rotation, LAWRENCE W. BAGGETT, Pacific Northwest Geometry Seminar University of Colorado, Boulder. The speakers are Brian Amrine, Brendt Brenken, Robert P. Boyer, Ray On Saturday afternoon, April 30, and Sunday Fabec, Henry Helson, Joe Jenkins, Kathy Merrill, and morning, May 1, a meeting of the Pacific Northwest Arlan Ramsay. Geometry Seminar will be held. The Saturday session Noncommutative ring theory, KENNETH R. will meet in Room 101 of the James C. Fletcher GOODEARL and T. LENAGAN, University of Building; the Sunday session will be held in Room 335 Utah. The speakers are John A. Beachy, H. H. of the John A. Widstoe Building. The speakers will Brungs, Joe W. Fisher, Joel K. Haack, T. J. be R. Barlow, J. Carrell, M. Green, and M. Vitulli. Hodges, Hans-Christian Mez, S. Montgomery, W. K. Nicholson, Stanley S. Page, Sudarshan Seghal, S. P. Registration Smith, Kalathoor Varadarajan, and Roger Wiegand .. The registration desk will be located in the foyer of Nonlinear elliptic and parabolic partial differ­ the Lecture Hall Rotunda in the James C. Fletcher ential equations, KLAUS SCHMITT, University of Building and will be open from 1:00 p.m. to 5:30 Utah. The speakers will be Wolfgang Alt, Peter W. p.m. on Friday, and from 8:00 a.m. to 1:30 p.m. Bates, Jerrold W. Bebernes, Stephen Cantrell, Shui­ on Saturday. The registration fees will be $6 for Nee Chow, Chris Cosner, Steven R. Dunbar, Hans members of AMS or MAA, $8 for nonmembers, and Engler, Frank Hoppensteadt, F. A. Howes, C. Jones, $2 for students and unemployed mathematicians. Gene A. Klaasen, Jean L. Mawhin, H. G. Othmer, D. Sather, Andrea Shiaffi.no, and James Ward, Jr. Aeeomodations Four-dimensional topology, RONALD J. Although rooms have not been blocked at the STERN, University of Utah. The speakers will be following motels and hotels, they are included here

323 ...., ~

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·~. )p~/'~~,- w rf'. -~ , . ..,./ -.)· University of Utah, Salt Lake City for information purposes and participants are advised Social Events to mention the mathematics meetings when making There will be a banquet in the Tanner dining room reservations. All hotels and motels are in Salt Lake of the Alumni House on Friday evening, April 29. The City within two miles of the University of Utah featured speaker at the banquet will be RICHARD campus, and on direct bus lines. For information D. ANDERSON, the immediate past president of on local bus routes, call 801-263-3737. Prices shown the MAA. His topic is School mathematics .in the below do not include the applicable 9% tax. calculator-computer age. Desert Inn The entree at the banquet will be New York 50 West 500 South, 84101 cut steak; the cost will be $15.50 (including tip). Telephone: 801-532-2900 Participation is limited by the size of the dining Single $27; Double $30 (one bed), $33 (two beds) room. Those wishing to reserve a place or places Howard Johnson's should do so by writing to Karen Edge, Department 122 West South Temple, 84101 of Mathematics, University of Utah, Salt Lake City, Telephone: 801-521-0130 Utah 84112. A check or money order for $15.50 per Single $35 up; Double $38 up person should be enclosed and must arrive prior to Hotel Utah April 19, 1983. South Temple and Main Streets, 84111 Local Information Telephone: 801-531-1000 or 800-323-7500 Single $45 up; Double $45 up The Cniversity of Ctah is located in Salt Lake Little America City, about two miles directly east of the downtown 500 South Main Street 84101 district. Salt Lake City is served by a number Telephone: 801-363-9928 of airlines, including American, Continental, Delta, Single $38 up; Double $46 up Eastern, Frontier, Republic, Cnited, Western, and Wien. ~ost of the major hotels provide limousine Motel6 service to and from the airport. In addition, the city 176 West 600 South, Salt Lake City 84101 center can be reached from the airport by city bus Telephone: 801-521-3280 (50 cents) or taxi ($6). Single $16; Double $20, 3-4 persons $23 Persons arriving by automobile can most easily Salt Palace TraveLodge reach the city center by taking the Sixth South exit 215 West North Temple Street, 84103 off I-15. If driving from the east, leave I-80 at the Telephone: 801-532-1000 or 800-255-3050 Foothill exit and proceed along old route 40 until Single $28; Double $33 (one bed), $35 (two beds) reaching 13th East; turn right onto 13th East and Scenic Motel proceed to 2nd South. The University will be on the 1345 Foothill Drive, Salt Lake City 84108 right. There is no public parking available on campus Single $25.50; Double $28 (one bed), $30 (two beds) Friday, but a one-day permit may be purchased for 50 cents. Parking in all lots is available on Saturday.

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

Al-Daft'a, A. A., 84 *Culler, M., 64 *Jenkins, J., 61 *Petersen, C., 14 *Aifeld, P., 9 *Dunbar, S. R., 30 *Jensen, T., 38 *Ramsay, A., 60 *Alt, w., 79 *Eaton, W. T., 46 *Jones, C., 28 *Ruberman, D., 4 *Amrine, B., 25 *Edmonds, A. L., 7 *Klaasen, G. A., 52 *Rushing, T. B., 42 *Ancel, F. D., 39 *Engler, H., 78 Klement, E. P ., 83 * Salhaushas, K., 37 *Arner, P. R., 11 *Fabec, R., 24 *Kochevar, P., 36 *Sather, D., 50 •Arveson, W., 32 *Farin, G., 35 Kohlmayr, G. F., 82 *Sato, N., 6 *Barnhill, R. E., 8 *Fintushel, R., 18 *Lawson, T., 41 * Scharlemann, M. G., 16 *Bates, P. W., 49 *Fisher, J. W., 59 *Lee, Y., 17 * Schiaffino, A., 80 *Beachy, J. A., 72 *Gilmer, P.M., 2 *Livingston, C., 40 * Schorow, D. J., 69 *Bebernes, J. W., 77 *Gompf, R. E., 66 *Mawhin, J. L., 48 *Sehgal, S., 57 *Bleiler, S., 20 *Gordon, C. MeA., 65 *Merrill, K., 22 •Small, L. W., 63 *Boyer, R. P., 62 *Haack, J. K., 54 *Mez, H.-C., 55 *Stead, S. E., 10 *Brenken, B., 23 *Handel, M., 15 *Millett, K., 43 *Taylor, L. R., 44 *Brungs, H. H., 73 *Haas, J., 67 *Montejano, L., 70 * Trace, B., 5 *Cannon, J. W., 3 *Helson, H., 21 *Montgomery, S., 58 * Varadarajan, K., 76 *Cantrell, S., 29 *Hempel, J., 19 *Nicholson, W. K., 74 *Ward, J., Jr., 51 *Chow, S.-N., 27 *Herron, G., 13 *Nielson, G. M., 33 *Whelan, T., 12 *Cochran, T., 68 *Hodges, T. J., 71 *Oertel, U., 1 *Wiegand, R., 75 *Coram, D., 47 *Hoppensteadt, F., 31 *Othmer, H. G., 26 * Worsey, A. J., 34 *Cosner, C., 81 *Howes, F. A., 53 *Page, S. S., 56 *Wright, D. G., 45

325 Program of the Sessions

The time limit for each contributed paper in the AMS general sessions is ten minutes. In the special sessi9ns, 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 April 1983 issue of Abstracts of papers presented to the American Mathematical 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, 9:00a.m. Special Session on Four-dimensional Topology, I JWB, Room 335 9:00- 9:20 (1) Incompressible branched surfaces. Preliminary report. ULRICH OERTEL, Michigan State University (804-57-47) 9:25- 9:45 (2) Ribbon concordance, Alexander polynomials and 8-equivalence. PATRICK M. GILMER, Louisiana State University, Baton Rouge (804-57-49) 9:50-10:10 (3) Quasiconformal structures on topological spheres. JAMES W. CANNON, University of Wisconsin, Madison (804-57-10) 10:15-10:35 (4) Equivariant cobordism of knots in 8 4 . DANIEL RUBERMAN, Courant Institute of Mathematical Sciences, New York University (804-57-62) 10:40-11:00 (5) Ribbon disks bounded by the unknot. Preliminary report. BRUCE TRACE, University of Utah (804-99-81) 11:05-11:25 (6) Manifold links. NOBUYUKI SATO, Texas A&M University, College Station (804-57-16) 11:30-11:50 (7) Periodic knots. ALLAN L. EDMONDS, Indiana University, Bloomington (804-57-17)

Friday, 1:00 p.m. MAA Invited Speaker JWB, Room 335 1:00- 2:00 Some recent results in infinite dimensional topology. RICHARD D. ANDERSON, Louisiana State University, Baton Rouge

Friday, 2:10 p.m. Special Session on Computer-aided Geometric Design, I JFB, Room 102 2:10- 2:30 (8) Survey of surface interpolation methods. R. E. BARNHILL, University of Utah (804-65-23) 2:35- 2:55 (9) Multivariate derivative estimation by functional minimization. PETER ALFELD, University of Utah (804-65-22) 3:00- 3:20 (10) Smooth surfaces using generalized multiquadrics. SARAH E. STEAD, Brown University (804-65-30) 3:25- 3:45 (11) Examples of multistage approximations. PAUL R. ARNER* and R. E. BARNHILL, University of Utah, and S. STEAD, Brown University (804-65-21) (Introduced by Peter Alfeld) 3:50- 4:10 Discussion 4:15- 4:35 (12) Triangular Shepard surface method. TRACY WHELAN, University of Utah (804-65-31) 4:40- 5:00 (13) Smooth closed surfaces with discrete triangular interpolants. GARY HERRON, University of Maine, Orono (804-65-18) (Introduced by Peter Alfeld) 5:05- 5:25 (14) Contouring of three- and four-dimensional surfaces. CHIP PETERSEN, University of Utah (804-65-27) (Introduced by Peter Alfeld)

Friday, 2:10p.m. Special Session on Four-dimensional Topology, II JWB, Room 335 2:10- 2:30 (15) New proofs of some results of Nielsen. MICHAEL HANDEL*, Michigan State University, and WILLIAM P. THURSTON, Princeton University (804-57-68) 2:35- 2:55 (16) The genus two PL Schoenfties theorem. MARTIN G. SCHARLEMANN, University of California, Santa Barbara (804-57-57) 3:00- 3:20 (17) Classifying torsion-free subgroups of the Picard group. ANDREW BRUNNER, YOUN LEE*, NORBERT WIELENBERG, University of Wisconsin, Parkside, and MICHAEL FRAME, University of South Florida, New College (804-57-48) 3:25- 3:45 (18) Equivariant self-dual connections and pseudo-free actions on 8 5. Preliminary report. RONALD FINTUSHEL*, Tulane University, and RONALD STERN, University of Utah (804-57-70)

326 3:50- 4:10 (19) A trace inequality for simplicity of loops in surfaces. Preliminary report. JOHN HEMPEL, Rice University (804-57-69) 4:15- 4:35 (20) Unknotting classical knots. Preliminary report. STEVEN BLEILER, University of Texas, Austin (804-57-60)

Friday, 2:10p.m. Special Session on Theory and Applications of Cocycles for an Irrational Rotation, I JWB, Room 208 2:10- 2:30 (21) Cocycles and choice. HENRY HELSON, University of California, Berkeley (804-28-59) 2:40- 3:00 (22) Cohomology of step functions under irrational rotations. Preliminary report. KATHY MERRILL, University of Colorado, Boulder (804-43-42) 3:10- 3:30 (23) Cocycles associated with some irreducible representations of the irrational rotation algebra. BERNDT BRENKEN, Dalhousie University (804-47-53) 3:40- 4:00 (24) Fibered skew product extensions of group actions. Preliminary report. RAY F ABEC, Louisiana State University, Baton Rouge (804-22-38) 4:10- 4:30 (25) Some problems about cohomology of decomposable unitary operators with respect to an irrational rotation. Preliminary report. BRIAN AMRINE, University of Kansas (804-47-52) 4:40- 5:00 Discussion

Friday, 2:10p.m. Special Session on Nonlinear Elliptic and Parabolic Differential Equations, I PSB, Room 305 2:10- 2:30 (26) Some bifurcation phenomena in coupled oscillators. Preliminary report. D. ARoNSON, University of Minnesota, Minneapolis, E. DOEDEL, Concordia University, and H. G. 0THMER*, University of Utah (804-35-71) 2:40- 3:00 (27) Perturbation of singularities of free boundaries in the obstacle problem. Preliminary report. SHUI­ NEE CHOW*, AHMAD DIZAJI, and JOHN MALLET-PARET, Michigan State University ( 804-35-33) 3:10- 3:30 (28) A dynamical systems approach to bifurcation from the continuous spectrum. C. JONES*, University of Arizona, and T. KUPPER, Universitii.t Dortmund, Federal Republic of Germany (804-34-44) 3:40- 4:00 (29) On strongly coupled nonlinear elliptic systems. STEPHEN CANTRELL, University of Miami (804-35-08) 4:10- 4:30 (30) Periodic traveling waves in diffusive predator-prey equations. Preliminary report. STEVEN R. DUNBAR, University of Utah (804-35-56) 4:40- 5:00 (31) Large scale systems of voltage controlled oscillators. Preliminary report. FRANK HOP­ PENSTEADT, University of Utah (804-99-80)

Friday, 5:00p.m. AMS Invited Address JWB, Room 335 5:00- 6:00 (32) Quasitriangular operator algebras. WILLIAM AR.vESON, University of California, Berkeley (804-99-78)

Friday Evening Banquet and Special Address Tanner Dining Room of the Alumni House 7:30- Banquet School mathematics in the calculator-computer age, RICHARD D. ANDERSON, Louisiana State University, Baton Rouge

Saturday, 8:00a.m. Special Session on Computer-aided Geometric Design, II JFB, Room 102 8:00- 8:20 (33) Some practical aspects regarding Kriging and related methods for interpolation of scattered data. Preliminary report. RICHARD FRANKE, U.S. Naval Postgraduate School, and GREGORY M. NIELSON*, Arizona State University (804-65-20) (Introduced by Peter Alfeld) 8:25- 8:45 (34) Bivariate monotone interpolation. ANDREW J. WORSEY, University of Utah (804-65-29) (Introduced by Peter Alfeld) 8:50- 9:10 (35) Subdividing arbitrary polyhedra to generate surfaces composed of triangular B-splines. GERALD F ARIN, Dettenhausen, Federal Republic of Germany (804-65-24) (Introduced by Peter Alfeld) 9:15- 9:35 Discussion

327 9:40-10:10 (36) Computational aspects of multivariate B-splines in CAGD. PETER KOCHEVAR, Boeing, Seattle (804-65-26) (Introduced by Peter Alfeld) 10:15-10:35 (37) Weighted C1 splines for interactive interpolation. KESTUTIS SALHAUSHAS, University of Calgary (804-65-28) (Introduced by Peter Alfeld) 10:40-11:00 (38) Modeling symmetric surfaces. Preliminary report. THOMAS JENSEN, Evans and Sutherland Computer Corporation, Salt Lake City (804-65-25) (Introduced by Peter Alfeld)

Saturday, 8:00a.m.

Special Session on Four-dimensional Topology, III JWB, Room 335 8:00- 8:20 (39) Oribts of embeddings of 8 2 in 8 3 . Preliminary report. FREDRIC D. ANCEL, University of Oklahoma (804-57 -76) 8:25- 8:45 (40) Symmetric representations of knot groups and branched covers. ALLAN EDMONDS and CHARLES LIVINGSTON*, Indiana University, Bloomington (804-57-79)

8:50- 9:10 (41) Detecting the standard embedding of RP2 in 8 4 . TERRY LAWSON, Thlane University (804-57 -46) 9:15- 9:35 (42) {3-homotopy equivalences are a-cellular. LUIS MONTEJANO, Universidad Nacional, Mexico, and T. B. RUSHING*, Institute for Advanced Study (804-57-67) 9:40-10:00 (43) Maps of one orientable surface into another. Preliminary report. KENNETH MILLETT, University of California, Santa Barbara (804-57-75) 10:05-10:25 (44) Relative Rochlin invariants. LAURENCE R. TAYLOR, University of Notre Dame (804-57-72)

10:30-10:50 (45) Taming compacta in E 4 . JOHN J. WALSH, University of Tennessee, Knoxville, and DAVID G. WRIGHT*, Utah State University (804-57-61) 10:55-11:15 (46) Complements of connected 3-manifolds in 8 4 . WILLIAM T. EATON, University of Texas, Austin (804-57-66) 11:20-11:40 (47) Images of 8 3 under approximate fibrations. DONALD CORAM* and PAUL DUVALL, Oklahoma State University (804-57-63)

Saturday, 8:00a.m.

Special Session on Nonlinear Elliptic and Parabolic Differential Equations, II PSB, Room 305 8:00- 8:20 (48) Periodic solutions of some partial differential equations. JEAN L. MAWHIN, Universite de Louvain, Belgium (804-35-15) 8:30- 8:50 (49) Containment for weakly coupled parabolic systems. PETER W. BATES, Texas A&M University, College Station (804-35-06) 9:00- 9:20 (50) Subcritical solutions of Benard-type convection problems. GEORGE H. KNIGHTLY, University of Massachusetts, Amherst, and D. SATHER*, University of Colorado, Boulder (804-76-17) (Introduced by Klaus Schmitt) 9:30- 9:50 (51) On the ranges of some semi-linear gradient operators. JAMES WARD, JR., University of Alabama, Tuscaloosa (804-35-19) 10:00-10:20 (52) Steady state solutions for a system derived from the FitzHugh-Nagumo equations for nerve conduction. GENE A. KLAASEN, University of Tennessee, Knoxville (804-35-07) 10:30-10:50 (53) Perturbed elliptic problems with essential nonlinearities. F. A. HOWES, University of California, Davis (804-35-03)

Saturday, 8:00a.m.

Special Session on Noncomutative Ring Theory, I JFB, Room Bl 8:00- 8:20 (54) Subdirect products of serial rings with self-duality. JOEL K. HAACK, Oklahoma State University (804-16-12) 8:30- 8:50 (55) Combinatorial extensions of radical, nil and locally nilpotent algebras. HANS-CHRISTIAN MEZ, University of California, Irvine (804-16-13) 9:00- 9:20 (56) Semiperfect FPF rings. STANLEY S. PAGE, University of British Columbia (804-16-11) 9:30- 9:50 (57) Torsion units in integral group rings. SUDARSHAN SEHGAL, University of Alberta (804-16-45) 10:00-10:20 (58) Galois theory of prime rings. S. MONTGOMERY*, University of Southern California, and D. S. PASSMAN, University of Wisconsin, Madison (804-16-37) 10:30-10:50 (59) Special Jordan rings with fixed-point-free group actions. Preliminary report. JOE W. FISHER, University of Southern California and University of Cincinnati (804-16-54)

328 Saturday, 9:00a.m. Speeial Session on Theory and Applications of Cocycles for an Jrratrional Rotation, II JWB, Room 208 9:10- 9:30 (60) Applications of cocycles to group representations. Preliminary report. ARLAN RAMSAY, University of Colorado, Boulder (804-43-64) 9:40-10:00 (61) Minimal eigenvalues for positive Rockland operators. ANDRZEJ HULANICKI, Polish Academy of Sciences, Wroclaw, and JOE JENKINS*, State University of New York, Albany (804-22-36) (Introduced by Lawrence W. Baggett) 10:10-10:30 (62) Character theory of inductive limit groups. ROBERT P. BOYER, Drexel University (804-46-39)

Saturday, 11:00 a.m. MAA Invited Speaker JFB, Room 101 11:00-12:00 Approximating regular n-gons by folding paper: An interplay of analysis, geometry and number theory, JEAN J. PEDERSEN, Univerity of Santa Clara

Saturday, 11:00 a.m. Computer-aided Geometric Design Workshop JWB, Room 120

Saturday, 1:00 p.m. AMS Invited Address JFB, Room 101 1:00- 2:00 (63) Finitely-generated algebras satisfying a polynomial identity. LANCE W. SMALL, University of California, San Diego (804-16-55)

Saturday, 2:10p.m. Special Session on Four-dimensional Topology, IV JWB, Room 335 2:10- 2:30 (64) Incompressible surfaces and surgery on knots. MARC CULLER* and PETER B. SHALEN, Rice University (804-99-84) 2:35- 2:55 (65) Planar surfaces in three-manifolds. Preliminary report. CAMERON MeA. GORDON* and R. A. LITHERLAND, University of Texas, Austin (804-57-73) 3:00- 3:20 (66) Three exotic R 4 's and other anomalies. ROBERT E. GOMPF, University of California, Berkeley (804-57-65) 3:25- 3:45 (67) Minimal surfaces in foliated manifolds. JOEL HAss, University of Michigan, Ann Arbor (804-57 -50) 3:50- 4:10 (68) Four-manifolds which embed in R 6 but not in R 5 . Preliminary report. TIM COCHRAN, Massachusetts Institute of Technology (804-57-04) 4:15- 4:35 (69) Dihedral branched covers of knots in 8 3 . Preliminary report. DAVID J. SCHOROW, University of California, Berkeley (804-99-82) (Introduced by Ronald J. Stern) 4:40- 5:00 (70) A quick proof of Singhof's cat(M X 8 1) = cat(M) + 1 theorem. LUIS MONTEJANO, lnstituto de Matematicas, U.N.A.M., Mexico (804-99-83) (Introduced by Ronald J. Stern)

Saturday, 2:10p.m. Speeial Session on Noneommutative Ring Theory, II JFB, Room B1 2:10- 2:30 (71) Global dimensions of simple factors of the enveloping algebra of semi-simple Lie algebra. Preliminary report. T. J. HODGES*, University of Utah, and S. P. SMITH, University of Southern California (804-16-51) (Introduced by K. R. Goodearl) 2:40- 3:00 (72) Stability over Noetherian rings. Preliminary report. JOHN A. BEACHY, University of Washington (804-16-58) 3:10- 3:30 (73) Skew power series rings and derivations. H. H. BRUNGS*, University of Alberta, and G. TORNER, University of Duisburg, Federal Republic of Germany (804-16-02) (Introduced by K. R. Goodearl) 3:40- 4:00 (74) On tensor products and extended centroids. Preliminary report. W. K. NICHOLSON*, University of Calgary, and J. F. WATTERS, University of Leicester, England (804-16-05) 4:10- 4:30 (75) Cancellation of finitely generated modules over one-dimensional rings. ROGER WIEGAND, University of Nebraska, Lincoln (804-13-43) 4:40- 5:00 (76) The fundamental theorem of projective geometry over certain non-commutative rings. KALATHOOR VARADARAJAN, University of Calgary (804-16-01) 5:10- 5:30 Discussion

329 Saturday, 2:10 p.m.

Special Session on Nonlinear Elliptic and Parabolic Dift'erential Equations, ill PSB, Room 305 2:10- 2:30 (77) Invariance for functional parabolic systems. JERROLD W. BEBERNES, University of Colorado, Boulder (804-35-35) (Introduced by Klaus Schmitt) 2:40- 3:00 (78) Convergence to the mean in Neumann problems for parabolic integra-differential equations. Preliminary report. HANS ENGLER, Northwestern University and University of Heidelberg, Federal Republic of Germany (804-35-41) 3:10- 3:30 (79) Stability results for diffusion-aggregation systems. WOLFGANG .ALT, Heidelberg, Federal Republic of Germany (804-35-32) (Introduced by Klaus Schmitt) 3:40- 4:00 (80) Monotone methods for a class of quasi-linear parabolic equations. Preliminary report. ANDREA SCHIAFFINO, University of Utah and Universita di Roma, Italy (804-35-40) (Introduced by Klaus Schmitt) 4:10- 4:30 (81) Steady states and threshold behavior for some nonlinear differential/difference equations. JONATHAN BELL, State University of New York, Buffalo, and CHRIS COSNER*, University of Miami (804-35-14) 4:40- 5:00 Discussion

Saturday, 2:10p.m. Session for Contributed Papers JWB, Room 208 2:10- 2:20 (82) Can a computer refute the existence of a Dedekind complete ordered field? GERHARD F. KOHLMAYR, Glastonbury, Connecticut (804-03-74) 2:25- 2:35 (83) FUzzy measures assuming their values in the set of fuzzy numbers. ERICH PETER KLEMENT, University of Cincinnati, Cincinnati, and Johannes Kepler Universitii.t, Austria (804-28-34) 2:40- 2:50 (84) Quadrature and cubature in medieval Near Eastern mathematics. Final report. ALI A. AL­ DAFFA * and JOHN J. STROYLS, University of Petroleum & Minerals, Saudi Arabia (804-01-09)

Saturday, 2:10p.m. MAA Session for Student Papers JFB, Room 102

Hugo Rossi Salt Lake City, Utah Associate Secretary

Pacifl.e Northwest Geometry Seminar

Saturday, 2:10p.m. First Session JFB, Room 101 2:10- 3:00 A simply connected surface of general type with pg = 0. REBECCA BARLOW, University of Utah 3:40- 4:30 A decomposition theorem for the integral homology of a variety. JAMES CARRELL, University of British Columbia

Sunday, 9:40a.m. Second Session JWB, Room 335 9:40-10:30 Semi-normal rings and weakly normal varieties. MARIE VITULLI, University of Oregon 11:00-11:50 The period map for hypersurfaces of high degree on an arbitrary variety. MARK GREEN, Unversity of California, Los Angeles

330 On Some Mathematical Questions in Biology-Muscle Physiology 1983 Symposium, Detroit, Michigan, May 31, 1983

The seventeenth annual Symposium on Some The program has been arranged by an organizing Mathematical Questions in Biology will be held on committee consisting of H. Thomas Banks (Brown May 31, 1983, in the Brule A Room of the Westin University), Joel E. Cohen (Rockefeller University), Hotel, Detroit, Michigan, in conjunction with the Frank C. Hoppensteadt (University of Utah), Joseph annual meeting of the American Association for B. Keller (Stanford University), Donald A. Ludwig the Advancement of Science. The symposium is (University of British Columbia), Robert M. Miura sponsored by the American Mathematical Society, (University of British Columbia), chairman, Garrett the Society for Industrial and Applied Mathematics, M. Odell (Rensselaer Polytechnic Institute), and and Section A of the American Association for the Charles S. Peskin (Courant Institute, New York Advancement of Science. Cosponsor: Section G. University). Details regarding registration and local arrangements The theme of the symposium is Muscle Physiology, for the AAAS meeting appeared in the February 25th the study of the mechanisms which cause muscle issue of Science. contraction. There will be two half-day sessions, each including three one-hour lectures.

