Notices of the American Mathematical Society

june 1986, Issue 248 Volume 33, Number 3, Pages 441-592 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 the press. The summer and annual meetings are joint meetings of the Mathematical Association of America and the American 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 supplementary 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 the meeting. Abstracts should be submitted on special forms which are available in many departments of mathematics and from the headquarters office of the Society. Abstracts of papers to be presented at the meeting must be received at the headquarters of the Society in Providence. Rhode Island. on or before the deadline given below for the meeting. Note that the deadline for abstracts for consideration for presentation at special sessions is usually three weeks earlier than that specified below. For additional information consult the meeting announcements and the list of organizers of special sessions. ABSTRACT MEETING# DATE PLACE DEADLINE ISSUE August 3-11. 1986 Berkeley, California EXPIRED (International Congress of Mathematicians) 828 October 10-11, 1986 Logan, Utah tAugust 18 October 829 October 17-18, 1986 Charlotte, North Carolina tAugust 20 October 830 October 31-November 1, Denton, Texas tAugust 25 October 1986 831 January 21-24, 1987 San Antonio, Texas October 15 January (93rd Annual Meeting) April 3-4, 1987 Kent, Ohio April 25-26, 1987 Newark, New Jersey January 6-9, 1988 Atlanta, Georgia (94th Annual Meeting) August 8-12. 1988 Providence, Rhode Island (AMS Centennial Celebration) January 11-14, 1989 Phoenix, Arizona (95th Annual Meeting) t Please note change from March 1986 calendar.

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David Harris, in his feature article on "The Mordell Conjecture," explains the significance of Faltings' proof of the conjecture in relation to other important conjectures in the theory of abelian varieties. Page 443.

The proposed NSF budget for FY 1987 is examined for the areas related to the mathematical sciences and is compared with the actual funding levels over the past ten years. Page 450.

The Board on Mathematical Sciences, in a recent report entitled "Math­ ematical Sciences: A Unifying and Dynamic Resource," assesses and monitors some trends in pure and applied mathematics. Page 462.

The Treasurer of the AMS reports on the financial status of the Society for 1985. Page 480.

Joseph Buhler's column contains an article by Avner Ash which reviews the software package "Matlab," a high-level language used to manipulate two­ dimensional matrices. Page 482.

Kenneth Hoffman, in his Washington Outlook column, recaps the accomplish­ ments of the mathematical community since the issuance of the David Report in 1981. Page 486.

The ICM-86 current schedule of plenary and 45-minute lectures is posted for the August meeting in Berkeley. Page 488.

Peter Lax, Herbert Simon, and Antoni Zygmund are awarded the National Medal of Science for 1986. Page 497.

National Mathematics Awareness Week put mathematics into public view and demonstrated the influence of the mathematical community. Page 509.

The NSF report on graduate science/engineering enrollments alerts the scientific community to the pending shortage of Ph.D.'s. Page 511. The Mordell Conjecture David Harris

1. The basic results. On 19 July 1983, the New solutions of an equation transforms the algebraic York Times printed an article [NYT] on the problem of solving a Diophantine equation into solution by a West German mathematician named the geometric problem of finding points on its Faltings of "one of the outstanding mathematical locus of solutions. The rest of this article deals problems of the century" (in the opinion of Serge with the problem of integer and rational points Lang). Spencer Bloch of the University of Chicago on curves. §3 describes the pioneering work of said that the German's achievement "answers Mordell, Weil, and Siegel in launching the subject. questions that seemed absolutely unanswerable In §4, we describe the early attempts to deal with before." The most eye-catching part of Faltings' Mordell's Conjecture, climaxing in the work of paper [Fal] is a proof of the so-called Mordell the Soviet school. In §5, the work of Faltings is Conjecture, a fundamental question in the theory sketched. Finally, in §6, we discuss the problem of of Diophantine equations. effectively using Faltings' work to solve real-world Until recently, most experts had believed Diophantine equations. that the Mordell Conjecture would not soon be SIEGEL'S THEOREM. If Cis an affine curve settled. The Faltings paper is a difficult exercise defined over a ring R that is finitely generated in algebraic geometry, employing the theory of over Z and if its genus is 2: 1, then C has only moduli, the theory of semiabelian varieties, a finitely many points in R. theory of heights for abelian varieties, and other More precisely, for an irreducible equation such abstruse subjects. With this in mind, we f(x, y) = 0 to have infinitely many solutions are supplying these remarks with the hope they (x,y) where for a fixed natural number c, ex will serve to explain the significance of Faltings' and cy are integers in some number field, it is work on the Mordell Conjecture, its relation to necessary and sufficient that the equation can the solution of several important conjectures in be identically satisfied by two rational functions the theory of abelian varieties, and the historical x = P(t), y = Q(t) of the form context. 1 We begin by listing for reference five basic P(t) = antn + an-1tn- + · · · a_nt-n, results, all now proven. We shall refer to these Q(t) = bntn + bn-1tn-1 + · · · b_nCn results by these names throughout this article. which are not both constant, which is true if and In the next section, we shall give a general in­ only if the corresponding projective curve is of troduction to the problem of solving Diophantine genus 0 and has at most two poles for the function equations. The basic results deal with Diophan­ tine problems such as how many points with lxl + IYI· integer or rational coordinates lie on a curve. As (Siegel in fact extends this to curves given by discussed below, the usual trick of graphing the n- 1 equations inn unknowns; cf. [Si2]).

The article above is the thirteenth in the series of Special Articles published in Notices. The author, David Harris, received his Ph. D. from Harvard University in 1972, under the direction of David Mumford. In the mid-seventies, he survived a five-year-long fight with cancer, thanks to the chemotherapists of the Memorial Sloan Kettering Cancer Center. Since 1979, he has worked for the Department of Defense at Fort George G. Meade, Maryland. The series of Special Articles was created to provide a place for articles on mathematical subjects of interest to the general membership of the Society. The Editorial Committee of Notices is especially interested in the quality of exposition and intends to maintain the highest standards in order to assure that the Special Articles will be accessible to mathematicians in all fields. The articles must be interesting and mathematically sound. They are first refereed for accuracy and (if approved) accepted or rejected on the basis of the breadth of their appeal to the general mathematical public. Items for this series are solicited and, if accepted, will be paid for at the rate of $250 per page up to a maximum of $750. Manuscripts to be considered for this series should be sent to Ronald L. Graham, Associate Editor for Special Articles, Notices of the American Mathematical Society, Post Office Box 6248, Providence, Rhode Island 02940.

443 Thus, only a very special type of affine curve (In characteristic p approximately the same can have infinitely many integer points on it, and theorem is true, except for c_urves defined over a only a very special kind of Diophantine equation finite field which can have fimtely many countable of the sort that has locus of zeroes a curve can families of rational points (cf. [Grl, Mal, Pal, have infinitely many integer solutions. Sa2]).) This theorem is a function field-theoretic Let K be THE MORDELL- WElL THEOREM. analogue of the Mordell Conjecture for number prime field. Let a field finitely generated over its fields. It was an offshoot of the work of Lang on over K (i.e., a A be an abelian variety defined the restatement of the Mordell Conjecture, and embeddable in a complete variety defined over K, was first proven by Manin and Grauert. The work a group structure K-projective space, and having done by Soviet mathematicians on this theorem group of K­ defined over K). Let AK be the led to the ultimate idea of Faltings' proof. This generated rational points. Then AK is a finitely depends on noticing that Mordell's Conjecture abelian group ( cf. [Wel]). follows from: case, on elliptic curves defined As a special THE SAFAREVIC CONJECTURE FOR K-rational points, these points over K possessing CURVES. Let K be a finite extension of Q. group. Over the ration­ form a finitely generated There are only finitely many isomorphism classes part of this group can only be one als, the torsion of smooth curves X/ K of genus g 2: 2 which have abelian groups. But, the rational of fifteen finite good reduction outside a particular finite set of infinite abelian part. So, points can have a free places S of K. elliptic curves can have infinitely many rational points, but only finitely many integer points (in One may want to take a nonsingular variety affine coordinates). defined by equations over Z and produce a variety If a projective curve of genus zero has even a over a finite field by reducing the coefficients single rational point, one can show by elementary modulo p. A good reduction is one in which means that its rational points are in one-one such a method yields a nonsingular variety over correspondence with P1 (Q). So curves of genus the finite field. A variety has good reduction zero have either no rational points or infinitely at a prime p if there is some embedding of the many rational points. Thus, a conic either has variety which has good reduction at p ( cf. [Pal, no rational points or infinitely many that can Fal]). This may be related to the idea of testing be produced from a single one via the method whether a Diophantine equation has solutions in of sweeping lines. On an elliptic curve, there the integers by testing the necessary condition may not be any rational points, but the group that it have solutions modulo every prime p. structure does allow one in theory to construct all Also relevant to Faltings' work is the related the rational points from finitely many of them via Safarevic Conjecture for Abelian Varieties. This the group action, the so-called chord and tangent states that the set of K-isomorphism classes of method. Faltings work shows that no such method polarized abelian varieties defined over K of given for constructing rational points exists on curves dimension g and polarization degree d > 0 and of genus > 1. For more information on the low possessing good reduction outside a finite set of genus situation, see [Zal]. places S is finite. The polarization of an abelian variety involves the way one chooses to embed THE MORDELL CONJECTURE FOR NUM­ it in projective space. A refinement of Torelli's BER FIELDS. Let K be a finite extension of Q. can be used to show that if the Safarevic Let X/ K be a smooth curve of genus g 2: 2. Then theorem for abelian varieties is true, then so is the set of K-rational points on X. X(K) is finite Conjecture one for curves. Faltings established that both ( cf. [Fal]). the these conjectures are true in the course of settling (Cassels' paper [Cal] shows that the assump­ the Mordell Conjecture. tion that X is smooth is not serious. The theories 2. The general problem of solving Diophantine irreducibil­ of birational equivalence and absolute equations. Let f(xl, x2, ... , Xn) be a polynomial crucial.) Thus, ity show that smoothness is not in n variables with integer coefficients. Sometimes can have in­ only a very special type of curve we will consider such polynomials with coefficients it and only a very finitely many rational points on that are integers in a number field. When one equations special kind of system of Diophantine seeks solutions of the equation f(x) = 0 that are a curve can of the sort that has locus of zeroes either integers or rationals (or in the number have infinitely many rational solutions. field), the equation is called a Diophantine equa­ THE MORDELL CONJECTURE FOR FUNC­ tion. One talks of integral or rational solutions of TION FIELDS. Let K be a regular extension of the the Diophantine equation. Solutions with values field k of characteristic 0 and let C be a smooth in a field K are called K-rational solutions or curve of genus 2: 2 defined over K. Then either K-rational points. its set of rational points C(K) is finite, or C is A Diophantine equation is homogeneous if definable over k and all but a finite number of every term in it has the same degree. In this points in C(K) are in the image of C(k). case, if the vector x is a solution, then so is

444 ex for every constant c. Thus, for homogeneous curves (or, more generally, to systems of Diophan­ equations, it pays to "identify" two solutions that tine equations whose solution sets form curves in are nonzero multiples of one another, and look higher-dimensional spaces). In this way one is only for distinct solutions. For a nonhomogeneous led to restate the problem of solving Diophantine equation of degree d, there is always a related equations as the problem of finding all the rational homogeneous equation gotten by adding one more and integer points on curves. It is with this form variable to the set of n variables, and throwing of the problem that we are primarily concerned. into each term of the old equation enough copies A related problem, whose solution is used of the new variable to make each term have degree by Faltings, involves smooth projective varieties d. Then, solutions of the nonhomogeneous equa­ defined over finite fields. The numbers of rational tion correspond to solutions of the homogeneous points such a variety has in the various finite equation for which the new variable has value 1. extension fields can be expressed using a sort In practice, one looks for integer solutions in the of generating function, called the zeta function nonhomogeneous case, and rational solutions in of the variety. The Wei! Conjectures [We2] give the homogeneous case. The relationship between information about this zeta function that allows it homogeneous equations in n + 1 variables and in theory to be computed with enough mechanical nonhomogeneous equations in n variables may work. The last of these Wei! Conjectures was remind the reader of the relationship between finally proven by Deligne in 1973. The Wei! projective geometry and affine geometry (for good Conjectures on abelian varieties are rather simpler reason). to prove (cf. [Lal]), and are used in a crucial way Mathematicians ever since the ancient Greeks in Faltings' proof. have sought to solve particular Diophantine equa­ tions, and dreamed of finding a general solution. 3. The work of Mordell, Wei!, and Siegel. Let us The most infamous example is Fermat's work on consider the nonhomogeneous equation x2 +y2 = 1 the equations xn + yn = zn. Hilbert asked about and the corresponding homogeneous equation the existence of a general algorithm for the solu­ x2 + y2 = z2 . The affine equation clearly has tion of Diophantine equations. This is Hilbert's only finitely many integer solutions. The projec­ so-called Tenth Problem. It was finally solved by tive variety has infinitely many distinct integer Matijasevic in 1970 (cf. [Dal]). He showed that solutions (even after identifying solutions that are no such algorithm can be found that tells whether nonzero multiples of one another as mentioned be­ an equation has a solution in rational integers. It fore). Namely, the Pythagorean triples give such is still unknown if there is an algorithm to tell if integer points. The solutions in the projective an equation has a solution in rational numbers. plane include the solutions of the affine equation A different approach to Diophantine problems as special cases. But they also yield infinitely attempts to use geometry. Each Diophantine many rational solutions of the affine equation. equation defines as its locus of solutions over Thus, we are led to consider two related but the complex numbers a certain complex algebraic inequivalent problems. The problem of whether variety. If the equation is nonhomogeneous, its the number of integer points on an affine curve solutions form an affine variety. If the equation is finite we may call Siegel's Problem, after the is homogeneous, the solutions lie on a projective man who solved it. The problem of whether the variety. The relation described above between a number of rational points (after identifying those nonhomogeneous equation and its homogeneous that are nonzero multiples of one another) on version just corresponds to a certain affine variety a projective curve is finite is called the Mordell and the projective variety gotten by taking its Problem. Of course, these theoretical problems closure in the proper projective space (i.e., the may not help to find a bound on the number of projective variety gotten by adding points "at solutions (even when the number has been proven infinity"). finite) and certainly may not help us find even The study of solutions of Diophantine equa­ one solution. This is called the problem of getting tions then becomes the problem of finding integer an effective solution. points and rational points on complex varieties Historically, these problems arose out of the defined by equations with coefficients in number work of Mordell. For the general history of the fields. Thus, the equation that Fermat studied problems see [La4]. In his 1922 paper [Mol], is the equation of a nonsingular curve in the Mordell used direct methods to study rational projective plane of genus (n- 1)(n- 2)/2. In points on elliptic curves, that is, nonsingular general an irreducible homogeneous equation of curves of genus 1. In the following year, he degree n in three variables corresponds to a curve wrote the seminal paper [Mol] on the subject. in the projective plane of genus (n- 1)(n- 2)/2 He showed that the rational points on an elliptic minus a correction for the singular points of the curve form a finitely generated group, while the curve. One can hope that geometry may make affine curve has only finitely many integer points. some Diophantine problems more tractable. The He conjectured that affine plane curves of higher obvious first step is to try to restrict one's atten­ genus that are hyperelliptic also have only finitely tion to Diophantine equations that correspond to many integer points, and that smooth projective

445 curves of genus at least 2 always have at most Other mathematicians attempted to approach finitely lilany rational points. the Mordell Conjecture either through the piece­ These conjectures proved the inspiration for meal analysis of particular families of curves, or a continuing stream of research papers lasting to through careful use of the theory of heights. Ex­ this day. The hyperelliptic conjecture was proven amples of such work may be found in [Ca2, De2, anonymously by Carl Siegel in 1926 [Si1]. The Mo3, Mul]. conjecture on projective curves of genus at least The first major breakthrough came in the 2 became the Mordell Conjecture, now finally 1960s with the proof of the function field-theoretic established in Faltings' paper. The proof that the equivalent of the Mordell Conjecture. It is rational points on an elliptic curve form a finitely standard operating procedure in algebraic number generated group inspired Andre Wei! to write his theory to ask, for any theorem or conjecture over first great paper, the proof of the Mordell- Weil number fields, whether the analogous statement Theorem. This theorem helped to inspire much is true for function fields, and vice versa. Such of the algebraic geometry and number theory analogies are inspired by general theory about done since it was published in 1928 ( cf. [Wel, how to prove statements about fields of finite We3]). Weil's work led to the theory of abelian type over prime fields. Neron showed the two varieties, combining group theory with algebraic Mordell Conjectures establish the corresponding geometry and analysis in a vast generalization of conjecture for any finitely generated field over Q. the nineteenth-century theory of elliptic integrals. Manin and Grauert both managed to prove the For more information, see [La4, Zal]. function field analogue of the Mordell Conjecture We may take this opportunity to refer the stated above, without having any clear insight reader to the sources for more recent work on beyond that already in Lang's book [La4] on how the problem of rational points on elliptic curves. to proceed in the number-theoretic case. For the The basic survey of the subject is [Cal]. Since details, see [Grl, Mal, Sal]. Cassels' paper, Barry Mazur has done important The work of Manin was a part of a con­ work using the theory of moduli: see [Ma2, Ma3, siderable Soviet research effort on the Mordell Ma4, Ma5, Sel]. Conjecture. The central figures were Manin, At the heart of Mordell's work was his insight Safarevic, Demjanenko, and Parshin. In retro­ on the usefulness of the theory of heights, as later spect, the work of Parshin and Safarevic (cf. exemplified in the Thue-Siegel-Roth Theorem. [Pal, Pa2, Sal]), centered about the Safarevic Siegel, in his masterpiece [Si2], saw how to put Conjecture stated above and its relationship to the existing theory of heights together with the the function field version of the Mordell Conjec­ Mordell-Wei! Theorem and a primitive version ture, was to be the crucial' inspiration for the of what was to become the theory of abelian work done by Faltings fifteen years later. Parshin varieties to solve what we have called the Siegel showed that if the Safarevic Conjecture could be Problem. His result, Siegel's Theorem, is quoted established, then the Mordell Conjecture could be above. This paper contains much else of value, proven. including the famous Siegel's Lemma, important More mathematics had to be developed before to transcendence theory. For more information Parshin's ideas could be applied to the number­ on Siegel's Theorem see [La2, La4]. A theory of theoretic Mordell Conjecture. These included the heights for abelian varieties is central to Faltings' development of the theory of moduli for abelian work. Such a theory of heights can be considered varieties and curves and the development of a a modern version of the famous method of infinite sophisticated theory of abelian varieties using descent introduced by Fermat. representation theory. The latter is embodied in 4. Early attempts at the Mordell Conjecture. As the modern theory of Tate modules, as described the theory of abelian varieties developed, it be­ in [Lal] and [Cal]. came increasingly clear that Mordell's Conjecture 5. A sketch of Faltings' paper. Let K be a finite should be understood as a statement about cer­ extension of the rationals Q with algebraic closure tain abelian varieties containing curves of genus K*, A an abelian variety defined over K. Let 2 or more. On such abelian varieties, called Ja­ 1r = Gal(K* / K) be the absolute Galois group of cobians, one could hope to use the Mordell-Wei! K, l a prime. Then 1r operates on the Tate module Theorem to get information about the rational points of the embedded curves. Such methods Tt(A) = limA[In](K*). date back to Chabauty (cf. [Ch1, Ch2]) in the <---- 1930s and 1940s, but were made quite concrete by Lang in [La4]. One restatement of the Mordell More precisely, Tt(A) is the set of all vectors Conjecture by Lang conjectures that a curve ly­ (a1,a2, ... ,an, ... ), where n > 0, with an a K*­ ing on an abelian variety has only finitely many point on A whose order is a power of l, and points in common with any subgroup of finite such that lan+l = an for every n, and la1 = 0. type of the abelian variety. He showed that the Addition of these infinitely long vectors is done Mordell Conjecture could also be considered as a componentwise. Then Tt(A) can be made a conjecture on algebraic families of curves. module over the l-adic integers Zt.

446 The Tate group is a torsion-free module over Faltings discusses the behavior of heights z,, and is a module of finite type of dimension under isogenies. Here, the Weil Conjectures are 2 dim A over z,. The Tate module can be extended needed. Faltings proves (1) and (2). These to a vector space Et(A) over the l-adic numbers are then used to prove that two abelian varieties Q,, E,(A) = 1!(A) ®z, Q,. This vector space is are isogenous iff their extended Tate modules are called the extended Tate group and 1r operates on isomorphic as 'If-modules. This settles the isogeny it. Faltings' basic results are: conjecture, an important result in itself. Faltings (1) The representation of 1r on Et(A) is proves (3) from the previous results together semisimple. with another application of the Wei! Conjectures. (2) The map EndK(A)®z z,-+ End,..(11(A)) Finally, one gets the Safarevic Conjecture for is an isomorphism. curves, and so-as explained before-the Mordell (3) LetS be a finite set of places of K, d > 0. Conjecture. Then there are only finitely many isomorphism 6. Some final remarks. Now that the Mordell Con­ classes of d-fold polarized abelian varieties over K jecture has been solved, one obvious question is: which have good reductions outsideS. what about those systems of Diophantine equa­ Result (2) says that certain types of well­ tions whose solution sets do not form curves? behaved endomorphisms of Tt(A) correspond to What can be said about integral and rational the endomorphisms of A. This settles a version points on higher-dimensional varieties? Some of a conjecture of Tate. For similar results on the early attempts have been made to study such use of the Tate module to study maps between questions. In [Ch3] and [Ch4], Chatelet studies abelian varieties, compare Lang's book on abelian cubic surfaces. In [La3] and [La5], there are a varieties [Lal]. The Safarevic Conjecture for series of conjectures extending in various direc­ abelian varieties is (3), and so the Safarevic tions the results now known for curves. But Conjecture for curves follows from the theory of any real understanding of the situation in higher Jacobians of curves. dimensions remains a distant goal. After obtaining the Safarevic Conjecture, Even more important is the question of the Faltings can now employ Parshin's earlier argu­ effectiveness of the existing results. By this we ment from [Pa2] to prove the Mordell Conjecture. mean not merely their theoretical usability to We sketch the argument briefly. Suppose X is a solve Diophantine problems, but their practical smooth curve over K of genus at least 2 and S applicability. If a problem can be solved only is a set of places. For each K-rational point x, with an impossibly large amount of work, its one can construct a covering Y (x) of X, of genus theoretical solution may not be of any value. g', defined over a finite extension of K, which Unfortunately, this appears to be the situation is ramified exactly at x, and so Y(x) has good with the Siegel Problem, the Mordell Problem, reduction outside a set T. Here g', T, and the and perhaps the related problem of using the field extension depend only on X and S, but not Mordell-Wei! Theorem to find the group structure on x. By $afareviC's Conjecture, there are only of the rational points on an abelian variety. finitely many isomorphism classes of such Y(x). (There is an unproven algorithm for effectively If the set of rational points x were infinite, there using the Mordell-Wei! Theorem. Its correctness would be infinitely many distinct maps from some is equivalent to a 1929 conjecture by Wei! [We3] Y(x) to X. This contradicts a classical result on that remains open.) Quantitative results are Riemann surfaces of genus > 1. difficult to obtain (cf. [La4]). The proofs of The proof of (3) follows from the finiteness the Mordell-Wei! Theorem and of the Mordell of the set of isogeny classes (isogeny = onto ho­ Conjecture are highly nonconstructive. It is hard momorphism with finite kernel) of such polarized to estimate the number of rational or integral abelian varieties. This and (1) and (2) follow points or to get generators for the Mordell-Wei! from similar arguments using heights, the theory group. of moduli of curves and abelian varieties, and the In his paper [Si2], Siegel states that there Weil Conjectures on abelian varieties. . should be effective ways to estimate the number Faltings begins with the theory of semiabelian of integral points, even if the size of the integral varieties, which extends the concept of an abelian points cannot be estimated. Thus, we might variety over a field to a variety over a scheme. He have an idea of the number of integral points, discusses some results about the moduli spaces but none of how large a space must be searched for stable curves and principally polarized abelian to find them exhaustively. In practice, it is not varieties. Then, he develops a theory of heights clear that there are practical ways to estimate the applicable to semiabelian varieties. The basic the­ number of integral points on a curve. The general orem shows that given a constant c, there are only consensus among knowledgeable number theorists finitely many isomorphism classes of semiabelian appears to be that we are a long way from having varieties with certain properties whose heights are truly effective means of finding the integral (or bounded by c. This principle of bounded height rational) points on curves of genus > 1. may turn out to be the most useful result in Faltings' work is currently highly ineffective. Faltings' paper. It is the basis for all his results. For example, there is no known algorithm for

447 determining whether there are any rational points [Fa2] G. Faltings, Arakelov's theorem for at all on a curve (the rational point version of the abelian varieties, Invent. Math. 73 (1983), 337- theorem on integer points for which Matijasevic 348. gave a negative answer-so it is hard to have much [Grl] H. Grauert, Mordells Vermutung iiber optimism that such an algorithm will be found). Punkte auf algebraischen K urven und Funktio­ Fermat's Last Theorem is just one example of nenkorper, Inst. Hautes Etudes Sci. Publ. Math. how hard this problem must be. Assuming 25 (1965), 131-139. there are rational points on a curve, there is no [Lal] S. Lang, Abelian varieties, Interscience known computable upper bound for the number Publishers, New York, 1959. of rational points. Before Faltings' work, in [La2] S. Lang, Integral points on curves, lnst. fact, there was no curve X of genus 2 or more Etudes Sci. Publ. Math. 6 (1960), 27-43. for which it was known that X(K) is finite for Hautes all number fields. All this leads to a certain [La3] S. Lang, Some theorems and conjectures Math. Soc. pessimism about the practical usefulness of direct in diophantine equations, Bull. Amer. applications of Faltings' work to number theory, 66 (1960), 24Q-249. despite its considerable theoretical interest. Only [La4] S. Lang, Diophantine geometry, Inter­ time will tell whether this pessimism is justified. science Publishers, New York, 1962. Perhaps the future will show that the ulti­ [La5] S. Lang, Higher dimensional Diophan­ mate importance of Faltings' work will not lie tine problems, Bull. Amer. Math. Soc. 80 (1974), primarily in settling the Mordell conjecture. To 779-787. an algebraic geometer, his results on the Safarevic [Mal] J. Manin, Rational points of algebraic and Tate conjectures may open the way to signifi­ curves over function fields, Izv. Akad. Nauk SSSR cant progress. And perhaps Faltings' work on the Ser. Mat. 27 (1963), 1395-1440 (transl. in Amer. theory of heights for abelian varieties will prove Math. Soc. Transl. Ser. 2, vol. 50, 1966). even more important. Progress on Diophantine [Ma2] B. Mazur, Rational points on modular questions may come indirectly. Only time will curves, Lecture Notes in Math., no. 601, Springer­ tell. Verlag, 1977, pp. 107-148. [Ma3] B. Mazur, Modular curves and the BIBLIOGRAPHY Eisenstein ideal, lnst. Hautes Etudes Sci. Publ. [Cal] J. Cassels, Diophantine equations with Math. 47 (1978), 35-193. special reference to elliptic curves, J. London [Ma4] B. Mazur, Rational isogenies of prime Math Soc. 41 (1966), 193-291. degree, Invent. Math. 44 (1978), 129-162. [Ca2] J. Cassels, On a theorem of Demja­ [Ma5] B. Mazur and J.-P. Serre, Points nenko, J. London Math Soc. 43 (1968), 61-66. rationnels des courbes modulaires Xo(N), Sem. [Chl] C. Chabauty, Sur les equations dio­ Bourbaki 1974/75, no. 469, Lecture Notes in phantiennes liees aux unites d'un corps de nom­ Math., no. 514, Springer-Verlag, 1976, pp. 238- bres algebriques fini, thesis, Annali di Mat. 17 255. (1938), 127-168. [Mol] L. Mordell, On the integer solutions of [Ch2] C. Chabauty, Sur les points rationnels the equation ey2 = ax3 + bx2 + ex + d, Messenger des varietes algebriques dont l 'irregularite est of Math 51 (1922), 169-171. superieure d la dimension, Compt. Rend. 212 [Mo2] L. Mordell, On the rational solutions (1941), 1022-1024. of the indeterminate equations of the third and sur [Ch3] F. Chatelet, Points rationnels fourth degrees, Proc. Cambridge Philos. Soc. 21 certaines courbes et surfaces cubiques, Enseign. (1923), 179-192. (2) 5 (1959), 153-170. Math. [Mo3] L. Mordell, Some quartic diophantine Chatelet, Points rationnels sur cer­ [Ch4] F. equations of genus 3, Proc. Amer. Math. Soc. 17 taines surfaces cubiques, Centre National de la (1966), 1152-1158. Recherche Scientifique, Paris, 1966, 67-75. A remark on Mordell's M. Davis, Y. Matijasevic, and J. [Mul] D. Mumford, [Dal] Amer. J. Math 87 (1965), 1007-1016. Robinson, Hilbert's tenth problem. Diophantine Conjecture, equations: positive aspects of a negative solution, [NYT] P. Boffey, German is Hailed in Math Proc. Sympos. Pure Math., vol. 28, Amer. Math. Advance, New York Times, Tuesday, 19 July 1983, Soc., Providence, R. 1., 1976. p. Al. [Del] P. Deligne, Preuve des conjectures [Pal] A. Parshin, Algebraic curves over func­ de Tate et Shafarevitch (d'apres G. Faltings), tion fields I, Math. USSR-Izv. 2 (1968) no. 5, Seminaire Bourbaki, exp. 616, November 1983. 1145-1169. [De2] V. Demjanenko, Rational points of a [Pa2] A. Parshin, Algebraic curves over func­ class of algebraic curves, Izv. Akad. Nauk SSSR tion fields, Soviet Math. Dokl. 9 (1968) no. 6, 30 (1966), 1373-1396. (Russian) 1419-1422. [Fa1] G. Faltings, Endlichkeitssiitze fiir [Sal] I. R. Safarevic, Algebraic number fields, abelsche Varietiiten iiber Zahlkorpern, Invent. Proc. Internat. Congr. Math. (Stockholm, 1962), Math. 73 (1983), 349-366. Amer. Math. Soc. Transl. (2) 31 (1963), 25-39.

448 [Sa2] P. Samuel, La conjecture de Mordell [Sz1] L. Szpiro, La conjecture de Mordell pour les corps de fonctions ( d 'a pres Man in et (d'apres G. Faltings), Sem. Bourbaki, exp. 619, Grauert), exp. 287, Sem. Bourbaki, vol. 1964/65, November 1983. W. A. Benjamin, New York, 1966. [We1] A. Wei!, L 'arithmetique sur les courbes [Se1] J.-P. Serre, Points rationnels des courbes algebriques, Acta Math. 52 (1928), 281-315. modulaires Xo(N) ( d'apres Barry Mazur), Sem. [We2] A. Wei!, Number of solutions of equa­ Bourbaki, vol. 1977/78, exp. 511, 1979, 89-99. tions in finite fields, Bull. Amer. Math. Soc. 55 [Si1] C. Siegel, The integer solutions of the (1949), 497-508. equation y2 = axn + bxn-l + · · · + k, Collected [We3] A. Wei!, Sur un thioreme de Mordell, works, vol. 1, Springer-Verlag, Berlin, 1966, pp. Oeuvres Scientifiques, Collected Papers, vol. I, 207-20; originally in J. London Math Soc. 1 Springer-Verlag, New York, 1979, pp. 47-56 (orig­ (1926), 66-68. inally published in 1929). [Si2] C. Siegel, Uber einige Anwendungen dio­ [Za1] D. Zagier, £-Series of elliptic curves, the phantischer Approximationen, Collected Works, Birch-Swinnerton-Dyer conjecture, and the class vol. 1, Springer-Verlag, Berlin, 1966, pp. 209-266; number problem of Gauss, Not. Amer. Math. Soc. orig. in Abh. Preuss. Akad. Wiss. (1929) no. 1. 31 (1984), 739-743.

449 National Science Foundation Budget Request for Fiscal1987

This article is concerned with material submit­ There is, however, a difference this year in ted to Congress last February and refers to a reor­ the emphasis of these reports; no longer do we ganization of NSF announced in March 1986. find details of interest regarding achievements Since February and March, however, there have and progress in scientific knowledge, but rather been changes in the anticipated budgetary support a significant increase in the amount of financial for Mathematical Sciences Research in fiscal1987. detail concerned with increases, cuts, and the Supplementary information available when the shuffling of funds from one part of the budget to June Notices was prepared for the printer will be another and from one year to the next. found in the section NSF News & Reports later in The good news is that the mathematical this issue. EDITOR sciences are expected to continue to fare better this year and next than in the recent past, Since April 1973, Notices has published annually although the funds involved are very small (still a report outlining the proposals contained in the less than 4% of the entire budget). White House's budget request to the Congress The press releases and budget summaries for the needs of the National Science Foundation. which have been distributed contain little direct This year, details of the NSF budget for FY 1987 information about the effects of the Gramm­ were distributed to the public under a cover letter Rudman-Hollings limitations on the 1987 NSF (excerpts below) from Erich Bloch, director of budget requests, but there are some rather re­ the Foundation. Last year's report in Notices vealing tables in the four supplementary excerpts appeared in the June 1985 issue, pages 346 to that follow. These indicate reductions between 364. This year's report differs in several ways National Science Foundation from previous ones. We have, again, the usual Washington, DC 20550 Tables I to III: Table I indicates what part of February 7, 1986 the entire NSF budget is allocated to research in I am very pleased to provide . . . a summary of the mathematical sciences; Table II shows how the National Science Foundation's fiscal year 1987 the funds for the Division of Mathematical and budget request, which has been sent to the Congress. Physical Sciences (MPS) are apportioned among The request is for almost $1.7 billion and represents an increase of eight percent over fiscal year 1986. the Division's several Sections; Table III is an In a year of dramatic federal budget reduc­ attempt to place the longer-range growth of the tions, the President's budget request for the Na­ Foundation's budget in perspective with regard tional Science Foundation contains a moderate, but to the inflation of the past decade or two. very important, increase for the basic research and In addition to these more familiar tables, we education activities of the Foundation. Science and give figures for the relatively new program sup­ engineering enrich the lives of all of us, and provide the basis for both economic competitiveness and na­ porting supercomputers, the Advanced Scientific tional defense. Investing in science and engineering Computing (ASC) program is only a year or two research and education is clearly a critical activity old at this point but it shows signs of vigorous that must be maintained under any economic con­ growth (last year and this, more is being spent ditions. in this program than for the entire Computer Given the extraordinary constraints on the Fed­ Research Section of MPS) and it may not be long eral budget, however, it will be difficult to convince the Congress of the need to provide for an increased before the ASC budget could exceed the budget investment in basic science and engineering. Our for research in Mathematical Sciences. A new line balanced program of support for research and ed­ has been added this year below Table II giving ucation in science and engineering is important to figures for the ASC program. enhancing our nation's economic productivity and In recent years these more general tables and our competitive position internationally. As a soci­ the accompanying commentary have been sup­ ety and as a government we must look to the future and make the investments necessary to prepare for plemented by excerpts from the Administration's it. Budget submission to Congress which describe I hope you find the enclosed summary useful activities and progress related to research in the and informative. . .. Mathematical Sciences, Computer Research, and Sincerely, Science and Engineering Education (SEE). This Erich Bloch year we also include the corresponding report Director from the ASC Office.