PROGRAM Chairman: Robert M. Miura 9:00a.m. Some Mathematical Questions in Biology-Muscle Physiology Presiding: ROBERT M. MIURA The molecular mechanism of muscular contraction. HUGH E. HUXLEY, University Medical School, Cambridge, England The relationship between muscle biochemistry and muscle physiology. EVAN EISENBERG, National Institutes of Health A model of muscle contraction based upon component studies. MICHAEL B. PROPP, Massachusetts Institute of Technology

1:30p.m. Some Mathematical Questions in Biology-Muscle Physiology Presiding: CHARLES S. PESKIN Mathematical methods for determining cross-bridge properties uniquely from mechanical experiments on macroscopic muscle. H. MICHAEL LACKER, New York University Sarcomere dynamics-some physical insights into cardiac contraction. JOHN W. KRUEGER, Albert Einstein College of Medicine Electrical field problems in muscle: meaning to muscle, mathematicians, and physiologists. ROBERT S. EISENBERG, Rush Medical College

331 1983 Summer Research Institute Nonlinear Functional Analysis CONTEMPORARY MATHEMATICS and Applications {ISSN 0271-4132) July 11-29, 1983 UMBRAL CALCULUS AND HOPF ALGEBRAS University of California, Berkeley edited by Robert Morris The thirty-first Summer Research Institute spon­ CONTENTS: sored by the American Mathematical Society will S. A. joni and G.-C. Rota, Coa!gebras and bia!ge­ be devoted to nonlinear functional analysis and will bras in combinatorics take place at the University of California, Berkeley, Warren Nichols and Moss Sweedler, Hopf algebras from July 11-29, 1983. Members of the Organiz­ and combinatorics ing Committee include Haim Brezis, Felix Browder This book represents a unique blending of two (chairman), Tosio Kato, J. L. Lions, Louis Nirenberg, fields only recently recognized as related. On one and Paul Rabinowitz. It is anticipated that the hand lies the field of combinatorics with roots (at institute will be supported by a grant from the least immediately traceable via generating functions National Science Foundation. to umbral calculus, the speciality at hand) in the The topic was selected by the 1982 Committee 19th century writings of Boole on operator calculus. on Summer Institutes, whose members were Morris Both the foundations and much of the history of W. Hirsch, Robert Osserman (chairman), George C. the umbral calculus are explored in great clarity in Papanicolaou, Wilfried Schmid, Harold Stark, and previous papers by Rota and others. On the other Stephen Wainger. hand is the field of Hopf algebras, which is usually During the past two decades there has been a traced to a paper of Milnor and Moore but whose strong upsurge of activity in the study of nonlinear first general exposition by Sweedler is little more problems in analysis, involving both the development than a decade old. of general methods for the study of relatively broad Both Rota and Sweedler were pleased when the classes of problems and the attack on many significant University of Oklahoma was able to support their classes of special problems. The primary objective joint appearance at a conference funded by the of the institute is to bring together the specialists j. C. Karcher Foundation in May, 1978. The confer· in the various branches of this development to foster ence centered on lectures they gave, with S. A. J oni a unified view of what has been achieved and the assisting Rota. Sweedler lectured first on elementary principal directions of future development, as well as coalgebra theory aimed at combinatorists, Rota on the current state of the art in the applications of elementary combinatorics aimed at the algebraists. general methods to problems of importance in applied Both lectures converged toward those who were or mathematics and mathematical physics. There would work at the intersection. Sweedler and Warren has been no broad-scale meeting surveying these Nichols prepared notes of Sweedler's talks and Joni developments either in the United States or on an and Rota of Rota's and a mimeographed version was international level during the past decade. The circulated by the Oklahoma Mathematics Department. institute will have two levels of organization, the first The present volume makes these more accessible. centering on the general levels of methodology in the The Sweedler notes here are essentially unchanged field, and the second a more extensive development from those distributed by Oklahoma. They aim, in a of the application of these methods to various broad direct and elementary way, to give the reader suffi­ classes of important problems, centering primarily cient knowledge of coalgebra theory to understand upon nonlinear partial differential equations. the coalgebra formulation of special sequences of Housing accommodations will be available in the polynomials. campus residence halls for participants and their The Rota notes are reproduced from Studies in families, and daily meals will be served in the Applied Mathematics, volume 61, and represent a adjacent dining hall. Facilities are fully accessible reworking of the original, with corrections and a to the handicapped. In the spring a brochure will few additions. They contain detailed applications be mailed to all who are invited to attend. It not only to umbral calculus, but to partition stud­ will include information about the scientific program, ies, incidence algebras, lattice theory, and other the residence and dining facilities, firm room and traditional spheres of combinatoric interest. The board rates, as well as travel and local information notes form a broad survey for anyone who would and a reservation form to be used for obtaining like detailed and concrete examples of the areas accommodations at Berkeley. Each participant will already known to be amenable to a coalgebraic ap­ pay a social fee to cover the cost of refreshments proach. served at breaks and for social events, in additon to 1980 Mathematics Subject Classifications: 16A24, 05B20, a meeting registration fee of $45 (reduced to $15 for 05-02, 16A20 students). Funds for support will be limited and, Volume 6, viii + 84 pages (soft cover) therefore, it will be necessary for many participants List price $9, institutional member $7, to obtain their own funds. Anyone who wishes to individual member $5 receive an invitation to participate in the institute ISBN 0·8218-5003-2; LC 81-22756 and/ or be considered for financial assistance should Publication date: February 1982 write to Mrs. Dorothy Smith, American Mathematical To order, please specify CONM/6N Society, P. 0. Box 6248, Providence, R.I. 02940 prior Prepayment is required for all AMS publications. to April 15, 1983. The Organizing Committee will Order from AMS, PO Box 1571, Annex Station, then consider all requests and successful applicants Providence, Rl 02901, or call 800-556-7774 will be informed sometime thereafter if funds are available to provide support. to charge with VISA or MasterCard.

332 Albany Meetings, August 8-11, 1983 First Announcement

The August 1983 Joint Mathematics Meetings, including 87th Summer Meeting of the AMS the 87th summer meeting of the American Mathematical Society, the 63rd summer meeting of the Mathematical August 8-11, 1983 Association of America, and the 1983 annual meeting of Pi Mu Epsilon, will be held August 8-11, 1983 (Monday­ Colloquium Lectures Thursday), at the State University of New York, Center at Albany. The meetings will be preceded by the AMS It is hoped that a series of four Colloquium Short Course on August 6 and 7 (Saturday and Sunday). Lectures will be presented at 1:00 p.m., Monday Sessions will take place on the campus of the State through Thursday, August 8-11. The name of the University of New York, Center at Albany. lecturer and title of the lectures will be announced The members of the Local Arrangements Committee are Lindsay N. Childs (publicity director), William W. later. Fairchild, Richard Z. Goldstein (chairman), Timothy L. Steele Prizes Lance, Violet Lamey, William J. LeVeque (ex officio), David P. Roselle (ex officio), Hugo Rossi (ex officio), B. The 1983 Leroy P. Steele Prizes will be awarded at David Saunders, Edward C. Turner, Nura D. Turner, and a session at 4:45p.m. on Wednesday, August 10. Edward S. Thomas. Invited Addresses WHERE TO FIND IT PAGE By invitation of the AMS Program Committee, PREREGISTRATION AND HOUSING 334 there will be eight invited one-hour addresses. The SUMMER MEETING OF THE AMS 333 list of speakers, dates and times of the talks, and Colloquium Lectures, Prizes, Invited Addresses, some of the titles follow: Special Sessions, Contributed Papers, Council SELMAN .AKBULUT, Michigan State University, and Business Meetings, Other AMS Sessions 2:15 p.m. Wednesday; JAMES EELLS, University AMS SHORT COURSE 336 of Warwick, England, Harmonic maps of Riemann SUMMER MEETING OF MAA 335 surfaces, 8:30 a.m. Monday; ROBERT C. GUN­ Business Meeting, Board of Governors, Hedrick NING, Princeton University, 5:00 p.m. Monday; Lectures, Minicourses, Contributed Papers, Banquet for 25-Year Members LEO A. HARRINGTON, University of California, OTHER ORGANIZATIONS 338 Berkeley, 8:30 a.m. Wednesday; IRA HERBST, AWM,IIME University of Virginia, The Schrodinger equation, TIMETABLE 343 9:45 a.m. Wednesday; HERVE JACQUET, Colum­ bia University, 2:15 p.m. Tuesday; JOHAN H. OTHER EVENTS OF INTEREST 338 Book Sales, Summer List of Applicants, B. KEMPERMAN, University of Rochester, Func­ Exhibits, MATHFILE tional equations over groups, and the mean value INFORMATION FOR PARTICIPANTS 339 property, 3:30 p.m. Wednesday; and WEN-CH'ING University Housing, Food Services, Hotel WINNIE LI, Pennsylvania State University, Univer­ Accommodations, Registration at Meetings sity Park, 11:00 a.m. Monday. REGISTRATION DESK SERVICES 346 AMS/MAA Information, Audio-Visual Aid, Special Sessions Assistance, Comments and Complaints, By invitation of the same committee, there will Baggage and Coat Check, Check Cashing, be eleven special sessions of selected twenty-minute Local Information, Lost and Found, Mail, Personal and Telephone Messages, papers. The titles of these special sessions, and Transparencies, Visual Index the names and affiliations of the mathematicians MISCELLANEOUS INFORMATION 348 arranging them, are as follows: Athletic Facilities, Book Stores, Camping, Child Care, Crib Rental, Libraries, Local The Albany meetings will repeat an experiment Information, Medical Services, Parking, tried for the first time at the Toronto meetings in Social Events, Travel, Weather August of 1982, in that the program will be "joint" CAMPUS MAP 340 in a stronger sense than before. AMS and MAA sessions will run concurrently and simultaneously, IMPORTANT DEADLINES and the meetings have been shortened from five AMS Abstracts, days to four. In order to evaluate the effects For consideration for special sessions April26 Of contributed papers May 17 of this new type of program, a questionnaire will MAA Abstracts be distributed to all registrants at the Albany Of contributed papers May 16 meetings requesting opinions and comments. A Summer List of Applicants July 1 central location will be established for the collection Preregistration and Housing July 1 Housing cancellations (full refund) July 15 of these questionnaires at the meeting registration Housing caneellations (partial refund) After July 15 desk. It is hoped that all participants will find time Motions for AMS Business Meeting July 10 to respond, since this information will be a valuable Preregistration cancellations (50% refund) August 5 guide in future planning of summer meetings.

333 PREREGISTRATION AND HOUSING - MATHEMATICS MEETINGS HOUSING BUREAU

Preregistration. Preregistration for these meetings must Those who preregister for the AMS Short Course will be be completed by July 1, 1983. All those wishing to able to pick up their badges and other material in Albany preregister must complete the form which appears at the after 11:00 a.m. on Saturday, August 6, during the hours back of this issue and submit it along with the appropriate the AMS Short Course registration desk is open. preregistration fee( s) to the Mathematics Meetings Housing Those who preregister for either the Joint Mathematics Bureau in Providence by July 1. Meetings or the MAA Minicourses or both will be able to Preregistration for the meeting and full payment of pick up their badges and other material in Albany after room/board charges is a requirement in order to obtain 4:00p.m. on Sunday, August 7, during the hours the Joint confirmed residence hall accommodations at SUNYA Mathematics Meetings registration desk is open. through the Mathematics Meetings Housing Bureau as Summer List of Applicants. The AMS will publish a outlined in the column to the right. Summer List of mathematical scientists seeking employ­ Checks for preregistration fee(s), housing payments and ment for distribution at the Albany meeting. fees for social events should be made payable to the AMS. To be included in the list, applicants should complete Canadian checks must be marked for payment in U.S. the special applicant preregistration form found at the funds. Those who preregister for the AMS Short Course back of this issue. The completed form should be mailed and/or Joint Mathematics Meetings pay fees which are with the meeting preregistration form. The deadline for 30 percent lower than those who register at the meetings. receipt of applicant forms is the same as for the Joint The preregistration fees are as follows: Meetings (July 1, 1983). See the section on OTHER AMS Short Course EVENTS OF INTEREST for more details. Mathematics Meetings Housing Bureau. The form for Student/Unemployed $ 5 university residence hall accommodations will All Others $25 requesting be found at the back of this issue. The use of the Joint Mathematics Meetings services offered by the Mathematics Meetings Housing Member of AMS, MAA, liME $38 Bureau requires preregistration for the meetings. Persons Emeritus Member of AMS, MAA $ 9 desiring confirmed residence hall accommodations should Nonmember $58 complete the form, or a reasonable facsimile, and send it Student/Unemployed $ 9 to the Mathematics Meetings Housing Bureau, Post Office MAA Minicourses #1 through #6 $20 each Box 6887, Providence, Rhode Island 02940, so that it will arrive no later than July 1, 1983. DO NOT SUBMIT MINICOURSE FEE(S) Please read carefully the section on University Housing WITH PREREGISTRATION FORM. before completing the form. Forms sent to the wrong There is no extra charge for members of the families address and thus incurring delay in delivery to the Housing of registered participants, except that all professional Bureau until after the deadline cannot be accepted. All mathematicians who wish to attend sessions must register residence halls reservations with full prepayment for independently. room/board will be confirmed by the housing bureau. All full-time students currently working toward a All reservation requests must be received in writing and degree or diploma qualify for the student registration fees, be processed through the Housing Bureau in Providence. regardless of income. Please do not contact the university directly. Telephone The unemployed status refers to any person currently requests will not be accepted. unemployed, actively seeking employment, and who is not Housing assignments are made on a first-come, first­ a student. It is not intended to include persons who served basis, so participants desiring specific types of have voluntarily resigned or retired from his or her latest accommodations are urged to get their housing requests in position. as early as possible. Housing requests received after the The emeritus status refers to any person who has been deadline of July 1 most surely cannot be honored. a member of the AMS or MAA for twenty years or more, Participants who are able to do so are urged to and is retired on account of age from his or her latest share a room whenever possible. This procedure can position. be economically beneficial. The housing form should be A $4 charge will be imposed for all invoices prepared fully completed to ensure proper assignment of rooms. when preregistration forms are submitted without accom­ Participants planning to share accommodations should panying check(s) for the preregistration fee( s) or are provide the name(s) of the person(s) with whom they plan accompanied by an amount insufficient to cover the total to occupy a room. Each participant should, however, payments due. Preregistration forms received well before complete a separate preregistration form. Parties planning the deadline of July 1 which are not accompanied by to share rooms should send their forms together in the correct payment will be returned to the participant with a same envelope if possible. request for resubmission with full payment. Please make all changes to or cancellations of residence A 50 percent refund of the preregistration fee( s) will hall reservations with the Housing Bureau in Providence be made for all cancellations received in Providence no before July 15, 1983 in order to receive full refund of later than August 5, 1983. No refunds will be granted for housing payment. After that date, cancellations should cancellations received alter that date, or to persons who be made with the Housing Bureau in Providence up until do not attend the meetings. August 5, 1983, at which time a partial refund (amount The only exception to this rule is someone who paid minus one night's room/board) will be made. No preregisters for the Joint Mathematics Meetings only in cancellations can be made between 5:00 p.m. on August 5 order to attend an MAA Minicourse, and is too late and 4:00 p.m. on Sunday, August 7, after which changes to obtain a slot in the Minicourse. In this case, full or cancellations may be called in to Mary Coccoli at the refund will be made of the Joint Mathemtics Meetings Telephone Message Center number in Albany. Changes in preregistration fee, provided the preregistrant has cheeked reservations may be made at any time by notifying the the box on the preregistration form that this was his or Housing Bureau. her intent. Individuals who preregister for both the Joint N.B.: Place your AMS or MAA mailing label on the Meetings and a Minicourse and who intend to attend the preregistration/housing form where indicated. If you Joint Meetings, even if the Minicourse is not available, do not have a label readily available, please supply should, of course, not check the box on the preregistration complete name, address, and AMS or MAA member form. In this case, the Joint Meetings Preregistration will code. be processed.

334 The calculus of variations in the large and its Contributed Papers applications, MELVYN S. BERGER, University There will be sessions for contributed papers on of Massachusetts, Amherst; Dynamical systems, Monday afternoon from 2:15 to 6:00 p.m., Tuesday LOUIS BLOCK, University of Florida; Ergodic morning from 8:00 to 10:45 a.m., Tuesday afternoon theory concerning point transformations with finite from 2:15 to 4:15 p.m., Wednesday afternoon from invariant measure, NATHANIEL A. FRIEDMAN, 1:00 to 4:30 p.m., Thursday morning from 8:00 SUNY, Center at Albany; Tensor products and p­ to 10:45 a.m., and Thursday afternoon from 1:00 summing operators in harmonic analysis, COLIN to 5:30 p.m. Abstracts should be prepared on the C. GRAHAM, Northwestern University, and BERT standard AMS form available from the AMS office M. SCHREIBER, Wayne State University; Applica­ in Providence or in departments of mathematics, tions of algebraic topology, TIMOTHY A. LANCE, and should be sent to the American Mathematical SUNY, Center at Albany; Automorphic functions Society, Post Office Box 6248, Providence, Rhode and automorphic representations, WEN-CH'ING Island 02940, so as to arrive by the abstract WINNIE LI, Pennsylvania State University, Univer­ deadline of May 17. Members are reminded that a sity Park; Several complex variables, R. MICHAEL charge of $12 is imposed for retyping abstracts that RANGE, SUNY, Center at Albany; Summability are not in camera-ready form. methods, BILLY E. RHOADES, Indiana Univer­ Late papers will not be accepted. sity, Bloomington; Solutions of operator equations Council Meeting and fixed points, V. M. SEHGAL, University of Wyoming, and S. P. SINGH, Memorial Univer­ The Council of the Society will meet at 5:00 p.m. sity, Newfoundland; Differential analysis in infinite on Sunday, August 7 in the Squire Room of the dimensional spaces, SRINIVASA SWAMINATHAN, Ramada Inn. Dalhousie University; Topological methods in com­ Business Meeting binatorial group theory, EDWARD C. TURNER, The Business Meeting of the Society will take SUNY, Center at Albany. place immediately following the Steele Prize Session Most of the papers to be presented at these on Wednesday, August 10. The secretary notes the special sessions will be by invitation; however, following resolution of the Council: Each person who anyone contributing an abstract for the meeting who attends a Business Meeting of the Society shall be feels that his or her paper would be particularly willing and able to identify himself as a member appropriate for one of these sessions should indicate of the Society. In further explanation, it is noted this clearly on the abstract, and should submit it that each person who is to vote at a meeting is by April 26, three weeks earlier than the normal thereby identifying himself as and claiming to be deadline for contributed papers, in order that it a member of the American Mathematical Society. may be considered for inclusion. For additional information on the Business Meeting, please refer to the announcement titled Committee Committee on the Agenda for Business Meetings on the Agenda for Business Meetings. The Society has a Committee on the Agenda for Business Meetings. The purpose is to make Business Meetings orderly and effective. The committee does 63rd Summer Meeting of the MAA not have legal or administrative power. It is intended that the committee consider what may be called "quasi­ August 8-11, 1983 political" motions. The committee has several possible courses of action on a proposed motion, including but Business Meeting not restricted to (a) doing nothing; The Business Meeting of the MAA will take place (b) conferring with supporters and opponents to arrive at 4:30 p.m. on Tuesday, August 9 at which the Carl at a mutually accepted amended version to be circulated B. Allendoerfer, Lester R. Ford, and George P61ya in advance of the meeting; awards will be presented. (c) recommending and planning a format for debate to suggest to a Business Meeting; Board of Governors (d) recommending referral to a committee; The MAA Board of Governors will meet at 9:00 (e) recommending debate followed by referral to a a.m. on Sunday, August 7 in the Squire Room of the committee. There is no mechanism that requires automatic Ramada Inn. submission of a motion to the committee. However, if a Hedrick Lectures motion has not been submitted through the committee, it may be thought reasonable by a Business Meeting to The 32nd Ea~le Raymond Hedrick Lectures will be refer it rather than to act on it without benefit of the given by ELIAS M. STEIN of Princeton University. advice of the committee. The title of this series of three lectures will be The committee consists of Everett Pitcher (chairman), Marian B. Pour-El, David A. Sanchez, and Guido L. announced later. Weiss. Minicourses In order that a motion for the Business Meeting of August 10, 1983 receive the service offered by the The MAA is planning six Minicourses, as follows: committee in the most effective manner, it should be in Minicourse #1: PASCAL for Mathematicians, the hands of the secretary by July 11, 1983. organized by HARLEY FLANDERS, Florida Atlan­ Everett Pitcher, Secretary tic University. Given from 7:30 p.m. to 9:30 p.m.

335 American Mathematical Society Short Course Series Population Biology Albany, New York, August 6-7, 1983

The American Mathematical Society, in conjunction with its eighty-seventh summer meeting, will present a one and one-half day short course titled Population Biology on Saturday and Sunday, August 6 and 7, 1983, at the State University of New York, Center at Albany. The program is under the direction of Simon A. Levin of Cornell University. Population biology is probably the oldest area in mathematical biology, but remains a constant source of new mathematical problems and the area of biology best integrated with mathematical theory. The need for mathematical approaches has never been greater, as evolutionary theory is challenged by new interpretations of the paleontological record and new discoveries at the molecular level, as world resources for feeding populations become limiting, as the problems of pollution increase, and as both animal and plant epidemiological problems receive closer scrutiny. The purpose of this course is to acquaint the participant with the mathematical ideas that pervade almost every level of thinking in population biology. The mathematical methods to be discussed include dynamical systems theory, partial differential equations, stochastic processes, matrix algebra, control theory and optimization, game theory, and differential geometry. Synopses of the talks and accompanying reading lists appear in this issue of the Notices, pages 350 and 351. A basic knowledge of ordinary and partial differential equations, linear algebra, and probability theory will be assumed. Other mathematical methods to be discussed will be drawn from stochastic processes, control theory and optimization, game theory, and differential geometry. Background reading may be found in Studies in mathematical biology, Part II: Populations and communities, Simon A. Levin, editor, Volume 16 in Studies in Mathematical Biology, MAA, 1978. The reading lists also give a variety of sources for study prior to the course. The course will consist of six 60-minute lectures and interpolatory material by Simon A. Levin. Ethan Akin (City College, CUNY) will speak about evolution, game theory and differential geometry, James C. Frauenthal (Bell Laboratories, Holmdel) about population dynamics and demography, Wayne M. Getz (University of California, Berkeley) about optimal control theory in population biology, Thomas Nagylaki (University of Chicago) about mathematical population genetics, George Sugihara (Oak Ridge National Laboratory) about graph theory and food webs, and James A. Yorke (University of Maryland) about epidemiology. Simon A. Levin will supplement with material on models of population dispersal, and on the theory of evolution of interacting populations. The short course is open to all who wish to participate upon payment of the registration fee. There are reduced fees for students and unemployed individuals. Please refer to the sections titled Preregistration and Housing-Mathematics Meetings Housing Bureau and Registration at the Meetings for details. The short course was recommended by the Society's Committee on Employment and Educational Policy, whose members are Lida K. Barrett, Lisl Novak Gaal, Irwin Kra, Robert W. McKelvey, Donald C. Rung (chairman), and Barnet M. Weinstock. The short course series is under the direction of the CEEP Short Course Subcommittee, whose members are Stefan A. Burr (chairman), Robert W. McKelvey, Cathleen S. Morawetz, Barbara L. Osofsky, and Philip D. Straffin, Jr.

336 on Monday, August 8, and on Tuesday, August second session Dr. Cline will present his solution of 9. The course is aimed at mathematicians who do the problem. Any alternate approaches suggested not presently (or can barely) program in PASCAL, by the participants will be discussed. Dr. Cline will but who have access to PASCAL on a microcom­ describe generalizations of this particular problem, puter or larger system. No previous knowledge of and he will also be available to discuss the place PASCAL or programming will be assumed. The of the mathematically trained individual in industry, talks will cover the whole PASCAL programming and the curricular needs implied by employment in language with emphasis on solving mathematical industry. programming problems. There will be special em­ Minicourse #3: An introduction to the mathe­ phasis on recursion, linked memory allocation, and matical techniques and applications of computer scientific uses of the unique data structure flexibility graphics is being organized by JOAN WYZKOSKI of PASCAL. A wide range of applications will of Bradley University, and will be given from 8:30 be demonstrated, including matrix manipulations, a.m. to 10:30 a.m. on Monday, August 8, and integration, differentiation, differential equations, in­ Wednesday, August 10. Graphs and illustrations of variant factors of integer matrices, set theory, series geometrical objects are useful tools in the teaching inversion, etc. The Minicourse will be conducted in of mathematics. Computer graphics simplifies the a lecture-demonstration mode. The block structured production of these teaching aids. This Minicourse nature of PASCAL makes it particularly flexible for will present some of the mathematical techniques used programming intricate mathematical computations, to produce realistic pictures on graphics terminals. and easier to read than most other popular program­ Emphasis will be on the use of these techniques ming languages. It is becoming the first programming to complement mathematics instruction. Some of language taught in many colleges and universities, the topics to be discussed are curve and surface but the emphasis in almost all current courses is sketching, 2D and 3D transformations, perspective on data processing applications. Probably mathe­ drawing and hidden line removal. Since personal matics departments should offer PASCAL courses for computers will be available for demonstrations and mathematics, sciences, and engineering students. in-class implementations, programming experience is Minicourse #2: Problems from industry for use in necessary. the undergraduate classroom, is being organized by Minicourse #4: Uses of computers in under­ MARVIN S. KEENER and JEANNE L. AGNEW, graduate mathematics instruction (CONDUIT) is and will be given from 8:30 a.m. to 10:30 a.m. on being organized by DAVID A. SMITH of Duke Monday, August 8, and Wednesday, August 10. This University, and is being held from 7:30 p.m. to 9:30 Minicourse is based on problems obtained under the p.m. on Monday, August 8, and Tuesday, August direction of the organizers from representatives of 9. It is a Minicourse intended for college teachers. industries who were willing to share their expertise Uses of existing microcomputer software to enhance in order to help provide the undergraduate student instruction in full courses in the undergraduate cur­ an opportunity to deal with a real-world problem in riculum will be demonstrated. Included will be its raw form. These problems have arisen in the software for use in single- and multi-variable calculus, work of the industry presenting them, and can be differential equations, and topics at the lower division solved, at least in part, using only undergraduate college level. Presentations will be given by math­ mathematics. They have been written up by the ematicians who have developed the software and organizers in the format suitable for classroom use. A have had extensive experience with its use in their catalog of these industrial problems will be available courses. It is planned for participants to have the to the participants. The Minicourse will focus on opportunity to work with the software themselves on a discussion of selected problems already developed, microcomputers. and on the identification and development of a new Minicourse #5: Uses of computers in under­ problem with the help of an industrial representative. graduate mathematics instruction (NON­ The cooperating representative will be Dr. Jerry Cline from McDonnell Douglas Astronautics in St. CONDUIT). This Minicourse will be held from Louis. The first part of the first session will be 8:30 a.m. to 10:30 a.m. and 2:00 p.m. to 4:00 p.m. on Tuesday, August 9 and will be identical to devoted to a brief description of the creation of the problem collection, and a discussion of several of Minicourse #4 with the exception that only materials the problems, their solution, and the ways in which produced by individuals and organizations other than they can be used in the classroom. Each participant CONDUIT will be demonstrated. will receive in advance one of the problems to be Minicourse #6: Combinatorics. This Minicourse discussed in detail. Following these discussions, has been only tentatively scheduled for 8:30 a.m. to Dr. Cline will present a possible new problem for 10:30 a.m. on Monday, August 8, and Wednesday, development, giving its background and the reasons August 10. Further information will be available why it is important to McDonnell Douglas. Between later. sessions the participants will have an opportunity The Minicourses are open only to persons who to work on the solution of Dr. Cline's problem. have registered for the Joint Mathematics Meetings He will be available during this time to discuss it and paid the Joint Meetings registration fee. informally with interested participants. If necessary, Please note that participants registering for one computing facilities will be available. During the or more of the Minicourses should not include

337 the registration fee(s) with their preregistration. twenty-five years or more. The banquet will take A reservation will be made for participants who place Wednesday evening, August 10. A cash bar preregister for the Minicourses and a confirmation will will be operated from 6:00 p.m. to 6:45 p.m. in the be sent once the preregistration has been processed. Patroon Lounge in the Campus Center. Dinner will The Minicourses have separate registration fees of be served at 7:00 p.m. in the Patroon Room, one $20 each, and are limited to thirty participants each. flight up. The menu for the dinner is as follows: Payment of the fee(s) must be made at the meeting Chilled wedge of Saratoga melon, Iced relishes, registration desk in Albany two hours prior to the Roast prime rib of beef au jus, Rosette potatoes, beginning of the Minicourse or the reservation will be Fresh native green beans with mushrooms, Romaine relinquished to someone on the waiting list. When salad with vinaigrette dressing, Brioche and crescent making payment, the participant should present the rolls with butter, Minted chocolate mousse with confirmation to the cashier. "Standby" reservation Chantilly cream, Ladyfingers and after-dinner mints, confirmations will be issued to participants whose Wine, coffee, and tea preregistration was received after the Minicourse was Please note that all tickets for this banquet filled. These individuals should check with the must be purchased through preregistration, since meeting cashier one hour prior to the Minicourse to a guarantee must be given to the caterer much see if any openings have occurred. earlier than usual. Tickets are $16.05 each If the only reason for registering for the Joint and interested participants should complete the Meetings is to gain admission to a Minicourse, this appropriate section of the preregistration form. Also, should be indicated by checking the appropriate spaces are provided on the preregistration form for box on the preregistration form. Then, if the ticket purchasers to indicate their seating preference, Minicourse is full, full refunds can be made of if desired. the Joint Mathematics Meetings preregistration fee. Otherwise, the Joint Meetings preregistration Activities of Other Organizations will be processed, and then be subject to the 50 percent refund rule. Pi Mu Epsilon (IIME) will hold its annual meeting Contributed Papers on Wednesday and Thursday, August 10 and 11. The Papers are being accepted on four topics in J. Sutherland Frame Lecture will be given at 8:30 collegiate mathematics for presentation in contributed p.m. on Wednesday, August 10. The name of the paper sessions at the MAA Summer Meeting in lecturer and the title of the talk will be announced Albany. The topics are: later. • The undergraduate mathematics curriculum The Association for Women in Mathematics • Special concerns: Remediation, articulation, (AWM) will hold a panel discussion on grants at and math anxiety 9:10 a.m. on Wednesday, August 10. The AWM membership meeting will follow the panel discussion • The use of computers in undergraduate at 10:10 a.m. on the same day. mathematics instruction • Classroom notes Presentations are normally limited to ten minutes, Other Events of Interest although selected contributors may be .given up to twenty minutes. Book Sales Individuals wishing to submit papers for any of Books published by the AMS and the MAA these sessions at Albany should send the following will be sold for cash prices somewhat below the information to the MAA Washington office (1529 usual prices when these same books are sold by Eighteenth Street, NW, Washington, DC 20036) mail. These discounts will be available only to before May 16. registered participants wearing the official meeting 1. Title badge. VISA and MASTERCARD credit cards will 2. Intended session be accepted for book sale purchases at the meeting. 3. A one-paragraph abstract (for distribution at the The book sales are open the same days and hours meeting) as the Joint Mathematics Meetings registration desk, 4. A one-page outline of the presentation and are located in the meeting registration area. 5. A list of special equipment required for the Exhibits presentation (e.g., computer, film projector, The book and educational media exhibits will be videotape player). located in the Lecture Center Vestibule and are open Late papers will not be accepted. from 1:00 p.m. to 5:00 p.m. on Monday, August This information will be sent to session leaders 8, and from 8:30 a.m. to 4:30 p.m. on Tuesday who will arrange for refereeing. Selection of papers and Wednesday, August 9-10. All participants are will be announced by July 1. encouraged to visit the exhibits during the meeting. Banquet For 25-Year Members MATHFILE The MAA is planning a banquet for individuals MATHFILE, the computerized version of Mathe­ who have been members of the Association for matical Reviews, will be demonstrated in the exhibit

338 area during regular exhibit hours. Sample searches University Housing will be performed on two terminals connected to Participants requesting housing on campus during computers of the vendors who offer MATHFILE: the meeting will be assigned to Indian Quadrangle, BRS (800-833-4707, in New York 518-783-1161) and located in the southeast corner of the uptown campus. DIALOG (800-227-1960, in California 800-982-3810). Colonial Quadrangle will be used as an overflow MATHFILE is also available during evening hours dormitory, if necessary. The check-in desk for Indian at greatly reduced rates on "BRS After Dark". This Quad is located in the lobby of Mohawk Tower, system uses a simplified command language and is which can be identified by a weather station on especially designed for the end-user with a home its roof. The check-in desk for Colonial Quad is computer or a terminal. located in the lobby of Livingston Tower (located Summer List of Applicants diagonally across the campus from Mohawk Tower on the northwest side). Those participants who At the direction of the AMS-MAA-SIAM Com­ are arriving via Western Avenue (New York Route mittee on Employment Opportunities, which is 20) will enter the campus from the south entrance charged with operation of the Employment Register and follow signs leading them to Indian Quad. and with the publication Employment Information Those arriving on Washington Avenue, bear left after in the Mathematical Sciences, the Society will entering one of the northern entrances and proceed publish a Summer List of mathematical scientists along the perimeter of the campus to Indian Quad. seeking employment for distribution at the Albany The entrance to the lobby faces the inside of the meeting in August 1983. quadrangle. The check-in desk will be in operation Copies of the 1983 summer list will be available at daily from 9 a.m. until 11 p.m. with the exception of the Transparencies section of the registration desk for August 7 and 8, when the hours are 9 a.m. until 3 $2. Following the meeting, they may be purchased a.m. the following morning. Temporary parking will from the AMS office in Providence for $3. This be allowed during check-in. (See section on Parking.) list should prove useful to employers who have last Participants desiring confirmed reservations for on­ minute openings in the latter part of the summer or campus housing must preregister and send required in the fall. payment( s) to the Mathematics Meetings Housing The deadline for receipt of applicant forms to Bureau prior to the July 1, 1983 deadline. (Please appear in this summer list is July 1, 1983. refer to sections below titled Room Rates Including Instead of an Employment Register at the Summer Meal Plans and Meal Plans Only.) Meeting in Albany, there will be an opportunity Dormitories at SUNYA are not air-conditioned. for posting of both applicant resume forms and Each quad consists of a high-rise tower and eight low­ employers' announcements of open positions in or rise units. In addition to the usual dormitory room near the main meeting registration area. There setup with communal bathrooms, there are a number will be no special room set aside for interviews. of suites available containing one single and either No provisions will be made by the Society for one or two double-bedded rooms and bath. There interviews: arrangements will be the responsibility of are no cots available in the domitories. each employer and applicant. Messages may be left Children aged ten and over will be charged the in the message box located in the registration area. same rates as adults. Children nine and under will Special applicant and employer forms will be be charged a reduced rate for room and board (must available at the Transparencies section of the purchase either Plan C or D). registration desk both for applicants to post resumes Parents will be allowed to bring portacribs or and for employers to post forms announcing positions. small cribs for infants and toddlers. These children Applicants who submit an applicant form, but do may stay in the same room with parents at no not plan to attend the meeting will appear on the charge. It is not a requirement that food plans be printed list only. There is no provision made for purchased for children four years of age and under. posting resumes for participants who do not attend For additional information regarding families, please the meeting. writ-e to the Mathematics Meetings Housing Bureau, P. 0. Box 6887, Providence, Rhode Island 02940, or call 401-272-9500, extension 239. Information for Participants Beds will be made up in advance for preregistered participants. In addition to bed linen, two towels are provided. A small bar of soap and plastic glass will Information which appears below in the sections be provided upon request. Since this may prove to on University Housing and Food Services was be inadequate, it is advised that participants bring furnished by SUNYA. Although firm information an additional bar of soap as well as a face cloth, if was requested, because of the uncertainty of necessary. Each room is equipped with floor lamp, the State of New York budget, many of the desk and dresser. No clothes hangers are provided, statements in these two sections may be subject to and it is suggested that participants bring their own change between now and the time of the meetings. supply. Bathrooms will cleaned periodically. Every attempt will be made to provide updated There will be a limited number of rooms information in future announcements. available for those without confirmed reservations;

339 State University of New York, Center at Albany

CAMPUS GUIDE Colonial Quadrangle • T Livingston Tower ~ Academic Podium I Indian Quadrangle 1 Social Science T Mohawk Tower 2 Humanities 3~~ 3 Education Service Buildings rl 4 Campus Center Motor Pool 5 Physics Security Office 6 Chemistry Commissary CDCD 7 Biology Plant Office \ .,o 8 Library ~,~. 9 Theatre-Music Motor Inns 10 Earth Science G) Ramada Inn 11 Fine Arts "'\I 'C.- @ Thruway House \f'I'C.si

~ PARKING ~

;i ~

LAKE

lf

~

HEALTH SERVICE

340 however, beds in these rooms will not be made up who request housing, with either an appropriate in advance. Participants arriving without prior refund or request for additional payment. reservations will be given a set of bed linens, pillow, Note: Rooms on campus are not available without and blanket in order to make up their own beds as a meal plan. well as two towels, and upon request a plastic cup and soap. Again, it is recommended that additional The following adult package plans offered by soap, a face cloth and clothes hangers be brought by SUNYA include 7 percent state and local sales tax. the individual. Plan A: Single room with breakfast and lunch Keys will be issued to each participant staying in $27.60 per day the residence halls. These keys open the outside door of the residence hall, the door to the suite (if Plan A: Double room with breakfast and lunch applicable), and the door to the individual room. $22.60 per day per person There is no deposit requirement for keys; however, Plan B: Single room with breakfast, lunch and there will be a penalty of $20 payable to the dinner $34.40 per day SUNYA Key Account by the individual participant Plan B: Double room with breakfast, lunch and should keys be lost to cover cost of replacement dinner $29.40 per day per person of both locks and keys. The adult rates for lodging and food apply There are heat detectors in hallways and lounges. to children 10 years of age or older. The rates There are also heat and smoke detectors in individual indicated below apply to children up to and including bedrooms. There are three elevators in the dormitor­ the age of 9. ies, one of which stops at every floor. All will be Plan C: Breakfast and lunch $14.90 appropriately marked. Stairways are unlocked. There is one kitchen unit in the basement of Plan D: Breakfast, lunch, and dinner $20.25 Mohawk Tower as well as in most of the low­ The above options are available by prepayment rise buildings on Indian Quad containing a small only. The following payments must accompany refrigerator, range-top and oven. No utensils are the preregistraton/housing form and must be provided. It is suggested that participants who wish received in Providence by the deadline of July 1, to avail themselves of this convenience bring their 1983. own kitchen articles. 1.) Preregistration fee(s) There are coin-operated laundry facilities in the 2.) Full payment for room/board tower basement as well as in the basement of each 3.) Fee(s) for tickets (if applicable) low-rise building. There is a 35 cent charge for the Any form received without an amount sufficient use of washers; dryers are free. Participants are to cover the above items will be returned, which requested to bring their own laundry materials. will delay processing of the housing request. Any Vending machines dispensing soft drinks, candy, form received with an amount which is more than etc., are located in the tower. required will be processed and an appropriate No pets are allowed in the residence halls. Alcoholic refund will be issued. beverages are not prohibited; however, residents are Meal Plans Only required to exercise moderation. For preregistered participants not staying in There is no telephone service in any of the residence residence halls, commuters, or those staying in hotels, hall rooms. Public telephones are located in the lobby there is a modified plan whereby meals only can of the towers. be purchased. Meal tickets must be purchased Check-in Locations and Times through preregistration, and the daily rate per person is given below: The check-b. desk will be in operation in the lobby of Mohawk Tower, Indian Quad, daily from 9 Adults and Children 10 years of age and over: a.m. to 11 p.m. with the exception of the nights of Plan E: Breakfast and lunch $ 6.60 August 7 and 8 when the desk will remain open to accommodate late comers until 3 a.m. As indicated in Plan F: Breakfast, lunch, and dinner $13.40 the text above, there is a possibility that Livingston Children 9 years of age and under: Tower (Colonial Quad) may be used as an overflow Plan G: Breakfast and lunch $ 3.90 dormitory. This tower is situated on the northwest corner of the campus diagonally across the quadrangle Plan H: Breakfast, lunch, and dinner$ 9.25 from Mohawk Tower. Note: Meals on a cash-as-you-go basis are not available in the cafeterias; however, the Campus Room Rates Including Meal Plans Center Snack Bar offers a limited menu for cash between the hours of 7:30 a.m. and 1:00 p.m. Note: As of this writing, the State University of New York central administration has yet to Lake George Cruise approve projected rates for the coming year as A cruise on Lake George aboard the Ticonderoga listed below. H' a change in these rates should has been arranged for Sunday, August 7. Buses occur, notification will be sent to preregistrants will depart from the Indian Quadrangle at 4:45 p.m.

341 and return to Indian Quadrangle at 10:15 p.m. The only through preregistration due to deadlines set buses will be marked "Lake George Cruise". This bySUNYA. cruise sails the beautiful island-dotted narrows of Food Services Lake George, with live music on board. The cost Food service in the cafeteria will begin with is $15 per person for those not on any meal plan, breakfast on Saturday, August 6, and will end or those on either meal Plans A, C, E, or G. For with lunch on Thursday, August 11. No food those on meal Plan B or F, the ticket price is $8.20. service will be available after lunch on Thursday. For children up to and including 9 years of age on Only prepaid meals will be served in the cafeteria meal Plan D or H, tickets are $9.65. (Since children in the basement of Indian Quad. Since dinner will under 10 are offered meals at a reduced rate, the not be available on Thursday, August 11, be sure cost of their tickets for the cruise is higher.) This to include rate for Plans A, C, E, and/ or G for box dinner replaces the regularly scheduled meal for the night of August 10 only. (Plans B, D, F, and that night for those taking the cruise. Individuals H do not apply for that night.) Cafeteria hours of who do not wish to participate in the cruise may operation are: still obtain their evening meal in the cafeteria. Breakfast 7:00 a.m. to 8:30 a.m. Since the Lake George Steamboat Company has Lunch 11:30 a.m. to 1:00 p.m. set a deadline for reservations, this cruise will be Dinner 5:00 p.m. to 6:30 p.m. offered through preregistration only. If a sufficient A sample breakfast served in the dining room number of preregistrations is not received by July 1, would be various fruits and juices, eggs (two styles), sausages, pancakes with syrup, sweet rolls, and interested participants will be notified of cancellation assorted jams and jellies. A typical lunch would be and appropriate refunds will be made. soup, hamburger steak with creole sauce, french fried The above prices include the cost of the bus, ·potatoes, salad variations, assorted breads, whipped the cruise, all taxes and gratuities, and a boxed fruited jello and hot dogs, hamburgers and grilled dinner consisting of fried chicken, potato salad, fruit, cheese sandwiches. Dinner menus include cream of brownie, soft drink, and relishes. mushroom soup, browned roast chicken with peach Picnic half or ham and roast beef cold platter, baked potato, buttered peas, rye bread, ice cream, lemon cake and At 6:15 p.m. on Monday, August 8, there will be fresh apples. Second helpings will be allowed for all a picnic outside the Mohawk Tower cafeteria. This meals. meal replaces the regularly scheduled dinner for that Hotel Accommodations night for all participants on meal Plans B, D, F, Blocks of rooms have been set aside for use by and H. There is no additional charge for participants participants at the hotels listed below. Participants on either of these plans for the picnic. For those should make their own reservations early with the not on any meal plan as well as those on meal Plan Albany Meetings Travel SuperPhone. (See box on A, C, E, or G, the cost of the picnic ticket is $7. page 344.) Reservations at these hotels will not be Because the guarantee must be given to the caterer available by calling the hotel directly. The rates in advance of the meeting, the picnic will be offered listed below are subject to change, and to an 8 through preregistration only. The menu will feature percent sales tax. barbecued chicken, hot dogs, hamburgers, assorted In all cases, a one night's deposit including the 8 salads, corn, baked beans, watermelon, brownies, percent tax is required to guarantee a room. This can soda, coffee, tea, and milk. In case of rain, this event be charged to a credit card when booking through will be moved to the cafeteria. the Albany Meetings Travel SuperPhone. If an emergency occurs and you find you must cancel your Beer Party reservation, this must be done prior to 6:00 p.m. A beer party will be held at 7:00p.m. on Tuesday, on the day of your arrival, or you will be charged August 9 outside the Indian Quadrangle. Tickets are for that day. The cut-off date for reservations at $5.35 per person and must be purchased through all hotels is July 25, 1983. After this date none of preregistration. In addition to beer, white and red the hotels are obligated to provide a room at these wines, soft drinks, potato chips, pretzels and peanuts special rates. will be served. In case of rain, this event will be The estimated walking distance from the hotel to the meetings is given in parentheses following the moved to the cafeteria. telephone number. The number after the name of MAA Banquet the hotel is the number it carries on the map. For those individuals not on any meal plan, as The following codes apply: AC = Air Conditioned; well as those on meal Plan A and E, the cost of the CL =Cocktail Lounge; FP =Free Parking; GR = ticket for the MAA 25-year banquet on Wednesday Game Room; RT = Restaurant; SP = Swimming Pool; SU = Sauna; TV = Television. is $16.05. There is an additional charge of $9.25 for In all cases, children 17 years of age and under participants on Plan B and F who plan to attend the can stay in a parent's room at no charge. The rates banquet. for cots are noted below for each hotel, as they vary. N.B.: These additional amounts must be In all cases "Single" refers to one person in one bed; included when completing the preregistra­ "Double" refers to two persons in one bed; "Twin" tion/housing form. All stated elsewhere in the refers to two persons in two beds. A rollaway cot meeting announcement, tickets for the MAA ban­ for an extra person can be added to double or twin quet, picnic, beer party, and cruise are available rooms only.

342 TIMETABLE The purpose of this timetable is to provide assistance to registrants 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

SATURDAY, August 6 POPULATION BIOLOGY

11:00 a, m. - 4:00 p.m. REGISTRATION 2:00 p.m. - 2:20 p.m. Introduction to mathematical population biology Simon A. Levin 2:20 p.m. - 3:20 p.m. Population dynamics and demography James Frauenthal 3:30p.m. - 4:30p.m. Epidemiology James Yorke 4:40 p.m. - 5:00 p.m. Summary Simon A, Levin

BUNDA Y, August 7

8:00a.m. - 2:00p.m. REGISTRATION 9:00 a, m. - 10:00 a.m. Population genetics Thomas Nagylaki 10:10 a, m. - 11:10 a.m. Evolution, game theory, and differential geometry Ethan Akin 11:20 a.m. - 11:45 a.m. Coevolution Simon A, Levin 1:30 p.m. - 2:30 p.m. Optimal control in population biology Wayne Getz 2:40 p.m< - 3:40 p.m. Graph theory and food webs George Sugihara 3:50p.m, - 4:00 p.m. Summary Simon A. Levin 4:00p.m. - 4:45p.m. General discussion

JOINT MATHEMATICS MEETINGS

SUNDAY, August 7 American Mathematical Society Mathematical Association of America

9:00a.m. - 4:00 p.m. \ Board of Governors Meeting 4:00p.m. - 8:00 p.m. REGISTRATION 4:00p.m. - 8:00 p.m. AMS Book Sale MAA Book Sale 5:00p.m. - 10:00 p.m. Council Meeting

MONDAY, August 8 AMS MAA 8:00 a.m. - 4:30 p.m. REGISTRATION 8:00 a.m. - 4:30p.m. AMS Book Sale \ MAA Book Sale 8:15 a.m. - 8:25 a.m. WELCOME ADDRESS 8:30 a.m. - 9:30 a.m. Invited Address Harmonic maps of Riemann surfaces James Eells 8:30 a.m. - 10:30 a.m. Minicourse Ji2 Problems from industry for use in the undergraduate classroom 8:30 a.m. - 10:30 a.m. Minicourse Ji3 An introduction to the mathematical tech­ niques and applications of computer graphics 8:30 a.m. - 10:30 a.m. Minicourse #6 Combinatorics (tentative) 8:40 a.m. - noon Contributed Paper Session The use of computers in undergraduate mathematics instruction 8:40 a, m. - noon Contributed Paper Session Classroom notes 11:00 a, m. - noon Invited Address Title to be announced Wen-Ch'ing Winnie Li 1:00 p.m. - 2:00 p.m. Colloquium Lecture I Title and speaker to be announced 343 Albany Meetings Travel SuperPhone 800-556-6882 INSTANT ... PERSONALIZED ... IMMEDIATE CONFIRMATION OF TRAVEL AND HOTEL RESERVATIONS FOR THE JOINT MATHEMATICS MEETINGS IN ALBANY, NEW YORK One free call does it all! No forms to mail and no waiting when you use your major credit card (VISA, MASTERCARD, American Express or air travel card.) TRAVEL SUPERSERVICE: - Fly to Albany with US61R. the official carrier for the Joint Mathematics Meetings and get 30 percent or more off! No restrictions on reservations purchased at least 14 days in advance. This special offer is available ONLY through Mathematics SuperPhone! Call SuperPhone toll-free today at 800-556-6882 and SAVE! - Unbiased advice on the most direct, economical air routes. - Fast, computerized reservation, ticketing, invoicing, itinerary. - Guaranteed lowest possible air fares for your city. - Automatic "Fare Check" at regular intervals before your departure to assure the most economical rate. If lower fares develop, your ticket will be automatically rewritten at savings to you. HOTEL CONFIRMATION: Along with your travel arrangements, ask for immediate confirmation of your hotel accommodations. All hotels for this meeting must be confirmed through this number. N.B.: University accommodations must be obtained through the Mathematics Meetings Housing Bureau. CALL NOW: 800-556-6882 (In Rhode Island and outside the Continental U.S. call401-884-9500.) Hours of Operation: 9:00a.m. to 7:00p.m. EST, Monday through Thursday, Fridays until 6:00p.m.

Ramada Inn (#1 on the campus map) expected to register, and should be prepared to show 1228 Western Avenue their meeting badge, if so requested. The fees for Albany, New York 12203 registration at the meetings, which are 30 percent Telephone: 518-489-2981 (10 minutes) more than the preregistration fees, are listed below. Single: $34 Double: $39 AMS Short Course Twin $46 Rollaway cot $5 Student/Unemployed $10 Extra person in room: No charge All Other Participants $30 Code: AC, CL, FP, GR, RT, SP, SU, TV, One-day Fee (Second Day Only) $15 Free transportation is provided to and from the Joint Mathematics Meetings airport, bus station, and AMTRAK station. Upon arrival, contact the hotel and request transportation. Member of AMS, MAA, liME $49 Thruway House (#2 on campus map) Emeritus Member of AMS, MAA $12 1375 Washington Avenue Nonmember $75 Albany, New York 12206 Student/Unemployed $12 Telephone: 518-459-3100 (10 minutes) MAA Minicourses #1 through #6 Single: $34 Double: $40 All Participants $20 each Twin $40 Rollaway cot $6 Do not submit Minicourse fee(s) Extra person in room: $6 (18 and over) with preregistation form. Code: AC, CL, FP, RT, SP, TV Registration fees may be paid at the meetings in Free van service is provided. Contact the hotel cash, by personal or travelers' checks, or by VISA upon arrival at the airport. or MASTERCARD credit cards. Canadian checks TraveLodge (#3 on campus map) must be marked for payment in U.S. funds. 1230 Western Avenue There will be no extra charge for members of Albany, New York 12203 the families of registered participants, except that Telephone: 518-489-4423 (10 minutes) all professional mathematicians who wish to attend Single: $29 Double $35 sessions must register independently. Twin $38 Rollaway cot $4 All full-time students currently working toward a Extra person in room: No charge degree or diploma qualify for the student registration Code: AC, FP, SP, TV fees, regardless of income. Registration at the Meetings The unemployed status refers to any person currently unemployed, actively seeking employment, Meeting preregistration and registration fees only and who is not a student. It is not intended to include partially cover expenses of holding meetings. All a person who has voluntarily resigned or retired from mathematicians who wish to attend sessions are his or her latest position.

344 TIMETABLE

MONDAY, August 8 American Mathematical Society Mathematical Association of America

1:00 p.m. - 5:00 p.m. EXHIBITS 2:15 p.m. - 3:15 p.m. I MAA - Invited Address 2:15 p.m. - 6:00 p.m. Special Sessions Sessions for Contributed Papers 5:00 p.m. - 6:00 p.m. Invited Address Title to be announced Robert C. Gunning 6:15 p.m. PICNIC 7:30 p.m. - 9:30 p.m. Minicourse #1 PASCAL for mathematicians 7:30 p.m. - 9:30 p.m. Minicourse #4 Uses of computers in undergraduate mathematics instruction (CONDUIT)

TUESDAY, August 9 AMS MAA and Other Organizations

8:00a.m. - 10:45 a.m. Special Sessions Sessions for Contributed Papers 8:30 a.m. - 4:30 p.m. REGISTRATION 8:30 a.m. - 4:30 p.m. AMS Book Sale [ MAA Book Sale 8:30 a.m. - 4:30 p.m. E:\.'ll IE ITS 8:30 a.m. - 10:30 a.m. ;\iAA - Minicourse #5 Uses of computers in undergraduate mathematics (nonCONDUIT) 8:40 a.m. - 9:40 a.m. MAA - Invited Address 9:50 a.m. - 10:50 a.m. MAA - Invited Address 11:00 a, m. - noon MAA - Hedrick Lecture I Title to be announced Elias M. Stein 1:00 p.m. - 2:00 p.m. Colloquium Lecture II Title and speaker to be announced 2:00 p.m. - 4:00 p.m. MAA - Minicourse #5 Uses of computers in undergraduate mathematics (nonCONDUIT) 2:15 p.m. - 3:15 p.m. Invited Address MAA - Invited Address Title to be announced Herve Jacquet 2:15 p.m. - 4:15 p.m. Sessions for Contributed Papers 4:30 p.m. - 5:30 p.m. MAA - Business Meeting 7:00 p.m. - 10:00 p.m. Pi Mu Epsilon - Reception 7:00 p.m. BEER PARTY 7:30 p.m. - 9:30 p.m. MAA - Minicourse Ill PASCAL for mathematicians 7:30 p.m. - 9:30 p.m. MAA - Minicourse #4 Uses of computers in undergraduate mathematics instruction (CONDUIT)

WEDNESDAY, August 10 AMS MAA and Other Organizations 8:30a.m. - 9:30 a.m. Invited Address Title to be announced Leo A. Harrington 8:30 a.m. - 4:30 p.m. REGISTRATION 8:30 a.m. - 4:30 p.m. AMS Book Sale I MAA Book Sale 8:30 a.m. - 4:30 p.m. EXHIBITS 8:30 a.m. - 10:30 a.m. MAA - Minicourse 112 Problems from industry for use in the undergraduate classroom 8:30 a.m. - 10:30 a.m. MAA - Minicourse #3 An introduction to the mathematical tech­ niques and applications of computer graphics 8:30 a.m. - 10:30 a.m. MAA - Minicourse #6 Combinatorics (tentative)

345 Persons who qualify for emeritus membership in Assistance, Comments and Complaints either the Society or the Association may register at A log for registering participants' comments or the emeritus member rate. The emeritus status refers complaints about the meeting is kept at the Trans­ to any person who has been a member of the AMS parencies section of the registration desk. All or MAA for twenty years or more, and is retired on participants are encouraged to use this method of account of age from his or her latest position. helping to improve future meetings. Comments on all Nonmembers who register at the meetings and phases of the meeting are welcome. If a written reply pay the $75 nonmember registration fee are entitled is desired, participants should furnish their name and to a discount of the difference between the member address. registration fee of $49 and the nonmember registration Participants with problems of an immediate nature fee of $75 as a $26 credit against dues in either the requiring action at the meeting should see the meeting AMS or MAA, or both, provided they apply for manager, who will try to assist them. membership before September 11, 1983. Audio-Visual Equipment Nonmember students who register at the meetings Rooms where special sessions and contributed and pay the $12 registration fee are entitled to paper sessions will be held will be equipped with an a discount of the difference between the student overhead projector, screen, and blackboard. preregistration fee of $9 and the registration fee of Presenters of ten- or twenty-minute papers are $12 as a $3 credit against dues in either the AMS or strongly urged to use the overhead projector rather MAA or both, provided they apply for membership than the blackboard for their presentation in order before September 11, 1983. to obtain maximum visibility by all members of the who thus Nonmembers and nonmember students audience of the material being presented. qualify may apply for membership at the meetings, A member of the AMS /MAA staff will be available or by mail afterwards up to the deadline. to advise or consult with speakers on their audio­ Registration Dates and Times visual requirements. AMS Short Course Baggage and Coat Cheek Lecture Center Vestibule Provision will be made for participants checking Saturday, August 6 11:00 a.m. to 4:00 p.m. out of the residence halls early to leave baggage in Sunday, August 7 8:00 a.m. to 2:00 p.m. the registration area until leaving the campus. Joint Mathematics Meetings Check Cashing [and MAA Minicourses (until filled)] The meeting cashier will cash personal or travelers' Lecture Center Vestibule checks up to $50, upon presentation of the Sunday, August 7 4:00 p.m. to 8:00 p.m. official meeting registration badge, provided there Monday, August 8 8:00 a.m. to 4:30 p.m. is enough cash on hand. Canadian checks must be Tuesday, August 9, marked for payment in U .8. funds. and 8:30 a.m. to 4:30 p.m. Local Information Wednesday, August 10 This section of the desk will be staffed by members Assistance and Information Desk of the Local Arrangements Committee and other Lecture Center Vestibule volunteers from the Albany mathematical community. Thursday, August 11 8:30 a.m. to 1:30 p.m. Lost and Found Please note that the Joint Mathematics Meetings See the meeting cashier. registration desk will not be open on Thursday, Mail August 11, and that the telephone message center will not be in operation that day. Other services All mail and telegrams for persons attending the provided during the meeting at the registration desk meetings should be addressed to the participant, will also no longer be available (see section below on c/o Joint Mathematics Meetings, Department of Registration Desk Services). There will, however, Mathematics, SUNY, Center at Albany, Albany, be a small desk set up in the Lecture Center Vestibule, New York 12222, Mail and telegrams so addressed where local information will be available and where may be picked up at the mailbox in the registration a staff member will provide limited assistance to area during the hours the registration desk is open. participants. No registration or cash transactions will First class mail not picked up will be forwarded after be possible at this desk. the meeting to the mailing address given on the participant's registration record. Personal Messages Registration Desk Serviees Participants wishing to exchange messages during the meetings should use the mailbox mentioned AMS /MAA Information above. Message pads and pencils are provided. It is Information on the publications and activities of regretted that such messages left in the box cannot both organizations may be obtained at this section of be forwarded to participants after the meeting is the registration desk. over.