450 Table I. National Science Foundation

1983 1984 1985 1986 1987 (Millions of dollars) Actual Change Actual Change Actual Change Plan Change Request ( 1) Mathematical Sciences Research Support I 34.8 18.4% I 41.2 15.8% I 47.7 8.4% s 51.7 15.7% $ 59.8 (2) Other Research Support (Note A) 960.6 17.9% 1131.0 14.2% 1293.6 3.1% 1334.0 8.4% 1446.3 (3) Education, Information, Foreign Currency Program (Note B) 24.7 169% 66.5 40.9% 93.7 3.8% 97.3 4.4% 101.6 (4) Program Development and Management ("Overhead") (NoteC) ~ 0.9% 66.3 8.6% 72.0 0.7% 72.5 7.6% 78.0 (5) Totals 11085.8 20% 11305.1 15.5% $1507.0 3.2% 11555.5 8.4% Sl685.7 (6) (1) as %of (1) and (2) 3.50% 3.52% 3.56% 3.73% 3.97% (7) (1) as % of (5) 3.21% 3.16% 3.17% 3.33% 3.55% 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 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.

Table II. Division of Mathematical and Physical Sciences

Section 1983 Actual 1984 Actual 1985 Actual 1986 Plan 1987 Request Mathematical Sciences $ 34.8 (11.5%) $ 41.2 {11.6%) $ 47.7 {12.0%) $ 51.7 {12.6%) $ 59.8 (13.3%) Computer Research 29.3 ( 9.7%) 33.7 ( 9.5%) 39.1 ( 9.8%) 40.2 ( 9.8%) 44.4 (10.0%) Physics 89.1 (29.5%) 104.4 (29.4%) 115.8 (29.2%) 118.6 (28.9%) 126.6 (28.2%) Chemistry 67.6 (22.4%) 79.4 (22.4%) 87.6 (22.0%) 89.8 (21.9%) 101.0 (22.5%) Materials Research 81.1 (26.9%) 96.3 (27.1%) 107.0 (26.9%) 109.6 (26.8%) 117.4 (26.1%) ------Totals $301.9 $355.0 $397.2 $410.0 $449.2 Advanced Scientific Computing $ 6.0 $ 41.4 $ 45.2 $ 53.6 last year's budget requests for FY 1986 and this of Information Science and Technology (currently year's plans for current spending, involving cuts in the Directorate of Biological, Behavioral and ranging from about 2% for ASC to just over 6% Social Sciences), the Office of Advanced Scientific for SEE. On the other hand, the FY 1987 budget Computing (now in the Office of the Director), requests exceed the FY 1986 planned expenditures and "some programs from the Directorate for by amounts ranging from just over 2% for SEE to Engineering"; (3) changing the name of Astro­ about 19% for ASC. nomical, Earth and Ocean Sciences (AAEO), after Over the decade and more in which these Astronomy is removed, to Directorate of Geo­ reports have been prepared for Notices, the in­ sciences (DOG). formation on which they have been based has There has been one phenomenon which may changed almost year-by-year. The detail reported not have received adequate coverage in these has evolved significantly, the accounting categories annual reports on the NSF budgets in Notices. have changed regularly, to such an extent that For the most part, we have reported what the it has often been difficult to present the figures Foundation says it wants to spend on the various in a form comprehensible to any but professional programs which it supports, and what it has accountants. actually spent after the year closes. These figures This situation continues. For example, an are referred to in the jargon as the "Obligational NSF press release dated March 24, 1986, an­ Authority" for which NSF seeks congressional ap­ nounces a reorganization involving a new round proval. Those familiar with budgets, and other of "musical chairs" within the Foundation which predictions, can understand that certain inaccu­ will involve (1) moving Computer Research out racies are inevitable ("foresight" vs. "hindsight") of MPS and moving Astronomy in; (2) creating and that what is actually spent during the fiscal a new Directorate of Computer and Information year will probably not be what was anticipated. Science and Engineering (CISE) to include the The various units of NSF appeared to finish Computer Research Section of MPS, the Division each year pretty close to their budgets, usually

451 Table III. Ten-Year Compilation of the NSF Budget

1978 . 1979 1980 1981 1982 1983 (Millions of dollars) Actual Actual Actual Actual Actual Actual ( 1) Mathematical Sciences Research Support $ 21.4 $ 22.8 $ 25.0 $ 28.3 $ 30.5 $ 34.8 1967 dollars* 10.2 9.7 9.3 9.6 10.2 11..') (2) Other Research Support 702.8 761.0 804.3 873.7 875.7 960.7 1967 dollars 33.'>.9 322.7 299.2 29.'>.3 292..'> 317.0 (3) Education, Information, Foreign Currency Program 84.3 88.4 87.7 80.6 29.8 24.6 1967 dollars 40.3 37.S 32.6 27.2 9.9 8.1 ( 4) Program Development and Management ("Overhead") 48.7 54.7 58.2 59.2 63.2 65.7 1967 dollars 23.3 23.2 21.6 20.0 21.1 21.7 ------(5) Totals $857.2 $926.9 $975.2 $1041.8 $999.1 $1085.8 1967 dollars 409.7 393.0 362.8 3.'>2.1 333.7 3.'>8.3

1984 1985 1986 1987 Increase Increase Actual Actual Plan Request 1978-1985 1978-1987 ( 1) Mathematical Sciences Research Support $ 41.2 $ 47.7 $ 51.7 $ 59.8 123% 179% 1967 dollars 13.2 1S.S S2% ( 2) Other Research Support 1131.0 1293.6 1334.0 1446.0 84% 106% 1967 dollars 364.8 419.1 2.'>% (3) Education, Information, Foreign Currency Program 66.5 93.7 97.3 101.6 11% 21% 1967 dollars 21.4 30.4 -2.'>% (4) Program Development and Management ("Overhead") 66.3 72.0 72.5 78.0 48% 60% 1967 dollars 21.3 23.3 0% (5) Totals $1305.0 $1507.0 $1555.5 $1685.7 76% 97% 1967 dollars 420.2 488.3 19% * Current dollars are converted to 1967 dollars using the wholesale/producer index. with "slide-rule accuracy", as some of us used to tion requested over $50 million for education and say, meaning a deviation of 5% or less from the for 1985 over $75 million, a figure which Congress target, but there has recently been a significant then raised to $82 million. The SEE Obligational departure from this pattern. Authority has been at about that level since. In the first term of the present Administra­ The problem, however, is that SEE has tion, the SEE effort was essentially terminated (cf. not managed to spend that much money and Notices, April 1982, pages 287 to 289). After a has consistently "carried over" (or anticipated decade or so of support at a level of $60 to $80 carrying over) about $30 million each year for the million a year, in 1982 and 1983 expenditures were past three years. reduced to, respectively, $21 and $16 million when It is surely understandable that, when a there was support for nothing but the Graduate major program has been abolished and its staff Research Fellowships which had been previously dispersed, it must take years to reconstruct what contracted. By 1984, however, the folly of this was there before. L. K. DURST had apparently been recognized and the Founda-

452 The following text was prepared by the and support for research conferences and work­ staff of the Mathematical Sciences Division of shops. Moreover, researchers are encouraged to NSF and was submitted to Congress as a part join colleagues in obtaining funds for computing of the Administration's Budget Request for the equipment that no single project could justify. Fiscal Year 1987. Activities of this kind will continue to augment the SRPS awards. Mathematical Sciences Subactivity During the last few years major structural problems in the support for research have become $59,800,000 visible. These problems resulted, in part, from Summary of Request a gradual attrition in support levels over several years. While this attrition was most pronounced The FY 1987 Budget Request of $59.8 million for in the funding levels for graduate students and the Mathematical Sciences subactivity represents postdoctoral associates, it has also affected sup­ an mcrease of 11.6% over the FY 1986 Current port for investigators. This problem has been Plan. (Millions of dollars) documented in the 1984 report from the National Research Council entitled, Renewing U.S. Mathe­ FY 1985 FY 1986 FY 1987 Activity Actual Plan Request matics: Critical Resource for the Future. As the only Federal agency with responsibility for sup­ Classical Analysis $ 4.55 $ 5.02 $ 5.77 port of research across the entire spectrum of the Modern Analysis 4.66 5.09 5.86 mathematical sciences, NSF plays a pivotal role Geometric Analysis 4.52 4.95 5.69 attempting to address these structural problems. Topology & Foundations 5.70 6.16 7.08 The issues are complex, but significant gains have Algebra & Number been made since FY 1984. A determined effort, Theory 7.91 8.58 9.87 sustained over a period of years, is required to re~ Applied Mathematics 6.42 7.03 8.91 align support for mathematical sciences research Statistics & Probability 5.60 6.17 7.44 with that of other sciences. Special Projects 8.37 8.74 9.18 -- -- The chart below illustrates the crucial role Total $47.73 $51.74 $59.80 played by NSF in the Federal support of basic research in the mathematical sciences. Percent­ Scientific Overview ages are based on a total Federal expenditure of The Mathematical Sciences Subactivity fosters the $92.86 million in FY 1985. For the past decade, creation and development of mathematical ideas, the mission-oriented research agencies have fo­ methods, and techniques and promotes their use cused their support on applied mathematics and in improving our understanding of physical, bio­ statistics. New programs have been aimed at logical, engineering, and social phenomena. To such emerging fields as dynamical systems and achieve these objectives, support provides for sig­ discrete mathematics. In all areas of the math­ nificant research in the mathematical sciences, and ematical sciences, Foundation-supported research serves to ensure the continuing vitality and long­ generally involves a broader range of topics than range health of the discipline. Each subdiscipline the more project-oriented research sponsored by of the mathematical sciences, from those with the the mission agencies. sharpest intrinsic focus to those that reach out to other areas of knowledge, is supported in such Federal Academic Mathematics Funding a way as to encourage their interaction and to FY 1985 provide a healthy balance among them. The core of the Foundation's program of research support in this area has been the Sci­ entific Research Project Support (SRPS) awards. Within the SRPS activities, significant emphasis has been given to increasing the level of support for the young mathematicians to whom the field owes so much for its vitality. This emphasis has received significant enhancement since FY 1984. The mathematical sciences community recognizes NSFIOTilER a need for greater opportunities for communica­ ,.4% tion between mathematical scientists in the same ONR or related fields, as well as with scientists in other 12.9% disciplines. To provide for this need the subactiv­ ity has a Special Projects Program which includes funding for research institutes at the University of 9.2% Minnesota and in Berkeley, California, postdoc­ toral and visitor support at three other research NSF coordinates its support of research in the centers, a program of postdoctoral fellowships, mathematical sciences with its counterpart Fed-

453 eral agencies through the Interagency Committee access to computers is small in dollars but large for Extramural Mathematics Programs. This in percentage. The increase reflects increasing use group meets regularly to share information on of computational equipment in all areas of the policies of support and to discuss areas of em­ mathematical sciences. phasis and of unusual scientific opportunity. Ex­ tensive day-to-day contact between program of­ FY 1987 Budget Highlights ficers at various Federal agencies is maintained (Millions of dollars) by telephone and personal visits. Conferences, workshops, and activities FY 1986 FY 1987 Percent at research centers are Activity Plan Request Change sometimes supported jointly by the Foundation and one or more of the other agencies. Classical Analysis $ 5.02 $ 5.77 14.9% Modern Analysis 5.09 5.86 15.1 Changes Between the Geometric Analysis 4.95 5.69 15.0 FY 1986 Budget Request and the Topology & Foundations 6.2 7.1 14.8 FY 1986 Current Plan Algebra & Number (Millions of dollars) Theory 8.58 9.87 15.0 Applied Mathematics 7.03 8.91 26.7 FY 1986 FY 1986 Percent Activity Request Plan Change Statistics & Probability 6.17 7.44 20.6 Special Projects 8.74 9.18 5.0 Classical Analysis $ 5.36 $ 5.02 -6.3% -- -- Modern Analysis 5.49 5.09 -7.3 Total $51.74 $59.80 15.6% Geometric Analysis 5.22 4.95 -5.2 The increase of 15.6% for the Mathematical Sci­ Topology & Foundations 6.63 6.16 -7.1 ences subactivity is partially in response to the Algebra & Number recommendations of the recent report from the Theory 9.11 8.58 -5.8 National Research Council entitled Renewing U.S. Applied Mathematics 7.64 7.03 -8.0 Mathematics: Critical Resource for the Future and Statistics & Probability 6.69 6.17 -7.8 the subsequent National Science Board resolution Special Projects 8.55 8.74 +2.2 on the need to bolster resources for the mathemat­ Total $54.69 $51.74 -5.4% ical sciences. The central role of the mathematical The FY 1986 Current Plan is 5.4% below the sciences in the computational approach to science FY 1986 Budget Request. This decrease results and engineering accounts for the remainder of the from the unspecified Congressional reduction. The increase. decrease was distributed across the programs The mathematical sciences are engaged in in accordance with perceived opportunities and the computational approach to science and en­ needs. The modest increase in special projects gineering in three distinct ways. First, as in reflects adjustments in support of postdoctoral most other disciplines, the computer is becom­ researchers and equipment. ing increasingly important as a tool within the The FY 1986 Current Plan maintains the mathematical sciences themselves. Second, math­ Foundation's increases in the support of graduate ematical scientists, particularly numerical ana­ students and postdoctoral researchers at some­ lysts and computational statisticians, work with what more modest levels than originally planned. scientists and engineers from all disciplines in Where possible, these increases are located in the integrated development and application of the SRPS programs. The table below describes computational mathematical models, algorithms the position with respect to critical infrastructure and efficient computational software. Finally, items. Maintaining progress on the personnel the mathematical sciences provide the necessary aspects of the infrastructure was held to have theoretical underpinnings for the development of the highest priority. Within the senior investi­ computational methods. gator category, the emphasis will be on younger The FY 1987 Budget Request provides $3.0 investigators. The increase for equipment and million for increased support for all three modes of activity with emphasis on the second. These (Millions of dollars) funds have been distributed across the Subac­ FY 1985 FY 1986 FY 1986 tivity according to expected use. The greater Actual Request Plan involvement of the programs of Applied Mathe­ Graduate Students $ 6.04 $ 8.60 $ 7.25 matics and Statistics and Probability in the area Increa:!e over F Y 1985 42.5% 20.1% of computational science and engineering results Postdoctoral Researchers 6.23 7.60 7.00 in larger increases for these programs. Funds as­ Increa:!e over F Y 1985 22.0% 12.4% sociated with this computational initiative will be SRPS Senior coordinated on a Subactivity-wide basis, with the Investigators 30.03 32.38 32.05 development of multidisciplinary teams providing lncrea:!e over FY 1985 7.8% 6.7% a focus for the research supported. Equipment & Access Since FY 1984, the budget of the Mathemat­ to Computers 1.72 2.78 2.19 ical Sciences subactivity has included increments Increa:!e over F Y 1985 61.4% 27.2% dedicated to the support of graduate students

454 and postdoctoral associates. This emphasis is The following text was prepared by the continued in the FY 1987 Budget Request. The staff of the Computer Research Division of table below summarizes the impact of the request NSF and was submitted to Congress as a on various items. part of the material which accompanied the (Millions of dollars) Administration's Budget Request for the Fiscal FY 1986 FY 1987 Percent Year 1987. Plan Request Change Graduate Students $ 7.25 $ 9.90 36.6% Postdoctoral Researchers 7.00 8.06 15.1 Computer Research Subactivity SRPS Senior $44,440,000 Investigators 32.05 35.62 11.1 Equipment & Access Summary of Request to Computers 2.19 2.85 30.1 Support for Computer Research is increased by • All support for graduate students will be $4.27 million, or 10.6%. located in the SRPS programs. The number (Millions of dollars) of graduate students supported will increase FY 1985 FY1986 FY1987 from about 750 in FY 1986 to about 875 in Activity Actual Plan Request FY 1987, and more of the students will be Theoretical Computer supported as full-time research assistants. Science $ 4.86 $ 5.15 $ 5.58 • The increase in funds for postdoctoral re­ Software Systems Science 4.24 4.80 5.47 searchers will come almost entirely in the Software Engineering 3.57 3.85 4.93 SRPS programs. The number of Mathe­ Intelligent Systems 5.22 5.35 5.67 matical Sciences Postdoctoral Research Fel­ Computer Systems lowships awarded by the Special Projects Design 3.60 3.50 3.47 Program will be held constant at 30. The Coordinated Experi- total number of postdoctoral researchers sup­ mental Research 15.12 15.10 14.40 ported is expected to increase from about 240 Special Projeets 1.02 0.92 3.42 in FY 1986 to about 255 in FY 1987 while the Computer Research time supported per postdoctoral researcher Equipment 1.49 1.50 1.50 increases significantly on SRPS awards. Total $39.12 $40.17 $44.44 • Funds for the support of senior investigators include support for the Mathematical Sci­ Scientific Overview ences aspects of Foundation-wide activities The new academic discipline of computer research such as the Presidential Young Investigator has its roots in the mathematical sciences, electri­ awards Research Opportunities for Women, cal engineering, and logic. The use of computers Resear~h in Undergraduate Institutions, Mi­ stimulates the growth of computer research as a nority Research Initiation, and the Exper­ discipline; in turn, computer research is build­ imental Program to Stimulate Competitive ing the knowledge base for improving computer Research. It is expected that the number technology. Unlike the physical sciences, which of investigators supported will increase from deal with the intrinsic properties of physical mat­ about 1300 in FY 1986 to about 1330 in ter, computer research discovers the concepts and FY 1987. laws governing problem-solving procedures and • The increase for computer equipment and develops computing systems which test and uti­ access to computing facilities is large in per­ lize them. The universe for study consists of centage, but small in amount. The increase is strategies and algorithms for solving problems, in response to the increasing needfor this re­ methods of representing and transforming infor­ source in all of the areas of the mathematical mation, programs for carrying out computation sciences. procedures, and machines for executing programs. Funding patterns in computer research are often difficult to discern because it is common to confuse research in other sciences using comput­ ers or applied research on computer technology performed by industry or the mission-oriented agencies with computer research. The com­ puter industry, paced by the constant demand for new or improved products, has concentrated on short-term requirements for most research and development projects. Some basic research is carried out, principally in a few of the largest companies, and companies also provide limited support for academic research. This is mainly in the form of discounts or gifts of equipment

455 but there are also some faculty study contracts, NSF programs such as the Computer Engineering graduate student fellowships or forgivable loans, Program within the Design, Manufacturing, and and a small number of unrestricted departmental Computer Engineering Subactivity and with the grants. The total investment by industry at aca­ programs of other agencies such as DARPA and demic institutions has been increasing but there is DOE. heavy emphasis on educational use of computers, concentration of research support at a few selected FY 1987 Budget Highlights academic centers, and product orientation. Both academic and industrial computer re­ (Millions of dollars) search are supported by several agencies and, with FY 1986 FY 1987 Percent the Department of Defense Strategic Computing Activity Plan Request Change initiative, Federal investment has risen rapidly. Theoretical Computer Most Federal programs support directed research Science $ 5.15 $ 5.58 + 8.3% Software Systems that is relevant to their respective missions. As Science 4.80 5.47 + 14.0 with industrial support, there is concentration at Software Engineering 3.85 4.93 + 28.1 a few academic centers on selected topics. In Intelligent Systems 5.35 5.67 + 6.0 contrast, NSF responds to unsolicited proposals Computer Systems in all areas of computer research from all research Design 3.50 3.47 0.9 performing institutions. As a result, NSF sup­ Coordinated Experi- port is more widely distributed in both subject mental Research 15.10 14.40 4.6 and locale and plays a unique role in discover­ Special Projects 0.92 3.42 +271.7 ing knowledge and strengthening the country's Computer Research scientific potential. Equipment 1.50 1.50 0 Total $40.17 $44.44 10.6% Changes Between the FY 1986 Budget Request and the The FY 1987 Request is for an increase of $4.27 FY 1986 Current Plan million or 10.6%, above the FY 1986 Current Plan. The Computer Research budget in FY 1987 (Millions of dollars) reflects two new NSF activities. FY 1986 FY 1986 Percent • Computational Science and Engineering Activity Actual Plan Change {$1.01 million in the Software Engineering Theoretical Computer Program Element). This will support 3-5 in­ Science $ 4. 758 $ 5.15 + 8.4% Software Systems vestigator interdisciplinary research groups, Science 4.40 4.80 + 9.1 advanced computational facilities, and indi­ Software Engineering 3.85 3.88 0 vidual research projects to discover computa­ Intelligent Systems 5.40 5.35 - 0.9 tional methods appropriate for new computer Computer Systems architectures. Design 4.18 3.50 -16.3 • Research on Advanced Computers {$3.0 mil­ Coordinated Experi- lion; $2.0 million in the Special Projects mental Research 16.30 15.10 - 7.4 Program Element and $1.0 million in other Special Projects 1.06 0.92 -13.2 program elements). This provides focused re­ Computer Research sources to study advanced computer concepts Equipment 1. 79 1.50 -16.2 and systems. Several areas have been identi­ Total $41.73 $40.17 - 3.7% fied that need this type of support including The FY 1986 Current Plan is 3. 7% below the advanced computer architectures, software is­ FY 1986 Budget Request. This decrease cor­ sues arising from new computer architectures, responds to the unspecified Congressional Ap­ and information structure, cognition and for­ propriation reduction. This reduction was not mal languages. $3.0 million will enable NSF spread evenly across the programs. Increased to make a start toward developing these new proposal pressure, coupled with identified needs research directions. The large project portion for more support for research in the foundations of this activity, $2.0 million, will be adminis­ of computer science and computer engineering, tered through the Special Projects Program have led to increases in the more theoretical com­ Element and $1.0 million will be used to puter research project support programs and a increase the number of smaller faculty and resulting larger decrease in other programs. The graduate student science research projects in priorities are consistent with plans to emphasize these areas. standard research project support over infras­ Other science research project support will tructure support, with substantial reductions be­ be increased by $1.46 million, or 6.2%, to a total ing made in the Computer Research Equipment, of $25.03 million. This will allow limited expan­ Special Projects, and Coordinated Experimen­ sion of support for research on the foundations tal Research programs. The reduction to the of computer science and will provide increased Computer Systems Design Program reflects grow­ support for young faculty and graduate students ing cooperation and joint sponsorship with other in all research areas. These items have very high

456 priority in academic computer research at this (2) for research communities to collect and ana­ time. lyze data leading to new insights concerning key Because of the extreme need for faculty scientific problems; and (3) for graduate students research projects and graduate student support, to obtain training in the use of supercomputers. the Coordinated Experimental Research Program Effective development of supercomputer fa­ Element will be reduced by $0.7 million to $14.4 cilities requires the creation of a national scien­ million. This will impose a net decrease of one tific research network (NSFNET). Initially, points or two in the number of active CER awards. The of access will be added to existing networks total number will drop to about 20 or 21 with (ARPANET, BITNET, CSNET, MAILNET, three or four new or renewal awards replacing the MFENET) to provide access for NSF-supported five active awards terminating in FY 1987. researchers. At first, not all supercomputers will be directly accessible from all networks, but enough paths will be provided to assure effec­ tive access. Subsequently it will be necessary The following text was prepared by mem­ to try several experiments to provide information bers of the staff of the Office of Advanced Scientific Computing of NSF and was submit­ required for the development of NSFNET. ted to Congress as a part of the material which The challenge for NSF is to establish both accompanied the Administration's Budget Re­ centers and network access, to make efficient use quest for the Fiscal Year 1987. of existing resources and capacities, and to create facilities dedicated to meeting the computational requirements of the many science and engineering disciplines. The time allocation chart which Advanced Scientific Computing follows shows NSF experience to date in both $53,630,000 the range and the degree of utilization among Summary of Request disciplines. Supercomputer Allocation (Millions of dollars) By Program FY 1985 FY 1986 FY 1987 August 1984-November 1985 Activity Actual Plan Request Advanced Scientific Computing $41.40 $45.23 $53.63 The FY 1987 Budget Request for Advanced Sci­ entific Computing is $53.63 million, an increase of $8.40 million or 18.6% over the Current Plan of $45.23 million.

Scientific Overview The Advanced Scientific Computing activity pro­ vides high quality access to unique advanced com­ putational facilities for all segments of the science and engineering basic research community. HOURS The specific objectives are: • To provide the large scale computational NSF Role resources essential both to the research and NSF undertakings are consonant with similar ac­ to the training of scientists and engineers. tivities undertaken or planned by other more • To extend and develop networks and com­ mission-oriented Federal agencies, most notably munications systems which assure access to the Department of Energy, Department of De­ large scale computing facilities. fense, the National Oceanic and Atmospheric Ad­ • To create computer programs for making ministration, and the National Aeronautics and software more portable; making operating Space Administration, which support research in systems more user-friendly; providing more weapons development, fusion energy, aerodynam­ efficient programs for scientists; and devel­ ics and atmospheric science. The NSF activity, as oping means to handle, store and interpret it provides advanced computing facilities to the massive amounts of data. academic basic research community, will make Supercomputers are powerful instruments; maximum use of the facilities and the knowledge they have contributed and will continue to con­ of other Federal agencies. tribute significantly to the advancement of scien­ The centers program focus is on providing tific and engineering research. Access to new com­ large scale computational resources to all scien­ putational centers for use in basic research activi­ tific and engineering disciplines for basic research ties will increase opportunities (1) for researchers requirements via national NSF supercomputer to attack previously intractable problems through centers. NSF, with considerable supercomputer the development of complex numerical models; center experience, particularly in the atmospheric

457 sciences, occupies a key position for creating this University, Purdue University, Boeing Computer new science resource. Services, Ana' Bell Labs, and Digital Productions. The approach of creating an Internet, or NSF is currently establishing three national network of networks, will ensure connection of supercomputer centers: the San Diego Super­ the national supercomputer centers and the more computer Center (University of California at San than 30 academic institutions of the two super­ Diego, CRAY X-MP /48); the National Center computer center consortia with other users of the for Supercomputing Applications (University of expanded ARPANET. Steps have been taken to Illinois, CRAY X-MP/24); and the John Von design an interagency plan to interconnect exist­ Neumann Center for Scientific Computing (near ing individual agency networks. A. Networking Princeton University, CYBER 205 interim machine Working Group has been formed as part of the which will be replaced by an ETA 10). A Federal Coordinating Committee on Science, En­ fourth center, the Pittsburgh Center for Advanced gineering, and Technology in order to provide Computing in Engineering and the Sciences, is in coordination for this effort. the process of initiation. In addition to providing Through a new technologies effort NSF seeks supercomputer services to academic researchers, to provide unique supercomputer hardware, and the national centers also encourage collaborative to facilitate engineers' and scientists' access to projects among visiting researchers. high powered machines via ·both user-friendly During FY 1986 NSF has been working with computer interfaces and improved software. NSF the Department of Defense agencies, Department is especially capable of supporting such projects of Energy, NASA, and Department of Commerce because of its commitments to scientific and to develop plans for NSF NET. This will build on engineering disciplines, to computer research, to the experience of DARPA with ARPANET, NSF mathematical development, and to the interface with CSNET, and other government and private areas among these fields of science. networks as well as the academic community's network, BITNET. A medium speed "backbone" Changes Between the network connecting the national centers was be­ FY 1986 Budget Request and the gun in 1986, ensuring that supercomputer users FY 1986 Current Plan on campuses with ARPANET connections or mem­ (Millions of dollars) bers of consortia networks will be able to gain FY 1986 FY 1986 Percent access to all NSF supercomputer centers. Activity Request Plan Change In FY 1986 the experimental systems compo­ Advanced Scientific nent of the new technologies effort has focused on Computing $46.23 $45.23 -2.16% a pilot program in interactive graphics and user­ A reduction of one million dollars from the friendly, intelligent, and transparent operating FY 1986 Budget Request has been made in the interfaces between keyboard and supercomputer. centers program, and the funds reprogrammed to The program also has supported algorithm and support development of an advanced supercom­ software development. In addition, an experimen­ puter system at the National Center for Atmo­ tal supercomputer center was opened at Cornell spheric Research. University where access is provided to an ad­ vanced parallel system consisting of an IBM 3084 FY 1987 Budget Highlights QX mainframe with a number of attached FPS 264 scientific processors. Parallel processing schemes (Millions of dollars) effectively dissect complex mathmatical [sic] prob­ FY 1986 FY 1987 Percent lems into several independent calculations which Activity Plan Request Change can be performed concurrently rather than se­ Advanced Scientific quentially, thus providing considerable increases Computing $45.23 $53.63 18.57% in performance. The requested increase will support significant growth of activities at the national centers, and a continuing level of effort in ne,twork and experi­ FY 1987 Budget Highlights mental technology development. The Phase I resource center activity will be continued at reduced funding levels. Phase II Background Centers (Illinois, Princeton, and San Diego) would The plan of the centers program has entailed a two be maintained at an augmented capabilities level phase approach: 1) to immediately provide time on and the John von Neumann Center will carry supercomputers via lease agreements with existing out the planned upgrade from a CYBER 205 supercomputing facilities, and 2) to replace the to an ETA 10. The Pittsburgh Center will be lease plan with new national centers. In FY 1985- undergoing rapid growth. Resource Centers will 86, NSF acquired services from six organizations be supported only when they provide services not to fill the immediate large scale computational readily available at the national centers. needs of researchers. These centers are located The NSFNET component of the network­ at the University of Minnesota, Colorado State ing program remains essentially the same as in

458 FY 1986. This level of support allows the con­ The following text was prepared by the tinuation of the commitments to the consortia staff of the Science and Engineering Educa­ networks, the community wide area networks tion Directorate of NSF and was submitted (especially ARPANET), the NSF "backbone" net­ to Congress as a part of the Administration's work, and the existing networking pilot projects, Budget Request for the Fiscal Year 1987. plus limited support for network management and operations. No new networking projects will be begun. The local access component of the Science and Engineering Education networking program will remain the same as in FY 1986, allowing funding of additional campus $89,000,000 gateway systems so as to connect an additional Summary of Request 20-30 campuses to the ARPANET. The FY 1987 Budget Request for Science and The new technologies program will upgrade Engineering Education (SEE) is $89,000,000, an the Cornell center's IBM 3084 to an IBM 3090 increase of $1,955,675, or 2.2 per cent over the in order to provide more supercomputing power FY 1986 Current Plan of $87,044,325. to the user. Software technology and training will be furnished to accomodate a larger cross (Millions of dollars) section of scientists and engineers; workshops will FY 1985 FY 1986 FY 1987 Activity Actual Plan Request be supported to introduce investigators to the unique capabilities afforded by the facility. The Research Career Development $27.30 $27.34 $27.30 request also allows for the possible development of Teacher Preparation and a large scale, massively parallel computer system. Enhancement 25.19 27.00 27.00 This program will also continue to develop in­ Materials Development, ResearcJg and Informal novative software, numerical methods and graph­ Science ducation 22.72 25.00 25.00 ical techniques to meet the needs of the scientific College Science and engineering communities. Included are user­ Instrumentation 5.00 5.50 7.50 friendly and transparent interfaces between the Studies and Program user and the computer, transportable computer Assessment 1.75 2.20 2.20 programs and efficient mathematical and compu­ Total $81.96 $87.04 $89.00 tational methods. The table which follows displays the funding NSF Role aspects of these plans for FY 1987 and relates The responsibility of the Science and Engineering them to prior years. Education (SEE) Activity is to define and fund (Millions of dollars) programs and projects that support the educa­ FY 1985 FY 1986 FY 1987 tional aspects of the Foundation's mission. The Actual Plan Request magnitude of the educational effort in the United Centers States and the long lead times needed for new JVNC (Princeton) $ 8.3 $ 8.0 $11.5 programs, materials and methods require a con­ San Diego 4.7 9.1 11.5 tinuing and significant involvement in this area. Illinois 5.0 7.0 8.5 Sustaining this level of commitment, visibility, Pittsburgh 0.0 2.0 4.0 and continuity expresses the serious national con­ Resource Centers 10.2 3.4 2.3 ------cern with science, mathematics, and engineering Subtotal $28.2 $29.4 $37.8 education and will draw the best, most creative Networking people into the process. NSFNET $ 3.7 $ 5.8 $ 6.0 SEE fulfills this responsibility by conducting Local Access 2.2 1.0 1.0 leadership activities that inform and/or stimulate -- -- other sectors and by support of original work and Subtotal $ 5.9 $ 6.8 $ 7.0 other merit-based high-leverage activities that New Technologies serve as prototypes and models of excellence for Experimental Center the Nation. A major objective is to encourage ap­ (Cornell) $ 4.9 $ 6.3 $ 6.2 propriate cooperation among academic scientists, Experimental Systems 0.9 2.7 2.6 -- -- engineers, educators and the private sector for Subtotal $ 5.8 $ 9.0 $ 8.8 intellectual partnerships as well as for leveraging funds. The role of SEE is to help insure that: • a high-quality precollege education in science and mathematics is available to every child in the United States, sufficient to enable those who are interested and talented to pursue technical careers, especially in science and engineering, as well as to provide a base for understanding by all citizens.

459 • those who select scientific and engineering instrumentation in four-year colleges. Moreover, careers have available the best possible pro­ a special committee of the National Science Board fessional education in their discipline. While is now completing an in-depth analysis of the other NSF programs provide apprenticeship problems facing undergraduate education and will training for the next generation of scientists recommend an appropriate NSF response. and engineers through research support, SEE NSF continues its diverse activities designed concentrates on the educational structure to ensure a continuing flow of the best and leading to this stage; that is, the teachers, brightest students into careers in mathematics, students, laboratory, and classroom resources science and engineering. This kind of activity that remain important through undergradu­ has been a hallmark of NSF's role in science and ate and graduate studies. engineering education and will continue to be • opportunities are available at the college level prominent in its programs into the future. for interested nonspecialists to broaden their SEE has a special role in the NSF with science, mathematics and technology back­ regard to increasing the participation of women, grounds. The great majority of students need minorities and the physically disabled in science a technical perspective and understanding and engineering. Better educational opportunities to give them insights into and acquaintance and conditions at all levels are ways to success in with the principles, practices, techniques, and this important area, and these are sought by SEE limits of science. in all phases of its mission. • informal science education programs are SEE coordinates its activities closely with available to maintain public awareness of other relevant areas of NSF and with other agen­ and interest in scientific and technological cies, e.g., NASA, Department of Energy and the developments that may affect their lives. Department of Education (DoEd). Such coordi­ Particular attention is focused on the prob- nation is intended to promote maximum use of lems and strategy of assistance to precollege expertise in the agencies, to minimize program education-the area of greatest complexity and overlap, and to optimize the use of limited Fed­ NSF's most urgent responsibility. The scope and eral resources. NSF and DoEd, for example, have complexity of the problems are so great and so collaborated in supporting a number of projects, diverse that we cannot expect wholly adequate including the International Study of Mathematics or well-coordinated solutions to emerge sponta­ and the broadcast series 3-2-1 Contact and The neously and totally from local, or even state Voyage of the Mimi. institutions. Here, a Federal leadership role is particularly critical. Changes Between the At the precollege level, NSF support is now FY 1986 Budget Request and targeted toward resolving problems of: FY 1986 Current Plan • inadequate subject matter competence of cur­ (Millions of dollars) rent teachers; FY1986 FY1986 Percent • inadequate subject area pre-service teacher Activity Request* Plan Change preparation; Research Career • classroom materials that are outdated or Development $27.30 $27.34 0.2% poorly adapted to the needs of major popu­ Teacher Preparation and Enhancement 25.00 27.00 8.0 lations; Materials Development • ineffective use of advanced technology; Research and Informal • poor exchange of information about success­ Science Education 22.70 25.00 10.1 College Science ful techniques and programs; and Instrumentation 5.00 5.50 10.0 • insufficient opportunities for many talented Studies and Program young men and women to gain experiences Assessment 2.00 2.20 10.0 that are critical to their early understanding Total $82.00 $87.04 6.1% and appreciation of science and mathematics. * Includes $31.45 FY 1985 Deferral plus $50.55 in new A current major reflection of SEE's leadership role obligational authority. is the special emphasis on elementary and middle The SEE Activity received an appropriation of school levels of education. These are critical $55,550,000 for its activities in FY 1986. This stages in the educational process and present amount represents an increase of $5,000,000 from problems of great magnitude. the FY 1986 Budget Request of $50,550,000. The In informal science education, NSF will con­ FY 1985 carryover into FY 1986 was $31,494,598 tinue, but not expand, support for "open-air" results from a planned deferral of $31.45 million televison [sic] broadcasts, and will increase sup­ in the precollege areas and a small additional port to help museums expand their role as a focus carryover of $0.04 million in the Research Career and locale for science clubs, hobby groups and Development subactivity. The FY 1986 Current other activities that encourage science interest. Plan totals $87,044,325 and reflects the net effect At the undergraduate level, NSF has begun of the FY 1985 deferral and carryover and the to attack the serious problem of deteriorating FY 1986 new obligational authority.

460 (Millions of dollars) The FY 1987 SEE budget increase of $2.0 mil­ FY 1986 FY1987 Percent lion will be concentrated in the College Science Activity Request Plan Change Instrumentation subactivity, bringing it to a level Research Career of $7.5 million. Because of the severity of needs Development $27.34 $27.30 - 0.2% in the undergraduate engineering disciplines $3.0 Teacher Preparation and million will be targeted for strengthenin~ the Enhancement 27.00 27.00 0 instructional engineering laboratories in the Na­ Materials Development, tion's colleges and universities. The remaining Research, and Informal Science Education 25.00 25.00 0 $4.5 million will be used to continue strengthening College Science the laboratory instruction at four-year colleges as Instrumentation 5.50 7.50 +36.4 was done in FY 1985 and FY 1986. Studies and Program Assessment 2.20 2.20 0 Changes in Budget Structure Total $87.04 $119.00 + 2.3% For better compatibility and administrative effi­ The allocations for the Research Career De­ ciency, all precollege teacher program areas are velopment; Materials Development, Research, and now associated within one subactivity, and in­ Informal Science Education; Teacher Preparation formal science education is now associated with and Enhancement; and Studies and Program As­ materials development and research. Therefore, sessment subactivities are unchanged from the the former Materials Development and Research FY 1986 Current Plan, enabling them only to subactivity is now termed Materials Develop­ maintain their base programs. However, this will ment, Research, and Informal Science Education include 560 new graduate fellowship awards in and the former Teacher Enhancement and Infor~ Research Career Development, and additional so­ mal Science Education subactivity is now termed licitations for proposals targeted at specific prob­ Teacher Preparation and Enhancement. lems in science and mathematics in the Nation's elementary and secondary schools.

461 The project that is the subject of this report CATHLEEN S. MORAWETZ was approved by the Governing Board of the New York University National Research Council, whose members are RONALD PYKE drawn from the councils of the National Academy University of Washington of Sciences, the National Academy of Engineering, WERNER C. RHEINBOLDT and the Institute of Medicine. The members of the University of Pittsburgh committee responsible for the report were chosen SHIN-TUNG YAU for their special competences and with regard for University of California, San Diego appropriate balance. FRANK GILFEATHER, Staff Director This report has been reviewed by a group MEG KNEMEYER, Staff Assistant other than the authors according to procedures SEYMOUR SELIG, Staff Officer approved by a Report Review Committee con­ sisting of members of the National Academy of Commission on Physical Sciences, Sciences, the National Academy of Engineering, Mathematics, and Resources and the Institute of Medicine. HERBERT FRIEDMAN This study by the Board on Mathematical Sci­ National Research Council, Chairrrw.n ences was conducted under Contract No. DMS- CLARENCE R. ALLEN 8514639 with the National Science Foundation. California Institute of Technology Copies available from: The Board on Mathemat­ THOMAS D. BARROW ical Sciences, 2101 Constitution Avenue, N.W., Standard Oil Company, Ohio (retired) Washington, D.C. 20418. ELKAN R. BLOUT Harvard Medical School Panel on Mathematical Sciences BERNARD F. BURKE PHILLIP A. GRIFFITHS Massachusetts Institute of Technology Duke University, Chairman GEORGE F. CARRIER HYMAN BASS Harvard University Columbia University CHARLES L. DRAKE PETER BICKEL Dartmouth College University of California, Berkeley MILDRED S. DRESSELHAUS ALEXANDRE CHORIN Massachusetts Institute of Technology University of California, Berkeley JOSEPH L. FISHER RICHARD DUDLEY George Mason University Massachusetts Institute of Technology JAMES C. FLETCHER WENDELL FLEMING University of Pittsburgh Brown University WILLIAM A. FOWLER RONALD L. GRAHAM California Institute of Technology AT&T Bell Laboratories GERHART FRIEDLANDER DAVID KAZHDAN Brookhaven National Laboratory Harvard University EDWARD D. GOLDBERG CATHLEEN S. MORAWETZ University of California, San Diego New York University MARY L. GOOD RICHARD SCHOEN Allied Signal Corporation University of California, San Diego J. Ross MACDONALD MASAMICHI TAKESAKI The University of North Carolina at Chapel Hill University of California, Los Angeles THOMAS F. MALONE MICHAEL E. TAYLOR Saint Joseph College State University of New York at Stony Brook CHARLES J. MANKIN FRANK L. GILFEATHER, Staff Director The University of Oklahoma Board on Mathematical Sciences PERRY L. MCCARTY Stanford University PHILLIP A. GRIFFITHS WILLIAM D. PHILLIPS Duke University, Chairman Mallinckrodt, Inc. HYMAN BASS ROBERT E. SIEVERS Columbia University University of Colorado PETER BICKEL JOHN D. SPENGLER University of California, Berkeley Harvard School of Public Health RALPH A. BRADLEY GEORGE W. WETHERHILL University of Georgia Carnegie Institution of Washington HIRSH COHEN IRVING WLADAWSKY-BERGER IBM Corporation IBM Corporation E. F. INFANTE RAPHAEL G. KASPER, Executive Director University of Minnesota LAWRENCE E. MCCRAY, Associate Executive Director JOSEPH J. KOHN Princeton University Mathematical Sciences: A Unifying and Dynamical Resource

Report of the Panel on Mathematical Sciences, initiated by the National Research Council

Preface a fundamental driving force in the worldwide progress that is altering the economic, political, During the summer of 1985, a group of state­ and social balance among nations. It is essential of-the-art reviews was initiated by the National for the United States to maintain momentum in Research Council (NRC) at the request of the mathematics if we are to maintain our overall National Science Foundation. The purpose of competitiveness in other areas. these reviews is to assess and monitor world The National Research Council, the scientific trends, relative strengths, and competitiveness of community, and I are deeply indebted to the panel the United States in rapidly evolving areas of members, the many colleagues who assisted them, science and technology. Particular emphasis is to and the able and conscientious reviewers. Thanks be placed on developments that influence the rate also go to Patricia Kenschaft, a mathematician at at which these fields evolve. Three study areas­ Montclair State College, for her excellent mathe­ cell biology, pure and applied mathematics, and matical editorial work. The chairman is especially materials science-were chosen for review. grateful to the staff of the Board on Mathematical The study on mathematics was conducted by Sciences, which under the leadership of Frank L. the Panel on Mathematical Sciences under the Gilfeather provided throughout the project the auspices of the Board on Mathematical Sciences support so essential to its successful completion. of the NRC's Commission on Physical Sciences, Those responsible for guiding science policy Mathematics, and Resources. The Panel has in the Congress and the Administration are the described major trends in modern mathematics primary audience for this report. The report and then illustrated them with a few "vignettes." will be useful to other audiences, too: leaders of Through this case-study approach, the Panel il­ universities, the mathematical sciences research lustrates its assertions that we are in a dynamic community, and ctlso those who are inquisitive period of mathematical discovery and that math­ about the mathematical sciences, about its struc­ ematics is the fundamental discipline of science ture and its current directions. and hence a critical U.S. resource. International competition and competitive­ Phillip A. Griffiths, Chairman ness are concepts that traditionally are alien to Panel on Mathematical Sciences the study of mathematics, in fact, cooperation among individuals of different nations is a vital Executive Summary part of the pursuit of solutions to mathematical problems. Mathematics is intrinsically inter­ Trends. This report emphasizes four major trends national, with its own language cutting across in the mathematical sciences: barriers of geography and culture. Research and • Mathematical sciences research is strong applications in engineering and the physical sci­ worldwide, and the United States is main­ ences and, more recently, in business and the social taining the leading role. sciences rely increasingly on mathematics for their • Mathematics is unifying internally. basic structure, for the modeling of phenomena, • Applications of mathematics in both tra­ and as the basis of new computational directions. ditional and new areas are flourishing and Mathematics is a major field of application of involving more central areas of mathematics. the recent powerful computer advances and is the • Mathematics is the driving force behind new fundamental discipline underpinning both the de­ areas of computational science and is in turn velopment of the computer techniques themselves profoundly influenced by high-speed comput­ and the applications of these techniques in all ing. fields. These trends, which demonstrate the vitality Mathematics has become, even more directly, of the mathematical sciences as well as the chang­ the language and the foundation of science, tech­ ing nature of this critical discipline are illustrated nology, and social organization. Indeed, it is in a series of six vignettes given in Chapter 3.

463 There are several corollaries of the trends, maintain its vitality. The figures cited above for which are especially notable in view of the de­ Ph.D. production reveal a precarious situation, clining U.S. Ph.D. production .in mathematical and we believe that it may be worsening. In sciences, as seen in the table below. The expand­ addition, if we are to assure the quality of the ing number and sophistication of tools needed mathematics enterprise in the United States, the for successful research as areas of mathematics opportunities for postdoctoral training must be become intertwined now require protracted study significantly expanded. often beyond the Ph.D. degree. This corollary As we provide sufficient opportunities for development is especially critical for those work­ young investigators and students, we must also ing at high levels of applications of mathematics. maintain the strength and vitality of senior lead­ The difficulty in reaching mathematical research ership. That the United States maintains math­ frontiers with the requisite deep, broad range of ematical pre-eminence is due in part to the knowledge, without the opportunity for extended commitment and investment made in the past study, may partially explain the decreasing Ph.D. to an outstanding group of researchers. Unfor­ production. This decrease will continue if the po­ tunately, this commitment has been considerably tential talent does not feel that continuing, as well weakened over the past decade as was documented as entry level, research support will be available. in the 1984 NRC report Renewing U.S. Mathe­ Some of the vignettes point out that the matics: Critical Resource for the Future (National number of young researchers is of great concern Academy Press, Washington, D.C., 1984). The and that considerable talent must be imported challenge of current policy is to provide greater to keep the United States in the forefront of opportunities for the young while maintaining our mathematical research fields. With the continued current strength in leadership. The two goals are resurgence of European mathematics, this cannot inextricably related. be a permanent solution to manpower shortages. While the United States maintains leadership in Introduction most areas of mathematics, in some important fields new talent is only sparsely available. This expository report reviews world trends in Ph.D. Production and Support. These issues mathematics and assesses the position of Ameri- of manpower and infrastructure relate directly can mathematics in the world mathematical com- to how the United States stands in production munity. It was written in response to a request of new Ph.D.s and in support of postdoctoral from the National Science Foundation to the fellowships and graduate students in the sciences Board on Mathematical Sciences. Chapter 2 de- generally. The results as summarized below and lineates four current trends in the development of reported in Chapter 4 (Table 1) are dramatic. the mathematical sciences. Chapter 3 contains Challenges. Continued concern should be six short vignettes, illustrating some of the major expressed over the decline in the number of Ph.D.s efforts in current mathematical research. The in the mathematical sciences in the United States, topics of the vignettes are: and in view of the critical role of mathematics • D-modules this trend must be reversed. It is essential that • Computational Complexity the long-range attractiveness of the field and the • Nonlinear Hyperbolic Conservation Laws prospects for success within it be such as to • Yang-Mills Equations attract able researchers. The competition for • Operator Algebras students among the various fields of science is • Survival Analysis keen, and it is critical that mathematics be able The vignettes substantiate the trends and to provide sufficient inducements to candidates to illustrate the excitement and struggle of mathe- Comparisons of Federal Support for Postdoctoral and Graduate Students in Three Fields of Science Mathematical Chemistry Physics Sciences 1980 1984 1980 1984 1980 1984 Annual Ph.D. Production 1538 1765 862 962 744 698 Postdoctorals (P.D.) federally supported 2255 2473 1210 1150 57 132 Graduate Students (G.S.) federally supported 3700 4118 2900 3348 200 411 Ratio P.D./Ph.D. 1.47 1.40 1.40 1.20 0.08 0.19 Ratio G.S./Ph.D. 2.41 2.33 3.36 3.48 0.27 0.59 Sources: NRC Survey of Doctorate Recipients. Survey of Graduate Science and Engineering Students and Postdocs, NSF. Survey of Doctorate Recipients, NSF, unpublished, Table D-32.

464 matical research, indicating more accurately than a long list of important new efforts. The recent could any compilation of areas and results the solution of the deep Bieberbach Conjecture in emphasis and vitality of the discipline. Moreover, classical function theory showed that the tenacity since specific research results are the substance of of a senior individual investigator is still a vital living mathematics, our general observations are force in mathematics. Finally, the solution of made more meaningful through the six vignettes. Mordell's Conjecture places mathematicians on The report closes with a brief epilogue, which the threshold of proving the enticing Fermat's records the key recommendations of the NRC Last Theorem. report Renewing U.S. Mathematics: Critical Re­ Within all areas of mathematics a spectacular source for the Future and remarks on the response number of major but lesser known problems to and impact of this important report. have been solved recently, including problems illustrated in the vignettes in Chapter 3. Further Trends documentation is presented in the forthcoming report of the Board on Mathematical Sciences, Mathematics is unifying, and it continues to be Survey of U.S. Mathematical Sciences. a dynamic resource in science. Of the many Unity of Mathematics. The unification that indications of this, the following are outstanding: is taking place within mathematics is obvious to • Difficult problems that have been unsolved people in the field and will be apparent in the for many years are now being solved with vignettes about D-modules, computational com­ amazing frequency- a strong confirmation plexity, the Yang-Mills equations, and operator of the vitality of mathematics. algebras. Unification occurs when there is a • Mathematics is unifying internally. The confluence of seemingly independent phenomena, division between pure and applied mathe­ motivating cooperative study of the significant matics that developed in the first part of underlying patterns. One symptom of this ac­ the twentieth century and allowed for the celerating unification is the increasing difficulty rapid development of new fields is now disap­ that agencies are having in assigning proposals to pearing. Moveover, the distinctions between their discipline programs, which now substantially traditional areas of specialization have be­ overlap. The trend burdens young investigators come blurred. with a need to pursue increasingly broad train­ • Traditional areas of applied mathematics are ing, as is noted in several of the vignettes. In flourishing. Further, significant, deep inter­ mathematics it is becoming critical to lengthen actions are occurring directly between core the training period substantially. mathematics and other fields, including the An example of this confluence of areas is the natural sciences, engineering, and the social Korteweg-de Vries (KdV) equation Ut + uux + sciences. Increasing interactions are occur­ Uxxx = 0 [where u is an unknown function u(x, t) ring not only with the traditional areas of in one space dimension and time], which arises physics, chemistry, and engineering but also both as the simplest nonlinear dispersive equation with newer fields such as molecular genetics, in shallow-water wave theory and as the equation business, sociology, information sciences, and of isospectral evolution of the potential in the studies in policy analysis. Schrodinger equation '¢xx + (k2 - u)'¢ = 0 of • Mathematics is the underpinning of the revo­ quantum mechanics. The intensive study of the lutionary changes taking place in all scientific KdV equation during the past quarter century and engineering fields as a result of the has affected many major areas of mathematics. advent of powerful computers. The devel­ For example, about a year ago a young Japanese opment of scientific computing has not only mathematician, using a development in the study highlighted a host of critical new mathemat­ of KdV equations initiated a decade ago by ical problems it has introduced new tools for the Moscow school, solved a major problem in mathematicians. algebraic geometry that was first discussed more With the necessary support, mathematics will than a century ago by Riemann, a German. continue to flourish, to attract excellent minds, · Mathematics and Other Sciences. The sym­ and in the coming years to produce much essential biotic relationship between mathematics and its new mathematics on an international basis. The areas of application is ever deepening as more role of any particular country in this development areas of science and engineering become almost is hard to predict. Mathematical leadership will indistinguishable from subareas of mathematics depend on many factors, primary among them and this relationship is producing exciting and the support that individual nations give to basic intriguing new mathematics. Cross-disciplinary sciences in general and mathematical research in collaboration between mathematicians and profes­ particular. sionals in other fields is accelerating and deserves Recent Breakthroughs. Among recurring encouragement. An important number of in­ solutions to previously unsolved problems, several teractions between mathematics and science and examples merit mentioning. One of them is the engineering are not exactly interdisciplinary, but program to classify finite groups, which generated might be more accurately described as resonance

465 phenomena in that advances in one field spur the behavior of the solutions to the KdV equation development in another. Examples of this im­ mentioned above was first discovered numerically. portant trend are described in the vignettes on The mathematical theory, in turn, provides a Yang-Mills Equations and Operator Algebras in deeper understanding of the models, revealing Chapter 3. The broadening and deepening of phenomena that enable people to analyze and these applications, as noted in the vignette on test previous computational results and conceive Nonlinear Hyperbolic Conservation Laws, create of new computations that will facilitate further pressure for mathematicians to pursue significant theory. postdoctoral study. An insightful and illuminat­ Core mathematics now consists of three ba­ ing discussion of the interaction of mathematics sic operations-computation, abstraction, and with science and engineering is given in the 1984 generalization. Raw information leading to math­ NRC report Computational Modeling and Mathe­ ematical discovery comes from concrete examples, matics Applied to the Physical Sciences (National and it is increasingly the role of computation to Academy Press, Washington, D.C., 1984). provide such examples, although they are fre­ New Opportunities. The third trend includes quently formulated mathematically. Abstraction the tendency for mathematics to be incorporated is the process of distilling the essential features into the language not only of science and tech­ from such examples. Number and space are, re­ nology, as has been traditional, but also into spectively, abstractions of the process of counting new fields of social science. Mathematical models and of our experience in the physical world, and (descriptions of real-world events that use mathe­ the mathematical idea of functions similarly ab­ matical language) form the basis of econometrics stracts human ideas of measurement and motion. and health policy analysis. The Survival Anal­ In these contexts, ideas from one manifestation ysis vignette in Chapter 3 demonstrates this; it of the abstraction are often relevant in solv­ examines statistical and mathematical methods ing problems in seemingly dissimilar situations. used to provide a realistic analysis for problems in Generalization uncovers hidden analogies between medical research, reliability theory, actuarial com­ abstract patterns of mathematics and frequently putations, and demographic studies. Mathemati­ extends the range of applicability of such patterns. cal analyses contribute substantially to decisions Thus in the development of mathematics, pe­ about economic and health policies, which in turn riods of computation often alternate with periods have enormous financial and social consequences. of theorizing. During the former, new raw data There is no longer any question as to whether are generated and horizons expanded. Eventually mathematical analysis will substantially influence there is a plethora of information in need of an in­ discussions of public policy but only whether it tellectual framework on the basis of which masses will be used appropriately and effectively. It is of material can be comprehended simultaneously. essential that those making the decisions under­ As reunification occurs, seemingly disparate ex­ stand and influence the assumptions used to form amples are often revealed as different aspects of the mathematical model and that mathematicians the same phenomenon. comprehend the applications sufficiently well that Computation, abstraction, and generalization they address and solve the correct problem. In need each other to be meaningful. Recent math­ fields where mathematical models are not sub­ ematics has focused more on concrete problems ject to experimental verification-such as those than on abstractions as it revels in the computing with the most drastic consequences-it is espe­ power suddenly available to it, and evidence of a cially essential that the mathematics be critically new unification is now appearing. At the same scrutinized. time, our computing power has matured to the The Role of Computation. Computational point where it can be an enormous asset in the mathematics is an integral part of the mathemat­ investigation of the still more complex mathemat­ ics discussed in the vignettes about complexity ical examples that will inevitably be suggested and nonlinear hyperbolic equations, and there are by pending research. A beautiful amplification of two important observations worthy of emphasis. this theme on the development of mathematics is First, computational methods pervade almost all given by Arthur Jaffe in an Appendix to the NRC aspects of science, and mathematics is the founda­ report, Renewing U.S. Mathematics: Critical Re­ tion of these methods. Today's complex problems, source for the Future (National Academy Press, involving computational solutions, range from the Washington, D.C., 1984). design of computer architecture itself through A dramatic example of using the computer the mathematical modeling of physical, chemical, to gain valuable mathematical insight can be seen biological, and engineering processes. Mathe­ on the cover of this report, which represents a matics, the intellectual basis of computational problem in the geometry of surfaces of constant science, has been and will continue to be the key mean curvature, a subject with applications to to the dynamic revolution being created by the mathematical physics, polymer chemistry, archi­ computer in science and engineering. Second, tecture, and other sciences. The existence of computational results provide insight for the de­ a new complete embedded minimal surface with velopment of mathematical theory. For example, finite topology, the first to be found in more

466 than 200 years, was established using computer­ of the entire mathematical community. A more aided graphics. The complex representation of exhaustive, although still selective, survey of the a potential example was analyzed by numerical mathematical sciences, Survey of U.S. Mathemat­ approximation and three-dimensional computer ical Sciences, is currently being prepared under graphics. An unexpected symmetry was revealed the auspices of the NRC Board on Mathematical in the surface, which was shown to exist in Sciences. Together these reports can provide only the equations themselves. Further analysis us­ a glimpse of the scope of current mathematical ing both traditional and computational methods research. Our choice of specific mathematical established the existence of families of new ex­ topics reflects our effort to select a representative amples as well as a rich new theory to explain sample of the entire mathematical enterprise. their existence. The ability to create accurate Language of Mathematics. There is another computer images has greatly facilitated communi­ significant challenge that we must face in our cation among scientists. The computer methods endeavor to explain mathematics to those without developed in this research are proving to be use­ years of study in the field. At least three centuries ful in solving problems in related mathematical ago, mathematics developed a language of its subjects. own, which has become thoroughly distinctive Organization of Mathematics. In addition and international. Just as it takes years for an to these four major directions in mathematics American or Japanese youngster, for example, to itself, what might be called "The Sociology of become fluent in the spoken language of the other, Mathematics" reveals how mathematics develops any aspiring American or Japanese mathematician and accentuates the unification within mathemat­ spends years studying the common language of ics and its growing interaction with other fields. mathematics. As a result, each can open the More and more the mathematical community is other's mathematics books and recognize the now organizing itself by areas of interest instead topic under discussion (even if totally ignorant of of fields of traditional study. For example, the the other's verbal language). Yang-Mills vignette describes how a variety of The complex and dynamic language of math­ mathematicians educated in the disparate fields ematics that has developed over the past three of topology, algebra, differential geometry, alge­ centuries brings great satisfaction in terms of in­ braic geometry, several complex variables, and ternational understanding for those who are fluent partial differential equations is cooperating to in it, but it presents substantial obstacles to those solve some exciting equations. who have not spent years in its study. On the one It is well known that some areas of science hand, the history of mathematics is one of progres­ have become "big science," where research is ac­ sively less translatability for nonmathematicians, complished by teams frequently gathered around while, on the other hand, the past few centuries major pieces of scientific equipment or laborato­ have seen ever-broadening human intellectual en­ ries. Each team consists of at least one senior deavors subsumed into the language and models scientist and many, sometimes a great many, ju­ of mathematics. Because of the language barrier nior investigators, working on related problems. between mathematicians and nonmathematicians, Mathematicians are also often informally grouped the following vignettes cannot describe the tech­ around a common research interest such as Yang­ nical essence of the fields discussed but will focus Mills equations, operator algebras, or computa­ only on a field's development and interaction with tional complexity. However, such mathematical others. groups are usually geographically separated, and U.S. Competitive Position. We conclude with the analogue of access to a common major piece a few remarks about the competitive position of of scientific equipment is their ability to gather the United States with regard to the field of math­ together for sustained periods of collaboration. ematics. First, we emphasize that international Furthermore, mathematicians share the scientific cooperation is a long-standing tradition in math­ community's needs for a means to prepare and ematics. When the vignettes use the words "a motivate potential researchers, from undergradu­ Soviet mathematician" or "in the United States," ate through postdoctoral levels. Institutes such as it should be understood that the results being the two National Science Foundation Mathemat­ described often depend on a background devel­ ical Sciences Institutes (Mathematical Sciences oped by many investigators in diverse parts of the Research Institute and Institute for Mathematics world. and Its Analysis), as well as other institutes, in­ Although mathematicians, with their special­ cluding those supported by Department of Defense ized international language, view their community agencies and special year projects provide essential as an international one and think relatively lit­ opportunities for mathematicians, undergraduate tle about the comparative ranking of national and graduate students, and postgraduates to col­ contributions to their discipline, the writers of laborate. this report believe that few would disagree that The vignettes in Chapter 3 illustrate the gen­ the United States was dominant mathematically eral patterns discussed above. However, these can in the decades following World War II. Just as give but a small sample of the many endeavors mathematics has become the language of science,

467 technology, and a widening circle of other fields, area of geometry. Considerably more general ge­ English has become the verbal language in which ometric objects than loci of polynomial equations most mathematics is explained. are studied in algebraic topology; in the 1950s When combined with a federal policy of aux­ and 1960s this was an especially active field and iliary support, the unique American system of was discussed on a number of occasions in the combining academic teaching and research at all popular scientific literature. levels has been a key to our strength. The United Any geometric object has a group of sym­ States remains attractive to foreign mathemati­ metries. For example, a cube is invariant under cians because of our relatively high employment a finite set of rotations. Similarly, a sphere is opportunities and, more recently, our powerful, symmetric under (the infinite set of) all rotations relatively accessible computers. We still have and reflections around its center. A continuous a favorable balance of trade from the wonder­ symmetry group, such as the latter example, is fully dynamic infusion of immigrants to American called a Lie group. Lie groups can also be viewed mathematics. However, Europe is regaining its as certain groups of matrices with their usual former mathematical liveliness, now more on a matrix multiplication. This multiplication is not continental ,than a national basis. Separately, commutative; that is, in general XY -:/; Y X. the Soviet Union, despite its professional isolation The set of derivatives along curves through the and its loss of numerous superb mathematicians, identity matrix of a particular Lie group can continues its tradition of excellence. The strength be viewed as an additive set of matrices and is of Japan's mathematical community is surging, called a Lie algebra. The theory of Lie groups and there is little doubt that in time major and algebras, one of the great achievements of mathematical activity will develop in China. modern mathematics, originated from questions It is still true that the United States is the in differential equations, which has always been world leader in mathematics, but this is surely central in analysis. But "group" and (of course) less so than it was ten or even five years ago. The "algebra" are quintessentially algebraic notions. enthusiastic mathematical activity in other coun­ Thus Lie theory, based on geometric symmetries tries has created a tendency toward parity among and increasingly useful for physicists, incorporates the major mathematical communities. The fact aspects of algebra, analysis, and geometry. that half as many U.S. citizens received mathe­ Another place where these three fields meet is matical sciences Ph.D.s in 1985 as in 1973 bodes in D-modules, which have been developed recently ill for our future ability to compete in the world in Japan. A module (with or without the D) is an mathematical community (see Table 1 in Chapter algebraic structure consisting of a group such as 4). Certain areas of mathematics are becoming a vector space whose elements can be multiplied dependent on foreign-trained researchers as noted, by another set of mathematical objects such as for example, in the vignette on Operator Alge­ matrices. D-modules are modules whose vectors bras. It is certain that during the next decade can be multiplied by partial differential operators much excellent mathematics will be created. The with analytic coefficients. One motivation for major questions are, where and by whom. their study was to focus attention on the equations themselves rather than solutions to differential equations. They were investigated using methods Vignettes of algebraic geometry invented in France in the D-Modules. Core mathematics is broadly divided early 1960s. into analysis, algebra, and geometry /topology, The theory of D-modules is also related although all three subfields include extensive ap­ to the Riemann-Hilbert problem, posed in Ger­ plied mathematics components. (Geometry and many around the turn of the century. Sup­ topology are not synonymous, but each term is pose we have a linear, homogeneous system of often used for areas in their broad confluence.) n first-order ordinary differential equations for However, as the interplay of analysis, algebra, n functions fi, ... , fn of one complex variable and geometry /topology becomes ever more com­ z. It might be written !' = Af, where the plicated, even the division of core mathematics coefficient A is an n by n matrix whose ele­ into subfields (not to mention its distinction from ments are analytic functions of z. In general applied mathematics) seems artificial. Indeed, the system will have n independent solutions the essential unity of mathematics is vivid as (/ll,···,fin), (/21, ... ,/2n), ... , Unl,···,fnn) we review some important recently discovered such that each solution (g1, ... , 9n) can be writ- relationships among these traditional subfields. ten uniquely as a linear combination g; = ctfli + Algebraic geometry has been one of the most c2hi + · · · + cnfni for some constants Ct, ... , Cn· lively areas of research in algebra during recent But there may be singularities, points at which decades. It is the study of geometric objects one or more of the elements in A(z) tend to that are the loci of points satisfying polynomial infinity or are otherwise irregular. Not only may equations in two or more variables, such as the solutions be undefined at the singularity, but familiar conics from classical geometry. Mean­ when we follow a curve around the singularity, while, algebraic topology has become a leading one solution may be transformed into a different

468 one. Going around the given singularity (and no Such decision questions were formerly ad­ others) once in a clockwise direction has the effect dressed primarily by logicians, but computers of transforming a solution by multiplication by an have given new urgency to algorithmic questions. invertable matrix, depending only on the system Computers after all operate with only a few prim­ and not the path traversed. The Riemann-Hilbert itive tools, and the programs which instruct them problem, a question in analysis, was to show a are essentially algorithms. For most problems converse for the theorem just presented-that for that computers are asked to solve the existence any admissible map from a set of singularities into of some algorithm is usually evident; instead, the invertible matrices there is a system of n differen­ problem is to find algorithms that are efficient tial equations, as described, such that encircling and reliable. The mathematical analysis of these each singularity changes the solution by the cor­ practical considerations has spawned the field now responding matrix. The solution, now several called "complexity theory." decades old, has been extensively generalized. This field measures the complexity of an al­ Mathematicians working in this country and gorithm by the maximum number f(n) of basic France were investigating ostensibly a quite differ­ steps that the algorithm needs to solve those ent area-an area at the intersection of algebraic cases of the problem requiring n digits in their geometry and algebraic topology that focuses on statement. The theoretically tractable problems, the concept of duality in geometry. In 1977 called type P, are those for which there is an researchers in the United States conjectured the algorithm with f(n) growing no faster than some existence of a relation between this geometric the­ polynomial in n. There are adjustments to this ory and a fundamental problem from Lie groups. assertion, most notably in the well-publicized lin­ Their conjecture was then proved independently ear programming problem discussed below, where by two pairs of mathematicians, one in France the symplex algorithm, though exponentially long and the other in the Soviet Union. Their proof in the worst cases, is remarkably efficient in the surprised the mathematical world by using the vast majority of its application. generalized Riemann-Hilbert results, thus again Each generation of computer scientists has linking algebra with geometry through differential quickly encountered pressing problems that over­ equations. whelm available computational resources. Unable Papers about D-modules are impossible to to obtain exact answers, they resort to simulation, classify into the three traditional fields of analysis, approximation, and sampling, perhaps by using algebra, and geometry /topology, a problem for the Monte Carlo techniques. Such approaches are not editors of Mathematical Reviews. It also suggests always appropriate, as when a massive computa­ how very tentative any division of mathematics tion is used to make a momentous yes/no decision. into subfields may be. In the development of antiballistic missile software Computational Complexity. An algorithm is during the early 1960s scientists tried to cope with a procedure for solving a given class of problems such situations by the simultaneous use of many with a specified set of mathematical tools. In proce·ssors in a multiprocessor environment. It classical geometry the tools were straight edge was found that this could result in unpredictable and compass, and the ancient Greeks provided a and often serious deterioration in the performance simple algorithm for trisecting any segment using of the system as a whole. This discovery gen­ only these. Their extensive attempts to trisect a erated a serious study of such anomalies. The general angle were proved futile in the nineteenth resulting work in the mid-1960s provided rigorous century, when it was shown that there can be no bounds for these deleterious effects. Performance such algorithm. guarantees of this type, usually called ''worst-case If the tools in algebra are addition, multi­ bounds," remain a major focus of algorithmic plication, division, and taking radicals {square analysis. roots, cube roots, etc.), then there is a famil­ The fundamental class of NP-complete prob­ iar algorithm for solving any quadratic equation, lems was defined in 1971 independently by a ax2 + bx + c = 0. However, a question posed Canadian and by a former Soviet citizen who by Renaissance Italians was answered when it now resides in the United States. This class was proved by Abel, a Norwegian in the early includes thousands of basic computational prob­ nineteenth century, that there can be no such lems arising in computer science, mathematics, algorithm for solving equations of degree five physics, biology, economics, business, and the or more. The tenth problem posed by Hilbert social sciences. The NP-complete problems are at the 1900 International Congress of Mathe­ of equivalent complexity in the sense that if one maticians asked whether there is an algorithm of them submits to an algorithm of polynomial for deciding if a general polynomial equation, complexity, then so can all the others. Proving !(x~, ... , xn) = 0, with integer coefficients has a the widespread conviction that no such algorithm solution in the positive integers. In the 1960s a exists (that P f. NP) is considered the most Soviet mathematician, strongly incorporating the fundamental of the open problems in theoretical work of two Americans, proved that there is no computer science. It is known that any algo­ such algorithm. rithm for solving an NP-complete problem can be