346 TIMETABLE

WEDNESDAY, August 10 American Mathematical Society MAA and Other Organizations 8:40 a.m. - 10:50 a.m. MAA- Contributed Paper Session The undergraduate mathematics curriculum 8:40 a.m. - 10:50 a.m. MAA - Contributed Paper Session Special concerns: Remediation, articula- tion, and math anxiety 9:00 a.m. - 11:30 a.m. IIME - Contributed Paper Session 9:10 a.m. - 10:10 a.m. Association for Women in Mathematics Panel Discussion 9:45 a.m. - 10:45 a.m. lnvited Address The Schrodinger equation Ira Herbst 10:10 a.m. - 10:40 a.m. AWM- Membership Meeting 11:00 a.m. - noon MAA - Hedrick Lecture II Title to be announced Elias M. Stein noon - 1:00 p.m. IIME - Council Luncheon 1:00 p.m. - 2:00 p.m. Colloquium Lecture III Title and speaker to be announced 1:00 p.m. - 2:00p.m. l\'lAA - lnvited Address 1:00 p.m. - 4:30p.m. Special Sessions Sessions for Contributed Papers 2:15p.m. - 3:15p.m. lnvited Address Title to be announced Selman Akbulut 3:30 p.m. - 4:30p.m. lnvited Address Functional equations over groups, and the mean value property Johan H. B. Kemperman 4:45p.m. - 6:15p.m. Prize Session and Business Meeting Members 6:00p.m. - 8:30 p.m. l\'lAA - Banquet for 25-Year 6:30 p.m. - 8:15 p.m. IIME - Banquet 7:00p.m. - 9:33 p.m. MAA - Film Program projection 7:00p.m. Points of view: Perspective and to fill space 7:28 p.m. Dragon fold ... and other ways 7:39p.m. Caroms 7:52 p.m. Linear programming 8:04p.m. Turning a sphere ins ide out 8:27 p.m. Hypothesis testing, infer entia 1 statistics Part II a triangle 8:55 p.m. Journey to the center of 9:07 p.m. Symmetry and tesselations 7:00 p.m. - 10:00 p.m. MAA - Section Officers Meeting Lecture 8:30 p.m. - 9:30 p.m. liME - J. Sutherland Frame

Organizations THURSDAY, August 11 AMS MAA and Other

8:00 a.m. - 9:00 a.m. liME - Dutch Treat Breakfast 8:00 a.m. - 10:45 a.m. Special Sessions Sessions for Contributed Papers 8:30 a.m. - 1:30 p.m. ASSISTANCE & INFORMATION DESK 8:40 a.m. - 9:40 a.m. MAA - lnvited Address 8:40 a.m. - 10:10 a.m. MAA- Panel Discussion Paper Session 9:10 a.m. - noon liME- Contributed 9:50 a.m. - 10:50 a.m. MAA- lnvited Address III 11:00 a.m. - noon MAA - Hedrick Lecture Title to be announced Elias M. Stein 1:00 p.m. - 2:00p.m. Colloquium Lecture IV Title and speaker to be announced 1:00 p.m. - 3:10p.m. MAA - Session 347 Telephone Messages Handicapped A telephone message center will be located in There is a ramp located at the rear of Mohawk the registration area to receive incoming calls for Tower, Indian Quad, at lobby level which will participants. The center will be open from August facilitate entry into the building. The elevators 7-10 only, during the same hours as the Joint provide easy access to all floors. Bathrooms, however, Mathematics Meetings registration desk. Messages are not equipped for handicapped persons. will be taken and the name of any individual for Libraries whom a message has been received will be posted until the message has been picked up at the message At present, the campus library plans to be open center. The telephone number of the message center from 8:00a.m. to 6:00p.m. Monday through Friday. will be announced later. Participants may use the library facilities at no charge. Transparencies Local Information Speakers wishing to prepare transparencies in Albany, the state capitol of New York, is steeped advance of their talk will find the necessary in upper Hudson Valley lore, surrounded by fields materials and copying machines at this section of and farmlands. The city was originally settled by the the registration desk. A member of the staff will Dutch in 1609, as the trading post of Fort Orange. assist and advise speakers on the best procedures and methods for preparation of their material. There is a Some of the attractions in the area are Fort Crailo, modest charge for these materials. Please note that a Dutch home and fort built about 1704, and where this service will not be available on Thursday, the song Yankee Doodle was composed; Schuyler August 11. Mansion, built in 1761-1762, and housing a beautiful collection of 18th and early 19th century furnishings; Visual Index Historic Cherry Hill, built in 1787, and now a An alphabetical list of registered participants, museum; and Ten Broeck Mansion, built in 1797- including local addresses, arrival and departure dates, 1798. Also within driving distance is the Saragota is maintained in the registration area. racetrack, and a visit to the Empire State Plaza in downtown Albany is recommended. It is hoped that the free university bus which Miscellaneous Information runs from the campus to downtown will be operating during the meetings. If so, a special 10 will be Athletic Facilities required. Interested participants are advised to check At this time, it is not known whether the gym will at the Local Information section of the registration be open during the meetings. If it is, participants desk. may use the swimming pool and squash courts free of Medical Services charge. Tennis courts are free to participants upon The University Health Service is open for emer­ presentation of the meeting badge; they remain lit gency care and doctor referrals from 8:00 a.m. to until approximately 11:30 p.m. There is an off-road midnight, Monday through Friday. The on-campus running course around the perimeter of the university ambulance can be summoned by calling 457-8633. which is approximately three miles long. Nearby hospitals providing emergency medical care Book Store are: The University Book Store is open Monday through St. Peters Hospital, New Scotland Avenue & Friday from 9:00 a.m. to 4:00 p.m. Manning Boulevard, 454-1318 Camping Albany Medical Center Hospital, New Scotland There are no camping sites within 15 miles of the Avenue {Near South Lake Avenue), 445-3131 university. The closest camping sites are: The emergency number on campus which should Thompson's Lake State Park {20 miles) be used to report medical emergencies is 457-7616. Moreau Lake State Park {45 miles) Parking Child Care There is adequate parking for all participants in The Local Arrangements Committee will have a the lot east of Indian Quad. Temporary parking list of babysitters available at the Local Information at the front entrance will be permitted for 10-15 section of the registration desk during the meeting. minutes for check-in purposes. There are no parking fees or sticker requirements. Crib Rental Social Events The supply of rental cribs available is very limited. See the section on Meal Plans Only which appears Portacribs are available for $10 a week and full­ earlier in this announcement for full details on the size cribs for $15 {not including delivery). Interested participants are advised to contact the following as Lake George cruise, picnic, and beer party. soon as possible: Travel Richard Goldstein, SUNYA, Mathematics Depart­ In August, Albany is on Eastern Standard Time. ment, 1400 Washington Avenue, Albany, New York There is regular airline service to Albany County 12203 {518-457-3943). Airport by several major airline carriers.

348 TIMETABLE

THURSDAY, August 11 American Mathematical Society Other Organizations

1:00 p.m.-5:30p.m. Special Sessions Sessions for Contributed Papers 3:30p.m. - 5:00p.m. ITME - Contributed Paper Session

The airport in Albany is approximately five miles Greyhound and Trailways provide bus service from from campus. The Ramada Inn provides free Boston, New York City, and Buffalo. The station is transportation to and from the airport, bus station five miles from campus. The cost of the cab from the and AMTRAK station. Upon arrival call the hotel bus station to the university is $7 plus 50 cents per and request transportation. person extra. Pine Hills Yellow Cab Company, whose airport USAIR the official carrier for the Albany provides both taxis and number is 869-2258, meetings, has agreed to offer a 30 percent minimum service to the city of Albany and limousines in its discount to any participant purchasing tickets on Albany County Airport. The one-way rate from the its airline at least 14 days in advance through airport to the campus is $9.90 for a single passenger, the Albany Meetings Travel SuperPhone. Other person for two or more persons in a and $6.40 per fares will, of course, still be available after the are being asked to supply airline taxi. Preregistrants 14 days limitation. All participants are urged and times on the preregistration flight arrival dates to consider this organization for their airline and and housing form. If indications are that arrivals hotel reservations. This service (which has an 800 will be heavy on a particular flight, Pine Hills Yellow number) is described in a box contained elsewhere will provide limousines at a $4.90 per Cab Company in this announcement. person rate. It is approximately a fifteen-minute ride by taxi from the airport to Indian Quad at SUNYA. August Weather All participants using public transportation to Normal high 81°F reach the campus should instruct the driver to drop Normal low 59°F them off at the Indian Quad. Record high 99oF Albany is one-half mile from the intersection of Record low 37°F 1-90 (New York Thruway-Mass Pike) and 1-87 (New Average August rain 2-8 inches York Thruway and Adirondack Northway). Follow 1-90 East and take Exit 2 marked "Washington Probability of rain 35% Average humidity 73% Avenue, SUNY". AMTRAK has a station in Rensselaer, 6 miles from campus. There are connections to Boston, New Hugo Rossi York City, Montreal, Buffalo, and Chic11.go. The cost Salt Lake City, Utah Associate Secretary of a cab from the AMTRAK station to the university is $8 plus $1 per person extra.

LP Roundedness of solutions of strictly hyperbolic partial differential Fourier Integral Operators equations. R. Michael Beals CONTENTS 1. Multipliers e-i

349 American Mathematical Society Short Course Series

Population Biology (August 6 and 7, 1983) Synopses and Reading Lists

Evolution, Game Theory and Economics (Ethan to population biology than the elegance of the Akin). Ecologists have recently directed new attention mathematics they generate. to techniques from economics and game theory. The The list below provides the names of several human problems of allocation of scarce resources and books which introduce topics in the demography of conflicts of interest can be regarded as aspects of a population biology. biological fate which we share with other species. 1. J. C. Frauenthal, Introduction to population At the primary level of analysis, members of modeling, Birkhiiuser, Boston, 1980. any species face constraints imposed by limitations 2. N. Keyfitz, Introduction to the mathematics of of time, energy and resources. In the work of population with revisions, Prentice-Hall, Englewood MacArthur and others biological adaptation is the Cliffs, New Jersey, 1977. solution by evolution of problems of cost-benefit 3. R. M. May, Stability and complexity in model analysis, portfolio management and market response ecosystems, Princeton University Press, Princeton, under conditions of pure competition. In the game New Jersey, 1974. theory context these are games against nature: the 4. J. H. Pollard, Mathematical models for the individual faces a choice of behavioral strategies and growth of human populations, Cambridge University the environment responds passively, if at all, to the Press, London, 1973. choice. At the second level of analysis, the environment Optimal Control and Principles in Population is seen to contain other species or other members Management (Wayne M. Getz). Continuous time of the same species who respond with strategies of optimal control theory deals with a class of problems their own. The response of evolution to intraspecific that typically fall within the following formulation. conflict has lately been analyzed by Maynard Smith Maximize over a set of admissable controls (i.e. and others using game theory, and these results are vector functions u(·) taking values in U :J Rm currently being extended to interspecific conflict as and piecewise continuous on [0, T)) the objective well. functional As background reading for this discussion I especially recommend Morton Davis's Game theory: faT L(x, u, t)dt + F(x(T), T), A nontechnical introduction, Basic Books, 1970 and, in paper, Harper, 1974. For biological background, subject to the system constraint equations I recommend the Scientific American issue on dxjdt = f(x, u, t) Evolution (September, 1978) with special attention to the article by Maynard Smith. and boundary conditions Demography in Population Biology (James C. x(O) E Bo and x(T) E 81, Frauenthal). Demography is narrowly defined as the where the system variable x ERn and 00 , 81 :J Rn. study of the vital rates of the human population. Pontryagin's Maximum Principle provides a set of The talk will consider a much broader set of issues. necessary conditions that can be used to construct The unifying idea will be that of a self-replicating extremal (candidate) solutions to the above problem. collection of individuals. Demography will be taken Often these solutions embody a qualitative principle. to mean the kinematic relationship between birth and For example, the solution to driving an n-dimensional death and the future development of a population. linear system to a prescribed manifold in minimum Two intersecting areas will be considered in some time is a control vector whose elements are 'bang­ detail; populations in which age is accounted for as bang' with at most n - 1 switches between their a function of time, and populations which have their maximum and minimum values. vital rates affected by interaction with outside forces A wide variety of problems dealing with the (typically other population groups). Topics will be growth, management and exploitation of populations investigated at levels ranging from examples suitable have been formulated in an optimal control setting. to undergraduate pedagogy through opened research The solutions to some of these have yielded problems. general types of principles. In this lecture I will The mathematical areas encountered will include present a selection of problems that span a range ordinary and partial differential equations, difference both of applications and of specific mathematical and integral equations and stochastic processes. formulations. The applications will include the Topics will be included more for their relevance exploitation of both lumped and age-structured

350 populations, the management of pest populations, 1. T. Nagylaki, The strong-migration limit in the development of resistance to pesticides, questions geographically structured populations, Journal of relating to investment strategies, the implementation Mathematical Biology 9 (1980), 101-114. of alternative technologies, and the allocation of 2. T. Nagylaki, Geographical invariance in resources to growth and reproduction. Principles such population genetics, Journal of Theoretical Biology as the Turnpike Property of Maximum Sustainable 99 (1982), 159-172. Rent solutions, bimodal harvesting policies, pest 3. T. Nagylaki and T. Petes, Intrachromosomal control through augmentation and the role of singular gene conversion and the maintenance of sequence solutions will be discussed. Also, problems will homogeneity among repeated genes, Genetics 100 include free, fixed and infinite time formulations, (1982), 315-337. various types of boundary conditions on x(O) and 4. M. Slatkin, Fixation probabilities and fixation x(T), and systems with so-called state variable jump times in subdivided populations, Evolution 35 discontinuities. (1981), 477-488. The lecture will conclude with a discussion of future directions in the field. Graph Theory and Food Webs (George Sugihara). Preparation. Participants should have some The problem of characterizing the multidimensional familiarity with optimal control theory and ordinary structure of large community niche spaces will be differential equation population models. This can be discussed with a view to uncovering constraints that obtained by reading Chapters 1, 2, and 4 of either of might place the structure of real ecological ensembles the following two texts: within some narrow subset of possible topologies. A new architectural description of the niche will be 1. C. W. Clark, Mathematical bioeconomics: introduced to determine whether natural ensembles The optimal management of renewable resources, are packed densely as a simple solid, or whether Wiley, New York, 1976. they are constructed loosely to contain holes as in 2. B. S. Goh, Management and analysis of a multidimensional swiss cheese. This is viewed as biological populations, Elsevier, Amsterdam, 1980. the converse of the classical species packing problem Some Mathematical Problems in Population and asks if any minimal constraints come into play in Genetics (Thomas Nagylaki). Two areas of theoreti­ assembling communities. These results will then be cal population genetics, neutral models of geographi­ related to the problem of determining lower bounds cal variation and the evolutionary effects of gene on the dimensionality of niche space. Particular focus conversion, will be discussed, with emphasis on will be given to Cohen's (1978) provocative finding unsolved problems. The deterministic problems that nature contains an excess of food webs that may primarily involve functional iteration; the stochastic be represented as interval graphs. This excess will be ones concern Markov chains and diffusion processes. shown to be a formal consequence of a necessary and In the area of geographical variation, limiting results, sufficient assembly rule for constructing communities. invariance principles, and robustness will be treated. 1. R. H. Atkin, Mathematical structure in human Under the topic of gene conversion, most attention affairs, Crane-Russak. ~ew York. 19i4. will be devoted to evolution at a single locus. 2. J. E. Cohen. Food webs and niche space. Princeton l'niversity Press. Princeton. :\ew Jersey. 1978.

Induction Theorems for Groups lian groups which is natural with respect to induction of Homotopy Manifold Structures and restriction for a covering projection of finite index Andrew J. Nicas (and more generally for fibrations with compact mani­ fold fiber). Dress induction and localization are applied The work of Sullivan and Wall, as extended by to obtain induction theorems for STop(M, aM). Kirby and Siebenmann, showed the existence of an exact sequence of pointed sets for the surgery theory 1980 Mathematics Subject Classifications: 57R67, 57R65, of a compact oriented manifold with boundary 18F25, 20C99. (Mm, aM), m ;;;..6: Memoirs of the American Mathematical Society Memoir Number 267, vi + 108 pages (soft cover) STop(M, aM) --+nTop(M, aM)--+ Lm(11"1 (M)). List price $8, institutional member $6, individual member $4 ISBN 0-8218-2267-5; LC 82-11546 In this Memoir it is shown how to make this sequence Publication date: September 1982 and its extension to the left an exact sequence of abe- To order, please specify MEM0/267N

Prepayment is required for all AMS publications. Order from AMS, P. 0. Box 1571, Annex Station, Providence, Rl 02901, or call toll free 800-556-7774 to charge with Visa or MasterCard.

351 Call For Topics For 1985 AMS Conferences

Suggestions are invited from mathematicians, 1985 AMS Summer Institute either singly or in groups, for topics of the various conferences that will be sponsored by the Society in Summer institutes are intended to provide an 1985. The deadline for receipt of these suggestions, understandable presentation of the state of the art and the name and address of the person to whom in an active field of research in pure mathematics, the information should be sent, as well as some and usually extend over a three-week period. Dates relevant information about each of the conferences for a summer institute must not overlap those of are outlined below. The accompanying form (or the Society's summer meeting (not known at this a facsimile thereof) is to be used when submitting printing, but sometime in August) and, in fact, there suggested topic(s) for any of these conferences. should be a period of at least one week between them. Individuals willing to serve as organizers should be Recent topics have been Operator algebras and aware that the professional meeting staff in the applications (1980); Singularities (1981); Recursion Society's Providence office will provide full support theory (1982); and Nonlinear functional analysis and assistance, before, during, and after each of and its applications (1983). Proceedings are pub­ these conferences. Organizers should also note that lished by the Society as volumes in the series a member of the Organizing Committee must be Proceedings of Symposia in Pure Mathematics. willing to serve as editor of the proceedings volume Deadline For Suggestions: August 15, 1983 that will be published by the Society. Submit to: Professor Robert Osserman, Chairman, All suggestions must include (1) the names and AMS Committee on Summer Institutes, Department affiliations of proposed members and chairman of of Mathematics, Stanford University, Stanford, the Organizing Committee; (2) a two- or three­ California 94305. page detailed outline of the subject(s) to be covered, including the importance and timeliness of the topic; 1985 AMS-SIAM Symposium (3) a list of the recent conferences in the same or Some Mathematical Questions in Biology closely related areas; (4) the names and affiliations of the proposed principal speakers; and (5) a list of This one-day symposium is held in conjunction likely candidates who would be invited to participate with the Annual Meeting of the AAAS in May. and their current affiliations. Any suggestions as to Some recent. topics in these annual symposia sites and dates should be made as early as possible have been Theoretical and experimental studies in order to allow adequate time for planning. In the in cellular, developmental and population biology case of a Summer Research Conference, a one-, two-, (1980), Biomechanics and mathematical models in or three-week conference may be proposed. developmental biology (1981), Neurobiology, the study of the nervous systems of oraganisms (1982). 1985 AMS-SIAM Symposium In The 1983 Symposium will be on Muscle physiology. Papers from the symposium are published by the Applied Mathematics Society as volumes in the series Lectures on Mathematics in the Life Sciences. A two-day symposium in applied mathematics will henceforth take place in every odd-numbered year Deadline For Suggestions: August 15, 1983 in conjunction with a spring Eastern Sectional Meet­ Submit to Dr. Robert M. Miura, Chairman, ing. The 1983 symposium, whose topic is Inverse AMS-SIAM Committee on Mathematics in the Life problems, will be held in New York City on April Sciences, Department of Mathematics, University 12-13. The next such symposium will be held during of British Columbia, 121-1984 Mathematics Road, the two days preceding the 1985 spring meeting Vancouver, B.C., Canada. at a site that has not yet been selected. Some topics in recent years have been Computational fluid 1985 AMS-SIAM Summer Seminar dynamics (1977); Mathematical problems in frac­ ture mechanics (1978); and Mathematical psychol­ The goal of the summer seminar is to provide an ogy and psychophysiology (1980). Proceedings are environment and program in applied mathematics published by the Society as volumes in the series in which experts can exchange the latest ideas and SIAM-AMS Proceedings. newcomers can learn about the field. Recent topics Deadline For Suggestions: August 15, 1983 are Algebraic and geometric methods in linear systems theory (1979), Mathematical aspects of Submit to: Professor Alan C. Newell, Chairman, physiology (1980), Fluid dynamical problems in AMS-SIAM Committee on Applied Mathematics, astrophysics and geophysics (1981), Applications Program in Applied Mathematics, University of of group theory in physics and mathematical Arizona, Tucson, Arizona 85721. physics (1982), and Large-scale computations in

352 fluid mechanics (1983). Proceedings are published by the Society as volumes in the series Lectures in Applied Mathematics. Deadline For Suggestions: August 15, 1983 Submit to: Professor Alan C. Newell, Chairman, AMS-SIAM Committee on Applied Mathematics, Program in Applied Mathematics, University of Some Mathematical Questions Arizona, Tucson, Arizona 85721. in Biology- Neurobiology Robert M. Miura, Editor 1985 AMS Summer Research Conference Series The six papers presented in this book deal with three different aspects of neurobiology--the morph­ These conferences are similar in structure to ology of nerve cells, the analysis and study of action potential phenomena, and ion movements inside and and represent diverse those held at Oberwolfach, outside nerve cells. The papers are addressed to areas of mathematical activity, with emphasis on biologists, especially physiologists and neuroscien­ areas currently especially active. Careful attention tists, and mathematicians who arc interested in the is paid to subjects in which there is important applications of mathematics to neurobiology. They interdisciplinary activity at present. Topics for should also appeal to others who have general inter· the second series of one-week conferences, being ests in seeing the interactions between mathematics held in 1983, are Combinatorics and algebra, and experimental neurobiology. Applications of algebraic K-theory to algebraic The collection of papers contains experiments geometry and number theory, Axiomatic set and theory working together and leads to a better theory, Group actions on manifolds, Ordered understanding of neurobiology. Most of the earlier fields and real algebraic geometry, Microlocal books in this series have concentrated mainly on analysis, Fluids and plasmas, geometry and mathematics-a few on biology. Here there is a dynamics, Probability theory, partial differential healthy mix of the two. Four of the papers were equations and applications, Geometrical analysis written by experimentalists who also do mathematics. Readers should gain an appreciation of the synergy of singularities, and Kleinian groups. Proceedings between experiment and mathematics and a view of by the Society as are scheduled to be published the current state of basic research in mathematical volumes in the series Contemporary Mathematics. neurobiology. Deadline For Suggestions: August 15, 1983 These lectures were presented at the Sixteenth Submit to: Professor R. 0. Wells, Jr., Chairman, Annual Symposium on Some Mathematical Ques­ AMS Committee on Summer Research Conferences, tions in Biology at the AAAS meeting in Washing· Department of Mathematics, Rice University, Hous­ ton, D. C. in January 1982. It was jointly organized ton, Texas 77001. and sponsored by AMS, SIAM, and Section A, Mathematics of the AAAS.

1985 AMS Short Course Series CONTENTS Speakers and session chairmen The AMS Short Courses consist of a series of Robert M. Miura, Preface lectures and discussions ordinarily extending over a Charles F. Stevens, Quantitative specification of period of one and one-half days immediately prior to neuron form the Joint Mathematics Meetings held in January and john Rinzel, Neuronal plasticity (learning) August each year. Each of the courses is devoted to Richard E. Plant, The analysis of models for excit­ a specific area of applied mathematics or to areas of able membranes: An introduction C. Nerve pulse interactions mathematics used in the study of a specific subject Alwyn Scott, John A. Connor and Georgia Nikolakopoulou, Cal­ or collection of problems in one of the physical, cium diffusion and buffering in nerve cytoplasm biological, or social sciences. Topics in recent years Charles Nicholson and Joseph M. Phillips, Diffusion have been Networks (August 1981), Tomography in the brain cell microetwironment (January 1982), Stat1:stical Data Analysis (August 1982), Computer Communications (January 1983), 1980 Mathematics Subiect Classifications: 92; 34, 35, 42, and Population Biology (August 1983). Proceedings 58, 76, 94 series Lectures on Mathematics in the Life Sciences are published by the Society as volumes in the Volume 15, x + 122 pages (soft cover) Proceedings of Symposia in Applied Mathematics. List price $19; institutional member $14, individual member $10 Deadline for Suggestions: July 1, 1983 for ISBN 0·8218-1165·7; LC 82·18418 January 1985 course and December 1, 1983 for Publication date: November 1982 specify LLSCI/15N August 1985 course To order, please Prepayment is required for all AMS publications. Submit to: Professor Stefan A. Burr, Chairman, Order from AMS, PO Box 1571, Annex Station, AMS Short Course Subcommittee, Department of Computer Sciences, CUNY, City College, New York, Providence, Rl 02901, or call 800-556-7774 New York 10031. to charge with VISA or MasterCard.

353 American Mathematical Society Conferences Suggested Topic For 1985 Conference (check one)

D AMS-SIAM Symposium in Applied Mathe­ D AMS-SIAM Summer Seminar (In applied math­ matics (Held in conjunction with an Eastern ematics and held in June/July) Sectional Meeting) Submit this form before the August 15, 1983 Submit this form before the August 15, 1983 deadline. deadline. D AMS Summer Research Conferences (Series D AMS Summer Institute (In pure mathematics of six to ten week-long conferences held in and usually held in July/August) June/ July/ August) Submit this form before the August 15, 1983 Submit this form before the August 15, 1983 deadline. deadline. D AMS Symposium on Some Mathematical D AMS Short Course Series (Held in conjunction Questions in Biology (Held in conjunction with with Annual and Summer meetings) AAAS Annual Meeting) Submit this form before the July 1, 1983 and Submit this form before the August 15, 1983 December 1, 1983 deadlines. deadline.

Please print or type your responses. Return completed application to chairman of appropriate committee by deadline listed.

I propose organizing an AMS conference in 1985, as specified above. on the following topic:

I suggest the following members and chairman of the Organizing Committee. These individuals have D / have not D been requested to serve on that committee. (Prospective committee members, chairman, and their current affiliations.)

Date ______Submitted by------­ Address------

N.B.: Please attach a two- or three-page detailed outline of the subject(s) to be covered, including the importance and timeliness of the topic, a list of the recent conferences in the same or closely related areas, the names and affiliations of the proposed principal speakers, and a list of appropriate candidates for invitations to participate and their current affiliations.