469 modeled by an appropriate circuit. Until recently linear profit function, possibly depending on a the only established lower bounds on circuits for large number of variables, is to be maximized in a NP-complete problems were linear;· however, last region defined by linear constraints. This question year a Taiwanese mathematician now living in is geometrically equivalent to finding the highest the United States made a dramatic breakthrough point in some high-dimensional convex solid or by establishing the conjectured exponential lower polyhedron in n-dimensional space, where n can bound, but under the assumption that the number be quite large. The importance of these problems, of "levels" of the circuit is bounded. Since then which have many applications in such varied a Swedish graduate student in the United States areas as airline scheduling, meteorology, portfolio simplified and strengthened these arguments. Al­ management, and telephone traffic routing, was most simultaneously, two Soviet mathematicians first observed more than 40 years ago by Dantzig independently established an exponential lower in the United States and Kantorovitch in the bound assuming that the circuit is "monotone," a Soviet Union. Dantzig developed the very effective related but distinct development. This work has "simplex algorithm" for solving LP problems, been extended by an Israeli doing postdoctoral which examines first one vertex and then another, work in the United States, and many complexity moving along the outside edges of the high­ specialists now believe that the tools are finally dimensional polyhedron in such a way as to becoming available to prove the corresponding improve the function that is to be optimized at results for unrestricted circuits, thereby finally each step until the optimal vertex is reached. settling this fundamental problem. Even though these n-dimensional polyhedrons, As the difficulty of NP-complete problems which typically arise in practical problems, can was realized, attention shifted to other ap­ have exponentially many vertices, over 40 years of proaches. These included approximation algo­ experience with the simplex algorithm indicates rithms, average-case instead of worst-case perfor­ that only rarely are more than 4n or 5n vertices mance analysis, and randomized algorithms that tested before the optimum point is attained, give a confident guess rather than a firm answer. despite the fact that pathological examples can Timely examples of the latter are the twin prob­ be constructed that do indeed require that all lems of deciding if a large integer n is prime or, vertices be tested. if it is not, of factoring n. One such algorithm After the concept of NP-completeness was randomly selects an integer k less than n, per­ introduced in 1971, researchers struggled without forms a simple test, and announces either that n success to find either a polynomial-time algorithm is definitely composite or that the problem is still for LP or a proof that it wqs NP-complete. In undecided. About three quarters of the possible 1979 the polynomial "ellipsoid algorithm" was k's will establish that a composite n is not prime. produced in the Soviet Union, but the bound on Thus, performing the test with 100 independent this method grows as n6 , rendering it impractical k's that do not prove n to be composite justifies for large problems, e.g., those having hundreds the conclusion that n is prime with a mere one in of thousands of variables. Subsequently, however, 4100 chance of error. the ellipsoid algorithm has had a significant im­ The study of primes has long been central pact in the theory of combinatorial optimization. to number theory, a field that has been pursued In 1984 a young Indian mathematician in for its own splendid beauty but traditionally was the United States made a striking breakthrough considered to have few pragmatic consequences. when he discovered an iterative method that Recent innovations in cryptography have com­ plunges through the interior of the polyhedron, pletely reversed the latter perception. The secu­ transforming it nonlinearly at each step so as rity of important cryptosystems depends crucially to stay as far as possible from the boundaries on the belief that the problem of finding the prime of the changing solid. The number of steps factors of a random number with a thousand dec­ required by this method is on the order of n3·5 , imal digits or more is, and will be for decades, a significant improvement over the earlier n 6 , a computationally infeasible problem. However and a variety of applications have shown that faith in this belief is beginning to erode because when implemented cleverly, this algorithm seems of some recent unexpected advances in primal­ to perform significantly faster than the simplex ity testing and factoring. Essentially overnight, algorithm. a Dutch mathematician produced the currently Very recent efforts in understanding this new best factoring algorithm by using the theory of method indicate that an n-dimensional polyhe­ elliptic curves from algebraic geometry, a field dron can be equipped with a coordinate system mentioned in the previous vignette. It is not yet that transforms its interior into a certain quasi­ clear whether this will lead to even more effective hyperbolic geometry so that the trajectories to algorithms that will undermine the security of the optimal vertex form geodesics in this space. these cryptosystems. Obviously, this area is extremely active world­ This vignette concludes with a discussion of wide, and much more work is needed before a linear programming (LP), a subject that occurs full understanding can be achieved. Nevertheless, widely in discrete optimization. For example, a it is already apparent that complexity theory

470 addresses many practical problems and that so­ The Russian's construction has been gener­ phisticated core mathematics, including algebraic alized in several ways, and dramatic progress has geometry, number theory, and geometry, is vital arisen in the last three years from a combination to complexity theory. of these ideas. In particular, reliable and efficient Nonlinear Hyperbolic Conservation Laws. solutions of the equations of gas dynamics in any The development of both theory and numerical number of space dimensions are now available, methods for solving nonlinear hyperbolic con­ and they reveal and explain intricate physical servation laws (NLHCLs) has been an exciting phenomena that previously had been only dimly area of recent mathematical research. There are comprehended. Examples include the disclosure many applications of NLHCLs because they de­ of unexpected complexity in flows involving inter­ scribe many important physical systems, including acting discontinuities, the discovery of transition some in aerodynamics, meteorology, water waves, criteria from regular to Mach reflection for waves plasma physics, and combustion. In gas dynam­ impinging on a surface, and the revelation of insta­ ics these are the laws of conservation of mass, bility mechanisms for supersonic jets. Both theory momentum, and energy. and experiments in NLHCL have been aided by elaborate computations from experiments using The major technical obstacle to both solving new laser technologies. and analyzing these systems is the fact that their Recently, much computational activity has solutions are not smooth, that is, they do not been directed toward solving systems of NLHCLs have derivatives of all orders. Many standard ap­ that are structurally more complex than those of proximation procedures require smoothness, and, gas dynamics, in particular those that arise in furthermore, solutions tend to be unique only if combustion theory or in the analysis of flow of certain physical constraints (called entropy con­ porous media-a subject of great relevancy for ditions) are satisfied. Since there are other oil recovery. New methodologies have appeared important equations in hydrodynamics, relativity, involving front tracking, mesh and optics refinement, and that do not have smooth solutions, piecewise parabolic approximations. NLHCLs are providing Some of a testing ground for inno­ these are related to higher-order versions vations with potentially of the wide applicability. aforementioned Russian construction. The first systematic computational attempts Many important practical problems remain to solve NLHCLs, motivated by problems of jet open. For example, there is as yet no reliable propulsion and in the Manhattan Project, were numerical method for solving the equations of made in the United States during World War II. combustion theory in more than one space di­ John von Neumann and others provided a clear mension except in the low-Mach-number limit. formulation of the problem and introduced several The newer numerical methods are so complex crucial ideas, in particular artificial viscosity (a that computer science questions regarding their justifiable technique for smoothing the problem) implementation, similar to those discussed in the and a first analysis of stability. These ideas previous vignette, have become crucial. Also, per­ were greatly extended after the war and gave turbations of NLHCLs, for example by boundary rise to an elegant theory of weak solutions that layers, are beginning to be considered. treated the lack of smoothness without smoothing. Why numerical methods fail and why they In particular, the Lax-Wendroff scheme and its sometimes succeed so spectacularly are questions many variants yielded acceptable solutions for that have been studied successfully in recent years, many practical problems. However, many other especially through the re-examination of the pre­ problems remained unsolved, and the theory was cise role and possible forms of artificial viscosity. incomplete. New theoretical tools for understanding practical In the 1950s the relevance of the Riemann algorithms and new ideas, such as the notion of problem, well known to chemists and engineers variation diminishing schemes, are producing a working with the Riemann shock tube, became slow· confluence of algorithms rooted in disparate fully recognized. The Riemann problem contains a priori notions of what is important for practical the pathology of the general NLHCL problem calculations. but is in a more tractable form. An American New techniques of functional analysis, in mathematician gave a mathematical analysis of particular the compensated compactness method the Riemann problem, and then a Russian math­ that originated in France, have given new impetus ematician incorporated the American's analysis to existence theory, removing some of the ear­ into a numerical construction that described the lier limitations. The compensated compactness misbehavior of the solutions in a natural way. analysis is significant to the broader context of These developments led to a proof, developed in homogenization and order/disorder phenomena, the United States, that solutions exist for one­ two other areas in which the French have been in­ dimensional NLHCLs subject to some technical volved. In addition, partial existence and stability restrictions. This result gave rise to practical al­ results have appeared for nonsmooth solutions in gorithms that in turn generated sharper existence more than one space dimension for both convex results. and nonconvex systems.

471 The strong interaction between mathematics ential equations, index theory (or anomalies), and and practical applications in NLHCL is clear from even the representation of infinite-dimensional this account. All the major advances in compu­ groups, and it remains fertile research ground. tation have been anchored in theoretical devel­ The extended impact of Yang-Mills theory opments. Indeed, most of the more innovative does not yet involve the complete equations but practical algorithms are due to mathematicians, concentrates on their role as nonlinear extensions and more mathematicians have been involved than of Laplace's equations, which are well known to be is apparent. An explosion of knowledge could at fundamental to earlier mathematics, physics, and this time be safely forecast if there were more engineering. In two variables Laplace's equation high-caliber people active in the field. is related to the Cauchy-Riemann equations, part However, there are too few people with a of the foundation of complex analysis. One form combined understanding of the abstruse physi­ of the Yang-Mills equations, called the self-dual cal, mathematical, computer science, and related Yang-Mills equations (SDYM), is a four-variable aspects of NLHCLs. Such an understanding re­ analogue of the Cauchy-Riemann equations. The quires a broad education in several fields that is instantons are solutions of these equations, and not easily available in the United States. One ex­ they appear to have properties nearly as basic as planation for the strong role played by Israelis in solutions of the Cauchy-Riemann equations. The this field may be that in Israel many mathemati­ SDYM equations are in turn important in algebra, cians are exposed to engineering problems and to geometry /topology, and analysis, respectively. programming during a lengthy military service. At first glance, the SDYM equations seem ab­ In other countries, such as Japan, China, the So­ solutely intractable for writing explicit solutions. viet Union, and in much of Europe, most students Even for SU(2), a small essentially non-Abelian complete algebra by the seventh grade and soon Lie group, there are nine first-order equations begin calculus, leaving time during secondary with twelve unknown dependent variables as well school for those with mathematical and technical as the four independent variables on the space. talent to pursue advanced topics. Given that There are three extra degrees of freedom, due changes in precollegiate education will take a long to gauge symmetries, so it is not surprising that time to evolve, the most immediate solutions in insight from algebraic geometry had to precede the United States would seem to be an extension progress in topology. The Penrose twistor meth­ of the graduate student years through supporting ods were used to transform these equations in young investigators with adequate postdoctoral four-dimensional space to a problem concerning fellowships and providing enrichment for talented holomorphic bundles on a six-dimensional man­ undergraduate students. ifold. These methods from algebraic geometry Yang-Mills Equations. In 1954, Yang in the are surprisingly general and can lead in many United States and Mills in England constructed a directions, for example, toward Kac-Moody Lie nonlinear version of Maxwell's equations that in­ algebras and models with loop groups. corporated a non-Abelian group, typically SU(2), Learning about the structure of the space the group of two by two unitary complex matrices of instantons over four-dimensional manifolds has with a determinant one. [SU(2) is a three­ generated profound insight into the topological dimensional Lie group.] This was first conceived structure of general four-dimensional manifolds, as a classical theory transplanted to Lorentz space­ demonstrating the fundamental value of an equa­ time; but when it is used in quantum theory, it tion specific to a low dimension like four. This is convenient to use Euclidean space-time. The was helpful to physicists, who have since de­ theory has been incorporated into nearly every veloped the fundamental intuition of instantons model of particle physics since the construction as solitons, the wavelike solutions of KdV and in 1975 of instanton solutions and the more re­ Sine-Gordon that superimpose nonlinearly. cent construction of multi-instanton solutions in It is interesting to note that mathematicians four-dimensional space. Shortly thereafter, the had been able to deal with the structure of man­ Penrose twistor theory was shown to transform ifolds in dimensions two, three, five, and greater. an apparently very messy nonlinear system of The Cauchy-Riemann equations are used in two partial differential equations into an elegant prob­ dimensions, geometric methods are employed in lem in algebraic geometry. Then the equations three, and mathematicians find five and more themselves were noticed to be natural geometric dimensions amenable to standard methods of al­ objects. gebraic topology. The gap of the fourth dimension The Yang-Mills equations depict the cur­ appears to be filled by SDYM theory. vatures (fields) of connections (potentials) as a In analysis, one of the key properties of principal bundle over a Riemannian manifold. Yang-Mills theory is its conformal invariance. By now, Yang-Mills theory is prominent in pure Some of the basic equations about instantons can mathematics. It has already affected subjects as be formulated in the context of the calculus of diverse as differential geometry, algebraic geom­ variations. The conformal invariance of the theory etry, the topology of four-dimensional manifolds, implies that the variational problem does not the calculus of variations, nonlinear partial differ- satisfy the conditions that are required in order

472 to use the method of direct steepest descent, so of all linear transformations on Hilbert spaces of helpful in three-dimensional work. The attempt arbitrary dimensions are the building blocks of to understand the failure of the steepest descent the simplest operator algebras. However, con­ method for the Yang-Mills problem has led to sideration of proper subalgebras of the universal the development of new variational techniques, operator algebra over some Hilbert space reveals which are useful on a variety of problems. It is far more complex situations. This complexity interesting, and possibly significant, that the three can be slightly relieved and the study reduced to fundamental scale-invariant geometric problems three basic types of von Neumann algebras simply coincide with three basic models of quantum called Types I, II, and III. fields: the Yamabe problem (phi-four theory), During the early postwar years, both Ameri­ harmonic maps (sigma models), and the Yang­ can and French mathematicians made substantial Mills equations. progress in operator algebras, including some In any case, Yang-Mills theory is a beautiful important applications to the theory of infinite­ example of the intense bonds between current dimensional group representations. The French theoretical physics and all subfields of core math­ school became relatively inactive by the early ematics. Yang-Mills theory is a young discipline 1960s and re-emerged in the mid-1970s when that will undoubtedly attract many more mathe­ a young mathematician, subsequently a Fields maticians in the near future. Its results to date Medalist, began working in the area. During are primarily due to the efforts of English, Ameri­ the late 1950s and early 1960s, some Japanese can, and Soviet researchers. Although Americans mathematicians entered the field. Simultaneously, cannot claim to dominate the field, they have numerous theoretical physicists became involved certainly contributed significantly to its develop­ and provided valuable insights derived from their ment. The necessity of extremely broad training, physical applications. It was shown that there mentioned in the final paragraph of the preceding exist infinitely many distinct Type II and Type vignette, also applies to successful research in III factors. The latter, especially, remained mys­ this field. Increasingly, postdoctoral or protracted terious because they did not possess the special study is necessary in order to become a successful functionals called traces that were so fundamental researcher in many areas of mathematics. to studying the structure of the other two types. Operator Algebras. The area of operator When it was discovered that the physically inter­ algebras is currently very active and provides esting algebras are of Type III and certain classes another excellent example of unexpected interac­ were explicitly parameterized by physicists, parts tions between areas of core mathematics and the of the mystery began to unravel. natural sciences. Quantum physics originated the Major international conferences began having Heisenberg Uncertainty Principle, which forces a crucial influence on operator algebra theory in consideration of quantities P and Q satisfying the mid-1960s. In the late 1960s, a conference PQ- QP = h/27ri. It motivated a search for held in the United States removed a key obstacle appropriate mathematical systems containing in­ to analyzing Type III factors by displaying results finitely many such noncommuting variables. In proved independently by a Japanese mathemati­ the 1920s, M. H. Stone and J. von Neumann cian and some Dutch and German physicists. demonstrated that such systems require a general At a subsequent conference, the French Fields theory of algebras of operators on Hilbert spaces, Medalist, then a student, was motivated to work generalizations of finite-dimensional vector spaces. on the subject. He opened entirely new vistas by These were studied extensively in the 1930s by F. investigations that combined algebra and analysis Murray and von Neumann. in new ways and led to the classification of the The mathematics of operator algebras is algebraic structure of Type III factors. In the characterized by a profound blend of noncommu­ 1970s, researchers turned toward the geometric tative algebra and infinite-dimensional analysis. aspects of the subject. A new extension theory Although operator algebras exhibit a very rich of c· -algebras generated a successful synthesis of structure, no serious work following that of Mur­ gf;ometry and algebra, resulting in a unified view ray and von Neumann appeared until after World of features from these two subjects. At about the War II. same time, the fundamental Atiyah-Singer Index The simplest example of an operator algebra Theorem was extended from locally trivial families is the entire set of n by n dimensional matrices to the more general foliations by using operator for a specific n. A general operator algebra is algebras. Additionally, a Soviet mathematician a subset of the bounded linear transformations solved specific cases of a long-standing problem in on a (generally infinite-dimensional) Hilbert space differential topology concerning smooth deforma­ that is closed under addition, multiplication, an tions by developing powerful new techniques in adjoint operation, and a suitable limiting process. c· -algebras. If the system is closed only for the strongest Recently, an American mathematician, born limiting process, it is called a c· -algebra. If it in New Zealand and educated in Switzerland, has also contains the most general limits, it is said found a totally unexpected connection between to be a von Neumann algebra. The full algebras three apparently diverse fields-knot theory, the

473 classification of subfactors of Type II factors, and One approach to classifying two mathematical the theory of Heeke algebras. This activity has entities in some class is to assign each entity occurred within the past two years, a catalyst in the class some mathematical label (called being the fortuitous meeting of the Mathematical an invariant) that coincides on the two entities Science Research Institute at Berkeley in 1985, considered to be the same. Thus a search for sponsored by the National Science Foundation, appropriate invariants for links began. In the where experts in operator algebras were meeting 1920s polynomial invariants were developed by concurrently with those in low-dimensional topol­ studying the topology of the space remaining ogy, the field that contains knot theory. The ex­ when the link is removed from ordinary three­ citement of the connection between von Neumann space. The corresponding polynomial for the knot algebras and knot theory may be overshadowed 62 above is 1- 3x + 3x2 - 3x3 + x4 . by investigations of its utility to biologists in Another approach to knots first studied in describing large-scale structures of DNA. A knot the 1920s was a mathematical structure called is a closed curve in three-space, and a link is a the braid group. Braids can be spliced together (possibly interlocking) system of knots. Although thus forming a group, however splicing the top of they can be surprisingly complicated, knots and a braid to its bottom forms a knot or link, and links can be adequately represented by a pro­ indeed all knots and links may be so formed (in jection onto the plane. Two relatively simple possibly many ways). examples are:

The knot 62

The link 6~ (also called the Borromean rings)

Braids and their invariants have rewarded Although people probably have always used knots, investigators with considerable valuable insight the first known attempt to list and classify them over the years. However, their connection with mathematically (as opposed to mechanically) re­ operator algebras remained unnoticed until the sulted from an erroneous hypothesis of Lord proof of a deep result about subfactors of Type Kelvin in the late nineteenth century that atoms II factors required an analysis of one representa­ were knotted vortices in the ether. His vain hope tion of the braid group. Coincidentally, almost of deriving the periodic table by classifying knots the same representation, arising from a special stimulated scientific advances entirely different case of the Heeke algebras, had been discovered from his vision- in the characteristic but unpre­ by mathematical physicists in the 1960s as they dictable manner that pure research, stimulated partially solved the Potts model of statistical by attractive ideas, often yields unanticipated mechanics. The connection with operator alge­ harvests quite different from their original intent. bras generated a trace function for the braids Knot theory considers two knots or links to be that could be used to recover numerical infor­ the same if one may be deformed without crossing mation. The trace of a braid then provided a strings until it is identical to the other. It is new (Laurent) polynomial invariant for the asso­ exceedingly difficult (and obviously fundamental) ciated knot or link. It was more sensitive than to decide when two given links are the same. the earlier polynomial in that it separated knots Trial and error methods are rarely satisfactory. that were previously indistinguishable. It could

474 even distinguish knots from their mirror images. tions, and demographic studies. John Graunt Computers can be used to compute this poly­ initiated this field in 1662 with his invention of nomial for many links. For the knot 62 , it is life tables for analyzing English mortality data, x-1 -1 + 2x- 2x3 - 2x4 + x5 and, for the link 6~, but recent clinical studies are far more complex. it is -x-3 + 3x-2 - 2x-1 + 4- 2x + 3x2 - x3 . Typically, these begin with patients who have an Both operator algebra and topology already unpleasa~t disease (possibly at different stages) have produced substantial generalizations of this bemg assigned randomly to two or more different new invariant that have been used to solve many treatments; they are then followed until they die venerable problems of knot theory. Similar work or disappear or the study ends. Observers record is expected to shed new light soon on the Potts variables, called covariates, that might affect the model, von Neumann algebras, knot theory, sta­ survival of the patients, including some that do tistical mechanics, quantum physics, and possibly not change, such as sex and age at diagnosis, even basic structures of life via the DNA appli­ and some that do, such as blood pressure and cation. In any case, these developments empha­ glucose level. Statisticians use these observations size again the eternity of a good mathematical to study the relationships between the covariates result, the harmony of all mathematics, the un­ and the survival time of patients after contracting predictable relationships between fields, and the the disease, and especially the comparative merits many bridges across the humanly created gap of the treatments. One goal may be to predict between core mathematics and basic science. the effect of different treatment on life expectancy At present, only a few institutions in the of future sufferers of the disease, another may be United States offer training in operator algebras. to determine which factors prolong the patients' In this area, as with other areas of mathemat­ normal functioning as long as possible. Subtleties ics, the United States has become increasingly such as the role of the individuals who are still dependent on hiring foreign-trained students and alive at the end of the study and those who scientists. In addition, some related areas with withdraw or disappear complicate the analysis. potential for interaction with engineering have One model for the life history of a patient (or been underemphasized in the United States. This piece of equipment) includes a vector Z(t) of co­ is an ideal time to put a greater effort into variates that vary (possibly randomly) with time, providing the correct conditions for new young from zero, when the study begins, to the time T researchers and strong leadership in the United when the study is terminated, and a finite-valued States in the subject. process Y(t) designating the state of the depen­ Survival Analysis. Since problems in collect­ dent variable. These states might be "alive-and­ ing, analyzing, and interpreting data are universal, functioning," "alive-not-functioning," "dead" or it is not surprising that people in many countries "lost." The treatments appear as coordinat~s of (including England, France, Scandinavia, the So­ Z(t). Analysis of these models concentrates on viet Union, and the United States) have made the intensity of changes; these intensities corre­ significant contributions both to statistics and spond to a stochastic process J(t, y0 , y), where also to probability, the branch of core mathe­ J(t, Yo, y) dt represents the probability that Y matics that has until recently provided the major changes from Yo to y between time t and t + dt theoretical support for statistics. Expansions in assuming a past of Z ( t). The life histories of the technology have both motivated and enabled the n patients are regarded as a set of independent most striking progress in statistics during the observations that can be used to estimate J. past decade. Advances in instrumentation and Until the early 1970s, research concentrated communication have generated enormously com­ on the extremes of either relatively simple sit­ plex sets of data, and the growth of computing uations with few assumptions or vastly more power permits the collection and management of complicated parametric models with heavy as­ such data. The United States has been the un­ sumptions on J. An example of the former is questioned world leader in statistical computing, the setting for the product limit estimator for mainly because the availability and sophistication the probability of surviving beyond time t, when of its equipment has been unmatched elsewhere. no covariants are measured and only questions of A further discussion of these developments can life expectancy are of interest. The assumptions be found in the NRC report Computational Mod­ of the parametric models were found to be too eling and Mathematics Applied to the Physical unrealistic by experts in biostatistics. Sciences, and in the forthcoming NRC Board on In 1973 Cox proposed his "proportional haz­ Mathematical Sciences report, A Survey of U.S. ards model" for survival-time data, basing his Mathematical Sciences. work in part on a model developed during the 1960s at the National Cancer Institute. In this However, other countries have made signifi­ model cant contributions; a notable example from Eng­ land is David Cox's proportional hazards model J(t, Alive, Dead) = exp[OT Z(t)]>.(t) for life history data. Life history analysis is a body of statistical techniques useful in medical where >. 2: 0 is a (nonrandom) function and (J is a research, reliability theory, actuarial computa- vector of unknown parameters. J(t, Alive, Dead)

475 can be an essentially arbitrary function of Z( s ), important pattern in the field, which is illustrated 0 ::: s ::: t. here, is that foreign scientists and students come The attractive features of the Cox. model are to study, lecture, and meet in the United States, easily seen if we specialize to fixed covariates. The and subsequently many of the best remain as model permits an arbitrary lifetime distribution citizens or permanent residents. for a control population and postulates a linear approximation for the log of the ratio of intensities corresponding to two different values of Z. The Review of the David Report effects of the covariates can then be measured in The David Report, a several-year effort by a terms of the components of e. broad cross section of industry and university sci­ The inferential procedures proposed by Cox entists, has considerably raised the consciousness were quickly applied to heart transplant and other of the science community. The principal finding data by statisticians and biomedical scientists in of this 1984 National Research Council report, the United States. Evidence of these applica­ Renewing U.S. Mathematics: Critical Resource tions appears in numerous citations of the model, for the Future (National Academy Press, Wash­ mostly in the medical literature. Although the ington, D.C., 1984), was that "federal support for model seemed to give sensible answers in applica­ mathematical research is markedly out of balance tions, its highly nonlinear nature and the complex with support for related fields of science." The probabilistic structure of the data prevented rig­ Panel on Mathematical Sciences believes that it orous theoretical analysis for some time. is important to review the major findings, the The theory for the case of time-independent subsequent support, and the effect of the critical covariates was independently developed in the David Report. United States and Denmark in the mid to late The major findings were as follows: 1970s, but these methods could not handle the much more difficult case of time-varying covari­ • Over a 15-year span from 1968 to 1982 con­ ates. In 1975, the thesis of a Norwegian student stant dollar federal support of mathematics studying in the United States with a United research dropped by over 33 percent. States statistician who emigrated from France • During the same period, the number of people showed how to attack the analytic problems in in the field doubled in size (the influx from this area. He applied a multivariate counting the late 1960s). process framework to both survival analysis mod­ • The effects of the resulting paucity of funding els and more general life history models. Most are being severely felt at the research uni­ significantly, he exhibited applications to these versities where almost all basic mathematical models of the deep results in continuous time research is done. martingales and stochastic integrals, which were • Unless corrective action is taken, the health introduced by researchers in Japan, the United of the nation's mathematical research effort, States, and France. Then others, primarily in now still the strongest in the world, will be The Netherlands, Norway, Denmark, England, seriously weakened. and the United States, used these techniques to • The central importance of mathematics to analyze the heuristic suggestions of Cox and to science and technology, hence to the economy devise and investigate new inferential procedures and defense, makes its continued strength in in more complicated life history models. Despite the United States imperative. their flexibility, the Cox models are still burdened • To address this imbalance, basic needs by by the questionable assumption that the ratio 1990 include substantial increases in support of intensities for two individuals can be modeled for graduate and postdoctoral students, in parametrically. It is not clear that the counting addition to establishing a greater base of process techniques will prove adequate for the support among young and senior researchers. analysis of the newer, more flexible, semipara­ Since the David Report appeared in 1984 it metric models that have been proposed, but they has received wide acceptance in the universities should provide a good starting point. and at the federal policy level. The problem seems The analysis of life history data is a rapidly to be understood. Evidence of this includes: expanding field widely used in a variety of disci­ • The National Science Board (representing all plines, including biomedical science, demography, disciplines of science and engineering) passed and sociology, fields that reciprocate by continu­ a resolution "that a concerted effort should ally presenting statisticians with data for which be made by all funding agencies to increase previous methods are inadequate. Although the support for the mathematical sciences for interaction of theory and application and the in­ several years until a proper level of sustaining ternational nature of this work are hardly new support has been achieved." features of statistics, they have become more • A subcommittee of the Department of De­ prominent recently. fense (DOD)-University Forum reported that The United States and England are the there had been significant erosion in mathe­ primary world centers of statistical activity. An matics support, partially because the growth

476 of computer science had masked the mathe­ progress appears to be moderate increases in matics funding situation during the 1970s. support for graduate and postdoctoral students, • The University Research Initiative, part of while other investigator support appears static. the fiscal year 1986 DOD appropriation, Even graduate student support progress leaves states the critical need to address the im­ the discipline in a vastly unfavorable position at balance that currently exists with respect to a critical time, see Table 1. mathematics funding. The National Science Foundation continues Currently, the federal agencies primarily re­ to advocate for additional resources for the math­ sponsible for mathematics funding are making an ematical sciences, primarily with a significant effort to redress the problem. However, recent cross-disciplinary effort in computational science budget constraints severely threaten their efforts. and engineering. A new mathematics program The issue is how to reach this balance of resources at Defense Advanced Research Projects Agency between related fields in a climate of overall lim­ (DARPA) is potentially a bright spot. Without ited budget enhancements. The difficult burden this new DARPA program, the current 1986 and falls on the agencies which must set priorities in 1987 budgets for mathematical sciences research funding research. at the DOD agencies would be severely depressed Movement toward a proper level of sustaining (Table 2). Even with these enhancements, if they support in the mathematical sciences for graduate remain, and the best intentions of the policy mak­ and postdoctoral students, established investiga­ ers, the renewal of U.S. mathematics will require tors, and equipment is varied. The only real continued resolute effort.

477 TABLE I

Comparisons of Federal Support for Postdoctoral and Graduate Students In Three Fields of Science

Ph.D m D Postdoc support. federal 4500

4000

3500 G.S. support. federal -....,~. ~ c 3000 ,~I c ~~;- 0 '~I Ill 2500 ;-!~~ .. , I, c..CLl -, 2000 ;~! 1500 "

1000

500

0 chem-80 chem-84 phy-80 phy-84 math-80 math-84

Mathematical Chemistry Physics Sciences 1980 1984 1980 1984 1980 1984

University faculty and staff with primary or secondary activity in R&D 9800 9200 9200 9200 7400 7200 Annual Ph.D. Production U.S. citizens 1169 1332 611 659 520 407 Total Ph.D.s 1538 1765 862 962 744 698 Postdoctorals federally supported 2255 2473 1210 1150 57 132 Graduate Students federally supported 3700 4118 2900 3348 200 411 Ratio P.D./Ph.D. 1.47 1.40 1.40 1.20 0.08 0.19 Ratio G.S./Ph.D. 2.41 2.33 3.36 3.48 0.27 0.59

Sources: NRC Survey of Doctoral Recipients. Survey of Graduate Science and Engineering Students and Postdocs, NSF. Survey of Doctorate Recipients, NSF, unpublished, Table D-32.

478 TABLE 2

Federal Support of Basic Academic Research in Mathematical Sciences 120

110

100

90

80 c "' 70 ~ :i 60

50

40

30

20

10

1982 1983 1984 1985 1986 1987 Fiscal Year

Federal Support of Basic Academic Research in Mathematical Sciences (FY)

82 83 84 85 863) 874)

NSF1) 34.2 37.1 45.6 52.7 57.2 65.5

DOD2) (AFOSR, ARO, ONR, DARPA) 23.3 26.5 29.9 32.3 42.4 39.1

Other2) (DOE, NASA, NIH) 4.3 4.8 4.9 5.5 5.9 4.8

Total 61.8 68.4 80.4 90.5 104.5 109.4

I) DMS represents about 90%. 2) This is based on estimates of the mathematics extramural component of some programs. 3) These are pre Gramm-Rudman, SDI and University Research Initiative figures. Their eventual effects may cancel each other. 4) The President's budget. Estimates of ultimate program emphasis (see 2) and future effects of URI and SDI make these tentative, especially outside NSF.

479 Report to the Members of the Society Report of the Treasurer

I. Introduction Grants & Contracts 9% 1,063,315 Meetings 2% 242,446 Over the past few years, the Society has experi­ Other 6% 630,328 enced both good and bad financial health. The Society earned approximately $4 70,000 in both Totals 100% $11,278,980 1978 and 1979, or about eight percent of revenues. During the following four years, 198Q-1983, the Journals. Although journals provide the larg­ Society lost $2,654,699; in 1984 income of $106,404 est fraction of revenue, they have been operating was generated. This recovery was largely due to at a net loss for the past several years. Costs the introduction of cost saving measures and a are under constant scrutiny and cost-saving pro­ reconsideration of pricing policies. The recovery cedures are implemented where possible, for in­ has continued into 1985 and will allow the Society stance by placing more type on a page, reducing to set aside a substantial amount as a contribution the size of type, using new and cheaper methods to the Future Operations fund. The Long Range of composition, doing in-house printing and using Planning Committee of the Executive Committee lighter paper to reduce postage costs. As a result and Board of Trustees has recommended that the of these efforts, journals have been able to support Society build this fund to an amount equal to one themselves during 1985; and, in spite of contin­ year's operating budget. I expect the Society's ued subscriber erosion, the Trustees expect the 1985 audited financial statements to show 1985 in­ 1987 journal prices to increase by only a modest come of $1,734,356 before transfers to the Future amount. Operations fund, and $590,598 after transfers to Books. Included in this category are not only Future Operations. As of December 31, 1985, the books (monographs or collections of articles) but Future Operations fund will contain an amount review volumes and indexes to journals. Books, equal to approximately 19% of the 1985 operating exclusive of the latter, continue to be financially expenses. The $590,598 of income remaining is sound, and selling prices of AMS books compare approximately five percent of revenues. It is very favorably with other mathematical books. A worth noting that the Society's unrestricted fund vigorous marketing campaign for books is being balances now approximate their balances in 1980, developed. A successful campaign will increase the first year of the most recent series of deficits, the number of books being sold and could keep during which such balances decreased to less than the price of books from rising substantially. one million dollars, a dangerously low level. Review volumes and indexes have been a I would like to report to you that the financial financial burden to the Society. Prices have to future of the Society is bright and that no clouds be set very high because the costs of producing appear on the horizon, but I cannot. The current these extremely large publications are high. The budget-cutting frenzy in Washington will surely question of whether to publish such volumes is a be felt by the institutions which provide the difficult one; on the one hand the profession needs majority of the Society's revenues, and it is only such publications as research tools; and on the prudent to assume that at some time in the not other, the price needed to break even is too high too distant future the Society will again need to for smaller institutions to pay. In some cases, adjust its operations to cope with this problem. prices have been set artificially low to address this The Society's 1986 budget shows an expected problem. excess of revenues over expenses of approximately $300,000 (after reinvestment of investment income Dues. The Society has about 450 institutional in the Future Operations fund). Similar results members and 20,000 individual members. Of the are being planned for 1987, and it is hoped that latter, about 6,000 pay no dues because they are this can be achieved through cost cutting and student nominees, emeritus members, or reviewers modest price increases. without convertible currency. Individual member dues are two-tiered to provide some relief to lower paid members. Institutional dues are set, in part, II. Revenue by the number of papers published by authors I now turn to a description of the Society's employed by the institution. Increases in dues revenue, followed by explanations of each category. for individual members are set annually by a Revenues to support Society activities in 1985 cost-of-living index. came from the following sources: Grants and Contracts. The amount of money Journals 65% $7,294,810 available from the federal government has declined Books 9% 1,003,080 substantially over the years. Currently, support is Dues 9% 1,045,001 mainly for travel and subsistence for participants

480 in research conferences, institutes, and seminars, of the Society's operating funds exceeded its plus the Society's cost in preparing and running liabilities by $1,045,250. The Society's short-term these conferences. The money received from liabilities, those which must be satisfied during government agencies is reimbursement only, with 1986, amounted to $8,500,000, and were made no profit to the AMS. The Society does have up of subscriptions and dues received in advance contracts to perform services for other nonprofit ($6,750,000) and other obligations. organizations, and this helps to recover some fixed These liabilities will have to be satisfied costs. with the so-called "current assets" of the So­ Meetings. As in prior years, registration and ciety, consisting of cash and cash equivalent exhibit fees at the two joint meetings were not ($3,710,000), accounts receivable ($460,000), in­ sufficient to cover expenses. It will therefore be ventories ($1,510,000), and other assets ($380,000), necessary gradually to increase fees over the next together with other income to be received during few years to bring revenues and expenses into 1986. balance. At December 31, 1985, the Society's in­ vestment in property and equipment amounts to $3,487,000, consisting of assets with a book value Ill. Expenses of $4,908,000 and mortgages totaling $1,421,000. The expenses of the Society are not as easily The property and equipment consists of land categorized as revenues. Most expenses are and buildings (original cost $3,394,000) and data assigned to departments, and these are later processing and other equipment (original cost allocated to specific publications, meetings, or $4,456,000). From these costs, depreciation of other accounts. The three major expenses of $2,942,000 has been deducted and charged to the Society are personnel ($5,520,000 or 58%), operations. buildings and equipment and related accounts Besides the assets mentioned above, the ($1,140,000 or 12%), and direct publication costs Society has permanently invested funds includ­ ($1,230,000 or 13%). The remaining 17% or ing endowment funds ($1,554,000) and quasi­ $1,650,000 is made up of pass-through funds from endowment funds ($2,329,000). The use of en­ grants, direct costs, and other general expenses. dowment funds is restricted by the funds' donors. The quasi-endowment funds are treated as if they IV. Assets and Liabilities were true endowments with the exception that the restrictions as to use have been made by So far, this report has dealt with sources of the Trustees. These funds are held and man­ revenue and applications of expense. Another aged by Rhode Island Hospital Trust National aspect of the Society's finances is what it owns Bank. F. P. PETERSON and owes, or its assets and liabilities. It is es­ timated that at December 31, 1985, the assets

481 Mathematical Software Joseph P. Buhler

A Matrix Laboratory 0. Eps can be set by the user. A too small Avner Ash or too large value of eps may lead to a spurious result for the rank or nullspace of a matrix. Ohio State University Similarly, Matlab apparently believes all matrices Department of Mathematics are semisimple. For instance, when we ask it to Columbus, OH find the eigenvectors of the unipotent matrix PC-Matlab is a microcomputer version of a col­ lection of matrix routines which has long been available on mainframe computers. It runs on an IBM PC or compatible. Versions for other small computers may become available in the near it returns two linearly independent eigenvectors, future. What follows is a description solely of PC­ (1, 0) and (1, -0.000000000000000003)! This re­ Matlab, version 2.02. For more information, con­ sult persisted even when I set eps to w-33 . The tact The Math Works, Inc., 158 Woodland Street, Matlab manual also seems to assume every matrix Sherborne, MA 01770. Telephone: (617)653-1415. is diagonalizable, which I suppose is true up to 1. Matlab in the abstract. We can think of experimental error. Matlab as a very high-level, very specialized Matlab includes a few operations on general programming language. It has only one kind of polynomials, which are represented as the vector data structure, two-dimensional matrices. Vectors of their coefficients. Matlab can evaluate a pass as 1-by-n or n-by-1 matrices, scalars as 1-by- polynomial, convolve two polynomials, and find 1 matrices, and text as vectors whose components all the roots of a polynomial, real and complex. are letters. Certain useful matrices can be created in There are two types of commands in Mat­ one command, such as diagonal, constant, and lab: those that deal with matrices (including random matrices. The random elements can be vectors and scalars) and those that perform pe­ generated from either a uniform or a normal ripheral services, such as reading and writing distribution. data, creating loops, and managing the memory. Now we come to the rich list of algorithms Matrix functions implementing state-of-the-art al­ for various matrix factorizations. Thi:J is the core gorithms represent the heart of Matlab. of Matlab. With single commands, we can factor Arithmetical functions for matrices appear in a matrix into 1) lower-triangular * permutation * fairly standard notation. All matrices have com­ upper-triangular (LU decomposition); 2) unitary plex entries automatically, although real matrices * triangular (QR decomposition); 3) unitary * are displayed without their 0 imaginary parts. diagonal* unitary (singular value decomposition). Arithmetic can also be performed elementwise by Similarly we can find the Hessenberg and Schur preceding the operator sign with a dot. The forms of a matrix, which are upper-triangular or trigonometric and exponential functions are built almost upper-triangular matrices similar to the in along with functions that round fractions to given one. The unitary similarity transformation integers in various ways, find remainders, and will be given at the same time if requested. use continued fractions to find the best rational The Cholesky factorization of a positive-definite approximation to a real number. matrix is also available. Many important functions and decomposi­ Let's look at a sample algorithm. All of tions of matrices may be performed in Matlab the major algorithms used by Matlab are taken by issuing just a single command. These in­ from two collections of linear algebraic routines, clude finding the determinant, condition number, called LINPACK and EISPACK [1, 2, 3]. These rank, and norm of a matrix; finding the inverse, collections were written in FORTRAN, but PC­ pseudoinverse, null space, range space, and row­ Matlab has them rewritten in the C language, reduced echelon form of a matrix; finding the with slight modifications. characteristic polynomial and eigenvalues of a Suppose we want to find the eigenvalues of matrix; and finding the generalized eigenvalues of the n-by-n matrix X. Matlab does not compute A with respect to B, that is solving Ax= tBx, x the characteristic polynomial of X and find its a nonzero vector, t a scalar. Matlab will also list roots. Instead, it puts X into Hessenberg form, the eigenvectors. and then applies the QR method. Here is a sketch We should remark here how critical round-off of the method (see for example [4], p. 294 for more error is in some of these computations. There details). Using the Gram-Schmidt algorithm, it is a constant in Matlab called "eps," which is writes X = QR, with Q unitary and R upper the smallest positive number distinguishable from triangular. Then RQ is similar to X, so it

482 Matlab provides us with three kinds of loop­ ing: "if", "for", and "while". All data consists of complex matrices of floating-point numbers, so there is no need for data types or dimensioning statements. This language suffices to write short procedures which set up matrices and then act on them with Matlab's powerful commands. 2. Matlab in the concrete. PC-Matlab runs on an IBM PC or compatible computer, with at least one disc drive, and MS-DOS version 2.0 or higher. It requires at least 320K of random access memory (RAM) and an 8087 coprocessor chip. Under MS-DOS 2.1 or higher, it will utilize all RAM up to 640K. To take advantage of the graphics capability, some kind of graphics display is necessary. A single copy of PC-Matlab, which is copy­ protected, costs $695. There are discounts for multiple and academic purchases. For instance, one copy ordered by a university staff member goes for $395, while a classroom kit, which allows FIGURE 1 only matrices of size 10 by 10 or less, costs $500 for ten copies. has the same eigenvalues. Replace X by RQ, Along with two discs that contain the basic and repeat again and again. X will converge programs and some auxiliary files, PC-Matlab to triangular form and then we can read off its includes a manual, on-line help, and some demon­ eigenvalues from the diagonal. (A few extra tricks strations. The manual includes a tutorial, which help speed up the method.) Matlab does have a is good, and a reference section, which is well warning message if no eigenvalue is found after organized but not complete, as I will illustrate 30n iterations of the method: "Solution will not below. converge." How does PC-Matlab interact with the out­ Now suppose we want to find the roots of a side world? It possesses its own style for data polynomial P. Then Matlab constructs a com­ files, but it comes with a translation program panion matrix, whose characteristic polynomial which helps to transfer data between Matlab and equals P, and applies the method above to find other sources or targets. PC-Matlab can read its eigenvalues. in data from disc files or the keyboard. It can Matlab has three special abilities besides the write out data to disc files or the printer or the matrix manipulation outlined above. One is data CRT terminal. However, it allows the user very analysis. There are functions to sort a vector and little choice in formatting the report. Some of the to find simple statistical measures of its dispersion. formatting commands require knowledge of the However, Matlab is no substitute for a statistics relevant portions of the C language. package. A second specialty is signal processing. There are two ways of ordering PC-Matlab Here we have fast Fourier transforms and their around. In interactive mode, it responds immedi­ inverses, in one and two dimensions, and some ately to one logical line of instructions. For more elementary filter designs. Most users will probably extensive work, Matlab can compile a sequence of enjoy the third specialty: graphics. Matlab very commands stored in a special file. A "command speedily plots one variable against another, once file", when invoked, will execute the sequence as if it is given a pair of vectors of the same length. it had been entered in interactive mode, so that all Give it a matrix Z of values and it will draw a variables can acquire new values. A "function file" three-dimensional mesh diagram (in perspective will return an answer, which can contain more from above and to the side) representing the than one matrix. In this case, all variables used in graph of the function z(x, y) = the xyth entry the file are dummy variables, and any namesakes in Z. The ordinary two-dimensional graphs can outside the function file remain unaffected. There be superimposed on each other with various line is a provision for declaring global variables, but styles, and the axes can be scaled and labeled by the authors of Matlab do not recommend using the user. Polar and logarithmic coordinates can it. Both command and function files can accept be invoked by single commands. The sample plot sequences of parameters. shown in Figure 1 took ten seconds to appear fully By convention, the names of command and on the screen. It is a plot of the 14-by-14 identity function files end in .m, so they are called ".m matrix. A monkey saddle with one hundred and files." Many of the built-in single commands of twenty-one mesh points takes under five seconds Matlab, described in section 1, are actually .m to appear. (See Figure 2.) files, which are read and compiled at the time of

483 use. These .m files can only be read from disc, 25-by-25 real matrix in twenty-six seconds. Loops so the user cannot create them interactively and slow it up. It performed the loop "for i from 1 to try them out while stored in memory, as one can 1000, set a= 1" in eighteen seconds. with Turbo Pascal, for example. Instead, one The language APL also deals exclusively in must cycle between Matlab and an.editor of one's arrays, and a few comparisons might be helpful. choice. No editor is supplied with PC-Matlab. Matlab is not a full-scale programming language. This cycle is facilitated by the ability to call It does not possess the formatting resources of external programs, outlined below. APL, nor the wide range of elementary functions The possibility of writing personal .m files that act on arrays, nor the capacity to deal with turns PC-Matlab into a programming language, arrays of more than two dimensions. Any large or as the authors like to say, it makes PC-Matlab program would be inordinately cumbersome in extensible. Our .m files behave just like theirs, so Matlab. On the other hand, APL has none of we ourselves can add to the package. We can even the high-level matrix algorithms built in, and it add to the on-line help facility to explain our .m uses many special and confusing symbols in a files. special syntax. Matlab has few special symbols PC-Matlab allows the user to invoke any and its syntax is immediately comprehensible to DOS-executable file from the disc without exiting a mathematician. Matlab. Since it takes about forty-eight seconds 4. Matlab in action. I have used PC-Matlab to load Matlab from a floppy disk, this is a great for some computations of the homology of sub­ boon. Of course, there must be sufficient internal groups of finite index in SL(3, Z). There is a close memory beyond the required 320K to hold and relationship between such homology with coeffi­ execute the external file. cients in a finite dimensional rational complex 3. Evaluation. I have already mentioned some representation and the theory of automorphic negative features of PC-Matlab. Most of these forms for the group GL(3). In particular, let can be summarized by saying that Matlab is S(g) denote the SL(3, C)-module of homogeneous too rigid. For instance, three-dimensional plots polynomials in three variables of degree g, let cannot be scaled or rotated at will without some S(g)* be its dual, and let W(g) be the largest tricky preliminary transformation of the data. irreducible constituent of S(g) ® S(g)*. Let G be It is awkward to do computations in integer the group SL(3, Z). If g is even, it is easy to arithmetic. It is impossible to send the result of a produce a nontrivial class in H3 (G, W(g)), which computation directly to the printer without some corresponds to an "eisenstein series" in the theory acquaintance with the C language. of automorphic forms. Any additional classes, or The documentation is clear and readable, but any classes at all for g odd, would correspond to not complete. For example, the reference section "cusp forms" for GL(3) [5). Cusp forms, in this explains that [) represents the empty matrix, but context, are the interesting ones because, aside does not say what will happen when you try to from certain predictable ones which are "lifted" append new rows to it. Normally, the new rows from GL(2), their existence is obscure. The first must be the same length as the rows of the old "lift" occurs when g = 10 [6]. matrix. How long are the rows of the empty I wrote a program in Matlab to compute matrix? H3(G, W(g)) for g < 10. It just so happened Nowhere does the manual state any limits that g = 9 was the limit attainable by any simple on how large a matrix can be. An attempt to approach on account of the memory and variable create too large a matrix will elicit a message size limitations of PC-Matlab. To run g = 9 that matrices of more than 8188 entries are not required solving approximately three hundred and allowed. This could be bad news to a new sixty equations in sixty-six variables. To fit this purchaser of Matlab whose heart was set on some into Matlab, I had to divide the equations into calculations involving 100-by-100 arrays. batches and keep track of an ever shrinking null There is much to say on the positive side space. It ran overnight, and made 3,888,343 and I hope that my description will convince the floating point operations. The final result was reader that, on the whole, PC-Matlab is excel­ that there are no cusp forms for g < 10. lently thought out, and gives access to powerful I cannot say that I have proved this, nor that algorithms in linear algebra. It is efficient and I believe it without a remnant of doubt. Round­ easy to use. It seems to be very accurate, al­ off becomes very prominent in a calculation of though I have not tested that carefully. Of course, this length on PC-Matlab. If I set the tolerance round-off error and instability are inherent in com­ around 0 ( "eps") too small, I could not even putations with matrices. The authors of Matlab find the eisenstein series. If I set it too large, have tried to select the most stable algorithms for spurious solutions crept in. I based my conclusion each task. And PC-Matlab executes very quickly. on a purely empirical setting of eps. A "proof" For instance, it multiplied two random real 50- would require a careful analysis of the errors by-50 matrices in ten seconds, found the inverse and even then I suppose there are people who of a random real 50-by-50 matrix in twenty-three might withhold assent. To continue this series of seconds, and found the eigenvalues of a random calculations will require a mainframe computer,

484 but the experience gained in the unpressured environment of my own PC will be very helpful. I am also computing the homology of sub­ groups of finite index in G with trivial coefficients. As a preliminary I must compute the first homol­ ogy group of r 1 (p) with trivial coefficients, where r 1 (p) is Heeke's group of 2-by-2 integral matrices of determinant 1 where the second row is congru­ ent to (0, 1) modulo p. By the theorem of Eichler and Shimura, this homology group is naturally isomorphic to the vector space of holomorphic and antiholomorphic elliptic modular forms of weight 2 and level p [7]. As explained for instance in [8], the homology calculation comes down to finding the space of all functions f (x, y) of two variables x and y (not both 0) in Z/p such that (i) f(x,y) = -f(-y,x); (ii) f(x, y) + f( -y, x- y) + f(y- x, -x) = 0. Technical considerations allow the inclusion of a third condition, FIGURE 2 (iii) f(-x,y) = f(x,y), which essentially eliminates the classes corre­ only fourteen lines, exclusive of comments. I used sponding to antiholomorphic forms. Similarly, the singular value decomposition for finding null half the eisenstein series can be eliminated by spaces. PC-Matlab found all thirty-five solutions imposing for p = 29 in one hundred and forty-one seconds. (iv) f(x,O) = 0. REFERENCES Now (Z/p)* acts on the solutions f by 1. J. J. Dongarra, C. B. Moler, J. R. Bunch, (a· f)(x, y) = f(ax, ay), so we can diagonalize G. W. Stewart, LINPACK Users' Guide, Philadel­ with respect to this action. That is, fixing a phia: SIAM, 1979. character of (Z/p)*, we impose t 2. B. T. Smith, J. M. Boyle, J. J. Dongarra, (v) f(ax,ay) = t(a)f(x,y). B. S. Garbow, Y. Ikebe, V. C. Klema, C. B. Moler, I have computed only when p is prime, which Matrix Eigensystem Routines-EISPACK Guide, is much simpler to program than the general case. 2d ed., Lecture Notes in Computer Science, vol. A few instructions in PC-Matlab find the smallest 6, Springer-Verlag, 1976. primitive root, say r, modulo p. Then it is easy 3. B.S. Garbow, J. M. Boyle, J. J. Dongarra, to write down the character t, namely for some C. B. Moler, Matrix Eigensystem Routines­ n, t(r k) = e27rv'=Ink/(p-1). (Because of condition EISPACK Guide Extension, Lecture Notes in (i), we need only co~~ider even n.) In doing th~s, Computer Science, vol. 51, Springer-Verlag, 1977. the automatic capab1hty of PC-Matlab to work m 4. G. Strang, Linear Algebra and its Ap­ the complex numbers is very useful. plications, 2d ed., New York: Academic Press, Another small program prepares a list of 1980. powers of r, a logarithm function with respect 5. A. Ash, Cuspidal cohomology of SL(3, Z) to r, and a choice of a set of representatives for with higher weights, in preparation. the projective line over Zjp with respect to the 6. A. Ash and G. Stevens, Cohomology equivalence relation induced by (i) and (iii). These of arithmetic groups and congruences between two .m files together take up about two-thirds of systems of Heeke eigenvalues, J. Reine Angew. a page. Math., to appear. Another two-thirds of a page goes for the 7. G. Shimura, Introduction to the arithmetic program to prepare the equations to be solved. I , theory of automorphic forms, Pub. Math. Soc. view conditions (i), (iii), (iv), and (v) as defining Japan 11 (1971), 234. a set of variables, by choosing a set of pairs ( x, y) 8. A. Ash, Cohomology of congruence sub­ such that knowing f on those pairs determines f groups of SL(n, Z), Math. Ann. 249 (1980), completely. Then (ii) defines a set of equations on 55-73. the values off on those basic pairs. Now (Z/p)* and a group of order six both act on the set of pairs (x,y), leaving (ii) stable, so there are about p/6 equations in about p/8 unknowns. Because of fixed points, there are actually a few more equations and unknowns than that. My final .m file to compute the homology of r 1 (p) for all even characters t, for prime p, took

485 Washington Outlook Kenneth M. Hoffman

Increasing Awareness With Mathematics Awareness Week behind us, it seems like a good time to review how we are doing elsewhere on our awareness scorecard. A long-term effort to increase awareness of the and the needs of our importance of mathematics Executive Branch. Awareness of the need the activities surround­ community began with to rebuild our research funding was exhibited David Report, 1981-1984. ing preparation of the in February by Erich Bloch, Director of the mechanisms for pro­ Subsequently, two important National Science Foundation (NSF). Bloch went viding national leadership were put in place by to Congress for authorization to reprogram so Council at the instigation the National Research that NSF's Mathematical Sciences Division would of our community: the Board on Mathematical be exempt from the FY 1986 cuts made under the Sciences Educa­ Sciences and the Mathematical Gramm-Rudman-Hollings Budget Reduction Act. tion Board. These boards are already developing and amplifying priorities and directions in math­ The picture was not so rosy at the De­ ematics research and education. partment of Defense (DOD), where mathematical sciences funding took a pretty good pounding Since last fall, our ability to increase aware­ because of Gramm-Rudman. Building effective ness in Washington and around the country has working relationships between our community and been further enhanced by the decisions of the the powers-that-be at DOD remains demanding. Joint Policy Board for Mathematics (JPBM) to retain the services of Kathleen Holmay as a public Congress. On February 26, Dr. Edward E. information consultant and Jennifer Vance as a David, Jr., testified before the Science, Research general counsel specializing in congressional mat­ and Technology Subcommittee of the House Com­ ters. The addition of these two professionals to mittee on Science and Technology and presented our team has given us both a stronger presence in an update of the David Report as part of the NSF Washington and a national capability which we budget authorization hearings. He placed partic­ have never had. ular emphasis on the need to move mathematics more to the center of science policy considerations This was vividly demonstrated in the plan­ in Congress. He also noted that, as the field faces ning and development of National Mathematics an escalating shortage of talent, the manpower Awareness Week, held April 14-21. In a few short situation in the mathematical sciences is worse weeks, our new consultant-representatives devised now than it was two years ago. In the question and implemented strategies for producing a spec­ and answer period, he expressed grave concern trum of events, centered about the two main that big science and big projects may push out thrusts of the week: congressional sponsorship support for the individual researcher, on whose and participation, and public observation of the creativity the future of science depends. Dr. fractal-space public service announcement which David's was the first testimony before Congress appeared on about two hundred television sta­ on the overall needs of mathematics since the time tions around the country. These thrusts brought of the American Revolution. It will be followed by us the highest degree of congressional and public several more appearances which Jennifer Vance is visi hili ty our field has yet attained. arranging for this spring.

These are accomplishments of which our The Public. A major symposium on "Math­ community can be proud, especially because of the ematics in Modern Science" will be held at the participation of many mathematicians who wrote National Academy of Sciences in Washington on their representatives in Congress and planned May 12, the first day of National Science Week. supportive local events on very short notice. It Organized and sponsored by the Board on Math­ is important, however, not to view these as ematical Sciences, the symposium will feature isolated events. They are initial steps in long­ three Nobel Laureates in science: Allan Cormack term strategies planned to heighten awareness on (medicine), Steven Weinberg (physics), and Her­ Capitol Hill and throughout the country. If the bert Hauptman (chemistry), who will talk about efforts continue to get the kind of support from the significance of mathematics for their sciences. the mathematical community that they received The moderator will be I. M. Singer. You will see in March and April, we will be able to carry publicity for this event, as well as for other events out our long-range plans and make a significant and accomplishments of our community, amplified difference in how our field and its needs are through the work of Kathleen Holmay. perceived.

486 The Mathematical Community. As we ven­ of pure and applied mathematics, indeed, of all ture further into the worlds of congressional and the major branches of the discipline, developed at public affairs with the aim of increasing aware­ such a mind-boggling pace over the last thirty­ ness, we need to be aware that we will be the five years. Our field is forming into a giant group most enlightened in the process. That is as mathematical laser, to be trained on the basic it should be. We must change because our field problems of mathematics and engineering science. and its place in the scheme of things are changing. It is a time for reaching out, for letting our various We have entered an era in which the power of publics know that massive changes, especially in mathematics will surpass anything in history thus education, will be required if the emerging power far, principally because of the coming together of our discipline is to be properly utilized.

Errett Bishop: Reflections on Him and His Research Murray Rosenblatt, Editor

This book is the proceedings of the Memorial Meeting for Errett Bishop, held at the University of California, San Diego, 24 September 1983. During his early days as a mathematician, Errett Bishop made distinguished contributions in many branches of analysis-first in operator theory, then in the theory of polynomial approximation, and thence to his outstanding research in function algebras. This work in turn led him to his highly original approach to the theory of functions of several complex variables. About 1964 Bishop turned his interests toward the foundations of mathematics. Whereas L. E. J. Brouwer's intuitionism took as basic the integers and the real numbers, Bishop proposed that the integers are the only basic, irreducible mathematical construct. His remarkable 1968 book, "Foundations of Constructive Analysis", was devoted to the development of a large part of modern analysis, suitably modified, on this one concept. The object of the present book is to present a view of Errett Bishop, who died suddenly in 1983 at the age of 54, as a human being, a colleague, and a mathematician. An eloquent statement of his philosophy is contained in his paper, "Schizophrenia in Contemporary Mathematics", which resulted from his AMS Colloquium Lectures in 1973 and which occupies about one-third of the book. In addition, there are memorial articles by S. Warschawski, J. Wermer, J. Kelley, H. Royden, I. Glicksberg, A. Nerode/G. Metakides/R. Constable, and Metakides/Nerode/R. A. Shore. Contemporary Mathematics, Volume 39, April 1985, 99 pages (softcover) List price $16, Institutional member $13, Individual member $10. To order, please specify CONM/39N

Shipping/Handling: 1st book $2, each add'! $1, max. $26; by air, 1st book $6, each add'! $3, max. $100 Prepayment required. Order from American Mathematical Society, P.O. Box 1571, Annex Station Providence, RI 02901-1571, or call toll free 800-556-7774 to charge with VISA or MasterCard

487 INTERNATIONAL CONGRESS OF MATHEMATICIANS

Berkeley, California, U.S.A. August 3-11, 1986

The Public Information Committee for the 1986 visiting mathematicians. I was impressed with International Congress of Mathematicians (ICM-86) the extent to which he used his visit out­ has solicited the views of a number of distinguished side Russia to learn of the work of outside mathematicians on the significance of the Congress researchers. Finally we found a time late at for the furtherance of mathematics and mathemat­ night after most activities had slowed when ical activity. The following are extracts from some we could meet. What we discussed I do not of the responses we received. remember. It was not a private meeting; many other Russians were present, not all of whom and "Mathematics underpins both the natural seemed to be mathematicians. There Smirnov of our social sciences and is an essential part showed me a copy of the booklet Summary of is scientific culture. Research in mathematics Lectures and Seminars, Summer Institute on of conducted on a very wide front both in terms Set Theoretic Topology, Madison, 1955. The its content and in its geographical distribution. Chinese had printed the booklet and sent some with Moreover new ideas of great depth and of the copies to Russia. Rather than resent at long-term implications are being developed the pirating of our booklet by the Chinese I the present time. was flattered by it. Since that time Smirnov "All this makes international meetings impor­ and I have met many times. I admire his tant occasions for mathematicians from all over mathematical talent. I have had dinner at the world to find out what is happening in their his home in Moscow along with his mentor subject, to derive stimulus from the latest ideas Paul Alexandrov. We ex!=hange cards at the and to benefit from new developments. The Christmas season. He and I remain friends. 1986 International Congress in Berkeley will be a major event for the mathematical community, "It is my feeling that individual friendships with several thousand mathematicians from all between people enhance our chances for peace over the world expected to attend. While small and good will between nations." specialist meetings play an important role, the R H Bing grander occasions provided by the 4-yearly In­ in helping to ternational Congresses are vital "'Mathematics is basic to other sciences, yet overall unity of emphasize and maintain the has a strong independent existence', 'Mathe­ for the subject and to provide overall guidance matics is truly international', 'Mathematics is the direction of the world's mathematicians on a single subject'. I read these three statements current research." to which every mathematician will agree re-­ Michael Atiyah cently in an important committee report. The "I have met many interesting foreign math­ International Congress of Mathematicians is ematicians at past ICMs. One meeting in the place where these statements come to life. particular that stands out in my memory is The 16 invited one-hour expository lectures the meeting of the International Congress in and the approximately 130 invited 45-minute Edinburgh in the summer of 1958. There I saw lectures in the 19 sections are supposed to be the Russian mathematician Yu. M. Smirnov. understandable to nonspecialists and to show As was pointed out in the comments under the links between different fields. Some talks the photo in the April 1982 issue of the Amer­ will stress the applications outside of mathe­ ican Mathematical Monthly, Smirnov and I, matics. A survey of the most important results along with J. Nagata, had gotten independent of the last years will be given. Every partici­ solutions of the metrization problem. When pant will have the chance to get new insights Smirnov and I met in Edinburgh, we wanted for his own research work. Many mathemati­ to arrange a meeting to discuss mathematics. cians travel to many smaller meetings in many This was difficult since Smirnov had a busy countries every year, but at the ICM and­ agenda. Rather than seeing sites and taking perhaps-only at the ICM every member will excursions he had arranged his schedule to have the opportunity to meet mathematicians include mathematical discussions with other outside of his normal circles and to get a re-

488 ally broad view of mathematics and the whole "Do I attend the ICM? You bet I do: I went international family of mathematicians." to my first one, in Zurich, Switzerland, in F. Hirzebruch 1932: and since then I have missed only one, in 1950- "As a former invited speaker, and as a partici­ and I still kick myself for not having had pant, I have attended some of the International the courage to borrow the money to go. So do come and let us meet Congresses of Mathematicians (ICM) since the at the ICM in Berkeley, California, 1950 Harvard one. These meetings are of in August 1986." Bernhard the highest level from the standpoint of cur­ H. Neumann rent research in mathematical sciences and "Some benefits of ICM are The Proceedings its applications. Each such congress offers Books, which give us, every four years, a a wealth of excellent lectures by top experts well balanced view of what mathematicians in a great variety of fields, by far more than are working on at the moment. The list a single person can digest. As a Brazilian of speakers, which is obtained after a long mathematician, I am convinced that the ICM selection process (Committees, Panels, etc.) has a series of unique events to stimulate involving many different countries. The fact furthering research and university teaching in that it gives to the press an opportunity to mathematics in developing countries. They are speak about mathematics. This shows to the by far the most important meetings offered general public that mathematics is not a dead to the mathematical community of all parts subject (as many wrongly believe), and that of our world. Young and relatively young re­ there is a lot of work to be done there-just search mathematicians should particularly be as in physics and biology; this is important for encouraged and supported to attend, to have high school students, for instance." personal contact with speakers and leaders in Jean-Pierre Serre their fields present there, getting acquainted We believe that the views we have quoted with the frontier boundaries of our mathemati­ make it plain that the Congress is an important cal accomplishments. It is not at all possible to event for us, enabling us to learn more about do that otherwise through journals, preprints the developments in our subject and bringing us and less comprehensive gatherings. In view into contact with some of those who are doing of the wide use of mathematics in concrete very exciting work today. We hope that you applications that benefit our modern society, will be convinced that you should attend ICM-86. on the one hand, and its lasting value as an Information about the Congress and registration intellectual activity, on the other hand, the high may be obtained by writing to ICM-86, Post Office standard and the large size of the ICM make Box 6887, Providence, Rhode Island, U.S.A. 02940. them a major attraction to mathematicians." Leopoldo Nachbin Members "Mathematics is the work of mathematicians, of the Public Information Committee and most mathematicians work best in contact for ICM-86 with other mathematicians. Thus it is im­ Yousef Alavi, Chair portant that mathematicians meet, to discuss Western Michigan University their mathematics, to exchange ideas, to learn Donald J. Albers of new progress, or just to get to know one Menlo College another. William G. Chinn "The ICM is, true to its name, truly interna­ City College of San Francisco tional: you meet mathematicians there from all over the globe. Some, from countries that Jane M. Day could not afford to send them, come with San Jose State University travel grants from the International Mathemat­ Peter J. Hilton ical Union. Thus the ICM gives its participants State University of New York, Binghamton a unique opportunity to meet mathematicians Lester H. Lange whom they would not otherwise be able to meet. Sari Jose State University Jean J. Pedersen Santa Clara University

489 TENTATIVE SCHEDULE

The information on the ICM-86 program provided below is tentative, and subject to change. A copy of the final version of this timetable will be included in the Congress program. The number in parentheses following the name of the 45-minute speaker represents the section number.

Saturday, August 2

12:00 - 20:00 Registration

Sunday, August 3

08:00 - 17:00 Registration

09:00 - 10:30 Opening Ceremonies Greek Theater

11:00- 12:30 Addresses On Works Of Fields Medalists And Nevanllnna Prize Winner

14:00 - 14:45 45-Minute Lectures In Sections Speaker and title to be announced (1) Renormalization: from magic to mathematics Immersed tori of constant mean curvature in Krzysztof Gawedzki (13) R3 (A counterexample to a conjecture of H. Some questions related to numerical methods Hopf) for stiff nonlinear ordinary differential Henry C. Wente (4) equations A survey of recent developments in three- Germund G. Dahlquist (14) dimensional topology Andrew J. Casson (5) Fixed points of symplectic mappings and periodic solutions of Hamiltonian systems Eduard J. Zehnder (12)

14:00 - 18:00 Sessions for Short Communications

15:00 - 15:45 45-Minute Lectures In Sections Modern developments in invariant theory Speaker and title to be announced (9) Vladimir L. Popov (2) Vortex methods in the analysis of turbulent Sullivan's fixed point conjecture and flow homotopical representation theory Alexandre J. Chorin {14) Haynes R. Miller (5) Inner functions: results. methods. problems A. B. Alexandroff (7)

16:00 - 16:45 45-Minute Lectures In Sections Periods of abelian integrals. theta-functions. Dynamics of area preserving mappings and differential equations of KdV type John Norman Mather (12) Enrico Arbarello (6) Polyhedral combinatorics- Recent develop­ Boundary distortion under conformal mapping ments N. G. Makarov (7) Alexander Schrijver (15) Speaker and title to be announced ( 11)

17:00 - 17:45 45-Minute Lectures In Sections Milnor k-theory and Galois cohomology Pseudo-random sequences that pass all A. Merkurjev (2) polynomial time statistical tests... and other L 'approche geometrique du probleme de tales of randomness Schottky Manuel Blum (16) The geometric approach of the Schottky Verbal problems in arithmetic teaching problem Zbigniew Semadeni (19) Arnaud Beauville (6) A self-focusing solution to the Navier-Stokes equations with a speed-reducing external force Vladimir Scheffer ( 11)

17:00 - 18:00 International Commission on Mathematical Instruction Session

18:30 - 20:00 Chancellor's Reception Faculty Glade

490 Monday, August 4

08:00 - 17:00 Registration

09:30 - 10:30 Plenary Address Complexity aspects of numerical analysis Stephen Smale

10:00 - 18:00 Exhibits Pauley Ballroom. Student Union

11:00- 12:00 Plenary Address Quasiconformal mappings Frederick W. Gehring

14:00 - 14:45 45-Minute Lectures In Sections Spectral theory of automorphic functions and Speaker and title to be announced {9) recent developments in analytic number theory The problem of the regularity of minimizers Henryk lwaniec {3) Mariano Giaquinta {11) Complex geometry in convex domains Multi-level approaches to large scale problems Laszlo Lempert (7) Achi E. Brandt {14)

14:00 - 18:00 Sessions for Short Communications

15:00 - 15:45 45-Minute Lectures In Sections Homogeneous structures Estimates for number of negative eigenvalues Alistair Lachlan ( 1) of SchrOdinger operators with singular Kloosterman zeta functions for GL(n) potentials Dorian Goldfeld (3) Victor lvril (11) The Weii-Petersson geometry of the space of Computational complexity in polynomial Riemann surfaces algebra Scott A. Wolpert {7) D. Yu. Grigorev {16) Speaker and title to be announced (9)

16:00 - 16:45 45-Minute Lectures In Sections Definability problems in analysis Homoclinic bifurcations Alexander S. Kechris (1) Floris Takens (12) Quantum groups (Hopf algebras. deformation Intersection theorems for finite sets of universal enveloping algebras and integrable Peter Frankl (15) quantum systems) Equilibrium analysis of large economies V. G. Drinfeld (B) Werner Hildenbrand {17)

17:00- 17:45 45-Minute Lectures In Sections Some remarks on Peano arithmetic and its The problem of guaranteed accuracy in subtheories numerical methods of linear algebra A. J. Wilkie {1) Sergei Konstantinovic Godunov {14) Manifolds of nonpositive sectional curvature Speaker and title to be announced (15) and manifolds without conjugate points The origins of the representation theory of Werner BaUmann {4) semisimple Lie algebras Speaker and title to be announced {9) Thomas Hawkins {18)

17:00- 18:00 International Commission on Mathematical Instruction Session

19:00 - 21:00 VIdeotapes of Plenary Addresses

. ' Tuesday, August 5

08:00 - 17:00 Registration

09:30 - 10:30 Plenary Address Geometry of four dimensional manifolds Simon K. Donaldson

10:00 - 18:00 Exhibits Pauley Ballroom. Student Union

11:00- 12:00 Plenary Address Underlying concepts in the proof of the Bieberbach conjecture Louis de Branges

491 14:00 - 14:45 45-Minute Lectures In Sections Large cardinals and small sets (a survey) Mathematical. physical. and computational Menachem Magidor (1) aspects of turbulence Segal's Burnside- ring conjecture and the Steven A Orszag (14) homotopy limit problem Speaker and title to be announced ( 15) Gunnar E. Carlsson (5) Mathematical teaching. computers. and Quantum strings and algebraic curves calculators Yuri I. Manin (13) Jean-Pierre Kahane (19)

14:00 - 18:00 Sessions for Short Communications

15:00 - 15:45 45-Minute Lectures In Sections The two faces of infinity Free boundary problems and problems in W. Hugh Woodin (1) noncompact domains for the Navier-Stokes Algebraic groups, Hodge theory and equations transcendence Vsevolod A Solonnikov (11) G. Wiistholz (3) The centrality of mathematics in the history Index theorem and the heat equation of Western thought Jean-Michel Bismut (4) Judith V. Grabiner ( 19) Actions of semisimple groups and discrete subgroups Robert J_ Zimmer (12)

16:00 - 16:45 45-Minute Lectures In Sections Speaker and title to be announced (3) Generalized Hamiltonian systems and optimal Curves on higher dimensional complex control theory projective manifolds Ivan A K Kupka (12) Charles H. Clemens (6) Computer-assisted proofs in analysis Oscar E Lanford Ill (14) Speaker and title to be announced ( 19)

17:00 - 17:45 45-Minute Lectures In Sections Speaker and title to be announced (6) Speaker and title to be announced (15) Chirurgie sur les fonctions holomorphes? Geometric problems of robot motion planning Surgery on holomorphic mappings? Jacob T. Schwartz (17) Adrien Douady (7) Optimization of the ensured result for the dynamical systems Arkadii V. KryazhimskiT (12)

17:00 - 18:00 International Commission on Mathematical Instruction Session

19:00- 21:00 VIdeotapes of Plenary Addresses

Wednesday, August 6

08:00 - 17:00 Registration

09:30 - 10:30 Plenary Address Efficient algorithms in number theory Hendrik W. Lenstra. Jr.