354 Invited Speakers V. M. Sehgal and S. P. Singh, Solutions of operator equations and fixed points and Special Sessions Srinivasa Swaminathan, Differential analysis in infinite dimensional spaces Invited Speakers at AMS Meetings Edward C. Turner, Topological methods in com­ binatorial group theory The individuals listed below have accepted invita­ tions to address the Society at the times and places Oetober 1983 Meeting in Fairfield indicated. For some meetings, the list of speakers is Eastern Section incomplete. Deadline for organizers: March 15, 1989 Deadline for consideration: August 2, 1989 Albany, August 1983 November 1983 Meeting in San Luis Obispo Selman Akbulut Ira Herbst James Eells Herve Jacquet Far Western Section Robert C. Gunning J. H. B. Kemperman Deadline for organizers: March 15, 1989 Deadline for consideration: August 4, 1989 Leo A. Harrington Wen-Ch'ing Winnie Li Ian M. Anderson, Geometric methods for partial Fairfield, Oetober 1983 differential equations John Harer Daniel J. Kleitman E. Coutsias and J. Mueller, Singular perturbations John Hubbard Wlodek Proskurowski, Numerical partial differen­ San Luis Obispo, November 1983 tial equations Audrey A. Terras, Automorphic forms Peter B. Gilkey T. S. Lam Ahmed I. Zayed, Partial differential equations, Evanston, November 1983 theory and applications Eiichi Bannai Eugene B. Fabes November 1983 Meeting in Evanston Burgess J. Davis Reinhard E. Schultz Central Section Louisville, January 1984 Deadline for organizers: March 15, 1989 Deadline for consideration: August 8, 1989 Gunnar Carlson M. Susan Montgomery Persi Diaconis Donald L. Burkholder, Probability and related parts of analysis Organizers and Topics Carlos E. Kenig, Harmonic analysis and its of Special Sessions applications to partial differential equations John A. Nohel, Volterra integral and integra­ The list below contains all the information about differential equations Special Sessions at meetings of the Society available Vera S. Pless, Connections between codes and at the time this issue of the Notices went to the designs printer. Stewart B. Priddy, Algebraic topology The section below entitled Information for Organizers describes the timetable for announcing January 1984 Meeting in Louisville the existence of Special Sessions. Associate Secretary: W. Wistar Comfort Deadline for organizers: April15, 1989 August 1983 Meeting in Albany Deadline for consideration: October 12, 1989 Associate Secretary: Hugo Rossi April1984 Meeting in Notre Dame Deadline for organizers: &r:pired Deadline for consideration: April 26 Central Section Deadline for organizers: July 15, 1989 Melvyn S. Berger, The calculus of variations in the Deadline for consideration: To be announced large and its applications Louis Block, Dynamical systems Spring 1984 Meeting Nathaniel A. Friedman, Ergodic theory concerning Far Western Section point transformations with finite invariant Deadline for organizers: July 15, 1989 measure Deadline for consideration: To be announced Colin C. Graham and Bert M. Schreiber, Tensor Spring 1984 Meeting products and p-summing operators in har­ monic analysis Eastern Section Timothy A. Lance, Applications of algebraic Deadline for organizers: July 15, 1989 topology Deadline for consideration: To be announced Wen-Ch'ing Winnie Li, Automorphic functions and Spring 1984 Meeting automorphic representations Southeastern Section R. Michael Range, Several complex variables Deadline for organizers: July 15, 1989 Billy E. Rhoades, Summability methods Deadline for consideration: To be announced

355 Information for Organizers Send Proposals for Special Sessions to the Special Sessions at Annual and Summer meetings Associate Secretaries are held under the general supervision of the The programs of sectional meetings are arranged Program Committee. They are administered by by the Associate Secretary for the section in the Associate Secretary in charge of the meeting with question: staff assistance from the Society office in Providence. Some Special Sessions arise from an invitation to Far Western Section (Pacific and Mountain) a proposed organizer issued through the Associate Hugo Rossi, Associate Secretary Secretary. Others are spontaneously proposed by Department of Mathematics interested organizers or participants. Such proposals University of Utah are welcomed by the Associate Secretaries. Salt Lake City, UT 84112 The number of Special Sessions at a Summer or (Telephone 801-581-8159) Annual Meeting is limited to twelve. Proposals, Central Section invited or offered, which are received at least Paul T. Bateman, Associate Secretary nine months prior to the meeting are screened for Department of Mathematics suitability of the topic and of the proposed list University of Illinois of speakers, and for possible overlap or conflict Urbana, IL 61801 with other proposals (specific deadlines for requesting (Telephone 217 -333-4996) approval for Special Sessions at national meetings are Eastern Section given above). If necessary, the numerical limitation is enforced. W. Wistar Comfort, Associate Secretary Department of Mathematics Proposals for Special Sessions should be submitted Wesleyan University directly to the Associate Secretary in charge of the meeting (at the address given in the accompanying Middletown, CT 06457 (Telephone 203-34 7-9411) box). If such proposals are sent to the Providence office, addressed to the Notices, or directed to anyone Southeastern Section other than the Associate Secretary, they will have Frank T. Birtel, Associate Secretary to be forwarded and may not be received before the Department of Mathematics quota is filled. Tulane University In accordance with an action of the Executive New Orleans, LA 70118 Committee of the Council, no Special Session may (Telephone 504-865-5646) be arranged so late that it may not be announced in As a general rule, members who anticipate the Notices early enough to allow any member of the organizing Special Sessions at AMS meetings are Society, who wishes to do so, to submit an abstract advised to seek approval at least nine months for consideration for presentation in the Special prior to the scheduled date of the meeting. No Session before the deadline for such consideration. Special Sessions can be approved too late to provide Special Sessions are effective at sectional meetings adequate advance notice to members who wish to and can usually be accommodated. They are arranged participate. by the Associate Secretary under the supervision of the Committee to Select Hour Speakers for the Abstracts of papers submitted for consideration for section. The limitation on the number of sessions presentation at a Special Session must be received depends on the space and time available. The same by the Providence office (Editorial Department, restriction as for national meetings applies to the American Mathematical Society, Post Office Box deadline for announcing Special Sessions at sectional 6248, Providence, RI 02940) by the special deadline meetings: no Special Session may be approved too for Special Sessions, which is usually three weeks late for its announcement to appear in time to earlier than the deadline for contributed papers for allow a reasonable interval for members to prepare the same meeting. The Council has decreed that no and submit their abstracts prior to the special early paper, whether invited or contributed, may be listed deadline set for consideration of papers for Special in the program of a meeting of the Society unless an Sessions. abstract of the paper has been received in Providence prior to the deadline. Information for Speakers A great many of the papers presented in Special Sessions at meetings of the Society are invited papers, but any member of the Society who wishes to do so may submit an abstract for consideration for presentation in a Special Session, provided it is received in Providence prior to the special early deadline announced above and in the announcements of the meeting at which the Special Session has been scheduled.·

356 New AMS Publications

AMS Book Orders-Toll Free Number For Users of VISA, MasterCard. Individuals in the continental United States may order books published by the Society by calling 800-556-7774 and using a charge card. The number will be attended from 8:00a.m. to 4:15 p.m. Eastern Time, Monday through Friday except on holidays. When using a charge card for mail orders, please be sure to specify whether VISA or MasterCard and include the account number, expiration date, and signature. Order from AMS, P. 0. Box 1571, Annex Station, Providence, Rl 02901. A reminder: the individual member rate applies to the purchase of one copy of any book for personal usc, and is not an alternative means of lowering costs for libraries.

AMS SHORT COURSE SERIES factors, Singular value decomposition, Some Proceedings of Symposia in Applied Mathematics more bilinear models, Trilinear models, An (ISS N 0160-7634) application, Appendix 1-Rclationship among some models, Appendix 2-PARAFAC pre­ Statistical Data Analysis processing, Bibliography. Ram Gnanadesikan, Editor 5. J. R. Kcttenring, A Case Study in Data Analysis Introduction, The data, Factor analysis and This book is an outcome of the 1982 AMS Short analysis of variance models, Factor Analysis Course given at Toronto. Statistical data analysis has results (Two-factor model, Residuals, A robust been receiving a great deal of attention recently as fit, Parallel factors, Size effects, Recapitulation). evidenced by the fact that subsets of the authors of Analysis of variance results (Estimates of main the present volume have given workshops or short effects and two-way interactions, Decompo~i­ courses on this topic at various meetings in the last tions of the two-way interactions, Assessing the two years, including those of the Mathematical Associ­ significance of the decompositions, Decomposi­ ation of America and ICME-IV. The interest may be tion of the three-way interaction, Recapitula­ due to many things-practical importance of the topic, tion), Summary, perspective, and critique, challenging research problems in a relatively young Acknowledgement, Bibliography. field, need for ideas and material for teaching courses 6. R. Gnanadesikan, Summary and Conclusions on the subject. Bibliography. Clearly neither the short course nor this book can 1980 Mathematics Subject Classification: 62-07 provide enough details on all of the above facets of Proceedings of Symposia in Applied Mathematics interest. However, the different chapters jo address Volume 28, x + 142 pages (hard cover and soft cover) these aspects, although with varying degrees of em­ Hard cover prices: List $22, institutional member $17, phasis. One hope of all the authors in publishing this individual member $11 book is that others will use this material as a starting Soft cover prices: List $16, institutional member $16, point and, with the help of some of the references, be all individuals $8 ISBN 0-8218-0040-X; LC 82-24308 able to develop workshops, short courses and other Publication date: April 1983 educational forums on their own. To order, please specify PSAPMI28N (hard cover) The authors are all employed by Bell Laboratories PSAPMSI28N (soft cover) which provided support for the efforts of all of them. Contents TRANSLATIONS OF 1. R. Gnanadesikan, Introduction 2. P. A. Tukey, Graphical Methods MATHEMATICAL MONOGRAPHS Introduction, Looking at a single collection of (ISSN 0065-9282) numbers, Comparing two or more sets of num­ bers, Exploring distributional models for data, Systems of Quasilinear Equations Looking at relationships among variables, Plots and their Applications to Gas Dynamics for higher-dimensional data, Concluding re­ B. L. Roidestvenskii and N. N. janenko marks, Bibliography. 3. C. L. Mallows, Robust Methods This book is essentially a new edition, revised and Introduction, The regression problem, Estima­ augmented by results of the last decade, of the work tion of location: I, Outliers, Estim;;tion of loca­ of the same title published in 1968 by "Nauka." It tion: II, Location and scale, Robust regression, is devoted to mathematical questions of gas dynamics. Robust smoothing, Other areas, Bibliography. Contents 4. ]. B. Kruskal, Multilinear Methods 1. Foundations of the Theory of Systems of Quasi­ Introduction, A basic bilinear model, The rota­ linear Equations of Hyperbolic Type in Two I ndcpen­ tion problem, Restrictions used to aid compari­ dent Variables son, Restrictions to find the true underlying § 1. Basic definitions, §2. Characteristic directions

357 of a system of quasilinear equations, §3. The Rie­ §3. Investigation of the stability of difference mann invariants, §4. Transformations of systems schemes, §4. Analysis of the simplest difference of quasilinear equations, §5. Conservative systems schemes, §5. Methods of constructing difference of quasilinear equations, §6. Formulation of the schemes for the equations of gas dynamics, Cauchy type problem for a system of quasilinear §6. The method of characteristics, §7. Explicit equations of hyperbolic type, §7. The Cauchy running-count schemes, §8. Homogeneous differ­ problem for linear and semilinear systems, §8. The ence schemes. Schemes with pseudoviscosity, Cauchy problem for a system of quasilinear equa­ §9. Schemes in Euler coordinates and implicit tions, §9. The Cauchy problem for a single equa­ schemes, § 10. Singularities of a difference solu­ tion, § 10. The behavior of the derivatives of a tion solution of a system of quasilinear equations, 4. Generalized Solutions of Systems of Quasilinear § 11. Remarks on the mixed prob Iem, § 12. Ana­ Equations of Hyperbolic Type lytic methods of findirig solutions of systems of §1. Formulation of the Cauchy problem in a differential equations in two independent vari­ class of discontinuous functions, §2. One quasi­ ables, § 13. Group properties of differential equa­ linear equation, §3. A system of quasilinear equa­ tions. tions, §4. Applications of the general theory of 2. Classical and Generalized Solutions of One-dimen­ systems of quasilinear equations of hyperbolic sional Gas Dynamics type § 1. General remarks on the mathematical descrip­ Bibliography tion of the motion of compressible gases, §2. Inte­ 1980 Mathematics Subject Classifications: 35 L60, 35 L45, gral conservation laws. The hydrodynamic equa­ 76N15, 65M10; 35099 tions of one-dimensional flows, §3. Study of the Translations of Mathematical Monographs simplest plane, one-dimensional flows, §4. Discon­ Volume 55, xx + 676 pages tinuities in the one-dimensional flow of compres­ List price $140, institutional member $105, individual member $70 sible gases. Shock waves, §5. Study of the shock ISBN 0-8218-4509-8; LC 82-24488 transition. The width of a shock wave, §6. The Publication date: April 1983 problem of the decay of an arbitrary discontinuity, To order, please specify MMON0/55N §7. The interaction of strong discontinuities, §8. The interaction of shock waves with travel­ RECENT REPRINTS ling waves, §9. Analytic solutions of one-dimen­ A HIERARCHY OF FORMULAS IN SET THEORY sional gas dynamics. A. Levy 3. Difference Methods for Solving the Equations of Memoirs of the AMS, Number 57 Gas Dynamics (ISBN 0-8218-1257-2), 76 pages cover) 1. The Cauchy problem in a Banach space for 1965; reprinted 1983 (soft § List price $12, institutional member $9, systems of linear differential equations, §2. Basic individual member $6 concepts of the theory of difference schemes, To order, please specify M E/\10/57

Prepayment is required for all AMS publications. Order from AMS, P.O. Box 1571, Annex Station, Providence, Rl 02901, or call toll free 800-556-7774 to charge with Visa or MasterCard.

358 Miscellaneous Doctorates Conferred in 1981-1982 (Supplementary List)

The following list supplements the University of California, Wolfson, John Gordon, Minimal rurfaces list of thesis titles published in the Berkeley in complex manifolds. November 1982 issue of the Notices (26;20,0,0,0,6,0,0) Wood, Jay Alan, An algebraization theorem for local hypersurfaces in projec­ (see page 639 for an explanation of the MATHEMATICS numbers in parentheses). tive space. Avrin, Joel David, Hypercontractive semi­ CALIFORNIA groups and magnetic vector potentials. Beer, Walter Josef, On Morita equivalence GEORGIA Stanford University of G* -algebras. (16;0,0,0,2,5,0,9) Bergman, Clifford Harris, Concerning University of Georgia the amalgamation bases of congruence (1;0,0,0,0,0,1,0) ENGINEERING-ECONOMIC SYSTEMS distributive equational classes. Balson, William Edward, The founda- Budde, Paul Edward, Support sets and MATHEMATICS EDUCATION tions of the design of agreement: A Gleason parts of M(H""). Thompson, Patrick, A theoretical quantitative theory of arbitration. Chen, Matthew Shih Chang, Height one framework for understanding young Borison, Adam Bruce, Optimal electric differential ideals in polynomial rings. children's concepts of whole number utility generation e:z;pansion under uncer­ Choi, Hyeong In, Asymptotic Dirichlet numeration. tainty. problems for harmonic functions on Coene, Patrick, Co118'1J.mer behavior under Riemannian manifolds. stochastic rationing-an intertemporal deLaubenfels, Ralph Jay, Accretive KANSAS analysis. operators and scalar operators. Cohan, David, Forest management and Delgado, Alberto Luis, On edge-transitive University of Kansas timber ind'IJ.Stry market equilibrium. automorphism groups of graphs. (1;1,0,0,0,0,0,0) Deziel, Louis Bernard, Jr., Market Dray, Tevian Gordon, The asymptotic equilibrium for transportation equipment structure of a family of Einstein-Mazwell MATHEMATICS leases. solutions. Bixler, James Patrick, Lateral comple­ Dintersmith, Ted Robert, A separating Dzung, Baoswan, IJ' -theory of degene­ tions of lattice ordered groups. choice hyperplane algorithm for evaluating rate-elliptic and parabolic operators of multi-attribute decisions. second order. YORK Kang, Uchoon, Determinants of fast­ Ema.II~Y-Khanasary, Mohamad-Reza, NEW growing, low-productivity service employ­ On the number of vertices of a simple ment in Mexico. polytope generated by a given polytope. Polytechnic Institute of New York Khilnani, Arvind, Mathematical models Hughes, John Forbes, Invariants of (3;1,0,0,0,2,0,0) in fishery policy analysis rationale, for­ regular holotropy and bordism of low mulation, algorithms and solutions. dimensional immersions. MATHEMATICS Kolstad, Charles Durgin, Economic and John, Thomas, Strategies of Borel games. Carro, Anthony, On inverse problems for requ.larity efficiency in air pollution Ng, Ho Kuen, Finitely presented dimen­ the sinqu.lar Sturm-Liouville operator. control. sion of rings and modules. Cazes, Albert N., Inverse Sturm-Liouville Langlois, Richard, Knowledge, order and Pego, Robert Leo, Viscosities and problems and Hill's equation. technology: A study in the philosophy and linearized stability for shock profiles. economics of appropriate technology. Chari, Bhagyalakshmi, Inverse spectral Planchart, Enrique Aurelio, Analogies in problems of Hill's matrices. McEntire, Paul L., Diversification theory symplectic geometry of some results of for portfolio problems with independent Cartan in representation theory. assets. Roytburd, Victor, Hopf bifurcation in a OKLAHOMA Nakano, Russell Toshio, Energy research model of solid fuel combustion. and development: An adaptive strategy Oklahoma State University with learning. Huberman, Daniel, Doubly slice knots and the Casson- Gordon invariants. (l;0,0,0,0,0,0,1) Roughgarden, Jeffrey Douglas, Rules, choice, and cooperation. Sauer, Timothy DuWayne, Nonstable 3 MATHEMATICS Saunders, Harry Duston, Energy­ reftezive sheaves on P . economy interactions: Oil and the world Sethian, James Albert, An analysis of DiVali, Robert Allen, Hopf algebra of economy. flame propagation. class functions and inner plethysms. Scaramucci, Jose, Government requ.lations Sheard, Alec Michael, ill, Indecomposable ultrafilters over inaccessible cardinals. as a strategic variable in the theory of the TEXAS firm. Simon, Horst, The Lanczos algorithm for solving symmetric linear systems. Stokoe, Peter, An economic analysis of Rice University warranty law in markets with risk and Tam, Audrey, Optimal choice of directions imperfect co118'1J.mer information. for the reconstruction of an object from a (4;1,0,2,0,1,0,0) finite number of its plan integrals. MATHEMATICAL SCIENCES Wagner, Bruce Harvey, Automorphisms and derivations of certain operator al­ Liu, Chung-Tsong, Initial value method gebras. for two-point boundary-value problems Marks, Scott Chesser, Optimization of functional programs. Potempa, Thomas C., Finite element methods for convection dominated trans­ port problems. Thabit, Khalid Omar, Cache manage­ ment by the compiler.

359 Special Meetings

TillS SECTION contains announcements of meetings of interest to some segment of the mathematical public, including ad 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. (Iriformation 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 the No~ices 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. All communications on special meetings should be sent to the Editor of the 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 the 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.

1982-1983. Special Year in Lie Group Representations, 18-20. Second Annual Western States Mathematical University of Maryland, College Park, Maryland. (June Physics Meeting, California Institute of Technology, 1982, p. 373) Pasadena, California. (February 1983, p. 206) July 1, 1982-September 1, 1983. Special Year in Math­ 19-21. IEEE Infocom 83: Second Annual Joint Conference ematics Related to Energy, University of Wyoming, of the IEEE Computer and Communications Societies, San Laramie, Wyoming. (October 1982, p. 588) Diego, California. (October 1982, p. 589) September 1, 1982-August 31, 1983. Statistical and 21-22. Fourteenth Annual Modeling and Simulation Con­ Continuum Approaches to Phase Transition, Institute ference, University of Pittsburgh, Pittsburgh, Pennsyl­ for Mathematics and its Applications, University of vania. (October 1982, p. 589) Minnesota, Minneapolis, Minnesota. (January 1982, p. 74) 25-27. Fifteenth ACM Symposium on Theory of Comput­ October 31, 1982-0ctober 1, 1983. Mathematisehes Forseh­ ing, Boston, Massachusetts. (October 1982, p. 589) ungsinstitut Oberwolfaeh (Weekly Conferences), Federal 25-28. First International Conference on Computer Republic of Germany. (November 1982, p. 698) Applications in Production and Engineering, Amsterdam, 1983-1984. Academic Year Devoted to Problems on The Netherlands. Iteration in Classical Real and Complex Analysis, The Information: CAPE 83, Conference Secretary, Organization Mittag-Leffler Institute, Djursholm, Sweden. (February Bureau Amsterdam B.V., Europaplein, NL-1098 GZ 1983, p. 205) Amsterdam, The Netherlands. 1983. Special Year in Commutative Algebra and Al­ 26-May 20. Ecole de Printemps: Les methodes en theorie gebraic Geometry, University of Illinois, Urbana, Illinois. de bifurcation locale, Nice, France. (February 1983, p. 206) (November 1982, p. 699) 28-30. Conference on Number Theory, University of Alberta, Edmonton, Canada. (February 1983, p. 206) APRlL 1983 30-May 1. Conference on Differential Geometry, Purdue 5-9. Thirty-fifth British Mathematical Colloquium, Univer­ University, West Lafayette, Indiana. (January 1983, p. 78) sity of Aberdeen, Scotland, United Kingdom. (October 1982, p. 589) MAY 1983 8-9. Illinois Number Theory Conference, Illinois State University, Normal, Illinois. (January 1983, p. 78) 2-6. International Conference on Fronts, Interfaces, and Patterns, Los Alamos, New Mexico. 11-14. IEEE Southeastcon '83, Orlando, Florida. Topics: Front tracking, contour dynamics, asymptotic (November 1982, p. 700) analysis of combustion fronts, numerical techniques, 11-15. Tutorial Conference on Neural Modeling, Carefree, surveys of physical phenomena involving fronts, inter­ Arizona. (February 1983, p. 206) faces, and patterns. ;1.5-16. Seventh Conference on Undergraduate Mathe­ Invited Speakers: J. D. Buckmaster (University of Illinois), matics, Oklahoma State University, Stillwater, Oklahoma. P. Clavin (France), R. di Perna (Duke University), J. (January 1983, p. 78; February 1983, p. 206) M. Hyman (Los Alamos), M. D. Kruskal (Princeton University), B. Nicolaenko (Los Alamos), S. A. Orszag 16-17. The Thirteenth Midwest Partial Differential (Massachusetts Institute of Technology), B. Scheurer Equations Seminar, University of Wisconsin-Madison, (France), D. H. Sharp (Los Alamos), N. J. Zabusky Madison, Wisconsin. (University of Pittsburgh), P. Horn (IBM), R. Kee Invited Speakers: A. Bahri, F. Browder, C. Conley, G. Da (Sandia, Livermore), C. Lin (Massachusetts Institute Prato, D. DeFigueiredo, R. Phillips, L. Tartar, and J. of Technology), G. Oster (University of California, Vasquez. Berkeley), R. Ross (Stanford University), R. Sekerka Information: Robert Turner, Department of Mathematics, (Carnegie-Mellon University), J. John (Los Alamos), R. University of Wisconsin-Madison, Madison, Wisconsin Stenzel (University of California, Los Angeles), H. Yuen 53706, 608-263-6272. (TRW), D. Forslund (Los Alamos), L. Margolin (Los

360 Alamos), K.-H. Winkler {Max-Planck-Institute), and M. 24-June 12. First Southeast Asian Colloquium on Graph Moldover (NBS, Washington). Theory, National University of Singapore. (November InformatiO'n: Alan R. Bishop, Center for Nonlinear Studies, 1982, p. 700) Los Alamos National Laboratory, Los Alamos, New 30-June 3. International Colloquium in Honor of Mexico 87545, 505-667-1444. Laurent Schwart•, Ecole Polytechnique, Palaiseau, France. 9-11. Conferenee on Simulation in Engineering Scienees, (November 1982, p. 700) Nantes, France. {February 1983, p. 206) Invited Speakers: M. Atiyah (Oxford), J. M. Bismut {Paris XI), L. Hi:irmander (Lund), P. A. Meyer {Strasbourg), 11-13. Optimisation Days 1983, Ecole Polytechnique, Y. Meyer (Polytechnique), S. Mizohata (Kyoto), L. Montreal, Canada. {October 1982, p. 589) Nirenberg (Courant Institute), A. Pelczynski (Varsovie), 12-14. Colloquium on the 200th Anniversary of the Death G. Pisier (Paris VI), A. Wightman {Princeton). of Leonhard Euler, Technical University of Berlin, Federal Program: The following is a list of seminars tha:t Republic of Germany. {November 1982, p. 700) will be given at the colloquium: Equations aux derivees partielles (C. Goulaouic and Y. Meyer, 16--18. Short Course on Design and Analysis of Eftlcient Polytechnique); Geometrie des espaces de Banach Algorithms, The Citadel, Charleston, South Carolina. (B. Maurey, Paris VII and G. Pisier, Paris vr); Principal Lecturer: Sartaj Sahni, University of Minnesota. Probabilites, martingales sur les varietes ( J. M. Program: The short course will be an introduction to the Bismut, Paris XI and P. A. Meyer, Strasbourg); design and analysis of efficient algorithms to include Analyse non lineaire et mathematiques appliquees (F. lecturers on the topics of divide and conquer, greedy Mignot, Polytechnique); Physique matbematique (R. algorithms, backtracking, and dynamic programming. Seneor, Polytechnique); Analyse sur les varit!tt!s (J. P. The primary objective of the course is to enable faculty Bourguignon, Polytechnique). members, having knowledge of the basic fundamentals of programming, to acquire techniques of algorithmic 31-June 10. Berkeley-Ames Conferenee on Nonlinear development which may be applied throughout a Problems in Control and Fluid Dynamics, Bechtel computer science curriculum. The course will be limited Center, The University of California, Berkeley, Berkeley, to 30 participants. California. Information: David Cozart, Department of Mathematics Purpose: The purpose of the conference is to examine the and Computer Science, The Citadel, Charleston, South nonlinear problems arising in the control of modern Carolina 29409, 803-792-7896. aircraft and associated problems in fluid dynamics. Speakers: (Tentative.) Roger Brockett; Gary Chapman; 16-18. Conferenee on Inverse Scattering: Theory and Robert Hermann; Art Krener; Jerrold E. Marsden; Applications, Tulsa, Oklahoma. {February 1983, p. 206) George Meyer; Eli Reshotko; Murray Tobak. 16-18. Fifth Symposium on Mathematical Programming Support: NASA-Ames through the Association for Physical with Data Perturbations, George Washington University, and Systems Mathematics. Washington, D.C. (January 1983, p. 78) Information: L. R. Hunt, Conference Director, Department 16--20. Twenty-seventh Annual Meeting of the Australian of Mathematics, Texas Tech University, Lubbock, Mathematical Society, The University of Queensland, Texas; Clyde Martin, Conference Organizer, Department Brisbane, Australia. {February 1983, p. 206) of Mathematics, Case Western Reserve University, Cleveland, Ohio; or Joyce Martin, Conference Secretary, 17-19. Conference on Large Seale Scientific Computation, 2430 Woodmere, Cleveland, Ohio 44106. Mathematics Research Center, University of Wisconsin, Madison, Wisconsin. {February 1983, p. 206) JUNE 1983 20--21. Fifth Nestor M. Riviere Memorial Lecture, Univer­ sity of Minnesota, Minneapolis, MinneiiDta. (February 6-8. 1983 National Educational Computing Conferenee 1983, p. 206) {NECC 83), Baltimore Convention Center, Baltimore, 20-21. Meeting on Partial Differential Equations, Brighton, Maryland. (January 1983, p. 78; February 1983, p. 207) Great Britain. {February 1983, p. 206) 6-8. SIAM 1983 National Meeting, Denver Marriott Hotel­ 21. Tweli'th Annual State of Jeft'erson Mathematics City Center, Denver, Colorado. (January 1983, p. 78) Congress, Whiskeytown Lake, Redding, California. 6-16. Eighth International Conferenee on Operator Speakers: Jay Thomas (Naval Postgraduate School); Peter Theory, Timisoara, Romania. {February 1983, p. 207) Griffin (Sacramento State University); Kent Wooldridge {California State University, Chico). 7-16. International Conference on Multifunction• and Integrands: Stochastic Analysis, Approximatjon and Information: Buck Ware, Department of Mathematics, California State University, Chico, California 95929, Optimilation, University of Catania, Italy. {February 916-895-6329/6111. 1983, p. 207) 22-29. Third International Conferenee on ·Functional­ 9-11. Utah State University Conferenee on Matrix Theory Dift'erential Systems and Related Topics, Blazejewko, and Applications, Utah State University, Logan, Utah. Poland. {November 1982, p. 700) (February 1983, p. 207) 23-25. Eighteenth New Zealand Mathematics Colloquium, 9-12. Conferenee on the Geometry of Normed Spaces, Massey University, Palmerstown North, New Zealand. University of Illinois Conference Center, Allerton Park, (October 1982, p. 589) lllinois. {February 1983, p. 207) 23-28. International Symposium on Multiple-Valued Logic, 13-16. Tenth International Symposium on Computer Kyoto, Japan. Architecture, Stockholm, Sweden. {October 1982, p. 590) Information: Tadahiro Kitahashi, School of Information 13-17. 1983 Mathematical Sciences Lecture Series, Depli:rt­ and Computer Science, Toyohashi University of Tech­ ment of Mathematical Sciences, Johns Hopkins University, nology, Tempaku-cho, Toyohashi 440, Japan. Baltimore, Maryland. {February 1983, p. 207) 24-25. Joumees de Ia recherche operationnelle, Saint­ 13-17. Microcomputer Graphics, Salisbury State College, Etienne, France. {February 1983, p. 207) Salisbury, Maryland. {February 1983, p. 207) 24-27. Joumees de statistique, Lyon, France. (February 16--18. Short Course on Factoring and Primality Testing, 1983, p. 207) Kent State University, Kent, Ohio. (January 1983, p. 79)