10:00 - 18:00 Exhibits Pauley Ballroom. Student Union

11:00- 12:00 Plenary Address New developments in the theory of geometric partial differential equations Richard M. Schoen

14:00 - 14:45 45-Minute Lectures In Sections Description of the local automorphisms groups Speaker and title to be announced {14) of real hypersurfaces Markov random field image models and their N. G. Kruzhilin (7) applications to computer vision The supercuspidal representations ofGLn and Stuart A Geman (17) other p-adic groups Philip C. Kutzko (8) Gauge theories and nonlinear partial differential equations Clifford H. Taubes {11)

14:00 - 18:00 Sessions for Short Communications

492 15:00 - 15:45 45-l'v .• nute Lectures In Sections L-series and the Green's functions of modular Oscillations and concentrations in solutions to curves nonlinear partial differential equations Don B. Zagier (6) Ronald J. DiPerna (11) Rational approximation of analytic functions Renormalization group in statistical mechanics A. A. Gonchar (7) and field theory Beyond the affine Lie algebras G. Gallavotti (13) Igor B. Frenkel (8) Speaker and title to be announced (14)

16:00 - 16:45 45-Minute Lectures In Sections Heights and L-series Spatial stochastic growth models- survival Benedict H. Gross (3) and critical behavior Almost Lie groups Thomas M. Liggett (10) Ernst A. Ruh ( 4) Speaker and title to be announced (16) Recent results in potential theory Thomas Wolff (9)

17:00 - 17:45 45-Minute Lectures In Sections On p-adic Heeke algebras for GL(2) Stochastic flows and stochastic partial Haruzo Hid a (3) differential equations Singularities in some geometric variational Hiroshi Kunita (10) problems Phenomena of non integrability in Hamiltonian Robert M. Hardt (4) systems Geometrical aspects of representation theory V. V. Kozlov (12) V. A. Ginzburg (8) Reliable computation in the presence of noise Nicholas Pippenger (16)

17:00- 18:00 International Commission on Mathematical Instruction Session

19 00- 21 00 VIdeotapes of Plenary Addresses

Thursday, August 7

08:00 - 13:00 Registration

09:30 - 10:30 Plenary Address String theory and geometry Edward Witten

10:00 - 13:00 Exhibits Pauley Ballroom, Student Union

11:00- 12:00 Plenary Address Recent progress in arithmetic algebraic geometry Gerd Faltings

15:15- 22:00 Barbecue and Rodeo Cow Palace

Friday, August 8

09 00- 16:00 Registration

09:30- 10 30 Plenary Address Soft and hard symplectic geometry Mikhael Gromov

10:00 - 18:00 Exhibits Pauley Ballroom. Student Union

11:00 - 12:00 Plenary Address Representations of reductive Lie groups David A. Vogan. Jr.

14:00 - 14:45 45-Minute Lectures In Sections Some aspects of spectral geometry Ultraviolet stability problems in quantum field Jeff Cheeger (4) theory Concentration phenomena and linear structure Tadeusz Balaban (13) of finite dimensional normed spaces Zeros of Epstein zeta functions and the Vitali D. Milman (9) CRAY-1 supercomputer Conditionally positive definite functions in Dennis A. Hejhal (14) quantum probability A. S. Holevo (10)

493 14:00 - 18:00 Sessions for Short Communications

15:00- 15:45 45-Minute Lectures In Sections Fast algorithms for evaluating the Riemann Modifications of functions. homeomorphisms zeta function and other number theoretic of the circle and Fourier series functions A M. Olevskii (9) Andrew Odlyzko (3) A nonparametric framework for statistical Geometric representation theory of compact modelling Lie groups Charles J. Stone (10) Tammo tom Dieck (5) Parabolic equations in differential geometry Regularity of the a-Neumann problem Richard Hamilton (11) David W. Catlin (7)

16:00 - 16:45 45-Minute Lectures In Sections The what. where and why of almost split Applications of topology with control sequences Frank S. Quinn. Ill (5) Maurice Auslander (2) Vanishing theorems and positivity in algebraic Spectre de varietes riemanniennes et spectre fibre spaces de graphes Eckart E. Viehweg (6) Spectra of Riemanian manifolds and spectra of Base change for automorphic forms on general graphs linear groups Yves Colin de Verdiere (4) Laurent C/ozel (B)

17:00 - 17:45 45-Minute Lectures In Sections Representations of finite-dimensional algebras Theory permanents and proof of van der Peter Gabriel (2) Waerden conjecture A survey of Riemannian metrics with special Georgii Petrovich Egorychev (15) holonomy groups Recent studies on the history of Chinese Robert L. Bryant (4) mathematics Speaker and title to be announced (B) W.- T. Wu (1B) Diffusion and waves in random media George C. Papanicolaou (10)

17:00 - 18:00 International Commission on Mathematical Instruction Session

19:00- 21:00 VIdeotapes of Plenary Addresses

Saturday, August 9

09:00 - 16:00 Registration

09:30 - 10:30 Plenary Address Problems in harmonic analysis related to oscillatory integrals and curvature Elias M. Stein

10:00 - 15:00 Exhibits Pauley Ballroom. Student Union

11:00- 12:00 Plenary Address Random processes in infinite dimensional spaces A. V. Skorokhod

14:00 - 14:45 45-Minute Lectures In Sections Simple Lie algebras over fields of prime Corona problems. interpolation problems and characteristic inhomogeneous Cauchy-Riemann equations Robert Lee Wilson (2) John B. Garnett (9) Trees and hyperbolic geometry Axiomatising the observable behaviour of John W. Morgan (5) automata Robin Milner (16)

14:00 - 18:00 Sessions for Short Communications

15:00 - 15:45 45-Minute Lectures In Sections Reflection groups in Lobachevsky spaces and Speaker and title to be announced (11) algebraic surfaces Qualitative analysis of bursting oscillations in Viachevslav V. Nikulin (6) biological systems Integrable systems and infinite dimensional John Rinzel (17) algebras Tetsuji Miwa (B) Cyclic cohomology and noncommutative differential geometry Alain Cannes (9)

494 16:00 - 16:45 45-Minute Lectures In Sections Combinatorial methods in symplectic Schri:idinger operators with random or quasi geometry periodic potentials Ya. M. Eliashberg (4) Thomas Spencer (13) Representations of 3-manifold groups and The search for provably secure cryptosystems applications in topology Adi Shamir (16) Peter B. Shalen (5) Harmonic analysis and partial differential equations Carlos E. Kenig (9)

17:00- 17:45 45-Minute Lectures In Sections Local representation theory of finite groups Some new results in the theory of nonlinear Theorie locale des representations d'un groupe elliptic and parabolic equations fini N. V. Krylov (11) Michel BroUI§ (2) The concept of construction and the Recent progress on the geometry of surfaces representation of curves in seventeenth­ in R3 and using computer graphics as a century mathematics research tool Henk J. M. Bos (18) William H. Meeks Ill (4) Numerical geometry of algebraic 3-folds V. V. Shokurov (6)

17:00- 18:00 International Commission on Mathematical Instruction Session

19:00- 21:00 VIdeotapes of Plenary Addresses

20:00 - 22:00 Jazz Concert

Sunday, August 10

09:00 - 16:00 Registration

09:30 - 10:30 Plenary Address Algebraic K-theory of fields Andrei Aleksandrovich Suslin

11:00- 12:00 Plenary Address Equation solving in terms of computational complexity Arnold Schonhage

14:00 - 14:45 45-Minute Lectures In Sections Nilpotent orbits. primitive ideals. and The mechanism of Feigenbaum universality characteristic classes Jean-Pierre Eckmann (13) Walter Borho (2) Lower bounds for the monotone complexity of Nonlinear estimation and stochastic flows Boolean functions Mark H. A. Davis (10) A. A. Razborov (16) Justification of Feigenbaum's quantitative discovery in one-dimensional dynamical systems Dennis P. Sullivan (12)

14:00 - 18:00 Sessions for Short Communications

15:00 - 15:45 45-Minute Lectures In Sections Equivariant embeddings of homogeneous Quasiconvexity and partial regularity in the spaces calculus of variations Corrado de Concini (2) Lawrence C. Evans (11) A class of Markov random fields with finite Speaker and title to be announced (13) range Speaker and title to be announced (16) T. Arak (10)

16:00 - 16:45 45-Minute Lectures In Sections On the commutant of absolute Galois group Uniformity and irregularity G. V. BelyT (2) Joisef Beck ( 15) Speaker and title to be announced (8) Subfactors of type 11 1 factors and related topics Vaughn F. R. Jones (9)

495 17:00-1745 45-Minute Lectures In Sections The classification of hyperfinite von Neumann Face numbers of complexes algebras Anders Bjorner (15) Uffe Haagerup (9) Equations diophantiennes et /'evolution de The classical problem of plateau and the /'algebra geometry of the space of Riemann surfaces Diophantine equations and the evolution of Anthony J. Tromba (11) algebra Spectral properties of metrically transitive I. Bashmakova (18) operators Leonid A. Pastur (13)

17:00 - 18:00 International Commission on Mathematical Instruction Session

19 00- 21:00 VIdeotapes of Plenary Addresses

20:00 -22:00 Classical Concert

Monday, August 11

09:00 - 14:30 Registration

09:30- 10 30 Plenary Address Classifying general classes Saharon Shelah

11:00- 12:00 Plenary Address Analytical approaches to quantum field theory and statistical mechanics Jiirg M. Frohlich

12:15 - 13 30 Closing Ceremonies

496 News and Announcements

National Medals of Science Awarded Assistant Professor in 1949. Between 1949 and 1958 he advanced from Assistant Professor to full On March 12, 1986, President Reagan awarded the Prof~ssor. In 1963, he became the Director of the .\Tational Medal of Science to twenty individuals AEC Computing and Applied Mathematics Center in recognition of their achievements in scienc~, of the Courant Institute of Mathematical Sciences. mathematics, and engineering. Included m th1s He was Director of the Courant Institute from group were three mathematical scientists: Peter 1972-1980. Professor Lax was a staff member of D. Lax, Herbert A. Simon, and Antom Zygmund. the Los Alamos Science Laboratory in 1945. The National Medal of Science was estab­ Professor Lax was a Fulbright Lecturer in lished by Congress in 1959 to ~rovide spe~ial 1958 and a Sloan Fellow from 1959- 1963. He was recognition to individuals for the1r ou~stand1?g awarded the Chauvenet Prize of the Mathematical contributions to knowledge in the physical, biO­ Association of America in 1974 and the Norbert logical, mathematical, engineering, behavioral, or Wiener Prize in Applied Mathematics in 1975. social sciences. Selection is based on the total Also he is a member of the National Academy impact and importance of an individual's work of S~iences , the National Science Board ( 1980-- on the present state of his or her chosen. fie~d. 1986) and the Academie des Sciences. Professor In addition. achievements of an unusually sJgmfi­ Lax has served on many AMS committees and cant are considered and judged in relation natur~ as Member-at-Large of the Council ( 1962-1964), to their potential effects on the development of Vice President ( 1970-1971), and President ( 1979- scientific thought. 1980). Peter D. Lax was honored for his "outstanding, innovative, and profound contributions to ~he theory of partial differential equations, applied Cathleen S. Morawetz, of the Courant Insti­ mathematics, numerical analysis, and scientific tute of Mathematical Sciences, was asked by the computations. He has formed many of_ the Editors of Notices to comment on Lax's contribu­ tions. She responded: underlying principles in the field of nonlin~ar hyperbolic equations; his approach to scatter~ng theory has yielded applications to the propagatiOn Peter Lax has made deep contributions to of sound and light." both pure and applied mathematics. He Peter D. Lax was born on May 1, 1926, in has brought to bear, on several difficult and Budapest, Hungary. He received his Ph.D. in unwieldy problems arising in applications, 1949 from New York University, where he became the elegance and structure of mathematics and bent the results into useful answers. He has influenced a whole generation of young mathematicians, arousing their interest in mathematics and its application to practical problems. He has also made exceptional contributions to public service. As a member of the National Science Board, he chaired the "Lax Panel" which conducted a careful scrutiny of the computer power available to American scientists and called the country's attention to its extremely serious inadequacy.

Three major areas of research work have been nonlinear hyperbolic equations, scatter­ ing theory and nonlinear dispersive equations. Many of the underlying principles for nonlin­ ear hyperbolic equations, including t_h_e r~le of conservation laws and many stability IS­ sues in numerical models, are due to Lax. In scattering theory, Lax and Phillips' approach yielded a deeper understanding of the propa­ gation of sound and light and ultimately to new methods in hyperbolic geometry. In non­ linear dispersive equations, Lax discerned the intrinsic mathematical structure of the KdV­ Schrodinger relationship and led researchers Peter D. Lax to new insight in dispersive problems.

497 Herbert A. Simon was honored for his "funda­ Allen Newell of Carnegie-Mellon University mental contributions to our understanding of was asked by the Editors of Notices to comment human problem-solving behavior and decision­ on Simon's contributions. He responded: making within economic organizations. His work includes philosophy and methodology of science, Herbert A. Simon received the National economics, political science, applied mathematics, Medal of Science for his work in understand­ computer science, and cognitive psychology. The ing the nature of information processing in sheer breadth of his record, s panning so many humans. This has been a central concern areas and disciplines, would be pretentious were it of his for thirty years and one that led to not so innovative and scientifically distinguished. " his co-founding the field of artificial intelli­ gence in the mid-fifties. For the methods Herbert A. Simon was born on June 15, 1916, whereby humans solve novel problems is very in Milwaukee, Wisconsin. In 1943, he received much of a piece with how computers solve a Ph.D. in political science from the University novel problems. But this is only one of of Chicago. He has held research and faculty the fields in which Simon has worked, as positions at the University of California, Berkeley, attested by his Nobel Prize in economics. and at the Illinois Institute of Technology. He In fact, a quick enumeration reveals that has been a member of the faculty of Carnegie­ Simon has made contributions to at least Mellon University since 1949, where he is currently a dozen fields- administrative science, artifi­ the Richard King Mellon University Professor of cial intelligence, computer science, economics Computer Science and Psychology. and econometrics, history and philosophy of science, management science and opera­ Professor Simon's work was recognized by his tions research, philosophy, political science, election to the National Academy of Sciences in psychology (both individual and social), so­ 1967. He has received the Distinguished Scientific ciology, and statistics. In all of these the Contribution Award of the American Psycholog­ contributions were technical and often exten­ ical Association and the A. M. Turing Award sive. In many, they made major impacts of the Association for Computing Machinery and on the conduct of the field. In the social was elected Distinguished Fellow of the American sciences his work has uniformly moved the Economic Association. In 1978, he received the fields towards greater use of quantification Alfred B. Nobel Memorial Prize in Economics "for and mathematics. However, he has also been his pioneering work on decision-making processes deeply involved in the substantive content of within economic organizations." Professor Simon each discipline. For instance, he co-authored was also the Gibbs Lecturer in 1984. a leading textbook in public administration. Simon is indeed a scientist and intellectual for all seasons. But he will tell you that all of his work has been dedicated to a s ingle aim, namely, to understand how humans can get along and prosper in this world with their limited resources for processing information. It would appear, however, that he is speaking mostly about the rest of us.

Antoni Zygmund was honored for his "outstanding contributions to Fourier Analysis and its applica­ tions to partial differential equations and other branches of analysis. He created the strongest school of analytical research in the contemporary mathematical world." Antoni Zygmund was born in Warsaw , Poland on December 26, 1900. He received his Ph.D. from the University of Warsaw in 1923, and hon­ orary D.Sc. degrees from Washington University (1972), the University of Torun in Poland (1973), the University of Paris (1974), and the Univer­ sity of Uppsala in Sweden (1976). From 1922 to 1930 he was Instructor of Mathematics at Warsaw Polytechnical School, from 1926 to 1930 Privat Docent of the University of Warsaw, and during 1930- 1939 Professor at Wilno University. He spent 1939-1940 as Visiting Lecturer at the Mas­ Herbert A. Simon sachusetts Institute of Technology, then became

498 Assistant Professor at Mt. Holyoke College, where Elias M. Stein of Princeton University was he advanced to Associate Professor by 1945. He asked by the Editors of Notices to comment on was Professor at the University of Pennsylvania Zygmund's contributions. He responded: from 1945- 1947, then became Professor at the University of Chicago, where he has been Swift Antoni Zygmund is one of the great analysts Distinguished Service Professor of Mathematics of our time, whose deep and many-faceted since 1967. work and broad influence continue to have highly significant impact on many a reas of Professor Zygmund served as Member-at­ mathematics. Large of the Council of the American Mathemat­ ical Society from 1954- 1956, as Vice President Perhaps his most famous work is t he the­ from 1954 to 1955, and on many AMS commit­ ory of singular integrals which he began in tees. He gave an invited address at the 1943 the 1950s (working jointly with Calderon). Summer Meeting in New Brunswick, New Jersey. The results obtained have been of great He delivered the Colloquium Lectures at the 1953 importance in questions of Fourier analysis Summer Meeting in Kingston, Ontario, gave a in Rn, real-variable theory, and most no­ thirty minute address at the 1954 International tably in the theory of partial differential Congress of Mathematicians in Amsterdam, and equations, both in their analytic and topo­ spoke at the Symposium on Singular Integrals in logical aspects. Zygmund's earlier work (in Chicago in April 1966. He was also an organizer the 1930s and 1940s) dealt mainly with the of the Special Session on Differentiation Theory theory of one-dimensional Fourier theories in Evanston, Illinois in November 1968. (the Riemann theory of trigonometric series, and Littlewood-Paley theory), but even then Professor Zygmund held a Rockefeller Foun­ his interests in analysis in Rn had a lready dation Fellowship at Oxford and Cambridge Uni­ developed in work with Marcinkiewicz (his versities in 1929-1930 and a Guggenheim Foun­ student). Among their results were the the­ dation Fellowship in 1953-1954. He received the ory of the strong maximal function, with prize of the Polish Academy of Sciences in 1939. applications to multiple harmonic functions, In 1979 he was awarded the AMS Steele Prize. He and boundary behavior of functions of several is a member of the National Academy of Sciences, complex variables. the American Association for the Advancement of Science, the Polish A cademy of Sciences, the No description of the importance of A. Zyg­ Argentinian National Academy, the Spanish Na­ mund to modern mathematics would be com­ tional Academy in Madrid, and the National plete without mention of the strong school Academy of Sciences in Palermo, Italy. of mathematicians whose development he guided, and the major accounts he gave of Fourier analysis in h is Trigonometric Series (1959), a greatly expanded version of the first edition Trigonometrical Series (1935). This work will surely be considered one of the classics of twentieth century mathematics.

AMS Postdoctoral Research Fellowship The Society has awarded one Postdoctoral Re­ search Fellowship for 1986-1987. The recipient is Dinakar Ramakrishnan, w ho received his Ph.D. from Columbia University in August 1980. Ra­ makrishnan was an instructor at the University of Chicago ( 1980-1982), an assistant professor at the Johns Hopkins University ( 1982- 1985), and a member of the Institute of Advanced Study ( 1982- 1983). He is currently an assistant professor at Cornell University. Professor Ramakrishnan received a C olumbia University Faculty Fellowship (1975- 1977), and a Graduate Research Fe llowship (1977- 1979). He was awarded an Alfred P. Sloan Fellowship in February 1986. Professor Ramakrishnan plans to spend his AMS Fellowship year at the Institute for Advanced Study and the Mathematical Sciences Antoni Zygmund Research Institute.

499 struction {USCMI) will sponsor a series of invited survey talks on August 2, 1986. The series is meant to enhance the understanding and appreci­ ation of some of the major research-related work to be discussed at ICM-86. There is no regis­ tration fee for this USCMI Pre-Congress Session. It will take place from 2- 6 p.m. at the Wheeler Auditorium on the University of California at Berkeley campus. The presentation times, the speakers, and the titles of their talks are: 2 p.m., Robert Ed­ wards, Highlights of low dimensional topology; 3 p.m., Richard Karp, The polynomial-time fron­ tier: Recent developments in computational com­ plexity theory; 4 p.m. , Clifford Taubes, The physics and geometry of estimates in nonlinear partial differential equations; and 5 p.m., Andrew Ogg, Modular functions and number theory. For further information, contact the session organizer, Warren Page, New York City Technical College, 300 Jay Street, Brooklyn, New York 11201. Telephone: {718) 643-3637, or {718) 643- 2470. Dinakar Ramakrisbnan

The AMS Research Fellowship fund was es­ AMS Membership Category Changes tablished in 1973 in response to the need for funds for postdoctoral research. The fellowships There have been two changes in AMS member­ are awarded on the basis of mathematical merit ship categories which the Society would like to to persons who are five to ten years past the bring to the attention of its members. At the Ph.D., but below the academic rank of professor urging of the AMS President, Irving Kaplansky, (regardless of their age) . Fellows must be citizens the Council has approved life memberships. In or permanent residents of a country in North response to suggestions from the AMS member­ America. The awards are intended to support ship, the Trustees Committee on Membership has research fellows for a period of one year and, at formulated a scheme for multi-year memberships present, carry a stipend of $30,000 each with an which was recently approved by the Council. (See expense allowance of $1 ,000. The competition was the AMS Reports and Communications sections held under the supervision of the Society's Com­ in the October 1985 and March 1986 issues of mittee on Postdoctoral Fellowships, consisting of Notices.) J. William Helton, Peter J. Kahn, Neil I. Koblitz, Effective with the 1987 membership year, Stephen Lichtenbaum, Kenneth C. Millett, and AMS members who are at least sixty-two years of Ivar Stakgold {chairman). age and who have been members of the Society There were many strong candidates among for at least twenty years may elect to become life the forty-three applicants. The fellowship is members. The dues for this class of membership financed by the generous contributions of sup­ have been set as a one-time payment equal to five porters of mathematical research, supplemented times the higher ordinary member dues for the with Society funds appropriated according to a membership year in which the life membership matching formula. The continuation of the AMS becomes effective. This does not effect those Research Fellowship program depends on con­ persons who became life members before October tributions the Society receives. Every member 25, 1941 , when another version of life membership, of the Society is urged to contribute to the with no age restriction, was discontinued. Fund. Contributions are, of course, tax de­ Also effective with the 1987 membership year, ductible. Checks should be made payable to the individual AMS members may elect to prepay their American Mathematical Society, clearly marked dues for up to five years. The details for this "AMS Research Fellowship Fund," and sent to the option had not yet been set by the Executive American Mathematical Society, P.O. Box 1571 , Committee and Board of Trustees at the time this Annex Station, Providence, RI 02901. issue of Notices went to the printer. The option will not apply to journal subscriptions. USCMI Pre-Congress Session The request for early payment of your 1987 dues, which you will be receiving shortly, will con­ As announced in the March 1986 Notices , the tain details about these new options. If you wish United States Commission on Mathematical In- to take advantage of one of these new categories \; 500 of membership, please be sure to make the appro­ Sloan Research Fellowships Awarded priate change in your dues category when making your payment. If you have any questions about Ninety young scientists of extraordinary promise either of these dues categories, please do not hesi­ have been selected under a program known as tate to call the Society's membership department Sloan Research Fellowships to receive awards of at 800-556-7774, or write: American Mathemat­ $25,000 each, the Alfred P. Sloan Foundation ical Society, Attn: Membership Secretary, P.O. announced. The fellows are working at forty­ Box 6248, Providence, RI 02940. nine colleges and universities on problems at the frontiers of physics, chemistry, neuroscience, economics, and pure and applied mathematics. The Sloan Research Fellowships Program is by far the oldest of the Sloan Foundation and orie of the oldest fellowship programs in the country. Guggenheim Fellowships It began in 1955 as a means of encouraging basic research by young scholars at a time in their The John Simon Guggenheim Memorial Founda­ careers when teaching duties can be most pressing tion has announced the award of 272 fellowships in and when other support can be difficult to obtain. its sixty-second annual competition. This year's Many fellows have gone on to distinguished careers list of awards includes eight in the mathematical both in their disciplines and in public life. Ten sciences and related disciplines. Sloan Fellows have won Nobel prizes later in their Names of these recipients, their positions, in­ careers and hundreds have received other honors. stitutional affiliations, and their proposed studies This year's winners were selected from 450 are: Solomon Feferman, Professor of Mathemat­ nominees by a committee of recognized scientists ics and Philosophy, Stanford University, (Proof and economists. The ·mathematicians on the theory of formal systems for mathematics, and committee this year were Peter D. Lax, Courant editorial work on Kurt Godel's Nachlass); Har­ Institute of Mathematical Sciences, New York vey M. Friedman, Professor of Mathematics and University; Barry Mazur, Harvard University; Philosophy, Ohio State University (Studies in the and John Milnor, The Institute for Advanced foundations of mathematics); Mark H. Holmes, Study. Associate Professor of Mathematics, Rensselaer Twenty awards for mathematicians were Polytechnic Institute (Mathematical modeling of made to: GREG W. ANDERSON, University mechanoreceptors); Vaughan F.R. Jones, Pro­ of Minnesota; MICHAEL F. CHRIST, Prince­ fessor of Mathematics, University of California, ton University; LAURENT CLOZEL, University of Berkeley (Studies in algebra, analysis, and topol­ Michigan; J. BRIAN CONREY, Oklahoma State ogy); Victor Kac, Professor of Mathematics, Mas­ University; PETRE S. CONSTANTIN, University sachusetts Institute of Technology (Studies in of Chicago; MARC CULLER, Rutgers University; representation theory); Carlos E. Kenig, Pro­ LAWRENCE EIN, University of Illinois, Chicago; fessor of Mathematics, University of Chicago NICHOLAS M. ERCOLANI, Ohio State Univer­ (Studies in harmonic analysis and partial differ­ sity; ROBERT FRIEDMAN, Columbia University; ential equations); Charles D. Parsons, Professor DAVID GABAI, California Institute of Technol­ of Philosophy, Columbia University (Studies in ogy; HENRI A. GILLET, University of Illinois, the philosophy of mathematics); and Jeffrey Scott Chicago; JONATHAN B. GOODMAN, New York Vitter, Associate Professor of Computer Science, University; GLEN R. HALL, Boston Univer­ Brown University, on leave at the Mathematical sity; PHILIP J. HANLON, University of Michi­ Sciences Research Institute (Mathematical analy­ gan; JOHN L. HARER, University of Michigan; sis of algorithms and computational complexity). HIDEKI KOSAKI, Tulane University; GREGORY

Combined Membership List Shipping and Handling Charge

As a privilege of membership, individual members of the Society are entitled to receive a free copy of the Combined Membership List ( CML) in even numbered years. While the CML is free, the Society can no longer absorb the cost of shipping it to members. If you wish to receive the 1986-1987 CML when it is published, kindly remit $2 when you return your membership listing update, or include it with your membership dues. If you choose to add this remittance to your dues, please indicate this clearly when paying your bill.

501 experience. Candidates in medicine must have an M.D. or equivalent degree (e.g. D.D.S., O.D.) at News from the Mathematical the time of application. Sciences Institute Selection is based on the academic, profes­ Cornell University sional, and/or artistic excellence of the applicant, the validity and feasibility of the proposed study The Mathematical Sciences Institute, sponsored plan, the applicant's language preparation, and by the U.S. Army, announces a Workshop on personal qualifications. Preference is given to Fracture in Polymers and Metals to be held candidates who have not had prior opportunity August 4-8, 1986. Topics include: modeling for extended study or residence abroad. of time dependent crack growth; micromechan­ Application forms and further information ics of local failure; constitutive models for may be obtained from the Institute of Interna­ material breakdown; analytical and numerical tional Education, 809 United Nations Plaza, New methods; and interdisciplinary tutorials. The York, NY 10017, or from one of IIE's regional workshop organizers are C. Y. Hui, Theoreti­ offices in Atlanta, Chicago, Denver, Houston, or cal and Applied Mechanics, Cornell University, San Francisco. The deadline date for receipt of Ithaca, NY 14853 and S. L. Phoenix, Me­ completed applications is October 31, 1986. Re­ chanical and Aerospace Engineering, Cornell quests for application materials postmarked after University, Ithaca, NY 14853. October 15, 1986, will not be honored. -liE News Release F. LAWLER, Duke University; DINAKAR RA­ MAKRISHNAN, Cornell University; JAMES A. Fulbright Collaborative Research Grants SETHIAN, University of California, Berkeley; and STANISLAW J. SZAREK, Case Western Reserve The United States Information Agency (USIA) University. -Alfred P. Sloan Foundation News and the Institute of International Education (IIE) announce the May 1, 1986, opening of the com­ Fulbright and Other Grants petition for collaborative research grants abroad for teams of two or three U.S. graduate stu­ for Graduate Study Abroad dents or recent postdoctoral researchers under the The United States Information Agency (USIA) Fulbright Program. and the Institute of International Education (IIE) The Fulbright Collaborative Research Grants announced recently that the official opening of will be available for study in all countries in the the 1987-1988 competition for grants for graduate world, except most East European countries, the study or research abroad in academic fields and U.S.S.R., and Indochina, for the academic year for professional training in the creative and per­ 1987-1988. There are no restrictions on fields forming arts is scheduled for May 1, 1986. It is of study. Prospective applicants should check expected that approximately 700 awards to over with the IIE regarding country availability prior 70 countries will be available for the 1987-1988 to applying. academic year. Applicants must be U.S. citizens at the time The purpose of these grants is to increase of application, must have received the majority of mutual understanding between the people of the their high school and undergraduate education at United States and other countries through the educational institutions in the U.S., and must hold exchange of persons, knowledge, and skills. The a B.A. degree or equivalent before the beginning grants are provided under the terms of the Mutual date of the grant. Applicants with a Ph.D. at the Educational and Cultural Exchange Act of 1961 time of application may have obtained the degree (Fulbright-Hays Act) and by foreign governments, no earlier than June 1984. Applicants in medicine universities, corporations, and private donors. must have an M.D. degree or the equivalent Applicants must be U.S. citizens at the time (e.g., O.D., D.D.S.) at the time of application. of application, will generally hold a bachelor's Applicants in the creative and performing arts degree or its equivalent before the beginning date need not have a degree, but must have four of the grant and, in most cases, will be proficient years of relevant training and/ or experience. All in the language of the host country. Except for applicants must have sufficient proficiency in both certain specific awards, candidates may not hold the written and spoken language of the host the Ph.D. at the time of application. Candidates country to carry out research. for 1987-1988 are ineligible for a grant to a The statements of proposed research submit­ country if they have been doing graduate work ted by team members may be identical, comple­ or conducting research in that country for six mentary, or present different dimensions of the months or more during the 1986-1987 academic team's research. It is preferable that applications year. be submitted through a U.S. academic institu­ Creative and performing artists are not re­ tion or professional organization. In addition, quired to have a bachelor's degree, but they must evidence of affiliation abroad with a host country have four years of professional study or equivalent institution or with an ongoing project that will

502 oversee the research must be presented with the The first Rollo Davidson Prize was awarded application. in 1976. Sixteen prizes have been awarded in Grants will normally be for six to ten months all. The work of the Trust is supported by and will provide monthly fixed sum awards to each individual donations and by royalties associated member of the team. Grantees will also receive with the books Stochastic Analysis and Stochastic basic health and accident insurance coverage as Geometry, published in 1973 and 197 4 as memorial part of the award. It is expected that each tributes to Rollo Davidson. Communications member of the team will carry out their research relating to the work of the Trust should be in one foreign country for a minimum of six addressed to its Secretary, The Bursar, Churchill months during the same academic year, although College, Cambridge CB3 ODS, England. all members of the team do not necessarily have to concurrently conduct research in the host country. Additional information and application forms L. V. Kantorovich may be obtained from the U.S. Student Programs 1912-1986 Division, Institute of International Education, Leonid Vital'evich Kantorovich, of the Academy 809 United Nations Plaza, New York, NY 10017, of Sciences of the U.S.S.R. in Moscow, died April or from one of HE's regional offices in Atlanta, 7, 1986, at the age of 74. Chicago, Denver, Houston, or San Francisco. Kantorovich was born January 19, 1912, in Completed applications from all team members St. Petersburg (now Leningrad). At the age of 14, must be submitted to liE's New York headquarters he enrolled at Leningrad University, graduating in by January 16, 1987. -liE News Release 1930. By the age of 15, Kantorovich had already begun active research, obtaining important results Fulbright Teacher Exchange Program in the descriptive theory offunctions in 1927-1928. Since that time, Kantorovich published more than The United States Information Agency is pleased 260 articles and books on set theory, descriptive to announce the 1987-1988 Fulbright Teacher and constructive theory of functions, functional Exchange Program. analysis, numerical mathematics, the theory of The Teacher Exchange Program involves a extremal problems, mathematical economics, and one-on-one exchange for teachers at the elemen­ computer science. tary, secondary, and postsecondary levels with Kantorovich was appointed Professor of Math­ suitable teachers overseas. The 1987-1988 over­ ematics at Leningrad University in 1932, where seas exchange programs will involve Canada, the he worked until 1960. In 1935, he was awarded United Kingdom, France, the Federal Republic of the degree of Doctor of Physico-Mathematical Germany, Belgium/Luxembourg, Denmark, Nor­ Sciences for the totality of his scientific work. way, Switzerland, Colombia, and Argentina. The From 1930-1939, Kantorovich also taught at the number of exchanges available and the eligibility Leningrad Institute of Civil Engineers. It was requirements vary by country. there that he encountered a practical problem The program also provides opportunities for of optimal cutting of timber and plywood that teachers to participate in summer seminars from prompted his fundamental work in linear pro­ three to eight weeks in length. During the summer gramming. Kantorovich's first publications on of 1987, seminars will be held in Italy and in the this subject appeared in 1939. From 1940-1960, Netherlands. Kantorovich also worked at the Leningrad Branch Applications will be available in the summer. of the Steklov Institute of Mathematics (LOMI). The deadline for receipt of completed applications The Laboratory for the Application of Sta­ is October 15, 1986. For further information, write tistical and Mathematical Methods in Economics Fulbright Teacher Exchange Program, E/ ASX, was established in the framework of the newly cre­ United States Information Agency, 301 Fourth ated Siberian Division of the U.S.S.R. Academy of Street, S.W., Washington, D.C. 20547. Telephone: Sciences in 1958. The joint heads of the laboratory (202) 485-2555. were V. S. Nelmchinov and Kantorovich. In 1960, Kantorovich moved with the Lenin­ grad group of the laboratory to Novosibirsk where Rollo Davidson Trust the group was incorporated into the Institute of Rollo Davidson Prizes have been awarded to Mathematics as the Department of Mathematical Peter Hall of the Australian National University, Economics. Kantorovich was appointed Deputy Canberra, and to Jean-Fran<;ois Le Gall of the Director and head of the Department of Mathe­ Universite Pierre et Marie Curie, Paris. Peter matical Economics at the Institute. At the same Hall is being honored for his work on continuum time, he became the head of the Department of percolation. Jean-Fran<;ois Le Gall received the Computational Mathematics at Novosibirsk Uni­ award for his use of local-time arguments to obtain versity. He occupied these positions until he uniqueness theorems for stochastic differential moved to Moscow in 1971. equations and for his work on the multiple points From 1971-1976, Kantorovich worked at the of brownian motion. Institute for the Management of the National

503 Economy and, from 1976 until his death, at The AWM has designed a questionnaire for the Institute of System Research. He was also the directory so that entries in subsequent issues a member of the State Committee for Science will require little updating. Each listing will and Technology and chairman, . vice-chairman, include the woman mathematical scientist's name, and member of numerous scientific committees of address, and employer or institution; position; one various Ministries and State Committees. He was or two fields of mathematical interest; and titles awarded the Stalin Prize, now called the State of her two most important publications. Prize, in 1949. In 1965 he shared the Lenin Women who wish to be listed in the new Prize, and in 1975 he shared the Nobel Prize in directory should complete the questionnaire and Economics. send it immediately to the Directory of Women Kantorovich's scientific achievements were in the Mathematical Sciences, cjo R. Struik, widely recognized throughout the world. Many Mathematics Dept., Campus Box 426, University universities awarded him honorary doctoral de­ of Colorado, Boulder, CO 80309. grees and numerous scientific societies and acade­ mies elected him to membership. Mathematics into Type Directory of Women It is anticipated that Mathematics into Type by Ellen E. Swanson will be revised and updated in in the Mathematical Sciences the near future. Sinc;e the book's last revision In 1981 the AMS-MAA-SIAM Committee on in 1979, mathematical typesetting has undergone Women in Mathematics published a Directory numerous changes, due mostly to the availability of Women in the Mathematical Sciences with of computers to authors as well as typesetters. financial support from the AMS, the MAA, SIAM, The author is interested in comments from math­ and several publishers. It superseded the original ematicians or their representatives who have used Directory of Women Mathematicians which was her book in the preparation of mathematical published by the AMS in 1973-1974 and which manuscripts. She would like feedback as to what was followed by several supplements. The Associ­ sections of the book are most valuable, what ation for Women in Mathematics {AWM) has now areas need to be expanded, what areas, if any, assumed the responsibility for publication of an are not useful, etc. Please send your comments updated edition of the Directory of Women in the to Ms. Ellen Swanson, c/o Editorial Department, Mathemetical Sciences. American Mathematical Society, P.O. Box 6248, The questionnaire which appears in the back Providence, RI 02940. of this issue will form the basis for the new directory. Women who have earned a Ph.D. degree, or its equivalent, and candidates expecting Erratum to receive a Ph.D. at a later date will be kept on In "A New Knot Polynomial and von Neumann file. Algebras," by V.F.R. Jones, Notices, March The purpose of the directory is to make it 1986, Volume 33, Number 2, page 223, please possible for employers, individuals arranging col­ note the following correction. The example loquia, conferences or mathematics meetings, and ?trefoil = 21 2 - 14 + m212 should read ?trefoil = officers of professional organizations seeking nom­ -2[2 -14 + m2l2. inees for committees to find, in summary form, the curricula vitae of women in the mathematical sciences.