361 20-22. The Second West Coast Conferen.ee on Computing JULY 1983 in Graph Theory, University of Oregon, Eugene, Oregon. 3-20. Thirteenth Ecole d'Ete de Caleul des Probabilites, Program: The conference will feature a mix of instructional Saint-Flour, Cantal, France. (February 1983, p. 208) lectures by nationally renowned researchers, contributed papers by the conference participants, and informal 4-9. Tenth International Conference on General Relativity working sessions for more direct exchange of ideas. The and Gravitation, Fondazione Cini, Isola di San Giorgio, main theme of the invited talks will be algorithmic Venice, Italy. (October, 1982, p. 590) aspects of graph-theoretical models of communication 6-9. Conference on Physical Mathematics and Nonlinear networks. Partial Dift'erential Equations, West Virginia University, Speakers: (Tentative.) Stephen Hedetniemi (Clemson Morgantown, West Virginia. (January 1983, p. 79) University), Richard Ladner (University of Washington), 10-16. International Conference on Quasigroups and their Francis Yao (Xerox PARC). Applications, Universita Cattolica del Sacro Cuore, Milano, Deadline for Abstracts: Abstracts should include title, Italy. (February 1983, p. 208) author name and affiliation, and a 5-15 line summary; 10-16. Journees Arithmetiques 1983, Noordwijkerhout, abstracts should be received by April 29, 1983. The Netherlands. (November 1982, p. 700) Proskurowski, Information and Registration: Andrzej 11-15. 'lEX Users Group Meeting and Introductory 'JEX82 Science, Department of Computer and Information Users Course, Stanford University, Palo Alto, California. University of Oregon, Eugene, Oregon 97403, 503-686- (February 1983, p. 208) 4408. 11-15. Twelfth Conference on Stochastic Processes and 20-24. Linear Algebra & the Microcomputer, Salisbury their Applications, Cornell University, Ithaca, New York. State College, Salisbury, Maryland. (February 1983, p. 207) (February 1983, p. 208) 20-24. 19831nternational Symposium on the Mathematical 11-15. Symposium on Large Seale Systems Theory and Theory of Networks and Systems, Ben Gurion University Applications, Warsaw, Poland. (January 1983, p. 79) of the Negev, Beer Sheva, Israel. (October 1982, p. 590) 11-15. Twelfth Conference on Stochastic Processes and their Applications, Cornell University, Ithaca, New York. 20-25. Variational Methods for Equilibrium Problems of (November 1982, p. 700) Fluids (Plateau's Type Problems, Capillarity Problems, Rotating Fluids), Trento, Italy. 11-15. Ninth British Combinatorial Conference, University of Southampton, England. (October 1982, p. 590) Invited Lecturers: F. J. Almgren (Princeton), L.A. Caffarelli (Minneapolis), A. Friedman (Evanston), C. Gerhardt 11-16. Seventh International Congress of Logie, Methodol­ (Heidelberg), E. H. A. Gonzalez (Leece), S. Hildebrandt ogy and Philosophy of Science, Salzburg, Austria. (October (Bonn), U. Massari (Ferrara). 1982, p. 590; February 1983, p. 208)) Organizers: E. H. A. Gonzalez (Leece), I. Tamanini 11-22. Quadratic Forms and Hermitian K-theory, (Trento). McMaster University, Hamilton, Canada. (January 1983, Information: Secretary of C.l.R.M., University of Trento, p. 79) 38050 Povo (TN), Italy. 11-22. L.M.S.JS.E.R.C. Durham Symposium in Potential Theory, University of Durham and Grey College, Durham, on Machine 21-23. Ninth International Symposium United Kingdom. (October 1982, p. 590) Processing of Remotely Sensed Data, Purdue University, West Lafayette, Indiana. (January 1983, p. 79) 12-15. International Conference on the Teaching of Math­ ematical Modelling, Exeter University, Exeter, England. 21-24. Conference in Banach Algebras and Several (October 1982, p. 590) Riekart), New Complex Variables (in Honor of Charles 18-22. Tenth International Colloquium on Automata Connecticut. (February 1983, p. 207) Haven, Languages and Programming, Barcelona, Spain. 27-29. Symposium on Programming Language Issues in Information: ICALP 83, Fac. Inform., U.P.B., 31 Jordi Software Systems, San Francisco, California. Girona Salgado, Barcelona, Spain. Information: Laurence A. Rowe, Computer Science 18-22. International Conference on Mathematics in Biology Division, EECS, University of California, Berkeley, and Medicine, Bari, Italy. (October 1982, p. 590) California 94720. 18-23. Logie Colloquium '83, Technische Hochschule 27-29. ACM IEEE Twentieth Design Automation Con­ Aachen, Aachen, Federal Republic of Germany. (February ference, Miami Beach, Florida. (October 1982, p. 590) 1983, p. 208) 18-29. Ecole d'Ete d'Informatique, Clamart, France. 27-29. SIAM Symposium on the Applications of Discrete Speakers: R. W. Hockney (The University of Reading); J. Mathematics, Massachusetts Institute of Technology, Lenfant (Universite de Rennes/IRISA); R. H. Perrott Cambridge, Massachusetts. (October 1982, p. 590) (The Queen's University of Belfast). 27-July 13. Ecole d'Ete d'Analyse Numerique, Clamart, Information: Secretariat des Ecoles d'Ete, 1, avenue du France. General-de-Gaulle 92140 Clamart, France. Speakers: D. J. Bergman (Ohio State University); J. L. 18-August 12. Workshop on Latin Squares: Their Lions (I.N.R.I.A.); G. Papanicolaou (Courant Institute). Construction and Application, Simon Fraser University, Information: Secretariat des Ecoles d'Ete, 1, avenue du British Columbia, Canada. (February 1983, p. 208) General-de-Gaulle, 92140 Clamart, France. 19-22. Sixth European Congress on Operational Research (EURO VI), Vienna, Austria. (February 1983, p. 208) 27-July 15. Seminaire de Mathematiques Superieures­ NATO Advanced Study Institute on Topological Methods 25-29. Sixth International Symposium on Multivariate in Nonlinear Analysis, Universite de Montreal, Montreal, Analysis, University of Pittsburgh, Pittsburgh, Pennsyl­ Canada. (January 1983, p. 79) vania. (October 1982, p. 590; February 1983, p. 208) 25-August 2. Seminar on the History of Mathematics, 28-30. Thirteenth International Symposium on Fault­ University of Toronto, Toronto, Canada. (February 1983, Tolerant Computing, Milan, Italy. (October 1982, p. p. 208) 590) 31-August 6. Seventh International Conference on Near­ 30-July 10. Durham Symposium on Modular Forms, Grey Rings and Near-Fields, James Madison University, Har­ College, Durham, Great Britain. (January 1983, p. 79) risonburg, Virginia. (January 1983, p. 80)

362 AUGUST 1983 21-27. International Conference on Foundations of Com­ putation Theory, Linkiiping, Sweden. (February 1983, 1-5. CBMS Regional Conference on Yang-Mills Theory and p. 209) the Topology of Four-Manifolds, University of California, Santa Barbara, Santa Barbara, California. 22-26. Seventh International Conference on Structural Meehanics in Reactor Technology, Marriott Hotel, Principal Speaker: H. Blaine Lawson (SUNY, Stony Brook). Chicago, illinois. (October 1982, p. 591) Information: K. Millett, D. Moore, or M. Scharlemann, Department of Mathematics, University of California, 22-26. Universal Algebra, J6zsef Attila University, Szeged, Santa Barbara, Santa Barbara, California 93106. Hungary. (October 1982, p. 591) 1-10. Seventh International Congress on Mathematical 22-27. Tenth International Congress on Cybernetics, Physics, University of Colorado, Boulder, Colorado. Namur, Belgium. (October 1982, p.591) (February 1983, p. 209) 26-29. Polish Symposium on Interval and Fu•"Y Mathe­ 1-12. NATO Advanced Study Institute on Methods in maties, Poznan, Poland. (February 1983, p. 209) Ring Theory, Antwerp, Belgium. Information: F. van Oystaeyen, Department of Math­ 26-31. Conference on Combinatorial Groups, Busan ematics, University of Antwerp, U.I.A., 1 Univer­ National University, Busan, Korea. (November 1982, siteitsplein, B-2610 Wilrijk, Belgium. p. 701) 1-14. Workshop and Conference in Algebraic Topol­ 29-31. Conference to Honor the Sixtieth Birthday of J. ogy, Memorial University, Saint John's, Newfoundland. B. Keller, Northwestern University, Evanston, lllinois. (November 1982, p. 701) (February 1983, p. 209) 8-11. International Conference on Numerieal Methods in 30-September 8. International Conference on Singularities Laminar and Turbulent Flow, Seattle, Washington. & Dynarnieal Systems, Crete, Greece. Information: C. Taylor, Civil Engineering University, Information: Spyros Pnevmatikos, Department of Mathe­ College of Swansea, Swansea, SA2 8PP, Great Britain. matics, University of Crete, Iraklion Crete, Greece. 8-12. International Joint Conference on Artificial Intel­ ligence, Karlsruhe, Federal Republic of Germany. Information: Saul Amarel, Computer Science Department, SEPTEMBER 1983 Hill Center, Busch Campus, Rutgers University, New Brunswick, New Jersey 08903. 21-23. George D. Aiken Lecture Series: Technieal Sym­ 8-12. NSF-CBMS Regional Conference on Quasitriun­ posium on Acid Rain Transport and Transformation gularity and Analyticity in Operator Algebras, Texas Tech Phenomenon, University of Vermont, Burlington, Vermont. University, Lubbock, Texas. (February 1983, p. 209) Program: William B. Arveson, University of California, 23-24. Eleventh Annual Mathematics and Statistics Berkeley, will give ten expository lectures concerning Conference, Miami University, Oxford, Ohio. the following broad topics: compact perturbations, Program: The theme for this year's conference will be quasitriangularity, similarity, cohomology and automor­ "Operations Research and Mathematics as Applied in phism groups. Other talks are also planned. Business". Support: A grant from the National Science Foundation Speakers: William Lucas (Cornell University), Harvey under the CBMS Regional Conferences Program is Wagner (University of North Carolina), Tom Shriber providing some support for travel and living expenses. (University of Michigan). Infoimation: Gareth Ashton or Richard Saeks, Department Deadline for Abstracts: June 1, 1983. of Mathematics, Texas Tech University, Lubbock, Texas 79409. Information and Abstracts: Stan Payne, Department of Mathematics and Statistics, Miami University, Oxford, **9-13. Colloquium on Topology, Eger, Hungary. Ohio 45056. (February 1983, p. 209) **These dates have been changed from those previously announced.** 23-24. Ohio Delta Chapter of Pi Mu Epsilon Annual Student Conference, Miami University, Oxford, Ohio. 11-14. Eighth International Time Series Meeting, Toronto, Canada. Call for Papers: Undergraduate and graduate students are invited to contribute papers. Information: 0. D. Anderson, ITSM Toronto, 9 Ingheam Grove, Lenton Gardens, Nottingham, NG7 2LQ, Great Information and Abstracts: Milton Cox, Department of Britain. Mathematics and Statistics, Miami University, Oxford, Ohio 45056. 15-17. Symposium on Numerieal Analysis and Com­ putational Complex Analysis, Swiss Federal Institute of 25-0ctober 2. International Conference on Operator Technology (ETH), Ziirich, Switzerland. Algebras, Ideals and their Applications in Theoretieal Program: This symposium is dedicated to Peter Henrici on Physics, Naturwissenschaftlich-Theoretischen Zentrum, the occasion of his 60th birthday. Its intention is to Karl Marx University, Leipzig, German Democratic present recent developments in the fields of numerical Republic. (January 1983, p. 80) mathematics and applied and computational complex analysis. There will be one-hour invited talks and 20-rninute contributed talks. OCTOBER 1983 Invited Speakers: D. Gaier, W. Gautschi, W. B. Gragg, W. Liniger, G. W. Stewart, and V. Strassen. 24-26. 1983 Annual Conference of Association for Deadline for Abstracts: June 15, 1983. Computing Machinery, Sheraton Centre Hotel, New York, Information: Conference Office, Seminar fiir Angewandte New York. (February 1983, p. 210) Mathematik, ETH-Zentrum, CH-8092 Ziirich, Switzer­ land. NOVEMBER 1983 15-17. Fourth International Conference on Mathematieal Modelling, Ziirich, Switzerland. (October 1982, p. 591) 7-9. Twenty-fourth Annual IEEE Symposium on Founda­ 16-24. International Congress of Mathematicians, Warsaw, tions of Computer Science, Tucson Marriott Hotel, Tucson, Poland. (February 1983, pp. 150, 209) Arizona. (February 1983, p. 210)

363 7-11. Society for Industrial and Applied Mathematics1983 Fall Meeting, Norfolk, Virginia. Program: The program for this meeting includes three symposia on topics of current importance in applied mathematics: computational aerodynamics; control stabilization and optimization in distributed parameter systems; and parallel processing. The meeting will be held November 7 through 9; on November 10 through 11 there will be a conference on the potential of parallel processing for scientific computing. Speakers: William F. Ballhaus (NASA Ames Research Lev Leifman, Editor Center); John Lumley (Cornell University); Robert W. MacCormack (University of Washington, Seattle); Jerry Table of Contents S. South, Jr. (NASA Langley Research Center); Michael D. M. Galin, Versa/ deformations of linear Hamil­ Delfour (University of Montreal); J. S. Gibson (University tonian systems of California, Los Angeles); Roger W. Hockney 0. A. Olcinik and E. V. Radkevi~, On the analyticity (University of Reading); David Kuck (University of of solutions of linear second order partial differen­ lllinois-Urbana); Gary Rodrigue (Lawrence Livermore tial equations Laboratory); Jacob T. Schwartz (Courant Institute of V. M. Petkov, Necessary conditions for the Cauchy Mathematical Sciences). problem for nonsymmetrizable hyperbolic sys­ Deadline for Abstracts: May 30, 1983. tems to be well-posed Information: H. B. Hair, Society for Industrial and Applied P. R. Popivanov, On the local solvability of a class Mathematics, 117 South 17th Street, Suite 1405, of pseudodifferential equations with double char­ Philadelphia, Pennsylvania 19103, 215-564-2929. acteristics 10-13. American Mathematical Association of Two-Year A. N. So~itai~vHi, On bifurcation of topological type Colleges Annual Convention, Court of Flags Hotel, Orlando, Florida. (February 1983, p. 210) of singular points of vector fields depending on parameters 24-27. International Congress on Computational and D. A. Silaev, Construction of the solutions of a sys­ Applied Mathematics, University of Leuven, Belgium. tem of boundary-layer equations by the method (February 1983, p. 210) of lines with respect to time A. I. Suslov, On Prandtl's system of equations for DECEMBER 1983 the boundary layer with a surface of discontinuity 5-7. Second Latin American Conference on Applied M. I. Vi~ik and A. I. Komc~, On the solvability of Mathematics, Laboratorio de Computaltao Cientifica the Cauchy problem for the Hopf equation corre­ LCC/CNPq, Rio de Janeiro, Brazil. sponding to a nonlinear hyperbolic equation Sponsors: Laboratorio de Comput~ao Cientffica, Instituto ju. V. Egorov and C. V. Rangelov, On a class of Argentino de Matematica, Universidad de Chile, pseudodifferential equations with multiple charac­ Universidad Nacional Seccional de Medellin, Universidad teristics Nacional Aut6noma de Mexico, Universidad de Puerto ju. 5. ll'ja~cnko, The multiplicity of limit cycles Rico, Universidad Central de Venezuela. arising from perturbations of the form w' = Program: One-hour invited addresses and 20-minute con­ P2jQ1 of a Hamiltonian equation in the real and tributed papers from all areas of applied mathematics. complex domain The invited speakers will be announced later. T. F. Kalugina, A priori estimates for equations of Call for Papers: Contributed papers in applied mathe­ matics, both theory and methods, and in applications parabolic type of mathematics to the solution of problems of prac­ V. ju. Kisclcv, Almost periodic Fourier integral oper­ tical interest will be reviewed. These must not ators and some of their applications have been presented elsewhere and must deal with A. A. Lok~in, Fundamental solutions of quasihyper­ previously unsolved problems or present new solution bolic equations and polynomials of several vari­ techniques. Five copies of each paper (double spaced) ables with abstract (single spaced, 200-word maximum) should 0. A. Olcinik and N. 0. Maksimova, On the behavior be submitted. Accepted papers will be announced by September 30. Papers presented at the conference will of solutions of inhomogeneous elliptic systems in appear in the Proceedings. unbounded domains Deadline for Contributed Papers: June 15,1983. A. V. Fursikov, First integrals and integrability of Information and Papers: Andrew M. Olson, Coordinator, systems of quasilinear equations Puerto Rico-U.S. National Organizing Committee, M. A. Sub in, The density of states of selfadjoint Mathematics Department, University of Puerto Rico, elliptic operators with almost periodic coefficients Box BF, Rio Piedras, Puerto Rico 00931. 1980 Mathematics Subject Classifications: 34, 35, 42, 47, AUGUST 1984 and others Volume 118, iv +340 pages (hard cover) August 1984. Fifth International Congreu on Mathematics List price $70, institutional member $53, Education, University of Adelaide, Australia. (June 1982, individual member $35 pp. 331, 376) ISBN 0·8218-3072·2; LC 82-20595 Publication date: December 1982 To order, please specify TRANS2I118N Prepayment is required for all AMS publications. Order from AMS, PO Box 1571, Annex Station, Providence, Rl 02901, or call 800-556-7774 to charge with VISA or MasterCard.

364 & Communications CBMS REGIONAL CONFERENCE SERIES AMS Reports (Supported by the National Science Foundation) (ISSN 0160-7642} Recent Appointments BROWN-PETERSON HOMOLOGY: AN INTRODUCTION AND SAMPLER Committee members' terms of office on standing by W. Stephen Wilson committees expire on December 31 of the year unless The book is divided into three main parts. The given in parentheses following their names, first is a development of the basic properties of otherwise specified. Brown-Peterson homology assuming knowledge of the homotopy and homology of the Thorn space, MU. Proofs arc complete. Melvin Hochster {1984) and Elias M. Stein {1984) The second part is a highly personal sampler of have been elected to the Executive Committee of Wilson's work with BP in the last decade. This in­ the Council by the Council members. The other cludes stable homotopy, associated homology theo­ members of the Executive Comittee are Paul R. ries, Morava's structure theorem, Hopf rings, Morava Halmos {1983), Andrew M. Gleason (ex officio), K-theories and a study of the U-spectrum for BP. Everett Pitcher (ex officio), Julia B. Robinson {ex In this part a new computation of the homology of officio), chairman, and Paul J. Sally, Jr. {1983). Eilenberg-MacLanc spaces due to Ravenel and the author is presented as is Chan's proof of no torsion Dorothy M. Stone {1984) has been appointed to in the spaces in the U-spectrum for BP. Proofs arc the Committee on Prizes by President Julia B. rarely given in the sampler. Robinson. Continuing members of the committee The third part is a presentation of unpublished are Ivan Niven {1983), chairman, and Daniel Zelinsky work on unstable BP operations. For the student, a {1983). detailed computation is carried out. This book is primarily directed to graduate stu­ Peter D. Lax has been appointed by President dents interested in the field, and, also, to algebraic Julia B. Robinson to the ad hoc Committee to topologists who wish to learn something about BP. Select the Winner of the Bocher Prize of 1984. In addition, the calculation of the homology of Other members of the committee are A. P. Calderon, Eilenberg-Mac Lane spaces may be useful in teaching chairman, and I. M. Singer. advanced algebraic topology courses, and the section on unstable operations is directed at researchers John W. Petro {AMS, 1985) has been appointed by although it is a legitimate continuation of the sam­ President Julia B. Robinson to the joint AMS-MAA­ pler. SIAM Committee on Employment Opportunities. 1980 Mathematics Subject Classifications: 55N22; 55-02, Current members of the committee are Wilfred E. 55P20, 55P42. Barnes {MAA, 1984), James W. Daniel {SIAM, 1983), CBMS Regional Conference Series Calvin T. Long {MAA, 1983), chairman, and Donald Number 48, vi + 86 pages (soft cover) List price $10, individual $5 C. Rung (AMS, 1984), chairman designate. Terms on ISBN 0-8218-1699-3; LC 81-20619 this committee expire October 31. Publication date: May 1982 To order, please specify CBMS/48N Martin D. Davis {1986) has been appointed to be the Society's representative to the Section A of Prepayment is required for all AMS publications. the American Association for the Advancement of from AMS, PO Box 1571, Annex Station, Order Science by President Julia B. Robinson. The term of or call 800--556-7774 Providence, Rl 02901, the appointment is from June 1, 1983 through May to charge with VISA or MasterCard. 27, 1986.

Homology and Dynamical Systems John M. Franks This book is an exposition of a number of results This book would be appropriate for a graduate level dealing with the connections between algebraic topol­ course. Except for an assumed background in alge­ ogy and dynamical systems. For the most part proofs braic topology the material is largely self-contained. are included; where they are omitted a reference is There are numerous books on algebraic topology given. The topics covered include: Morse gradients, and many on dynamical systems. This is the only symbolic dynamics and subshifts of finite type, book devoted to the inter-relationships of these two Smale and Morse-Smale diffeomorphisms and flows, fields. and the zeta function and homology zeta function of a diffeomorphism. 1980 Mathematics Subject Classifications: 58F09; 57R50 The book is intended for graduate students or CBMS Regional Conference Series Number 49, vii + 120 pages (soft cover) researchers interested in the relationship between List price $14, individuals $7 topology and dynamical systems. It is especially ISBN o-8218-1 ;oo-o; LC 82-8897 appropriate for persons with a background in topol- Publication date: September 1982 ogy who want to learn about dynamical systems. To order, please specify CBMS/49N Prepayment is required for all AMS publications. Order from AMS, P.O. Box 1571, Annex Station, Providence, Rl 02901, or call toll free 800-556-7774 to charge with Visa or MasterCard. 365 Officers of the Society, 1982 and 1983 Except for the Members-at-Large of the Council, the month and year of the first term and the end of the present term are given. For Members-at-Large of the Council, the last year of the present term is listed. COUNCIL Presidents: Andrew M. Gleason 1/81-12/82 Associate Secretaries lulia B. Robinson 1 /83-12/84 Raymond G. Ayoub 1/77-12/82 Ex-president: Andrew M. Gleason 1/83-12/83 Paul T. Bateman 1/67-12/83 Frank T. Birtel 1/77-12/84 Vice Presidents W. Wistar Comfort 1/83-12/84 Michael Artin 1/82-12/83 Hugo Rossi 1/82-12/83 Paul R. Halmos 1/81 "12/82 Treasurer Calvin C. Moore 1/83-12/84 Franklin P. Peterson 8/73-12/84 Elias M. Stein 1/82-12/83 Associate Treasurer Secretary: Everett Pitcher 1/67-12/84 Steve Armentrout 7/77-12/84 Members-at-large All terms are for 3 years and expire on December 31 of the given year 1982 1983 1984 1985 Frederick W. Gfhring Donald L. Burkholder Peter A. Fillmore Peter L. Duren Lee Lorch Paul R. Halmos* Melvin Hochster Susan J. Friedlander RichardS. Millman Alan j. Hoffman Robert P. Langlands Robin Hartshorne Marian B. Pour-EI Linda Keen Susan Montgomery Michael Shub Mary Ellen Rudin* 0. Carruth McGehee Hector J. Sussman Olga Taussky-Todd David A. Sanchez Paul j. Sally, Jr. I. M. Singer* *Member-at-large, as provided for in Article 7, Section 4 (last sentence) of the Bylaws of the Society. Publications and Communications Committees Bulletin Editorial Committee Proceedings Editorial Committee Felix E. Browder 1/78-12/83 Thomas H. Brylawski 1/80-12/84 Meyer jerison 1/80-12/85 David M. Goldschmidt 1/82-12/85 Calvin C. Moore 1/82-1 2/84 William E. Kirwan II 1/80-12/83 Colloquium Editorial Committee David J. Lutzer 1/80-12/83 john W. Milnor 1/79-12/84 Reinhard E. Schultz 1/80-12/83 Louis Nirenberg 1/83-12/85 George R. Sell 1/83-12/86 Stephen Smale 1/78-12/83 j. jerry Uhl, Jr. 1/82-12/85 Elias M. Stein 1/77-12/82 Lawrence A. Zalcman 1/79-12/82 Mathematical Reviews Editorial Committee Representatives on American journal of Mathematics Robert G. Bartle 1/83-12/85 Spencer Block 1/83-12/85 Pau I T. Bateman 9/77-1 2/82 Victor W. Guillemin 1/78-12/83 Morton Lowengrub 1/82-12/84 Richard G. Swan 5/77-12/82 Carl M. Pearcy 3/78-12/83 Transactions and Memoirs Editorial Committee Mathematical Surveys Editorial Committee Michael Art in 1/79-12/82 Donald W. Anderson 1/79-12/84 Donald L. Burkholder 1/83-12/86 R. james Milgram 1/77-12/82 William B. johnson 1/82-12/85 JaneCroninScanlon 1/78-12/83 janMycielski 1/80-12/83 R. 0. Wells, Jr. 1/83-12/85 Walter David Neumann 1/82-12/85 Mathematics of Computation Editorial Committee Steven Orey 1/79-12/82 James H. Bramble 1/73-12/83 Paul H. Rabinowitz 1/80- 8/82 Carl de Boor 1/77-12/82 Linda Preiss Rothschild 1/83-12/86 Morris Newman 1 /81-12/83 Lance W. Small 1/83-12/86 Daniel Shanks 1/79-12/84 Joel A. Smaller 9/82-12/83 HughC.Williams 1/83-12/85 R.O.Wells,jr. 1/79-12/82 Chairman, Committee to Monitor Problems in Communication Robert G. Bartle 1/81-1 2/82 W. Wistar Comfort 1/83-12/83 BOARD OF TRUSTEES Steve Armentrout 7/77-12/84 Cathleen S. Morawetz 1/76-12/85 Frederick W. Gehring 1/83-12/87 Franklin P. Peterson {ex officio) 8/73-12/84 Andrew M. Gleason {ex officio) 1/81 "12/82 julia B. Robinson (ex officio) 1/83-12/84 Ronald L. Graham 1/82-12/86 Alex Rosenberg 1/74-12/83 joseph j. Kohn 1/78-12/82 P. Emery Thomas 1/80-12/84

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POSITIONS AVAILABLE The Department of Mathematics at Temple University is conducting a search for a Senior-level applied mathematician. Candidates must have a proven strong research background, LAMAR UNIVERSITY must be capable of assuming a leadership role within the MATHEMATICS DEPARTMENT HEAD Department and must be interested in guiding Ph.D. disserta­ Ph.D. in Mathematics or Mathematical Sciences: Record of tions. Appointments will be made at the Associate or Full professional accomplishments. Send vita and three references Professor level. Salary is competitive, based upon background to: Mary Katherine Bell, Chairman, The Search Committee, and experience. Department of Mathematics, P.O. Box 10047, Beaumont, Applications, including a resume and three letters of Texas 77710. An Equal Opportunity/Affirmative Action recommendation should be sent to Professor Marvin Knopp, Employer. Chairman, Personnel Committee, Department of Mathemat­ ics 038-16, Temple University, Philadelphia, PA 19122. Temple University is an Affirmative Action/Equal Oppor­ tunity Employer. FACULTY POSITION AVAILABLE LAMAR UNIVERSITY DEPARTMENT OF MATHEMATICS Ph.D. in Mathematics or Mathematical Sciences: emphasis KANSAS STATE UNIVERSITY on teaching, research, or computer assisted instruction. Send Applications are invited for several anticipated temporary vita to: Dr. M. A. Laidacker, Department of Mathematics, (9-month) positions at the Assistant Professor level, or above, P.O. Box 10047, Beaumont, Texas 77710. An Equal beginning August 1983. Salary commensurate with qualifica­ Opportunity Employer. tions. Candidates in all fields of mathematics are encouraged to apply. Strong interest in both teaching and research is expected. Ph.D. in mathematics or equivalent required. Start­ COLUMBIA UNIVERSITY ing date: August 15, 1983; Application deadline: May 1, Department of Computer Science 1983. Contact: R. Richard Summerhill, Department Head, Lectureships Department of Mathematics, Kansas State University, Man­ Positions as lecturer or senior lecturer will be offered to hattan, Kansas 66506, (913) 532-6750. KSU is an AA/EOE. excellent teachers with superior academic backgrounds. Faculty of all ranks in other disciplines, as well as computer ALBION COLLEGE science, are encouraged to apply for these non-tenured term Tenure-track position in COMPUTER SCIENCE, starting late appointments, whose duration will typically be 3 to 6 years. August 1983. Ph.D. in the mathematical sciences with a Pay is comparable to professorial levels. master's degree in computer science or the equivalent ex­ These lectureships provide an opportunity for teacher/ perience. Normally three courses are taught each semester, scholars with some substantial prior computing experience two preparations are possible. Good students, comfortable to make a transition to computer science. Columbia Uni­ working conditions, support from the Administration. Rank versity has an outstanding young research faculty and fa­ and salary commensurate with experience. Applications cilities. A five--million dollar office and research-laboratory from women and ethnic minorities are welcomed. Apply to building is now nearly completed. Lecturers will be en­ J. A. Wenzel, Chair, Albion College, Albion, Ml 49224; couraged to participate fully in departmental activities. (517) 629-5511, Ext. 287. Lecturers will teach two undergraduate courses each Albion College is an Equal Opportunity/Affirmative Action semester. They must be abie to make highly effective Employer. presentations to large classes. The University has several DEC 2060 computer systems dedicated to instructional use. l'vlany terminals for students are in dormitories and other Texas A&M University convenient locations. Knowledge of innovative uses of Several openings for 1983-1984 at all levels. Teaching technology for teaching is desirable. and Research. All areas considered. Tenure-track positions Send resume and names of three references to Professor require permanent residence or U.S. citizenship. Send vita, jonathan L. Gross, Vice-Chairman, Department of Compu­ three letters of recommendation, reprints to Professor H. E. ter Science, Columbia University, New York, NY 10027. Lacey, Head, Department of Mathematics, Texas A&M Uni­ Columbia University is an equal opportunity /affirmative versity, College Station, Texas 77843. AN EQUAL OPPOR­ action employer. TUNITY THROUGH AFFIRMATIVE ACTIONS EMPLOYER.

367 POSITION AVAILABLE MATHEMATICS. Assistant or Associate Professor. Tenure· track position. Doctorate in mathematics preferred; ABD will be considered. Seek person who can offer excellent Visiting Assistant Professor instruction for a broad range of courses in mathematics. Southern Methodist University Computer competence a definite plus. Send letter of appli­ a one-year appointment as Applications are invited for cation and resume to: Dwight C. Stewart, Vice President are sought in ap­ a visiting assistant professor. Candidates for Academic Affairs, Union College, Barbourville, KY plied mathematics who can supply evidence of excellence 40906. Deadline for receiving applications: April 8, 1983. in teaching. Union College is affiliated with the United Methodist Send resume and the names of three references to Church and is an equal opportunity/affirmative action G. W. Reddien, Chairman, Department of Mathematics, employer. Southern Methodist University, Dallas, Texas 75275. Affirmative Action/Equal Opportunity Employer UNIVERSITY OF CALIFORNIA, RIVERSIDE The Department of Mathematics of the University of THE UNIVERSITY OF MISSISSIPPI California, Riverside, encourages applications for one tenure­ CHAIRPERSON DEPARTMENT OF MATHEMATICS track position and one temporary position effective Fall The Department of Mathematics at the University of Mis­ 1983. Both positions are at the level of Assistant Professor. sissippi is seeking a chairperson with a distinguished record The Department prefers to fill the tenure-track position of research and scholarship. Administrative experience in a with a person who works in some aspect of Topology and doctoral-level academic department and ability to attract who can function effecti'vely in both our Mathematics and grant support would be desirable. The Chairperson is re­ Computer Science Instructional Program; however, we will sponsible for providing leadership for the Department, pro­ consider applicants from all research areas within Mathe· moting development of the Department and its programs, matics and Computer Science. Significant research accom· developing and administering its budget and ensuring high plishment or exceptional research potential and evidence of standards of teaching and research. The Department of good teaching ability are required. The temporary position seventeen faculty members offers degrees through the Ph.D. is open to applicants from all research areas within Mathe­ The University of Mississippi is situated in Oxford, a pleasant matics and Computer Science, who demonstrate in both re­ university community seventy-five miles from Memphis, search and teaching. When it is mutually agreeable the Tennessee. Salary and rank are negotiable. The starting date position is extendable to a second year. Please send resume, is negotiable but july 1, 1983, is preferred. Applications publications list, and a few selected reprints or preprints should include curriculum vitae and names, addresses, and and have at least 3 letters of recommendation sent to: telephone numbers of three references. To ensure consider· Albert R. Stralka, Chairman ation applications should be sumitted by April 15, 1983. Department of Mathematics Applications and nominations should be submitted to: University of California, Riverside Dr. Charles C. Alexander, Chairperson Search Committee, Riverside, CA 92521 Department of Mathematics, The University of Mississippi, The University of California, Riverside, is an equal oppor· University, MS 38677. EQUAL OPPORTUNITY EMPLOYER tunity/affirmative action empolyer.

The Mathematics/Computer Science Department, Eastern SACRED HEART UNIVERSITY Washington University, is accepting applications for three Applications are invited for positions beginning Fall 1983: positions beginning September 1983. Rank and salary com· 1) COMPUTER SCIENCE. One position at the assistant/ mensurate with experience and credentials. Ph.D. or equiva­ associate professor level. Applicants should have a Ph.D. in lent degree in a relevant discipline and specially in an area of Computer Science and demonstrate potential for excellence applied mathematics (statistics preferred), mathematics edu­ in teaching. cation or computer science required. Teach undergraduate 2) COMPUTER SCIENCE. Two positions at the rank of as­ and graduate-level courses, carry out research and curriculum sistant professor. Applicants should have a Master's degree development. Applicants in mathematics education need in Computer Science. Prior business or teaching experience K-12 experience. Send latter of application, resume and three is desired. A strong commitment to teaching undergraduates letters of recommendation to R. H. Dalla, Chairman, Depart· is necessary. ment of Mathematics and Computer Science, Eastern Wash­ 3) MATHEMATICS. Two positions at the rank of assistant ington University, Cheney, WA 99004. Credentials evaluation professor. Applicants should have a Ph.D. in Mathematics begins May 1, 1983. AA/EOE. and evidence of excellent teaching ability. Interested candidates should send a letter of application, SUSQUEHANNA UNIVERSITY transcripts, and 3 letters of recommendation to: Dr. jean T. The Department of Mathematical Sciences at Susque­ Sells, Chairman, Department of Mathematics, Operations hanna University has an opening commencing September Research and Computer Science, Sacred Heart University, 1983 in a faculty position teaching undergraduate courses P. 0. Box 6460, Bridgeport, CT 06606. AA/EOE. in Computer Science and Mathematics. Doctorate preferred, but others may apply. Strength in Computer Science is es­ COLLEGE OF CHARLESTON sential. Academic rank will be Assistant Professor, unless Department of Mathematics, Charleston, SC 29424 credentials are unusually strong. Salary will be commensu­ rate with qualifications. To apply, a resume, graduate tran· ASSISTANT/ASSOCIATE PROFESSORS. scripts, and three letters of reference should be sent to Two tenure-track positions, available 8-22-83. Teaching in a Professor john M. Reade, Chairman, Department of Mathe· liberal arts, state-supported institution of 5000 students matical Sciences, Susquehanna University, Selinsgrove, PA located in downtown Charleston, S.C. Requirements: 17870. Ph.D. in mathematics or related field, strong commitment Susquehanna has a Hewlett-Packard 3000 computer to undergraduate instruction and serious interest in contin­ and a strong and growing academic program in computer uing research. The College of Charleston is an Affirmative to the 24 students science with 104 majors in addition Action and Equal Opportunity Employer and specifically majoring in mathematics. Ability to contribute to the con­ encourages applications from women and minorities. The tinued development of a strong computer science major will be an important consideration in this search. salary depends on experience and is competitive in the national market. Please send a vita and three letters of Susquehanna University is an Equal Employment Op­ portunity, Affirmative Action, and Title IX institution. recommendation to Professor W. L. Golightly, Chairman.

368 POSITIONS AVAILABLE

CASE WESTERN RESERVE UNIVERSITY AMS SHORT COURSE SERIES Openings anticipated for Visiting Faculty, beginning Proceedings of Symposia in Applied Mathematics August 1983. Teaching experience and evidence of research ability required. Preference will be given to those with re· search interest in the areas represented in the department: GAME THEORY AND ITS APPLICATIONS dynamical systems, probability, differential geometry, control edited by William F. Lucas theory, algebra, statistics, numerical methods for pde. Appli· This volume contains the lecture notes prepared cants on sabbaticals welcome. by the speakers in the short course on Game Theory An equal opportunity employer and its Applications given in Biloxi, Mississippi in 1979. FLORIDA INTERNATIONAL UNIVERSITY Game theory has been a topic of broad interest MIAMI, FLORIDA as a purely theoretical subject which has relation­ ships to many other mathematical areas, and also as The Department of Mathematical Sciences anticipates having a subject widely used in applications over a large several tenure-track positions at the level of Assistant Profes· variety of problem areas. It is concerned with mathe­ sor starting in August 1983. Candidates should possess Ph.D. matical models for situations involving conflict and/ degree in Mathematics. Duties include teaching at the under· or cooperation. These arise in a fundamental way graduate and beginning graduate level, research and service. throughout the behavioral and decision sciences. Evidence of teaching ability and research potential is desir· Game theory has become a basic modeling technique able. Preferred areas of specialization are in combinatorics, in much of modern economic theory, political sci­ mathematical logic/recursion theory, numerical methods, ence, sociology, and operations research, and it has probability or algebra. Qualified candidates in other areas frequently been applied to many other fields. It is a will be considered. Salary is competitive. Applicants should subject highly suitable for joint research of an inter­ send vitae and three letters of reference to: disciplinary nature. Professor Robert C. Fisher This volume is concerned mostly with the n-per­ Department of Mathematical Sciences son theory (n ;;;. 3), although chapter 6 also describes Florida International University several basic two-person models. The first five ~hap Tamiami Campus, Miami, Fl 33199 ters deal for the most part with the multiperson co­ Florida International University is a member of the State o~erative games in the characteristic function (coali­ University System of Florida and is an Affirmative Action/ tional) form. The normal (strategic) form and the Equal Opportunity Employer. extensive (tree) form of a noncooperative game are stressed in chapter 6, although some basic definitions for the normal form do appear in an earlier chapter. FLORIDA INTERNATIONAL UNIVERSITY Selected applications of the theory which are covered ASSISTANT PROFESSOR here in some detail include economic market games, Tenure-track position for an Assistant Professor of Applied measuring power in political systems, equitable allo­ Statistics in the Department of Mathematical Sciences. cation of costs, and auctions. Many of the important recent uses of game theory have involved the n-per­ Duties include undergraduate and graduate teaching, research son cooperative models. and University service. Send r~sume and three letters of These lectures were presented to an audience of recommendation to: mature mathematicians. Nevertheless, this volume Dr. Toby S. Berk, Chairperson could also serve as a textbook for a general course Department of Mathematical Sciences in game theory at the upper division or graduate Florida International University levels. Tamiami Campus, Miami, FL 33199 Willidm F. Lucas, The multiperson cooperative games FlU is a member of the State of Florida University System William F. Lucas, Applications of cooperative games and is an Affirmative Action/Equal Opportunity Employer. to equitable allocation Louis j. Billera, Economic market games L. S. Shapley, Valuation of games L. S. Shapley, Measurement of power in political FOR SALE systems MATH. REVS. 1967 through 1975 complete, unbound. Robert J. Weber, Noncooperative games Write to C-3, c/o March Classified, AMS, P. 0. Box 6248, 1980 Mathematics Subject Classification: 900 Providence, Rl 02940. Volume 24, viii + 128 pages List price $13, institutional member $10, MATH SCI PRESS, 53 Jordan Rd., Brookline, MA 02146, individual member $7 617-738-0307. Announcing new series: SYSTEMS, INFOR­ ISBN 0-8218·0025-6; LC 81-12914 MATION AND CONTROL. First volume: Geometry and Publication date: October 1981 identification workshop, Eds., P. Caines and R. Hermann; $23. To order, please specify PSAPM/24N {soft cover) SALE: 40% reduction on titles in Lie Groups and Interdisci­ plinary Mathematics series. list in Oct. '82 Notices. Prepayment is required for all AMS publications. 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369 FUNDAMENTALS OF THE THEORY OF OPERATOR ALGEBRAS Volume I RICHARD V. KADISON and JOHN R. RING ROSE A. Volume in the PURE AND APPLIED MATHEMATICS Series This valuable work provides the student with a careful treatment of elementary functional analysis and uses this foundation to explain clearly the fundamentals of operator algebras. Topics covered in this-the first of two volumes-include: • Linear spaces • Hilbert space and linear operators • Banach algebras • Elementary C*-algebra theory • Elementary von Neumann algebra theory This text will be suitable for both graduate-level courses in functional analysis and operator algebras, and for self-study by graduate students and research workers who have a basic knowledge of point set topology and measure theory. 1983,416 pp., $29.50 ISBN: 0-12-393301-3

RENORMALIZATION EDWARD B. MANOUKIAN A. Volume in the PURE AND APPLIED MATHEMATICS Series This volume presents the basics of renorma:lization theory in a unified and mathematically rigorous manner. It is primarily designed for graduate students and researchers in quantum field theory and mathematical physics. Although the book's orientation is mathematical, readers with an undergraduate background in physics and mathematics will find the material accessible. Some familiarity with Feynman rules (e.g., the structure of a Feynman integral and a Feynman diagram) is, however, essential. The various chapters cover such topics as: • fundamental concepts of mathematical analysis and, in particular, the theory of multiple integrals and .Fubini's theorem • the theory of Feynman integrals as belonging to a very special class of functions, i.e., the class Bn functions • the study of asymptotic behavior of subtracted-out Feynman amplitudes in the Euclidean region 1983,224 pp., $36.00 ISBN: 0-12-469450-0

STUDIES IN APPLIED MATHEMATICS A VOLUME DEDICATED TO IRVING SEGAL EDITED BY VICTOR GUILLEMIN Volume 8 in the ADVANCES IN MATHEMATICS SUPPLEMENTA.RY STUDIES Series This book contains the proceedings of a conference on functional analysis held at the Massachusetts Institute of Technology in the Fall of 1979 in honor of Irving Segal on the occasion of his sixtieth birthday. The emphasis was on the applications of functional analysis and, hence, the contributions covered a wide spectrum of topics induding: • group representations • quantum field theory • operator theory • linear P.D.E.s • nonlinear P.E.D.s • scattering theory • Lagrangian analysis and quantum mechanics • the polymer problem in four dimensions 1983, 176 pp., $36.00 ISBN: 0-12-305480-x

370 SEMI-RIEMANNIAN GEOMETRY WITH APPLICATIONS TO RELATIVITY BARRETT O'NEILL A. Volume in the PURE A.ND APPLIED MATHEMATICS Series Here is an exposition of semi-Riemannian geometry (also called pseudo-Riemannian geometry) -the study of a smooth manifold furnished with a metric tensor of arbitrary signature. The principal cases are Riemannian geometry, where the metric is positive definite, and Lorentz geometry. Topics covered include: o manifold theory o Riemannian and Lorentz geometry o special relativity o symmetry and constant curvature isometries o Schwarzchild geometry Additional chapters deal with the calculus of variations and the remarkable results on Lorentz geometry (causality theory, singularity theorems) of S. Hawking and R. Penrose. 1983,480 pp., $45.00 ISBN: 0..12-526740-1

UNIFIED INTEGRATION E. J. McSHANE A. Volume in the PURE A.ND APPLIED MATHEMATICS Series The Lebesgue integral has long been acknowledged as an essential tool for the solution of physics and engineering problems. The complexity of the mathematical principles underlying this integral, however, has precluded its utility as a teaching tool in university courses save advanced mathematics. Instead, the more easily understood, but practically outmoded Riemann integral, is used in most university-level hard science courses. This book proposes the use of a different integral, a modification of one devised by J. Kurzweil, whose leve! of sophistication equals the Riemann integral and whose applications are as wide­ ranging and effective as those of the Lebesgue integral. The first chapter presents the integral in its simplest setting (real-valued functions of a single real variable). Other chapters include: o application to differential equations and to probability theory o integration in spaces of more than one dimension o line integrals and areas of surfaces The final chapter provides a more rigorous treatment of the integral that will be useful to students of advanced calculus. June/July 1983, c. 580 pp., price to be announced ISBN: 0-12-486260-8

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371 ...... How does a mathematician DIRECTOR get from Pafnuty Lvovitch Mathematical Sciences Tschebyscheff to a night Research Institute club in Jerusalem? Berkeley

'* '* * The Mathematical Sciences Research Institute seeks a Director to assume duties in Find out in August I 984 upon the retirement of the The Thread current Director, S. S. Chern at that time. The A Mathematical Yarn Director is the chief executive officer of the Institute and reports to the Board of Trustees of the Institute. The candidate should have the by qualifications to provide scientific and PHILIP J. DAVIS administrative leadership of the Institute. coauthor of The Mathematical Experience Tenure, salary and fringe benefits are negotiable. Send resume to Professor Hyman Bass, Chairman of the Board of Trustees, and

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~·. ~··~·. ~·. ~·· ~· •· •> ·•> ENGINEERS, PHYSICISTS, OR APPLIED MATHEMATICIANS LINEAR ALGEBRA AND ITS APPLICATIONS POSITIONS AVAILABLE AT Special Issue in Statistics SILVER SPRING, MARYLAND This is to announce a Special Issue of Linear Alge­ Salary Range: $24,508 to $38,185 (Salary Depends Upon Qualifications) bra and Its Applications devoted to Statistics. The Opportunities for innovative professionals to work in aim of this special issue is to focus attention on the Applied Mathematics Branch. Perform research those areas of statistics where methods from Linear and development on computational fluid dynamics Algebra are relevant (interpreted broadly). Con­ problems such as: • Explosion aerodynamics and blast wave tributions are invited. Papers should be suitable for propagation publication in the journal, and will be refereed in • Turbulent and separated flows the usual way. • lnternai and external missile aircraft aerodynam1cs The deadline for submission of papers is April 1, • Conservation laws. random choice method and 1984 with expected publication in early 1985. upwind schemes Papers should be sent to any member of the • Numerical analysis of singular perturbation problems Advisory Board: The positions offered require a B.S. Degree 1n Ingram Olkin, Department of Statistics, eng1neenng. physics. or applied mathematics and a Stanford University, Sequoia Hall, m1nimum of 3 years experience in the field. Stanford, CA 94305 Send resume to: Naval Surface Weapons Center C. R. Rao, Department of Mathematics and Attn: Mr. James R. Blakeley (Code P40) Statistics, University of Pittsburgh, White Oak Shenley Hall, Pittsburgh, PA 15260 Silver Spring, MD 20910 George P. H. Styan, Department of Mathematics, McGill University, Burnside Hall, 805 quest, rue Sherbrooks, Montreal, For further tnformat1on concerning the pos1t1ons. contact Dr. T. Zien on (202)394-2255 OUEibec, Canada H3A 2K6 AN E<~Ut.~ :·PPI)RT.JtJITY A~~iRMAT'VE ACTii. t, EMPL:)VER US CITIZENSHIP REQUIRED

372 EDITORIAL MANAGER

The M~rhem~tical Association of America has an opening for an Editorial Manager (EM), at its headquarters in Washington, D.C., beginning July I, !983, or as soon as possible thereafter. The EM will supervise an editorial department responsible for copy editing and layout for three journals and several hooks per year. In addition to supervising 2-3 editorial assistants the EM will be expected to participate in the editorial process and work closely with journal editors and authors. Qualifications for this position include mathematical training to the master's degree level or beyond, demonstrated writing skills, and professional experience, or its equivalent, in editing mathematical manuscripts. Salary will be commensurate with the experience and training of the candidate. Applicants should send a curriculum vitae and should arrange to ha\'e three letters of recommendation sent directly to: Dr. Marcia P. Sward, Associate Director MATHEMATICAL ASSOCIA TIC''-: OF AMERICA 1529 Eighteenth Street, N.W. Washington, D.C. 20036 The Machemarical As5ociacion of America i5 an equnl opporwnir• empln>cr.

r-NIVERSITY OF NORTH CAROLINA l U AT CHAPEL HILL DEPARTMENT OF MATHEMATICS Professorships The Whyburn and McConnell-Bernard University Distinguished Professorship in Mathematics The Mathematics Department seeks to fill the UNIVERSITY OF VIRGINIA position of University Distinguished Professor of The Gordon T. Whyburn Professorship of Mathematics. Applications for this senior position Mathematics and Joseph M. McConnell and are invited from outstanding research mathemati­ David M. Bernard Professorship of Mathe­ cians. Nominations for suitable candidates are also welcome. Although the Department has a particu­ matics. Two chaired positions for mathema­ lar interest in the area of Partial Differential Equa­ ticians with outstanding research a:-:d teach­ tions and their applications, distinguished research ing credentials. Applicants in probability I candidates in other fields of mathematics are en­ statistics are especially encouraged. Submit couraged to apply. will be expected to gen­ vita and the names of references to Chairman, A successful applicant erate activity and interact with faculty colleagues Chairman in the University. An established record of excel­ Department of Mathematics lence in research, demonstrated commitment to Mathematics-Astronomy Building teaching, and the ability to provide scientific lead­ ership are required. Cabell Drive Send application including Curriculum Vitae University of Virginia and names of at least three references to Charlottesville, Virginia 22903 Chairman, Department of Mathematics The University of Virginia is an Equal University of North Carolina Chapel Hill, N.C. 27514 Opportunity /Affirmative Action Employer. Applications will be accepted until the position is filled. UNC/CH is an Equal Opportunity/Affirmative Action Employer.

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Do you get JOURNAL OF INTEGRAL EQUATIONS? Editors: A.R. Bharucha-Reid, Georgia Institute of Technology, and M.Z. Nashed, University of Delaware The only journal devoted entirely to integral equations, this authoritative resource is 1 researchers, educators, engineers, and scientists in any valuable to mathematical DYes area where analytical mathematics ·is used. DNo Do you get LINEAR ALGEBRA AND ITS APPLICATIONS? Editor-in-Chief: Richard A. Brualdi and Hans Schneider, University of Wisconsin, Madison This is an internationally renowned outlet for research reports on analytic, algebraic, combinatorial, and numerical aspects of linear algebra and matrix theory. It is the DYes what is happening in linear algebra, and is indispensable for DNo leading source of mathematicians and all those who use linear algebra.

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1983 Subscription Information

Journal of Integral Equations Applied Mathematics and Computation: Modeling of the Environment Volume 5 (4 issues), $93.00 Volumes 12, 13 (2 volumes in 8 issues), $192.00 For air delivery In the U.S. and canada, add $9.00. Subscribers outside the For air delivery in the U.S. and Canada, add $13.00. Subscribers outside the U.S. should add: $7.00 for surfaoe mail, $15.00 for air delivery to Europe and U.S. should add: $11.00 for surfaoe mail, $30.00 for alrdaiivery to Europe and Mexloo, and $20.00 for air delivery to the rest of the world. Mexico, and $38.00 for air delivery to the rest of the -ld. Linear Algebra and Its Applications Studies in Applied Mathematics Volumes 49-55 f7 volumes in 21 issues), $399.00 Volumes 68, 69 (2 volumes in 6 issues), $96.00 For air delivery in the U.S. and Canada, add ~.00. Subscribers outside the For air delivery in the U.S. and Canada, edd $13.00. U.S. should add: $28.00 for surfaoe mail, $56.00 for air delivery to Europe and Subscribers outside the U.S. should add: $8.00 for surfaoe mall, $26.00 for Mexioo, and $70.00 for air delivery to the rest of the -ld. air delivery to Europe and Mexioo, and $32.00for air delivery to the rest of the wo~d. Subscription rates valid through December 31, 1983. Personal subscription orders must be prepaid. All subscriptions begin with the flrat Issue of the calender year. North-Holland

374 ttigvrian Mathematical Society c/o A. Olubummo, Mathematics Department University of lbadan lbadan, Nigeria

Authors Included in Volume 1 0. ABE, University of /badan ETHELBERT N. CHUKWU, Federal University of Technology, Yo/a, Nigeria HENERI A. M. DZINOTYIWEYI, University of Zimbabwe, and PAUL MILNES, University of Western Ontario G. 0. S. EKHAGUERE, University of lbadan J. 0. C. EZEILO, University of Nigeria PETER J. HILTON, Forschungsinstitut fur Mathematik and Battelle Research Centre, Seattle S.C. RASTOGI, University of Nigeria V. SINGH, Hindu College, Moradabad, India STANISLAW J. SURMA, University of Sokoto lfotnne 1• 19Bt BERTRAM YOOD, Pennsylvania State University

The Table of Contents and classifications of papers will appear in Current Mathematical Publications Volume 15, Number 8.