R H Bing 1914-1986 R H Bing, Professor of Mathematics, University of Texas at Austin, died on April 27, 1986, at the age of 71. He was President of the Society in 1977-1978. The news of his death reached the officers of the Society on the date that this issue of Notices was being sent to the printer. An article will appear in the August issue.

504 National Science Foundation News 8 Reports

Reprogramming NSF FY 1986 Funding The new directorate would include the Divi­ sion of Computer Research, now in the Directorate It was reported in the March issue of Notices that for Mathematical and Physical Sciences; the Divi­ the anticipated impact of the Gramm-Rudman­ sion of Information Science and Technology, now Hollings legislation on the Division of Mathemat­ in the Directorate for Biological, Behavioral, and ical Sciences would be a funding reduction of Social Sciences; the Office of Advanced Scientific approximately $2.2 million. This resulted from Computing, now in the Office of the Director; and a 4.3% reduction in the overall National Sci­ some programs from the Directorate for Engineer­ ence Foundation (NSF) budget and would have ing. been implemented through a variety of measures, The new directorate is being established including a 10% reduction in continuing and to create closer ties among such activities as new grant awards. Director Erich Bloch viewed computer science, information science, computer this funding decrease as particularly damaging engineering, supercomputers, and networking. to mathematics research and chose to request C. Gordon Bell, a widely known computer the reprogramming of the NSF budget to exclude architect, will head CISE when it is officially es­ the allocation to the Division of Mathematical tablished on July 1. Mr. Bell was chief technical Sciences from the reduction. This request was officer of Encore Computer Corporation, a. com­ approved by a congressional committee and the puter manufacturer in Marlboro, Massachusetts, $2.2 million slated to be cut was restored. A from July 1983 to January 1986. He helped de­ few other programs also received favorable treat­ velop the highly successful minicomputers of the ment, and as a result the remaining programs Digital Equipment Corporation (DEC), including in the Foundation will sustain a 4.8% cut. One the PDP-4, PDP-5, and PDP-8 as well as larger should remember that the constitutionality of the computers such as the PDP-6 and the DEC system Gramm-Rudman-Hollings Act is still in question 10. These computers established DEC as a leading and that the whole budget cutting process may manufacturer of minicomputers and of computers be a mental exercise. However, it appears that used in scientific research throughout the world. mathematics does not need to worry about the As vice president of engineering at Digital legality of the legislation as it affects FY 1986. Equipment Corporation, Mr. Bell led the design This preferential treatment may not be repeated work for the VAX-11 which set the world standard next year. in research laboratories and departments for sci­ It should be pointed out that the $4 mil­ entific and engineering calculations. From 1966 to lion increase (8.4%) over last year's budget in 1972, he was a member of the faculty of Carnegie­ the Division of Mathematical Sciences does not Mellon University where he conducted the first move rapidly toward solving the funding problems experiments on multiple processor architecture highlighted in the David Report. Statistics from and cooperated on fundamental work related to the AAUP indicate a 6. 7% increase in salaries at the classification of computational systems and Ph.D. granting institutions. As a result, salary their design. support may exhaust the budget increase to the Division of Mathematical Sciences, with little left for additional grants. Minority Graduate Fellowships Awarded The NSF recently announced the award of fifty­ five fellowships to minority students of outstand­ Computer-Related Activities ing ability for graduate studies in the sciences, to be Coordinated mathematics, and engineering. More than 670 American Indian, Black, Pa­ Erich Bloch, director of the NSF, plans to es­ cific Islander, and Hispanic students submitted tablish a new organizational unit, consolidating applications in a nationwide competition for these several computer-related activities into a Direc­ fellowships, which are awarded on the basis of torate for Computer and Information Science and merit. Engineering ( CISE). Panels of scientists, assembled by the N a­ In another organizational change, Mr. Bloch tiona! Research Council of the National Academy said the NSF's Division of Astronomy will be of Sciences, reviewed and evaluated the applica­ moved to the Directorate for Mathematical and tions, with final selections made by the NSF. In Physical Sciences (MPS) from the Directorate for addition to the fellowships awarded, NSF accorded Astronomical, Atmospheric, Earth, and Ocean Honorable Mention to 192 applicants. Sciences (AAEO). The latter unit's name will be Each fellowship provides a stipend of $11,100 changed to the Directorate of Geosciences (DOG). per year for full-time graduate study. An an-

505 nual cost-of-education allowance of $5,250 is also cost-of-education allowance for U.S. institutions provided to the U.S. institution by NSF in lieu was reduced from the previously announced level of all tuition fees. The cost-of-education al­ of $6,000 to accommodate the deficit reduction lowance for U.S. institutions· was reduced from provisions of the Gramm-Rudman-Hollings Act. the previously announced level of $6,000 to ac­ NSF Graduate Fellows may attend any ap­ commodate the deficit reduction provisions of the propriate nonprofit U.S. or foreign institution of Gramm-Rudman-Hollings Act. higher education. Each fellowship is awarded for NSF Minority Graduate Fellows may attend three years of graduate study. The fellowships any appropriate nonprofit U.S. or foreign institu­ may be used over a five year period to permit tion of higher education. Three years of graduate students to incorporate teaching or research as­ study are supported by each fellowship. The sistantships into their education during periods fellowships may be used over a five year period, in which they are not receiving their fellowship so students can incorporate teaching or research stipends. assistantships into their education during periods The new fellows come from 45 states and the in which they are not receiving their fellowship District of Columbia. Of the 505 award offers, support. 173 were made to women. By scientific discipline, The new Minority Fellows come from 25 the distribution of awards was as follows: 107 in states and the District of Columbia. Of the engineering; 18 in mathematics; 9 in applications 55 awards, 23 were made to women. The dis­ of mathematics; 33 in computer science; 40 in tribution among ethnic groups was as follows: physics and astronomy; 41 in chemistry; 26 in American Indians, 4; Blacks, 17; Hispanics and earth sciences; 134 in biological sciences, including Pacific Islanders, 34. There were 19 awards in the biochemistry; and 97 in the social sciences and behavioral and social sciences, 11 in engineering, psychology. 10 in the life and medical sciences, 15 in mathe­ The recipients in the mathematical and matics and physical sciences, including chemistry, computer sciences are listed below (institutions physics, and earth sciences. listed in parentheses are those awarding bach­ The 1987-1988 recipients in the mathematical elor's degrees, those listed outside the paren­ and computer sciences are listed below (institu­ theses are those at which graduate study will tions listed in parentheses are those awarding be pursued): GREGORY DOMINIC ABOWD (Uni­ bachelor's degrees, those listed outside the paren­ versity of Notre Dame), University of Illinois, theses are those at which graduate study will be Urbana-Champaign; ROBERT NEIL ASHCROFT pursued): RITA TERESE DURAN (Trinity Univer­ (Cornell University), Massachusetts Institute of sity), Northwestern University; SELMA MARIA Technology; SCOTT ELLIOT AXELROD (Rut­ GOMEZ (Harvard University), Harvard Univer­ gers University), Princeton University; BRENT sity; JACQUELINE M. HUGHES (University of ARNOLD BANISTER (Washington State Univer­ Cincinnati), University of California, Berkeley; sity), Washington State University; WALTER RICARDO A. PINCHEIRA (Vanderbilt Univer­ BRIAN BARNAS (Dartmouth College), Prince­ sity), University of California, Berkeley. ton University; CHARLES ANTHONY BIER (Rice -NSF News Release University), Princeton University; CHRISTINA LOUISE BLACK (Cornell University), Stanford Graduate Fellowship Awards Announced University; PABLO ALBERTO CALDERON (Uni­ versity of Buenos Aires), New York University; Five hundred and five outstanding college students JOHN BRUCE CARTER (Rice University), Rice are being offered fellowships for graduate study in University; MARA CHIBNIK (Brown University), the natural and social sciences, mathematics, and Harvard University; ROBERT EDWARD CYPHER engineering, the NSF recently announced. (Stanford University), University of Washington; Nearly 4,900 students submitted applications ART DUVAL (California Institute of Technol­ in the nationwide competition for the NSF Grad­ ogy), University of Wisconsin, Madison; MARK uate Fellowships, which are awarded on the basis ALAN EPSTEIN (University of Chicago), Uni­ of merit. versity of Chicago; STEPHEN JEFFRAY FROMM Panels of scientists, assembled by the Na­ (Princeton University), Massachusetts Institute of tional Research Council of the National Academy Technology; DOUGLAS AZIZ GALBI (Princeton of Sciences, evaluated the applications; final se­ University), University of California, Berkeley; lections were made by NSF. In addition to the ERANN GAT (Virginia Polytechnic Institute and fellowship awards offered, NSF awarded Honorable State University), Stanford University; RICHARD Mention to 1,417 applicants in recognition of their GRANT GIBSON (Central Missouri State Univer­ potential for scientific and engineering careers. sity), University of California, Berkeley; ADAM The fellowships provide a stipend of $11,100 MICHAEL GOTTLIEB (Harvard University), Mas­ per year for full-time graduate study. An an­ sachusetts Institute of Technology; MICHELAN­ nual cost-of-education allowance of $5,250 is also GELO GRIGNI (Duke University), Massachusetts provided by NSF in lieu of all tuition fees to Institute of Technology; EDWARD FRANKLIN the U.S. institution selected by each fellow. The GROVE (Brown University), University of Cal-

506 ifornia, Berkeley; KARL ROBERT G UTSCHERA University; JACOB ANDREAS WEGELIN (Uni­ (Harvard University), Princeton University; MAX versity of Washington), University of Chicago; HAILPERIN (Massachusetts Institute of Technol­ ELIZABETH SUSAN WOLF (University of Cali­ ogy), Stanford University; MARK DAVID fornia, Berkeley), Stanford University; DAVID S. HANSEN (Cornell University), Harvard Univer­ WOLFE (Cornell University), University of Cal­ sity; MICHAEL JOHN HANSON (University of ifornia, Berkeley; NEAL ERIC YOUNG (Cornell Washington), Stanford University; MARTHA University), University of California, Berkeley. JEAN HILLER (Massachusetts Institute of Tech­ -NSF News Release nology), Massachusetts Institute of Technology; EVERETT WILLIAM HOWE (California Institute of Technology), University of Illinois, Urbana­ Small Business Innovation Champaign; JOSEPH DAVID JACOBS (Northwest Missouri State University), Stanford University; Research Awards DANIEL ELLIOTT LOEB (California Institute of The NSF has made 124 initial phase awards of Technology), Massachusetts Institute of Technol­ up to $40,000 each under its Small Business In­ ogy; HAW-MINN Lu (Massachusetts Institute of novation Research (SBIR) program. The awards, Technology), Massachusetts Institute of Tech­ which total approximately $4.8 million, are for nology; PETER MATTHEW MAGYAR (Princeton advanced research in essentially all NSF science University), Harvard University; CHRISTOPHER and engineering areas. M. MALONE (Harvard University), Stanford Uni­ One hundred and seven firms in 27 states versity; RICHARD STUART MARGOLIN (Yale received the awards which were selected from 938 University), Princeton University; NICHOLAS proposals received under the 1985 SBIR solicita­ FREITAG MCPHEE (Reed College), Stanford tion. University; DEBORAH F. MINEHART (Harvard SBIR grants support high risk, potentially University), University of California, Berkeley; high payoff research in science-based or tech­ ROBERT DEVORE MITCHELL (Case Western nology firms with 500 or fewer employees. A Reserve University), University of Wisconsin, second objective is to accelerate the conversion Madison; CRETAN RAM MURTHY (Rice Univer­ of federally supported research into commercial sity), Cornell University; LEE AARON NEWBERG applications with private investment. (Massachusetts Institute of Technology), Prince­ ton University; MICHAEL DOWNES PETERSON SBIR is a unique approach to federal R&D (Princeton University), Massachusetts Institute designed at NSF in 1977. The program is intended of Technology; KAREN LEE PIEPER (Rice Uni­ to increase opportunities for small business to par­ versity), Stanford University; KEITH ANDREW ticipate in NSF research as well as to use small RAMSAY (University of Chicago), Princeton Uni­ high-technology companies to convert research versity; PAUL JONATHAN RESNICK (Univer­ into technology. This three-phase program be­ sity of Michigan), Carnegie-Mellon University; came the model for the Small Business Innovation RONALD ROSENFELD (Tel Aviv University), Mas­ Development Act (P.L. 97-219) which was signed sachusetts Institute of Technology; EDWARD into law by President Reagan on July 22, 1982. ERIC ROTHBERG (Stanford University), Stan­ The act requires all federal agencies with $100 ford University; JAMES ROBERT RUSSELL (Mas­ million or more in external R&D budgets to con­ sachusetts Institute of Technology), Stanford duct a similar SBIR program on their own R&D University; ERIC JOHN SCHWABE (Carnegie­ needs. During the first three years under the act, Mellon University), Massachusetts Institute of twelve participating agencies received more than Technology; DAVID SHAO (Michigan State Uni­ 35,000 proposals and have made approximately versity), Cornell University; STEPHANIE FRANK 3,500 awards. SINGER (Yale University), Stanford University; The 124 new NSF awards are for Phase MICHAEL JUDE SLIFKER (University of Pennsyl- . I of the three-phased SBIR program. These vania), Cornell University; ROBERT HAL SLOAN awards provide up to $40,000 and 6 months to (Yale University), Massachusetts Institute of determine, as much as possible within these lim­ Technology; LINCOLN TODD SMITH (Univer~ itations, whether the research idea is technically sity of Illinois, Urbana-Champaign), University of feasible and whether the small firm can perform Michigan; EDITH NELSON STARR (Harvard Uni­ high quality research before further government versity), Massachusetts Institute of Technology; support takes place. IAN LANCE TAYLOR (Yale University), Mas­ Phase II is the principal research effort for sachusetts Institute of Technology; CYNTHIA those projects that are most promising after the PAIGE THARPE (James Madison University), first phase. These awards at NSF have averaged Duke University; RICHARD ALARIC VAUGHAN $200,000 for 1-2 years. Phase III involves the (University of Virginia), Stanford University; follow-on private funding from venture capital or BYRON LEE WALDEN (Vanderbilt University), large industrial firms for product development and University of California, Berkeley; FRANK YU­ commercialization. No SBIR funds are provided HENG WANG (Princeton University), Harvard in Phase III.

507 Of the 124 awards ranging over 21 disciplines, NSB Cites Serious Flaws the following 9 awards were made in the math­ in Undergraduate Science, ematical and computer sciences. The winning Math, and Engineering Education firms, their locations, and project titles are: Weidlinger Associates, Palo Alto, Califor­ Undergraduate education in science, mathemat­ nia, Numerical Methods for Inverse Problems in ics, and engineering contains "serious deficiencies" Three-Dimensional Geophysical Modeling; XOX that are harming this country's economy and secu­ Corporation, Minneapolis, Minnesota, Intersec­ rity, according to a recent report of the National tion of Higher Dimensional Shapes for Geomet­ Science Board (NSB), the NSF's policymaking ric Modeling; Computer Technology Associates, body. Inc., Englewood, Colorado, A New Approach in The report cited three areas that require Automated Theorem Proving; Daniel H. Wag­ immediate attention "of the highest priority" by ner, Associates, Sunnyvale, California, Dealing the NSF and other federal agencies, by the states, With Uncertainty in Expert Systems; Datawise, and by the private sector. The areas of gravest Inc., Orlando, Florida, An Algorithm for Sup­ concern, according to the report, are laboratory porting Views in the Microcomputer Environ­ instruction, faculty preparation, and courses and ment; Incremental Systems Corporation, Pitts­ curricula. burgh, Pennsylvania, Incremental Semantic Anal­ The report was prepared by the NSB's Com­ ysis and Overload Resolution for Ada; Machine mittee on Undergraduate Science and Engineering Vision International, Ann Arbor, Michigan, In­ Education, chaired by Dr. Homer A. Neal, provost tegrated Computer Vision Research for Manu­ at the State University of New York at Stony facturing; Scientific Computing Associates, Inc., Brook. The Committee held four public hear­ New Haven, Connecticut, Building an Interactive ings at which industry, government, and academic Software System for Scientific Computation on leaders presented their views on undergraduate Supercomputers; and Scientific Computing Asso­ education. ciates, Inc., New Haven, Connecticut, Building a The Committee recommended that the NSF's Linda Compiler and Communication Kernel for annual expenditures at the undergraduate level Parallel Programming on Hypercube- Connected in science, mathematics, and engineering edu­ Multicompilers. -NSF News Release cation reach a "total of $100 million in new funds by FY 1989" with appropriate distribution among the following areas of highest priority: laboratory development, instructional instrumen­ U.S.-China Cooperative Science Program tation and equipment, faculty professional en­ The purpose of the U.S.-China Cooperative Sci­ hancement, course and curriculum development, ence Program is to promote opportunities for comprehensive improvement projects, undergrad­ cooperation in basic science between scientists of uate research participation, minority institutions the United States and of the People's Repub­ program, and information for long-range planning. lic of China on projects of mutual interest and "The Committee has concluded that the benefit. The underlying rationale of the pro­ Foundation's role must be strong leadership of gram is that scientific cooperation on the basis a nation-wide effort, an effort that will require of equality, reciprocity, and mutual benefit can participation by public and private bodies at all strengthen the scientific capabilities of both coun­ levels," the report says. tries, advance world scientific knowledge, promote According to the NSB report, responsibility understanding between the United States and for the academic health of undergraduate educa­ China, establish enduring relationships between tion lies primarily with the nation's colleges and American and Chinese scientists and institutions. universities and their governing bodies. Respon­ and, in general, contribute to broad national goal~ sibility for the financial health of the educational on both sides. institutions lies primarily with states, municipal­ ities, and the many supporters of private higher The program supports joint research projects education. of up to two years' duration in all fields of science Single copies of the NSB Committee's report (including social sciences) and engineering nor­ may be obtained from the Office of Legislative mally supported by NSF. Proposals should follow and Public Affairs, Room 527, National Science format and budgetary guidelines provided in the Foundation, 1800 G Street, N.W., Washington, program announcement (NSF 82-50: U.S.-China D.C. 20550. Telephone: (202) 357-9498. Cooperative Science Program) which may be ob­ -NSF News Release tained by calling (202) 357-7393 or writing to 1!.S.-China Cooperative Science Program, Divi­ SIOn of International Programs, National Science Foundation, 1800 G Street, N.W., Room 1214, Washington, DC 20550. It is the policy of the NSF to encourage minorities and women to compete in this program.

508 For Your Information

National Mathematics Awareness Week of his efforts on behalf of Mathematics Awareness Week. Domenici responded with a few comments During March, mathematicians across the coun­ on the problems facing mathematics due to bud­ try received a mailing asking for their assistance get restraints in Congress and expressed his desire in the passage of legislation that would estab­ to do more to stimulate mathematics and to im­ lish April 14-20, 1986, as National Mathematics prove young minds. Recognizing that the budget Awareness Week. This plea was answered by crisis is constraining new program development, an overwhelming response by the mathematical he went on to say that, with the return of financial community and resulted in the April lOth passage stability, it will be possible to act more vigorously of House Joint Resolution 519. This resolution and with less worry as to cost. was introduced to the House by Representatives Mathematics Awareness Week served as a Don Fuqua (D-Fla.), Manuel Lujan, Jr. (R­ focal point, highlighting the accomplishments of N.Mex.), Doug Walgren (D-Pa.), and Sherwood mathematics and American mathematicians. At Boehlert (R-N.Y.). An identical bill, Senate Joint this time of crisis for the mathematical profes­ Resolution 261, was introduced to the Senate by sion, public awareness is essential for regaining Senator Pete Domenici (R-N.Mex.) on January adequate funding for mathematics research and 29, 1986, and passage was subsequently secured for increasing the attractiveness of mathematics in February. as a professional career. The method by which such recognized days, weeks, and months become official is a significant Congressional Strategy process. Before legislation can be sent to the floor for the Mathematical Community of the House, a majority of members must sign on as cosponsors. Passage is usually obtained only Jennifer W !song Vance, Esquire a few days before the official starting date of the The Congress has shown a growing interest over event. As a result, only a fraction of the bills the past few years in many areas of science and requesting the recognition of special events ever mathematics: recruitment of students, supply of receive congressional approval. It was interesting college teachers, public understanding of science that, even with a spring recess for Congress in and mathematics, collegiate curriculum, and con­ early April, the required number of 218 cosponsors tinued support for mathematics research. was obtained on April 8 and, by the time of Congressional interest in these issues is im­ passage on April 10, 242 members of the House portant not only because it suggests future pos­ signed on to the bill. The bill was brought to sible roles for the mathematics community in the floor by Representative Robert Garcia (D­ congressional relations, but also because it may N.Y.), whose involvement was largely prompted signal the beginning of a shift in the way sci­ by the letters and phone calls he received from his ence and mathematics policy is developed in this constituency. Also instrumental in bringing this country. effort to completion was Representative William Ford (D-Mich.). Goals of the Mathematical Community The passage of this bill is a prime exam­ It is easy to identify specific goals of the math­ ple that the mathematical community can, by ematical community that directly support major working together, make its voice heard in the policy priorities of Congress-scientific advance­ chambers of Washington, and positively effect the ment, increased economic productivity, and a future well-being of mathematics. secure national defense. The impact of math­ ematical research on society, Activities during Mathematics Awareness the significance of mathematics education at all Week included the distribution of a public ser­ levels of training, and the general education of the citizenry bear vice announcement to over 200 television stations directly on these congressional across the country. This unusual and colorful goals. announcement opened and closed with fractal representations created by H. 0. Peitgen. Other r--lennifer Vance is an attorney specializing highlights of the week were the Natural History I in fed~ral practice in education and science who Museum's exhibit on the History of Mathematics has been retained by the Joint Policy Board Teaching in the U.S. at the Smithsonian Institu­ I for Mathematics for assistance in dealing with tion and the presentation of the Franklin Medal to congressional issues. From 1975 to 1982 she Benoit Mandelbrot for "his development of fractal ! was a Senior Legislative Associate for the House geometry." Committee on Education and Labor, and from 1982 to 1985 she worked on science education The week concluded with a congressional re­ policy and congressional liaison for the National ception on Capitol Hill at which Senator Domenici Science Foundation. was presented with a framed fractal in recognition

509 To support these goals, the mathematical legislative issues. Misunderstanding the congres­ community needs to document for Congress the sional process can create significant obstacles to consequences of the present crisis in training new achieving one's goals. mathematicians, the importance of recent devel­ Another very important factor is the role opments in mathematics-especially those that and cohesiveness of the mathematical community. bear on scientific computation-and the risks Recent examples, particularly in the area of as­ associated with continued under-funding of math­ tronomy, have shown that the Congress responds ematics research and mathematics education. well to field-initiated consensus and ordering of priorities. A few years ago the astronomers Goals of the Congress provided the Congress with a report in· which Whereas in the past the Executive Branch (e.g., they documented policy and project priorities, primarily the National Science Foundation) has thus enabling the Congress to understand bet­ been responsible for establishing priorities within ter the importance of various projects and the the general science budget, recent changes in relationship among them. the congressional process may alter the tradi­ Congress is able to devote only a small tion whereby Congress has deferred in budget amount of time to understanding any particular allocations to the scientific community. Serious issue. The clearer the mathematical community budget deficits, which resulted in the passage of can paint the global picture, the better Congress the Gramm-Rudman-Hollings amendment, may will understand the consensus of mathematicians. have a dramatic impact both on total funding These understandings will help enable Congress for science and mathematics and on the way the to resist pork barrel requests that distort the budget decisions are made. balance of funding and undermine sound national Members of Congress are growing more in­ policy. terested in the technical aspects of science policy The David Report, Renewing U.S. Math­ as they seek new ways to lower unemployment, ematics: Critical Resource for the Future, is an restore the balance of international trade, im­ excellent document that continues to be effectively prove the education of students, and revitalize used with Congress to expand their understanding the industrial work force. At the same time of the history of mathematics research funding. there has been a breakdown in the peer review Other shorter and more specific reports need to system, with repeated direct appeals to Congress be written to clarify funding needs and priorities by individual universities for support of specific within and among various branches of mathemat­ scientific projects. ics and mathematics education. The mathematical community may be sur­ prised to learn of the very strong interest within The Legislative Process the Congress in mathematics and science educa­ If one seeks to exert influence within the legisla­ tion. Close to $150 million has been appropriated tive process, it is important to understand some to the Department of Education for a mathe­ of the general patterns of activity on Capitol matics and science education program directed Hill. At the risk of being over-simplistic, the across the spectrum from the elementary to the following observations provide a framework for postsecondary level. Needs in mathematics are understanding specific strategies. specifically cited in the fiscal year 1986 Depart­ First, Washington is a town driven by in­ ment of Defense appropriations reports where formation and misinformation. Those able to it concerns the University Research Initiative-a carefully articulate their case in precise and eas­ new $100 million program to create science and ily understandable terms are most likely to be technology centers. And, more generally, there is understood-an important contribution to suc­ increasing interest in collegiate mathematics and cess in reaching their objectives. science education both within Congress and at Second, the federal budget deficit is the single the National Science Foundation. most important issue affecting the way Congress views policy. Congress is no longer thinking about Dealing with Congress the New Deal or the Great Society. Now it is Strong relationships with key congressional staff, primarily concerned with the ordering of priorities both in committees and in members' personal against a background of fiscal restraint. offices, are very important for building a base Third, although political "deals" continue to of informed understanding necessary for effective be part of the Congressional process, legislative action. Unfortunately, the tenure of congressional decisions will increasingly come under sharper staff is often short, so constant renewal of contact fiscal and intellectual scrutiny. is required to maintain continuity. Last, and most important, the legislative The congressional process itself is also very agenda is often crisis driven. What may be subject important. One must know the rules that gov­ to careful deliberation over several months in the ern the operation of the House and the Senate, private sector is often condensed on Capitol Hill to the authorization, appropriation and budget pro­ short one-page summaries prepared overnight. It cesses, and the ad hoc political nuances that affect is very important for the mathematical community

510 to develop the contact that will enable it to which have contributed to the slowdown in en­ respond in timely fashion when these crises occur. rollment growth are expected to continue, and an increase of less than 1 percent is estimated for Recommendations to the fall 1985 graduate S/E enrollment. By contrast, Mathematical Community the 1977-1983 yearly increase averaged 2 per­ The ability of the mathematical community to cent. The current leveling off follows the general reach its important long-term goals will require slowdown in the rate of expansion of academic consistent effort directed primarily at creating an enrollment at all levels. awareness among policy makers-virtually all of The leveling off in overall graduate S /E en­ whom are nonmathematicians-of the importance rollment occurred primarily as a result of smaller of mathematics to the well-being of the intellec­ enrollment among nondoctorate institutions cou­ tual, scientific, and technological infrastructure of pled with a slight drop in the enrollment growth the nation. The community must: rate in doctorate-granting institutions. • increase its understanding of the congres­ Graduate engineering enrollment rose by 2 sional authorization, appropriation, and bud­ percent to about 95,000 in 1984, compared to get process and the impact these have on a 5-percent average annual growth rate between mathematics budget and funding; 1977 and 1983. Graduate science enrollment was • help members of Congress and their staffs virtually unchanged from 1983, after an average understand the importance of mathematics increase of 2 percent per year during 1977-1983. to the well-being of the nation; Institutions. Graduate S/E • identify several key members of Congress who Doctorate-Granting enrollment in doctorate-granting institutions have shown particular interest in mathemat­ 363,000 in 1984, up 2 percent from ics and cultivate their support by increasing reached 1983. This compares to a 3-percent average their understanding of mathematics. annual growth rate over the previous six years. The problems of communicating the role growth reflects a 1-percent rise in the num­ of mathematics to the Congress are formidable. This ber of students enrolled full-time and a 2-percent But mathematics itself can carry the day: it is increase in part-timers since 1983, compared to recognized as useful, it is widely supported by average annual rates of increase of 2 percent and interested in educational reform, and it everyone 4 percent respectively, during 1977-1983. isn't burdened with controversial baggage that Among individual S/E fields, full-time grad­ would make politicians avoid it. What is needed enrollment in computer science continued is for the mathematical community to develop uate to increase at the fastest rate. The 9-percent on a legislative program and then a consensus to 1984 represented a sharp present it to Congress. increase from 1983 reduction from the 14-percent average annual in­ crease reported in the 1977-83 period. In view Graduate S/E Enrollment Shows of the slackening of the growth rate in total S/E Smallest Increase Since 1977 enrollment, however, the 1983-1984 increase in­ dicates that the market for computer scientists This report presents data collected in the Na­ is still relatively strong. Graduate engineering tional Science Foundation's (NSF's) Survey of enrollment growth also slowed, from a 7-percent Graduate Science and Engineering Students and average annual rise between 1977 and 1983 to 2 Postdoctorates, Fall 1984. Estimates are based percent in 1984. on responses from all 325 doctorate-granting in­ The number of women enrolled full time stitutions in the United States and a stratified grew by 2 percent in 1984, a rate of growth that random sample of the 299 U.S. institutions with was about one-half the 4-percent average of the one or more master 's-level programs in science 1977-1983 period. The number of men rose by and engineering (S/E) fields. Data are believed 1 percent, equalling the rate of the previous six to be accurate within ± 3 percent at the 95- years. percent confidence level. Responses were received Full-time S/E graduate students receiving from 93 percent of the departments surveyed, and their primary support from the Federal Govern­ estimates for nonresponse made up less than 6 ment increased by 1 percent, repeating the growth shown in this percent of the graduate S/E total rate of the previous year. The 48,000 total re­ report. Unless otherwise specified, "graduate S/E ported for 1984 is 10 percent below the 1980 peak either enrollment" includes all students enrolled of nearly 53,000. NSF and the National Institutes full- or part-time in programs leading to graduate of Health (NIH) were primarily responsible for the degrees. S/E 1983-1984 growth, up 4 percent and 3 percent, Highlights respectively. Full-time 8/E graduate students re­ ceiving their main support from their institutions All Graduate Institutions. The estimated 415,000 rose by 4 percent, and from other U.S. sources, S/E graduate students enrolled in all institutions by 6 percent. Decreases were reported in self and of higher education in fall1984 represented almost family support (1 percent) and foreign support no change from the prior year. Various factors categories (8 percent).