This moderately priced journal will be published in one volume of one issue late in 1983 as it was in 1982. Thereafter it is the expectation of the Nigerian Mathematical Society that it will become a quarterly. Journal of the nigerian Mathematical Society SUBSCRIPTION ORDER FORM VOLUME ONE 1982 0 Library or Institution $22 0 Member of the Nigerian Mathematical Society $10 0 Nonmember individual $11 VOLUME TW01983 ::J Please place an order and send an invoice when the price has been determined Ordered by: ______Mail to (if different):------

The American Mathematical Society has given assistance to the Nigerian Mathematical Society in starting this journal and will receive and forward to the editors subscriptions for the 1982 and 1983 volumes. Make checks payable to the Nigerian Mathematical Society and mail to either the address above or the address below.

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375 v~ 0 from the MAA Studies in Mathematics ... Studies in Computer Science Edited by Seymour Pollack Volume 22, MAA Studies in Mathematics 408pp. Cloth. List: $29.00 MAA Member: $22.00

Studies in Computer Science continues the tradition of excellence of the respected series MAA Studies in Mathematics. Written by computer scientists for mathematicians, the book presents a read­ able and balanced discussion of the role of mathematics in com­ puter science, and the contributions of computing to mathe­ matics. Opening with an historical overview of the development of computer science, the narrative continues with three articles on the nature of computer programs and the programming process, a discussion of computational complexity, and a concluding group of articles on the impact of computer science on artificial intelli­ gence, numerical analysis, statistics, and simulation. In the introduction the editor expresses his hope that the book will provide "an interesting look at an explosive field in the process of becoming." What an understatement! The process is well advanced; the impact of computer science on mathematics and its teaching is already immense. And this book is not just interesting, it is exciting and important as background reading for anyone interested in the interactions between mathematics and computer science. Teachers will find that it enriches their teach­ ing and students will find it rewarding supplementary reading. Order your copy today!

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376 Explore new theories f!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! w-ith Wiley books !!!!!!!!!!!!!!!!!!!!!!!!!!!!•

CRYSTAL SYMMETRY FUNCTION AND THEORY ON The Theory of Colour Crystallography PLANAR DOMAINS Maurice A. Jaswon, the City University, London; A Second Course in Complex Analysis & Maitland A. Rose, Henry Thornton School, Stephen D. Fisher, Northwestern Univ. London .\high len•! treatntt•nt of contple~ anal) sis collt't'll­ l 'niting both the descriptin• amltnathematieal trating on function tht•ory on a finite!) -l'Oilll('l'ted approaeht•s to erystallograph). this hook ineorporatt•s planar domain. Spl'cial etnphasis is gin•n to domai11s original and prt·,·iously unpuhlishl'd work 011 color hotmtkd hy a finitt• lliiiiiiH'r of disjoint analytil' sit11pll' spacl' groups. It distinguisht•s lwtwee11 the lllat'ro­ dosed curn•s. scopie and tnicroscopil' syn11ndry of t•ry,tals, and ( 1-1>7:31-1--1) \lat'l'h 1!:)1):3 apprm. -100 pp. $:3-1.!:),5 emphasizes tlw IIIOtif pattem as a fundatlll'lltal dellll'llt of mierosl'opil' synllndry. It also features a VARIATIONAL PRINCIPLES AND Ill'\\' systematil' notation for point-,group a11d spal'e­ FREE-BOUNDARY PROBLEMS group operators, as \\·ell as display spal't'-groups. Avner Friedman (0-27:35:3--1) l!:lll:3 appro~. 150 pp. $-17.!:)5 A l'tHnprPhensin>, up-to-dat<· treatllll'llt of ntriatio11al SURFACE TOPOLOGY ntdhods and their application to a Illnnlwr of fn•e• P.S. Firby & C. F. Gardiner, both of the Univ. of houndary prohil'ms. Ddailed statements of tlw Exeter standard theory of l'lliptic ami paraholil' operators an• inclndl'tl. This uniqm• hook draws on simple ideas in plant• ( 1-868-IH-:3) 1!:lS2 710 pp. $.52.50 geolltl'lry to e~amint• the topology of t·ontpact surfaces. It also stn•sst•s the n•lationship of surfat·t• PARTIAL DIFFERENTIAL topology to otlwr branches of matlwmatil's. In tht• EQUATIONS OF APPLIED finall'haptt•r, tht• authors link tlwir approach to tlw MATHEMATICS standard approach -algehraie topology. (0-27528-6) Hi82 216 pp. $5-1.!:)5 Erich Zauderer, Polytechnic lnst. of New York A lucid introduetion to techniques for soh·ing litwar QUEUES AND POINT PROCESS and nonlim•ar partial difh•n•ntial t•quations. Till' Peter Franken & Ursula Arndt, Humboldt Uni­ author discusses the construdion of physieal prol'eS> versitat, Berlin; Dieter Konig & Volker Schmidt, nwdels that gin• rise to I' DEs, the das.,ifil'ation of Bergakademie, Freiberg I' DEs. and their exact and approximatl' solution. 01 spt•cial interest is the in-depth cm·pragl' of pt•rturha• This unified treatment of bask prohil'ms in queueing tion and asymptotie mdhods. theory uses the theory of stationary random nwrkt•d (1-87517-1) :\pril ws:3 appro'\. t:iOO pp. $-I!:J.!:J,) proct:sst•s, as \\'l'll as ;, Ill'\\' class of stochastie pro­ l't'sses \\'ith embedded marked point prot·esst•s (1'\11'). METHODS OF REPRESENTATION (1-1007-1-!:l) 1!:)1):3 apprm. 2.50 pp. $2!:l.!:l5 THEORY INTRODUCTION TO THE With Applications to Finite Groups and Orders THEORY OF NUMBERS Charles W. Curtis, Univ. of Oregon, & Irving Harold N. Shapiro, New York Univ. Reiner, Univ. of Illinois at Urbana-Champaign lien· is a systt•matic, modern smTey of reprpsentation This sophisticated in n•stigation of tlw problems and theory, including cmTt•nt research, mdhods, and properties of integ<•rs relatt•s tlllllllJPr tlwory to otlwr applications. It pn•sents important Ill'\\' results that an•as of utathetnaties. Emphasis is on contemporary \\'en• availahlt• onh' in research articles or lecture n•sults, and complt•x functiou tht•ot•y is avoided. Solu­ notes. Each l'haptt;r is den•lo[H'd for e"plicit tions scatten•d throughout the text demonstrate the l'omputations ami applil'ations. application of tlwory ami techniques; special "do-it­ (1-lH!:l!:l-1--1) l!:ll>l S1H pp. $5!:l.!:l5 yourself' dmpters provide added opportunities for shaqwning mathematical thinking and skills. (1-867:37-:3) W8:3 approx. -11>0 pp. $:3!:l.!:l5 To order. 1\Titt• to \1. 1r\\'iii, Dt•pt. :3-!:J!:)tJ,5. Forthcoming JOI I:\.\\ !LEY & SO:\S, lnl'. t:i05 Third An·nm• AN INTRODUCTION TO THE THEORY :\t'\\' York, :\. Y. 10151; OF SMOOTH DYNAMICAL SYSTEMS In Canada: 22 \\'orct•stt•r Hoad, Ht•xdale. Ontario W. Szlenk, Univ. of Warsaw \(!:)\\' ILl (l-!:l0117-2) August HIS:3 In Press

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377 RECENT AMERICAN MATHEMATICAl SOCIETY PUBliCATIONS

CONTEMPORARY MATHEMATICS VI. Ramification of abelian extensions (ISSN 0271-4132) VII. The Weil groups of a local field VIII. Shafarevitch's theorem IX. The Herbrand distribution Advanced Analytic Number Theory 1980 Mathematics Subject Classifications: 1OD15, 12A65, Part 1: Ramification Theoretic Methods 12B10, 12B25, 12B27, 14G10, 10H99 Carlos J. Moreno Contemporary Mathematics Volume 15, viii + 192 pages (soft cover) ABSTRACT List J1rice $18, institutional member $14, $9 way all the ram­ individual member The book presents in a coherent ISBN 0-8218-5015·6; LC 82·22620 ification results from local fields which are necessary Publication date: January 1983 for an understanding of the new developments in To order, please specify CONM/15 C advanced analytic number theory. Chapter Ill gives an elementary introduction to Complex Representations of G L(2, K) /-adic Haar measures, a topic which has gained much for Finite Fields K importance in recent years due to the work of Mazur llya Piatetski-Shapiro and others. Chapters I, II, IV, V and VI are basic and could This book is composed from lecture notes of a very well serve as an introduction to local class field course given at Tel-Aviv University by Professor theory. Piatetski-Shapiro, who holds the Cissie and Aaron Chapter VII gives a detailed discussion of the rela­ Beare Chair in Algebra and Number Theory at that tive Weil group; it includes a proof of Grothendieck's institution. They were prepared for publication by local monodromy theorem which is the essential in­ Moshe Jarden. gredient in the construction of the Weii-Deligne Contents group. The chapter ends with the precise statement 1. Preliminaries: Representation theory; the general of the conjectural non-abelian local reciprocity law. linear group. (1) Linear representations of finite After some preparations in chapter VIII, chapter groups, (2) Induced representations, (3) The Schur IX presents a detailed study of the Herbrand distri­ algebra, (4) The group GL (2, K), (5) The conju­ bution which serves as a suitable substitute for the gacy classes of GL(2, K). Artin and Swan representations when dealing with 2. The representations of GL (2, K). (6) The repre­ conductor questions. The point of view is that of sentations of P, {7) The representations of B, (8) harmonic analysis and /-adic valued distributions. The Inducing characters from B to G, (9) The Schur main goal of the chapter has been to prepare the algebra of I nd~M. (1 0) The dimension of cusp ida I way for the development of the so-called "general­ representations, (11) The description of GL (2, K) ized explicit formulas of number theory" associated by generators and relations, (12) Non-decomposable to the L-functions of automorphic representations characters of L x, (13) Assigning cusp ida I represen­ and algebraic varieties over number fields. tations to non-decomposable characters, (14) The An introductory chapter outlines some of the correspondence between v and P.,, (15) The small most significant discoveries in algebraic number the­ Weil group and the small reciprocity law. ory dealing with the phenomenon of ramification, 3. r-functions and Bessel functions. (16) Whittaker notably the introduction by Hilbert of the higher models, (17) The r-function of a representation, ramification groups and the generalization to infinite (18) Determination of p by r , (19) The Bessel extensions by Herbrand. The same chapter also gives function of a representation, (20) A computation an introduction to the powerful technique of the ofrp(w) for a non-cuspidal p, (21) A computa· "explicit formulas" which leads to some far reaching tion of rp(w) for a cuspidal p, (22) The characters generalizations of the known prime number theorems. of G. Contents 1980 Mathematics Subject Classification: 20G05 0. Introduction Contemporary Mathematics I. Galois theory for infinite extensions Volume 16, viii + 72 pages (soft cover) II. Projective limits List price $13 , institutional member $10, individual member $7 Ill. Elementary theory of /-adic integration ISBN 0-8218-5019·9; LC 82-24484 IV. Ramification theory Publication date: January 1983 V. Multiplicative versus additive reduction To order, please specify CON M/16 C

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378 Publications of Heldermann Verlag Berlin

Research and Education in Mathematics edited by K. H. Hofmann and R. Wille Vall R. T. Rockafellar: The theory of subgradients and its appl. to problems of optimization. Convex and nonconvex functions; 28.00 DM NEW! Vol2 J. Dauns: A concrete approach to division rings; 78.00 DM NEW! Vol3 L. Butz: Connectivity in multi-factor designs ;_ 32.00 DM R&E is a lecture notes series that reports on new developments in mathematical research and education - quickly, informally, at a high level, and at a moderate pricc. Thl• type of material considered for publication includes notes arising from lectures or seminars, research oriented monographs, as well as proceedings containing new and original results. Manuscripts arc directly reproduced in a photomcchanic process.

Sigma Series in Applied Mathematics edited by V. En:B, R. D. Grigorieff and D. J. White NEW! Vall V. N. Lagunov: Introduction to differential games and control theory NEW! Vol2 F. J. Gould/1. W. Tolle: Complementary pivoting on a pseudo­ manifold structure with appl. in the decision sciences; 58.00 DM Books of this series introduce the reader to a field of mathematics with the aim to IC'ad him to q ucstions of actual research.

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379 Summer List of Applicants Instructions for Applicant Fonn on facing page The fonn. Forms submitted by job applicants who attend the August meetings in Albany will be posted. The first impression a prospective employer has of an AMS SHORT COURSE SERIES applicant may be based on the appearance of this form. Proceedings of Symposia in Applied Mathematics The forms should be carefully typed using a fresh black ribbon. The best results are obtained with a OPERATIONS RESEARCH: MATHEMATICS carbon-coated polyethylene film ribbon, but satisfac­ AND MODELS tory results may be obtained using a ribbon made of nylon o;: other woven fabric if suitable care is exer­ edited by Saul I. Gass cised. It is important that the keys be clean and make part of its educational activities, the American As a sharp, clear impression. Use a correcting typewriter Mathematical Society sponsors special topic short or correction tape or fluid if necessary. Submit the courses for the attendees of its national meetings. This original typed version only. Hand lettered forms are volume contains the revised lecture notes for the acceptable if prepared carefully. short course Operations Research: Mathematics and Models given on August 19-20, 1979 at the 83rd The summary strip. Information provided here will summer meeting held in Duluth, Minnesota. These be used to prepare a printed list of applicants for dis­ lectures emphasized specific areas of operations re­ tribution to employers. Please supply all information search and the mathematics used in modeling and requested, and confine your characters to the boxes solving the related problems. The topics and lecturers provided. Use the codes below. Circled letters identify were: corresponding items on the form and the strip. 1. Mathematical modeling of military conflict situa­ Address forms to the Mathematics Meetings Hous­ tions, Seth Bonder, Vector Research, Inc. ing Bureau, P. 0. Box 6887, Providence, RI 02940. 2. Queueing networks, Ralph L. Disney, Virginia The deadline for receipt is July 1, 1983. See pages 334 Polytechnic Institute and State University. and 338 for more information. 3. Practical aspects of fishery management modeling, Frederick C. johnson, National Bureau of Stan­ @ Specialties dards. 4. Mathematical modeling of health care delivery sys­ AL = Algebra AN = Analysis tems, William P. Pierskalla, University of Pennsyl­ BI = Biomathematics BS = Biostatistics vania. CB = Combinatorics CM = Communication 5. Operations research: Applications in agriculture, CN = Control CS = Computer Science Robert B. Rovinsky, U.S. Department of Agricul­ CT = Circuits DE = Differential Equations ture. EC = Economics ED = Mathematical Education 6. Mathematical modeling applied to the relocation FA = Functional Analysis FI = Financial Mathematics of fire companies, Warren E. Walker, The Rand FL = Fluid Mechanics GE = Geometry HM = History of Math LO = Logic Corporation. MB = Mathematical Biology ME = Mechanics Each lecturer attempted to make his presentation MO = Modelling MP = Mathematical Physics self-contained in terms of defining the application MS = Management Science NA = Numerical Analysis areas and mathematics employed. The reader of the NT = Number Theory OR = Operations Research resulting notes will find that the authors, in their PR = Probability SA = Systems Analysis desire to broaden the usefulness of the published ST = Statistics TO = Topology material, have, in some instances, stretched the mean­ ing of self-contained. Thus, the reader might find @ Career Objectives that a bit of perseverance, coupled with dipping into some subsidiary references, is required to obtain the AR = Academic Research AT = Academic Teaching full benefits of the written discussions. However, NR =Nonacademic R&D NC = Nonacad. Consulting even the casual reader will be able to ascertain how NS = Nonacademic Supervision the field of operations research has contributed to the resolution of important decision problems-and @ (!) Duties how the field of applied mathematics has flourished in the guise of operations research. T =Teaching U = Undergraduate 1980 Mathematics Subject Classification: 90-01 G =Graduate R =Research Volume 25, viii + 198 pages C = Consulting A = Administration List price $11, instrtutional member $8, S = Supervision IND = Industry individual member $6 GOV = Government DP = Data Processing ISBN 0-8218-0029·9; LC 81-10849 Publication date: September 1981 Location To order, please specify PSAPM/25N (soft cover) E =East S =South C =Central M =Mountain W =West 0 = Outside U.S. I = Indifferent Prepayment is required for all AMS publications. Order from AMS, PO Box 1571, Annex Station, © U.S. Citizenship Status Providence, Rl 02901, or call 800-556-7774 C = U.S. Citizen P = Permanent Resident to charge with VISA or MasterCard. T = Temporarily in U.S. N = Non-U.S. Citizen

380 Summer List of Applicants / Mathematical Sciences Employment Register August 1983 Albany, New York (Please see instructions on facing page)

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381 New and Recent Titles from-

Volumes in the MATHEMATICS AND ENGINEERING Series

APPROXIMATION OF NONLINEAR OBSERVERS FOR LINEAR EVOLUTION SYSTEMS SYSTEMS JOSEPH W. JEROME JOHN O'REILLY This volume develops various nonlinear evolution mod­ This book provides the first comprehensive and unified els as partial differential equation models with eorre­ theory of observers for finite-dimensional linear sys­ sponding approximations. Topics covered include: tems. Intended for graduate students and research • problem formulations and uniqueness for dissipa­ workers specializing in control, the topics covered tive parabolic models ; include: • convergent regularizations and pointwise stability • elementary system and observer theory of implicit schemes; • minimal-order state observers • nonlinear elliptic equations and inequalities; • linear state function observers • numerical optimality and the approximate solu­ • dynamical observer-based controllers tion of degenerate parabolic equations. • minimum-time state reconstruction of discrete 1983,296 pp., $46.00 systems February estimation for 0-12-384680-3 • observers and linear least-squares ISBN: stochastic systems. March 1983, e. 290 pp., price to be announced ISBN: 0-12-527780-6

FIRST ORDER ELLIPTIC STOCHASTIC MODELS, SYSTEMS ESTIMATION, AND CONTROL A FUNCTION THEORETIC APPROACH PETER S. MA YBECK JAMES L. BUCHANAN ROBERT P. GILBERT and These volumes are designed to foster a thoroul!:h un­ This book investigates systems of first order elliptic derstanding of the fundamental concepts incorporated partial differential equations-an area of much inter­ in stochastic processes, estimation, and control, and est. In seeking similarities to the prototypical elliptic to provide insights into applying the theory to realistic systems, ie., the Cauchy-Riemann equations, the au­ practical problems. thors consider many important questions, including: • Do integral representations analagous to the Cau­ Volume 3/CONTENTS: chy integral formula exist? • Dynamic Programming and Stochastic Control • Are the classical boundary value problems-the • Linear Stochastic Controller Design and Perform­ Hilbert and Riemann-Hilbert problems-appro­ ance Analysis priate? • Nonlinear Stochastic Controllers • Is the Liouville theorem valid? 1982, 304 pp., $39.50 ISBN: 0-12-480703-8 The book considers these central problems nut only for systems in the plane, but also in R'" and C11 • Volume 2/CONTENTS: February/March 1983, 296 pp., $45.00 • Optimal Smoothing ISBN: 0-12-283280.9 • Compensation of Linear Model Inadequacies • Parameter Uncertainties and Adaptive Estimation • Nonlinear Stochastic System Models • Nonlinear Estimation 1982, 304 pp., $39.50 ISBN: 0-12-480702-x

Send payment with order and save postage and handling. STOCHASTIC SYSTEMS Prices are in U.S. dollars and are subject to change GEORGE ADOMIAN without notice. From a pre-publication review: "Equations with stochastic coefficients occur in realis­ tic descriptions of many important processes.... Dr. Adomian has evolved new and very effective methods for treating these equations."-Richard Bellman, Uni­ versity of Southern California This volume presents a computable and useful method for the solution of the nonlinear stochastic operator ACADEMIC PRESS, INC. equations which model dynamical systems, including A Subsidiary of signal processing systems and filter and propagation Harcourt Brace Jovanovich, Publishers media. New York • London • Toronto • Sydney April1983, e. 345 pp., price to be announced San Francisco ISBN: 0-12-044370-8 111 FIFTH AVENUE, NEW YORK, N.Y. 10003 313317

382 PREREGISTRATION AND HOUSING FORM, ALBANY, NEW YORK AMS Short Course Joint Mathematics Meetings MAA Minicourses August 6-7, 1983 August 8-11, 1983 August 8-1 0, 1983 MUST BE RECEIVED IN PROVIDENCE NO LATER THAN JULY 1,1983 Please complete this form and return it with your payment to Mathematics Meetings Housing Bureau P. 0. Box 6887, Providence, Rhode Island 02940, Telephone: (401) 272-9500, Ext. 239 DEADLINES: PREREGISTRATION: July 1, 1983 for preregistration fee(s) RESIDENCE HALL ROOM PAYMENT: July 1, 1983 (partial refunds only after July 15.) CANCELLATIONS: Preregistrations may be cancelled until August 5 by writing or calling the Mathematics Meetings Housing Bureau. (See above.) 50% of the preregistration fee(s) will be refunded if notification is received by this date. Confirmed residence hall reservations may be cancelled until July 15 and full payment will be refunded; however, after that date, only partial refunds will be allowed. CHANGES: Changes in arrival and departure dates must be made with the Mathematics Meetings Housing Bureau by August 5. (See address and number above.) After this date, please call message center number at Albany meeting. NOTE: All dues and meeting registration fees paid to AMS or MAA by professional mathematicians are tax deductible. REGISTRATION FEES Preregistration At Meeting (by mail prior to 7/1) JOINT MATHEMATICS MEETINGS Member of AMS, MAA, llME $38 $49 *Student, Unemployed or Emeritus $ 9 $12 Nonmember $58 $75 AMS Short Course Please affix AMS or MAA label here. Member/nonmember If none, complete 1--3 below. $25 $30 *Student or unemployed $ 5 $10 MAA MINICOURSES #1 through #6 (per course) $20 $20 *All full·time students currently working toward a degree or diploma qualify for the student registration fees, regardless of income. The unemployed status refers to any person currently unemployed, actively seeking employment, and who is not a student. Jt is rot 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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384 Maximum Principle in Finite Element Models for convection-Diffusion Phenomena by TSUTOMU IKEDA, Kyoto University, Japan NORTH-HOLLAND MATHEMATICS STUDIES, 76 Lecture Notes in Numerical and Applied Analysis (4), Editors: H. Fujita and M. Yamaguti The finite element method is known to be very This monograph introduces five finite element flexible for numerical analysis of such phenomena schemes for convection-diffusion equations which in complex domains. The essential features which possess these good properties, as well as good good finite element schemes should possess are: stability limits. The basic idea is the introduction of the maximum principle and the mass conservation either the artificial viscosity or the up-wind type law. technique, with circumcentric mass lumpings. 1983. x + 160 pages. Price: US $38.251Dfl. 90.00. ISBN 0·444·86596·3 Published jointly with, and distributed in Japan by Kinokuniya Company Ltd., Japan.

Recent Progress of Algebraic oeometry in Japan edited by MASAYOSHI NAGATA, Kyoto University, Japan NORTH-HOLLAND MATHEMATICS STUDIES, 73 This volume contains survey reports on develop· (M. Sato). Deformations of Compact Complex ments of algebraic geometry in Japan during the Manifolds and Some Related Topics (M. Namba). 1970's. Birational Classification of Algebraic Varieties (K. Ueno). Algebraic Vector Bundles (M. Maruyama). and Logarithmic CONTENTS: Birational Geometry On Group Actions (M. Miyanishi). Abelian Varieties Class of Non­ Forms (S. litaka). On a Certain and Related Topics in Algebraic Geometry (T. Oda). Algebraic Non-Kahler Compact Complex Manifolds 1983. viii + 214 pages. Price: US $46.751Dfl. 110.00. ISBN 0·444·86469·5

Spectra and the Steenrod Algebra Modules over the Steenrod Algebra and the Stable Homotopy Category by H. R. MARGOLIS, Boston College, MA, U.S.A. NORTH-HOLLAND MATHEMATICAL LIBRARY, Vol. 29 The work is divided into three main parts. Part I is Part II is on the "surface structure" of the module an axiomatic treatment of stable homotopy theory category and the topological implications of this including a coherent presentation of most known structure. Part Ill is a systematic exposition of and many new results of a global nature. Parts II work, much previously unpublished, on the "deep and Ill are devoted to an examination of the global structure" of the category of modules over the mod structure revealed by the connection to the module 2 Steenrod algebra and the corresponding structure category provided by Zp · cohomology. The focus in in the stable homotopy category. 1983. about 500 pages. Price: US $65.001Dfl. 175.00. ISBN 0·444·86516-0

US $prices are valid only in the USA and Canada. In all other countries the Dutch Guilder price is definitive. NHIMATH/BK/0855 All prices are subject to change without prior notice. The Timeliest Information on the methods, applications and theory of dynamical systems!

THE LORENZ EQUATIONS numerous problems are a helpful addition. 1982. approx. 384 pp. 153 illus. cloth. $36.00 Bifurcations, Chaos And Strange (Grundlehren der Mathematischen Wissenschaften, Attractors Vol. 250) Colin Sparrow ISBN 0-387-90681-9 The research on Lorenz equations has progressed rapidly since the equations' formulation in 1963, as REGULAR AND STOCHASTIC applications to physics and engineering, as well as MOTION to meteorology have been developed. This is the A.J. Lichtenberg and M.A. Lieberman first book to provide a comprehensive survey of the behavior of solutions to Lorenz equations. It offers Using examples drawn from physics, astronomy, clear insight into the complicated theory of the mechanical engineering and fluid dynamics, this equations. No knowledge of Lorenz equations or practical guide brings together the vast body of the advanced theory of differential equations is research literature exploring the calculation of necessary. stochastic motion in nonlinear oscillator systems. 1982. 269 pp. 91 illus. Paper $19.80 Current techniques for determining the motion in (Applied Mathematical Sciences, Vol. 41) Hamiltonian systems. as well as in dissipative ISBN 0-387-90775-0 systems and the effect of external noise, are fully described. Regular and Stochastic Motion then compares these calculations to computer simulation NONLINEAR OSCILLATIONS, results. DYNAMICAL SYSTEMS, AND 1982. 499 pp. 140 illus. cloth. S36.00 (Applied Mathematical Sciences, Vol. 38) BIFURCATIONS OF VECTOR ISBN 0-387-90707-6 FIELDS J. Guckenheimer and P. Holmes GEOMETRIC THEORY OF Serving as a user's guide to the latest results in DYNAMICAL SYSTEMS dynamical systems theory, this mathematically rigorous volume introduces geometrical methods in An Introduction the study of differential equations and mappings Jacob Palis, Jr. and Welington de Melo which arise as models of physical problems. Topics Translated by A. K. Manning range from invariant manifold and normal form Recently translated from the Portugese, methods of dynamic systems and symbolic this introduction dynamics, local and global bifurcation theory to to the concepts of genericity and bifurcations of one and multi-dimensional maps, structural stability fills the gap between a standard chaos and strange attractors. course in ordinary differential equations and advanced research articles 1983. approx. 400 pp. 196 illus. cloth. S32.00 on dynamical systems. The main results presented (Applied Mathematical Sciences, Vol. 42) are the Hartman­ Grobman and the Stable ISBN 0-387-90819-6 Manifold Theorems, the Kupka-Smale Theorem and Peixoto's Theorem. The method of coverage is systematic, with GEOMETRICAL METHODS IN numerous examples, roughly 80 challenging THE THEORY OF ORDINARY exercises and 114 illustrations! Also provided is an account of recent results and open questions. DIFFERENTIAL EQUATIONS 1982. 198 pp. 114 illus. cloth. $28.00 V.I. Arnold ISBN 0-387-90668-1 Translated from the Russian by Joseph Szucs Differential equations in many instances, represent To order write: the link between mathematical theory and Springer-Verlag New York applications to natural sciences. Geometrical 175 Fifth Avenue Methods explores differential equations with an New York, N.Y. 10010 emphasis on structural stability, concentrating on the many applications to various asymptotic methods. Coverage includes the theory and z Springer- <( applications of perturbation theory, averaging, Verlag u Poincare normal forms, bifurcation theory; and New York • Heidelberg • Berlin • Vienna • Tokyo ~ 0::: u LU ·~ :::.;:: z <(