511 All support categories (research assistants, Full-time Graduate Students in Doctorate Insti­ teaching assistants, etc.) except "other types of tutions. The remainder of this report concen­ support" showed slight increases in 1984. The trates on full-time graduate S/E enrollment in 1-percent increase in the number of fellowships doctorate-granting institutions, which comprises and traineeships awarded was the first growth 59 percent of total graduate S/E enrollment in all since 1977. institutions. The 61,000 foreign students enrolled full The 247,000 graduate students enrolled full time in graduate S/E programs in doctorate­ time in programs leading to a master's or doc­ granting institutions in 1984 represented a 2- torate S /E degree represented an increase of 1 percent increase, down considerably from their percent over the 1983 total, compared to nearly 2 8-percent annual rise during the 1977-1983 period. percent in the number of part-timers. Both groups The number of U. S. citizens rose by 1 percent. showed substantial declines from their 1977-1983 average annual growth rates of 2 percent and 4 S /E Enrollment. Total graduate Total Graduate percent, respectively. enrollment in S/E programs in all institutions in the United States in fall 1984 is estimated at S/E Fields 415,000, virtually the same as in 1983 (Chart 1). 1 Although full-time enrollment in most S/E fields A slowdown in the rate of graduate enrollment was up slightly from the 1983 levels, the rates of growth was also reported by the Council of Gradu­ increase were significantly lower than those of the ate Schools, which showed a slight decline in 1984 1977-1983 period. Engineering enrollment, which in graduate enrollment in all fields at master's­ showed an average annual growth of 7 percent over granting institutions and an increase of less than 1 the previous three years, rose by only 2 percent percent in doctorate-granting institutions, for an in 1984. In the computer sciences, the 1983-1984 overall level of enrollment almost unchanged from growth was 9 percent, compared to 14 percent 1983. 2 Data from the Department of Education's per year from 1977 through 1983. Only in the Center for Statistics (CS) also show an increase of social sciences was an actual decline recorded-2 less than 1 percent in total graduate enrollment. percent-continuing a drop which began in 1981 (table 1). The slowdown in graduate engineering enroll­ ment growth may be partly attributable to the declining rates of growth in the number of for­ eigners enrolled. From 1983 to 1984, the number Chart 1. Graduate enrollment in the sciences and engineering of foreign graduate engineering students increased in all institutions at only about one-half the 3-percent growth rate two (Thousands) of U.S. citizens. As indicated below, about out of every five graduate students in engineering Percent change !rom oreced1ng year in recent years have been non-U.S. citizens; how­ 10,------,10 ever, the proportion has declined slightly since 1981. According to the American Society for En­ gineering Education, undergraduate enrollment fell by nearly 3 percent in 1984, primarily in Sctence the freshman and sophomore classes. 3 Graduate engineering enrollment may therefore level off or decline in the next few years. The patterns of growth by field differ signif­ icantly for men and women. For men, computer science was still the fastest growing field, up 10 percent since 1983; for women, however, engi­ 500 500 neering showed the highest 1983-1984 increase, Total up 7 percent. Both of these rates represent 400t_____ .J\:.------~400 sharp declines from the 1977-1983 period, when the number of men enrolled in graduate com­ 300 300 puter science programs grew at an average annual ------~----- and the number of women in Sc1ence rate of 12 percent 200 200 engineering rose by 16 percent per year. Engineering Sources and Types of Support 100 ...... \ • • • • • • • • • • • • • • • • 100 The number of full-time graduate students re­ ceiving their primary support from the Federal o~--L, __L_ _ _L__ L_ _ _L__ L__~o Government increased by 1 percent from 1983 to 1977 ·78' 79 ·so ·s1 ·s2 83 84 1984, the same rate as the previous year, an aver­ decline of 4 percent from the 52,900 in lnlerpo,aleCCa:a rr.as:er age annual SOURC~ \alior1ai SCIPrlCe foun,1at1on 1980. Growth in graduate students supported by

512 Table 1. Full-time graduate science/engineering enrollment Chart 2. Full-time graduate science/engineering enrollment in doctorate-granting institutions by field: 1984 in doctorate-granting institutions by source and type of major support

Percent By source of major support change, Average annual percent change Field 1984 1983-84 -10 -8 -6 -4 -2 10 Total, all S/E fields 246.800

Engineering 54.800 Total Sciences . 192.000 Federal sciences. 25.100 3 Physical support Environmental sciences . 11.300 -2 Mathematical sciences 10.600 Institutional Computer sciences ... 10, tOO support' Life sciences 66.200 Other U.S. 9.300 -1 sources 36.000 2 20,900 Foreign sources Psychology. 21.600 1 Social sciences . 47.100 -2 Self­ support SOURCE Na110nal St1ence Foundation -10 -8 -6 -4 -2 10

By type of major support -10 -8 -6 -4 -2 0 2 4 10 and NIH (3 percent) accounted for NSF ( 4percent) Total most of the rise in the federally supported grad­ uate student total. The largest relative growth, Fellowships fl percent, was reported in students supported by "other U. S. sources," including industry and nonprofit institutions. These sources, however, Traineeships of all still accounted for only about 7 percent Research full-timers. As in earlier years, institutional funds assistantships supported the largest number of S/E graduate Teaching students-two out of every five. The number rely­ assistantships ing primarily on self and family support declined those re­ All other 1 percent, the first drop since 1979, while types ceiving their primary support from foreign sources showed an 8-percent decrease in 1983-1984. -10 -8 -6 -4 -2 10

The number of graduate students relying alncludes support from State and local governments on fellowships increased for the second consecu­ SOURCE: NatiOnal Sc1ence Foundalton tive year, with institutional fellowships and those from other U.S. sources accounting for most of the increase. Students holding federally funded fellowships declined slightly, with increases in the almost all teaching assistants-99 percent of the to­ number of NSF and NIH fellowships offset by de­ tal reported-depended primarily on institutional creases in those supported by other agencies. The funding (chart 2). latest data on federal support for fellowships and traineeships indicate a 6-percent rise in the dollar Citizenship amount obligated for fellowships and traineeships Although the total number of foreign students from 1983 to 1984. 4 The growth in NSF -supported continued to rise, the 1983-1984 increase of 2 per­ fellows reflects a 36-percent rise in the amount cent was well below the 8-percent average annual budgeted for fellowships. 5 growth rate over the 1977-1983 period. By com­ Research assistants showed the largest rela­ parison, the 1983-1984 rise in foreign enrollment tive increase, 5 percent over the 1983 figure. This at all levels of higher education in all fields was increase is consistent with a 5-percent constant­ less than 1 percent. Factors contributing to the dollar rise in academic R&D expenditures from 1983-1984 reduction in the rate of growth include FY 1983 to FY 1984.6 Major growth came from the strength of the U.S. dollar, which has made it non-federal sources, with smaller growth in fed­ more costly for foreigners to come to the United erally supported research assistantships. On the States to study, and the decline in oil prices, basis of data presented in the FY 1985 and 1986 which has reduced the foreign exchange available budget documents, the number of NSF -supported to the OPEC countries. These nations accounted research assistants is expected to level off after for 21 percent of the foreign students enrolled steady growth since 1981.7 As in previous years, in American institutions of higher education in

513 1984.8 Foreigners were more heavily concentrated in graduate engineering programs than were U.S. COMBINATORIAL students but were underrepresent.ed in psychology METHODS IN and the life and social sciences in comparison to their proportion of total graduate enrollment TOPOLOGY AND (Chart 3). ALGEBRAIC GEOMETRY Chart 3. Foreign full-time graduate science/engineering John R. Harper and Richard enrollment in doctorate-granting institutions Mandelbaum, Editors as a percent of total by field This collection marks the recent Percent of total full-time enrollment resurgence of interest in combinatorial 50 methods, resulting from their deep and Field diverse applications both in topology Engineering 1984 and algebraic geometry. Nearly thirty mathematicians met at the University of Mathematical Rochester in 1982 to survey several of the areas where combinatorial methods Computer are proving especially fruitful: topology and combinatorial group theory, knot theory, 3-manifolds, homotopy theory Physical sciences and infinite dimensional topology, and four manifolds and algebraic surfaces. Social This material is accessible to advanced graduate students with a general course in algebraic topology along with some work in combinatorial group theory Environmental and geometric topology, as well as to established mathematicians with interests in these areas. For both student and Psychology professional mathematicians, the book provides practical suggestions for research 10 20 30 40 50 directions still to be explored, as well as the aesthetic pleasures of seeing the SOURCE. National Science Foundation interplay between algebra and topology which is characteristic of this field. Detailed Statistical Tables for Academic Sci­ ence/Engineering: Graduate Enrollment and Sup­ In several areas the book contains the port, Fall 1984 will be available early in 1986. first general exposition to be published For further information on the tables, contact the on the subject at hand. In topology, for Editorial and Inquiries Unit, Division of Science example, the editors have included M. Resources Studies, Telephone: (202) 634-4622. Cohen, W. Metzler and K. Sauerman's 1 article on "Collapses of K x I and group Because the sampling error at the total level is sig­ presentations" and Metzler's "On the nificantly larger than the apparent growth rate in total graduate S/E enrollment from 1983 to 1984, universe data Andrews-Curtis-Conjecture and related presented in this report should be used with caution. problems." In addition, J. M. Montesino 2 CGS Cammunicator, Special Report, Vol. XVIII, No. has provided summary articles on both 2, February 1985, p. 6. 3- and 4-manifolds. 3 "Engineering Enrollments, Fall 1984," in Engineering Educati.an, Vol. 76, No. 2, November 1985, p. 104. 4 1980 Mathematics Subject Classifications: National Science Foundation, Federal Support to Uni­ 14Jxx, 20Fxx, 55Pxx, 55Sxx, 57Mxx, and versities, Colleges, and Selected Nonprofit Instil:uJ:Wns, Fiscal Year others. 1984 (Washington, D.C., 1985), unpublished data. ISBN 0-8218-5039-3, LC 85-11244 5National Science Foundation, Justification of Estimates ISSN 0271-4132 of A~ to the Congress, Fiscal Year 1985 (Washing­ 376 pages (softcover), October 1985 ton, D.C., 1985), p. 43. List price $32, Institutional member $26, 6National Science Foundation, Academic Science/Elnr Individual member $19 gineering: R&D F'u.nds, Fiscal Year 1984 (Detailed Statistical To order, please specify CONM/44NP Tables) (in press) (Washington, D.C., 1986), table B-6. 7 National Science Foundation, Justification of Appropri­ the ations to Congress, Fiscal Year 1985, p. 17, and Fiscal Year Shipping/Handling: 1st book $2, each add'l $1, $25 1986, p. 20, op. cit. max. By air, 1st book $5, each add'l $3, $100 max. 8This proportion represents a sharp decline from the Prepayment required. Order from AMS, P.O. Box 24 percent in 1983/84 and 26 percent in 1982/83. See Insti­ 1571, Annex Station, Providence, RI 02901-1571, or tute of International Education, Open Doors: 1989/84 (New call 800-556-7774 to use VISA or MasterCard. York: Institute of International Education, 1984), p. 23. Open Doors: 1984/85 is currently in press.

514 Mathematical Text Processing Richard S. Palais

Technical Word Processing they work best by creating their papers from scratch on a word processor. Originally I had planned to review Macintosh Moreover they claim that by my unstated assumptions technical word processing software this month. I had omitted from discussion what they felt However, several important software packages are was a potentially large class of TWP users and that I "about to be released" and, since I would like had failed to give due consideration to to try out the released versions before writing software that was best adapted to their use. They my review, I have decided to postpone this also felt that this latter software was often simpler, review column until at least the August issue of less expensive, and easier to use than the Notices. Also planned for the August issue is an more sophisticated software I concentrated article by Larry Siebenmann on preprocessors on in my article; namely 'lEX software that was including, in particular, his own "Sweet- meant to produce publication 'lEX". quality copy. I would like to thank all those who have I still feel that all written or telephoned to me with comments, but a few mathematicians would be ill advised criticisms, questions, and information. I to choose this mode of will try creating their papers, to respond to your comments in later columns, but I would like to hear from others who find but now I would like to mention some comments that the method works for that took me entirely by surprise. them, and perhaps someone who does will write a short article on this approach. I have always taken it as axiomatic that one Please address correspondence to Richard should have in hand a reasonable first draft S. Palais, Department of of Mathematics, Brandeis a mathematical paper before sitting down University, Waltham, with a MA 02154. word processor. But two readers maintain that

LARGE-SCALE COMPUTATIONS IN FLUID MECHANICS Bjorn E. Engquist, Stanley Osher and Richard C. J. Somerville, Editors The purpose of the AMS-SIAM Summer Seminar on Applied Mathematics held at Scripps in 1983 was to bring together scientists interested in computational fluid mechanics and numerical analysts and mathematicians working in large-scale computations. The numerical modeling included geophysical problems of the atmosphere, ocean, and interior of the earth, and planetary, solar, and stellar atmospheres. Applications ranged from idealized turbulence in laboratory convection models to operational weather prediction. Engineering applications included aerodynamics, combustion, and flow in porous media. Recent advances in numerical analysis which have applications to these problems were stressed. These include shock capturing algorithms, spectral methods, boundary treatments, vortex methods; and parallel computing.

ISBN (Set) 0-8218-1122-3; (Part 1) 0-8218- PREPAYMENT REQUIRED. Order from 1129-0; (Part .II) 0-8218-1130-4, LC 84-24534, American Mathematical Society ISSN. 0075-8485 PO Box 1571 387 pp. (Part 1), 422 pp. (Part 2) (hardcover), July Annex Station 1985 Providence, RI 02901-1571 Set: lndlv. mem. 166, List $110, Inst. mem. $88 or call 800-556-7774 to use VISA Part 1: Indlv. mem. 140, List S66, lnst. mem. S53 or MasterCard. Part .11: Indlv. mem. 140, List $66, Inst. mem. $53 To order, please specify LAM/22NA (set), LAM/22.1NA (part 1), LAM/22.2NA (part 2) Shipping/Handling: let book $2, each add'! Sl, S25 max. By air, 1st book $5, each add'! S3, $100 max.

515 Letters to the Editor

Effective Text Editors for checking if a word or string occurs m a manuscript, The two articles about mathematical and tech­ nical word processing which appeared in the for moving to a particular location in a January 1986 Notices were quite interesting. As manuscript by searching for a string which is Richard Palais said in the second paragraph of his more or less unique to the location, article, he was writing from "the viewpoint of the working mathematician who would like to convert as part of search-replace to change occur­ handwritten mathematics into a computer file, rences of one string to another. This is and then use that file to produce printed copy." called global search-replace when the word The report of the Boston Computer Society com­ processor automates the process of changing pared technical word processors from pretty much repeated occurrences. the same viewpoint. Unfortunately the authoring and editing side of technical word processing is comparing IBM PC based technical receiving no consideration. This failure is partly While last spring I noticed that a word responsible for a counterproductive trend which is word processors tended to have good authoring and edit­ occurring with the spread of word processors into processor throughout its design or lack them mathematics departments. Many departments ing facilities One technical word processor, which are choosing word processors solely on the basis throughout. of departments have adopted for use of ease of use by secretaries and the quality of a number by their secretaries, searched many times more the technical output produced. More and more than some others. It also lacked global mathematicians in these departments are using slowly When facing the situation of in­ the same word processors to prepare their papers search-replace. numbered theorem early in a long paper from scratch as opposed to first creating hand­ serting a all the subsequent theorems, written (or typewritten) documents. They are and renumbering and internal references, a fast word pro­ using word processors which are not particularly formulas few other good authoring and editing well suited to the tasks of writing and editing. cessor with a could do in five minutes what this slow Ironically, the secretaries for whom the word pro­ features hours to do. Another exar.1ple is back cessors were chosen are being eliminated from the one took document preparation process. search, meaning search done from the current editing location in a manuscript back toward the Readers may wonder about the different word beginning. The slow word processor lacked back processor needs for authoring and editing as search. When creating a first draft of a paper the opposed to converting existing copy into printed current editing location is generally at or near the output. Here are examples related to the search end-so-far. If a word processor lacks back search capability of word processors. Search is extremely the writer must frequently move back toward the important for authoring and editing. For example top of the manuscript before searching. Back search is used: search is not so important for preparing finished output from existing copy. These are just two Policy on Letters to the Editor examples; there are many others. Letters submitted for publication in Notices are reviewed by the Editorial Committee, whose task is to determine Word processors are seeing increasing use for which ones are suitable for publication. The publication authoring, because once learned, they are excel­ schedule requires from two to four months between re­ lent writing tools. For example, even using the ceipt of the letter in Providence and publication of the slow word processor mentioned above, electronic earliest issue of Notices in which it could appear. Publication decisions are ultimately made by ma­ cut and paste is many times more convenient than jority vote of the Editorial Committee, with ample pro­ cut and paste by hand. Having to move toward vision for prior discussion by committee members, by the top of a document before searching is still mail or at meetings. Because of this discussion period, a lot more convenient than not having search at some letters may require as much as seven months before when working on a long paper. a final decision is made. all. Especially The committee reserves the right to edit letters. But the extra step of moving toward the top is Notices does not ordinarily publish complaints annoying if you are used to a word processor about reviews of books or articles, although rebuttals with back search. Good authoring and editing and correspondence concerning reviews in Bulletin of the features contribute enormously to the pleasure American Maih.ematicol Society will be considered for pub­ lication. and productivity obtained from a word processor. Letters should be mailed to the Editor of Notices, Moss Sweedler American Mathematical Society, Post Office Box 6248, IBM, Thomas J. Watson Providence, Rhode Island 02940, and will be acknowl­ Research Center edged on receipt. (Received January 27, 1986)

516 Presidential Young Investigators it be a separate section so that it is readily dis­ I served on the screening panel in mathematical carded. If material then winds up being repeated sciences this fall for the Presidential Young Inves­ in different sections, so what? That is allowed in tigators (PYI) awards. I was surprised by the low theses. number of applicants in pure mathematics. The We like to imagine that the young mathe­ PYI is a very important NSF award and it is im­ matician has years to prove herself or himself. portant that outstanding junior mathematicians Six years to develop credentials to get tenure is apply. I think there are some obstructions that the standard, but it only applies to those who need to be removed. have gotten the job. Careers can be and are (a) Under the present rules, the PYI is only permanently sidelined much earlier even when for applicants who have been out for a short the student does everything right, but does not number of years and involves a commitment by find three faculty members who are willing to the sponsoring institution. These rules remove understand the thesis work and write letters of from consideration new Ph.D.'s holding 3 year recommendation about the research accomplish­ terminal positions at leading institutions. This ments. I often see people being selected for-or should be addressed by the NSF as it impacts rejected from-assistant professorships or instruc­ unfairly on mathematics in contrast to some of torships before the thesis is finished. Many of the physical sciences. these positions come with heavy teaching loads, (b) The PYI awards are not sufficiently pub­ perhaps little hope for summer support, and per­ licized in the mathematical community. Some haps no experienced researchers in the field of potential applicants are discouraged by the (theo­ specialization to talk to. Any upgrading of that retical) requirement of obtaining matching indus­ job must be done quickly if at all. We all recognize trial support although in practice this requirement that good second jobs go preferentially to people can be waived. I would urge all departments to at the best universities. Publications and research consider nominations for the PYI. quality preprints can be the critical asset. Time can be wasted with impunity only in fields such Peter B. Gilkey as physics where people average perhaps 3 years University of Oregon (Received January 28, 1986) with postdoctoral fellowships. Let us teach our students how to write what it is necessary to write and not worry that a thin thesis will reflect Ph.D. Thesis Style poorly on us as advisors. Upon encountering once again a very lengthy James A. Yorke Ph.D. thesis that contained some excellent mate­ The University of Maryland (Received January 20, 1986) rial, some, but not that much, I felt some public discussion of thesis style was merited. I would like to address the advisors who believe that Funding for Computational Mathematics long-winded theses are a mandatory tradition, a The mathematical community is singularly unwill­ tradition that I feel is wasteful of time and people. ing to apply its analytical skills to itself. We have The degree recipient is at a difficult and crucial unconsciously made lack of funding an axiom. point in her or his career. Typically the recipient Our financial expectations are, on the whole, out will almost immediately be holding down a full of touch with the rest of science and engineering time job such as teaching a couple of courses that (salaries, grants, equipment, travel). Particularly he or she has not previously taught. Still that pernicious is the claim that the mathematical thesis must be turned into publishable papers. community has no large-scale funding needs such For most mathematicians, especially those with as the big dollar projects in physics, astronomy, no experience in writing papers for publication, etc. We greatly need to create a computational the task of converting the thesis into one or more infrastructure to support applied and computa­ papers will often take months. One of the most tional mathematics. important skills a budding research mathemati­ The recent article in Notices, "Future Direc­ cian must learn is how to write papers, and I feel tions" reveals the same assumed limits. All of our the time to begin that education is as a student, top schools should now have in place programs when one's writings are being supervised. I ask in Numerical Analysis, Parallel Computation and my students to write papers and to staple them Discrete Mathematics, with supporting computa­ together and call it a thesis. Of course pages tional facilities. Getting one small group at each must be renumbered and maybe the references of ten or twenty schools over the next few years is will be collected together before binding. A com­ a very limited goal! We need Numerical Analysts mon reason given for padding of a thesis with now, not in ten years. unpublishable material is that one must be sure The Notices survey claims average spending that the writer understands the details of what of $50,000 per Mathematics department per year. has been written, but that is as true of papers What is the corresponding figure for physics, as it is of theses. For those who feel additional chemistry, engineering? In industry, the typical elementary material must be included, please let workstation (for someone with less training than a

517 Ph.D. mathematician) goes for $50,000 plus (per might take several years to begin doing research person-well, maybe two or three). using a computer. To take advantage of the NSF's supercomput­ Conclusions: ing efforts, departments need a VAX-class mini­ ( 1) The David report understated our com­ computer as local link to the high speed network puter equipment needs by a considerable factor. accessing the supercomputers. Most Computer Funding is needed to build a computational infras­ Science, Physics, Chemistry, and Engineering tructure in mathematics, but we lack a consensus departments have several such VAX-equivalents. that this is, in fact, necessary. Most Mathematics departments have none. Elec­ (2) A mechanism is needed to get faculty tronic mail networks are becoming an important at undergraduate institutions involved in research means of communication (and publication). We've and instructional development. Researchers not just excluded ourselves from the realm of modern using computers must have a way to gain com­ scientific discourse! puter expertise without compromising their re­ search productivity or grants. How do faculty become aware of computa­ A tangential question: why do we have two tional methods? How do computers make their Institutes for Retraining in Computer Science, and way into the curriculum? In physics and other none for, say, Retraining in Numerical Analysis, fields, lab computers train the faculty in computer Scientific Computation, Statistical Computation, use, keeping them abreast of the state-of-the-art. Computational Combinatorics and Graph Theory, This eventually leads to classroom applications. or Mathematics and Computer Graphics? Does Mathematics departments are not acquiring com­ Numerical Analysis lack the hype or glamour of parable computer expertise. What expertise we Computer Science? have tends to be on out-of-date, cheap micro­ Thank you for the opportunity to raise these computers. We are not accumulating distilled issues in public. I'm quite aware that much of the knowledge in the form of software, nor making above may be contentious and controversial. My use of computers in the classroom. This further purpose is to initiate discussion. turns off gifted students, who perhaps are reacting to a perceived bias towards "pure" math. Peter J. Welcher U. S. Naval Academy This dearth of equipment is related to our (received February 7, 1986) historic lack of funding: we never asked for it. People rave about Mu-Math, which is expensive and inadequate compared to MACSYMA or SMP. The same is true of technical word processing soft­ ware, where even the official formatting system is arguably user-unfriendly. Much of the discussion reveals an acceptance of the status quo, math­ ematicians adjusting their expectations to what GROUP ACTIONS ON they have (Apple II computers). This is a shame, because computers finally have the power to do RINGS things of interest to mathematicians, if we could Susan Montgomery, Editor only obtain them. Within the year we'll see a lap-top portable Ring theorists and researchers in invariant IBM PC/AT compatible with internal hard disk. theory and operator algebra met at Within at most two years, the price will be Bowdoin for the 1984 AMS-IMS-SIAM Joint Summer Research Conference to in the $1500 range a student (or prof) can af­ exchange ideas about group actions on ford. Research-class machines will also be less rings and to stimulate general interaction expensive. Yet the mathematical community is among their various fields. Collecting relatively ignorant of this, because we lack a together the proceedings of that meeting, "critical mass" of 10% or more who are regular, this interdisciplinary volume reveals knowledgeable, computer users. Many mathe­ many topics common to the three fields: maticians purposely ignore anything related to K-theory, dual actions, semi-invariants, computers. and crossed products. A high percentage of the papers here are expository; of The Notices survey indicated relative satis­ particular significance are the papers faction with computer resources. Naive! No by the four survey lecturers, Formanek, graduate student is going to try to do research on Hochster, Passman and Reiffel. a non-existent or hard-to-access computer, with­ 1980 Mathematics SubJ'ect Classifications: out NSF and faculty support. Thus I am not 16, 13, 14L, 46L ISBN 0-8218-5046-6, LC 85-11242 surprised that the supply of computers seems to ISSN 0271-4132 roughly match demand. Not asked: would you 289 pages ( softcover), September 1985 use a computer if one were available? Note that it List price $27, Institutional member $22, is very hard to switch fields. Even in the presence Individual member $16 of an abundance of computational resources, one To order, please specify CONM/43NP

518 Queries Edited by Hans Samelson and Stuart Antman

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

Queries (b) Is the infinite product expansion 351. David Halprin (P. 0. Box 186, North Carl­ ton, Victoria 3054, Australia). Has any reader 1 - ! = (1 - !) .(1 + - 1-) seen reference to, or use of, the "Condition for e 2 3·2-3 Immobility of a Point," and also the "Condition 1 for Immobility of a Straight Line" (as derived and . ( 1 - -,-4-. 3-:----c· 2=--_- --:-4---::-3-+---,4) used by Ernesto Cesaro), appearing anywhere other than in Cesaro's own book, Lectures in . (1 + 1 ) ... intrinsic (natural) geometry, and in some of his 5·4·3·2-5·4·3+5·4-5 papers? Especially does any reader know of a better (more rigorous, yet simpler in derivation) correct? proof? It appears to be a very useful mathemati­ 354. R. N. Gupta (Centre for Advanced Study cal tool, no longer in use, but with much untapped in Mathematics, Panjab University, Chandigarh­ potential in geometry, differential geometry and I60014, India). Let VF be a vector space with calculus of variations. an infinite countable basis Vt, v2, ... , Vn, .... Let 352. Ashot E. Djrbashian (Inst. of Math. of Ar­ E = Hom~(V, V) acting on the left of V. It is menian Acad. Sci., Marshal Baghramian st. 24b, easily seen that, up to isomorphism, V is the Erevan 19, Armenian SSR, USSR, 375019). Con­ only simple left E module such that ann(V) = 0. sider a vector-valued function F = (h, ... , fn), Let I= {a E E 3 (aV:F) < oo}. Does there left module S such that defined in a region G C R n, for which the equa­ exist a unique simple E tions ann(S) =I? 355. David Shelupsky (Department of Physics, 3fk - 3fJ City College, CUNY, New York, NY 10031). A 1:::; j,k:::; n number of films recently shown on educational ax] axk' television refer to the results of using numerical models for the distribution of wind and tem­ perature in the Earth's atmosphere. Are there are satisfied. such readily available models compact enough for This is one of the possible generalizations of classroom presentation? Are there idealizations the notion of analytic function in Rn and it is (e.g., using an Earth whose surface is entirely natural to call the components of F conjugate covered with dry land or with water) that can harmonic functions (see, e.g., E. M. Stein and be treated analytically or, at least, with a small G. Weiss, Introduction to Fourier analysis on computer? Euclidean spaces, Princeton, New Jersey, 1971). Is anything known about conjugate kernels of the Poisson kernel in the unit ball of R n, n 2: 3? Responses The editors would like to thank all those who sent 353. Aldo P. Peretti (Murillo 1121-9° D, (1414) in replies. Buenos Aires, Argentina). (a) What is the expression for 280. (vol. 30, p. 491, August 1983, Howard Kleiman) Polynomial-time mapping of the travel­ n n ing salesman problem onto the hamiltonian circuit problem. Reply: If the directed (i.e., asymmetric) (1 +Xi+ YJ) II II traveling salesman problem can be solved in poly­ i=l J=l nomial time, then so can its subproblem consisting of the undirected (i.e., symmetric) traveling sales­ as a polynomial in the elementary symmetric man problem with distances restricted to {1, 2}. functions of the Xi and of the YJ? For an equivalence between this subproblem and

519 that of obtaining a hamilton circuit in an undi­ first reply in these Notices, vol. 33, p. 53.) Reply: rected graph, see pp. 35, 36 in M. R. Garey and D. See the report, Discretized operational calculus, S. Johnson, Computers and intractability: A guide I: Theory, by C. L. Lubich (Inst. F. Math. and to the theory of NP-completeness, W. H. Freeman Geom., Univ. Innsbruck, A-6020 Innsbruck, Aus­ & Co., New York, 1979. Note that the technique tria) and his paper Discretized fractional calculus in the reference is also valid for an equivalence (to appear in SIAM J. Math. Anal.). (Contributed between the directed traveling salesman problem by H. J. J. te Riele) 2} and that of with distances restricted to {1, 335. (vol. 32, p. 608, October 1985, Sherwood obtaining a hamilton circuit in a directed graph. Washburn) The diophantine equation 6y2 = Further insight is provided in E. L. Lawler et al. x( x 1) (2x 1). (See earlier response, vol. (editors), The traveling salesman problem, John + + 33, p. 336, March 1986.) Correction: The an­ Wiley & Sons, New York, 1985. (Contributed by nounced elementary solution by M. W. Bunder R. H. Warren) has a serious gap in it. It is in fact unlikely that 325. (vol. 32, p. 379, June 1985, Bertram Ross) Do an elementary solution is possible. (Contributed difference operators of fractional order exist? (See by J. J. Schaffer and J. C. Owings, Jr.)

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

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

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

520 Acknowledgement of Contributions

The officers and the staff of the Society acknowledge with gratitude gifts and contributions received during the past year. The inside cover of each issue of Mathemntical Reviews carries the names of the sponsoring societies which support that publication. Contributing members of the Society paid dues of $117 or more. In addition to contributions to the AMS Research Fellowship Fund, there were a number of unrestricted general contributions. Also listed this year are AMS members who have contributed to ICM-86. Some of the contributors have asked to remain anonymous. All of these gifts provide important support for the Society's programs. The names listed below include those whose contributions were received during the year ending March 31, 1986.

CONTRIBUTING MEMBERS

Abbott, James H Earle, Clifford J, Jr Johnson, Roy A Murphy, Donald P Seligman, George B Adams, Elliott T Eckhardt, William Sexauer, Norman E Akemann, Charles A Ecklund, Earl F, Jr Kamp, WilliamP Naimpally, Shabazz, Abdul Alim Amir-Moez, Ali R Epstein, D I Kaplan, Wilfred Somashekhar A Spencer, Joel H Kautzmann, lvar Andrews, George E Newman, Morris Stakgold, Frank N, III Nishiura, Togo Sternberg, David Apostol, Tom M Farrell, Roger H Fass, Arnold L Kelly, John B Nobel, John A Sullivan, Richard W Assmus, Edward F, Jr Kendall, Richard P Nummela, Eric C Swokowski, Earl W Galkowski, Jan T Kiefer, James E Babcock, William W Galuten, Aaron Killeen, John Olum, Paul Taylor, Laurence R Babikir, MA Glass, AMW Knuth, Donald E Orlik, Peter P Thomas, P Emery Baer, Robert M Gleason, Andrew M Kopala, Conrad Osofsky, Barbara L Tsao, Sherman Baumslag, Gilbert Gordon, Hugh Kunen, Kenneth Otermat, Scott C Uhlenbeck, Karen Beechler, Barbara J Gould, Henry W Grace, Edward E Laning, JH Palais, RichardS Unal, Aynur Bing, RH Leger, George F Graves, Robert L Palmer, Theodore W Bjorklund, Peter B Lemay, Papanicolaou, Warren, William E Greif, Stanley J WilliamH Booth, George W Lenard, Andrew George C Weintraub, Steven H Grimmer, Ronald C Bressoud, David M LeVeque, William J Paradise, Michael E Wendroff, Burton Gromov, Mikhael Lieb, Elliott H Pearson, Robert W White, George N, Jr Carson, Robert C Lieberman, David I Perry, William L Whitmore, William F Haddad, George F B Wilkins, J Ernest, Jr Chernoff, Paul R Peterson, Ronald J Halberstam, Heini Mamelak, Joseph S Wilson, Robert Lee Clifford, Alfred H Hartnett, WilliamE Mansfield, Maynard J Redheffer, Woeppel, James J Conner, Pierre E, Jr Hartzler, Stanley J Mathsen, Ronald M Raymond M Woolf, William B Cootz, Thomas A Hassinger, Bill, Jr Mattson, H F, Jr Reed, David L Wright, Robert K Corrigan, Hedlund, Gustav A Maxwell, James W Reedy, Christopher L Thomas Carney Hemstead, Robert J McCready, Robert R Reese, Matthias F Yahya, SM Herwitz, PaulS Meder, Albert E, Jr Riney, JohnS Yo he, James M Higgins, John C De Marr, Ralph E Mellender, James W Rosenblum, Marvin Hironaka, Heisuke Zelinsky, Daniel DeFacio, Brian Miles, E P, Jr Ross, Kenneth A Hochstadt, Harry Miller, WF Rovnyak, James L Ziebur, Allen D Defrancesco, Henry F Hochster, Melvin Mislin, Guido Zink, Robert E Demana, Franklin D Horrigan, Timothy J Moore, Richard A Sally, Paul J, Jr Dinneen, Gerald P Howe, Roger E Morawetz, Cathleen S Sam it, Jonathan Anonymous (8) Donaldson, James A Hufford, George A Moschovakis, Sawyer, Stanley A Donoghue, Hunt, Richard A Yiannis N Schori, Richard M William F, Jr Hutchinson, George A Muhly, PaulS Selfridge, John L

Corporate Members and Institutional Associates The Society also acknowledges with gratitude the support rendered by the following corporations, who held either Corporate Memberships or Institutional Associateships in the Society during the past year.

Corporate Members Institutional Associates Aerospace Corporation Princeton University Press AT&T Bell Laboratories D. Reidel Publishing Corporation General Motors Corporation Springer-Verlag New York Incorporated International Business Machines Corporation Daniel H. Wagner Associates Rockwell International

521 GENERAL CONTRIBUTIONS

Abbott, James H De Marr, Ralph E J a blow, Eric Robert Naroditsky, Smith, Roland F Adams, William W Dedecker, Paul Jackson, Stanley B Vladimir A Sonneborn, Lee M Adler, Irving Deeb, Waleed M Jaffe, Martha A Nevai, Paul G Sonnesso, Geraldine Spanier, Jerome Al-Medhadi, Nora Seif DeFacio, Brian Jeroslow, R G Newman, Morris Niino, Kiyoshi Speed, Terence Paul Amir-Moez, Ali R Deneen, Linda L Dian, Gloria S Nijenhuis, Albert Spielberg, Stephen E Antman, Stuart S Dollard, John D Kennison, Lawrence S Nishiura, Togo Starr, Norton Aroian, Leo A Doob, Joseph L Kent, George A Niven, Ivan Stock, John R Auslander, Bernice L Douglis, Avron Khalimsky, N ordhaus, Edward A Story, Helen F Downing, Reginald H Efim Davidovich Strout, John E Balser, Arienne S Duren, William L, Jr Kieval, Harry S Onishi, Hironori Struppa, Daniele C Dydak, Jerzy Kita, Hiroo Orloff, Leo N Sun, Hugo Sui-Hwan Bari, Ruth A Sunouchi, Gen-ichiro Klass, Michael J Bass, Robert James Palmer, Theodore P Swokowski, Earl W Baum, John D Eachus, J J Klein, Benjamin G Edrei, Albert Parker, Elwood G Birman, Joan S Klein, Edwin M Parker, Francis D Talmadge, Richard B Eisele, Carolyn Kleppner, Adam Blanche, Ernest E Eldridge, Klaus E Parlett, Beresford N Tan, Henry K Tatevossian, Leon H Bloom, Melvin Elliott, Sheldon E Kossack, C R Pearson, Robert W Borgman, Kyuno, Shoji Pell, William H Thomas, Marc Phillip Thomas, P Emery William M, Jr Fass, Arnold L Pereira da Silva, C Thorpe, James C Brauer, Alfred T Lander, Felix I Phillips, Ralph S Ferrari, Carlo Todor, Fabian Flanders, Harley Polachek, Harry Brauer, Fred Lapidus, Michel L Tolsted, Elmer Forman, John W Poor, Walter A Brauer, George U Lasher, Sim Porta, Harry J Tronel, Gerard Freier, Jerome B Brown, George B Law, Peter R Priest, Cynthia J Tugue, Tosiyuki Freitag, Herta T Browne, H Nelson, Jr Lawson, '\'alter R Protter, Philip E Frohlich, Jurg M Brzezinski, Juliusz Lebow, Arnold Przybylski, Thomas L Unai, Yasushi Unai Buianouckas, Utreras, Florencio Galbraith, A S Lichnerowicz, Andre Francis R Quade, Edward 8 Gantos, Richard L Livingston, Philip S Burkill, John C Quinn, Frank S, III Vance, Elbridge P Ghent, K S Lombardo, Charles J Vietsch, W Karel Giertz, Magnus Lubin, Clarence I Regan, Francis Canjar, Mike Gordon, B Brent Luther, C F Reid, James D Walters, Eleanor B Carmichael, Richard D Gosselin, John A Reissner, Eric Ward, John T Castore, Glen M Graver, Jack E Macphail, Moray S Reitwiesner, George W Warne, Ronson J Green, Simon Chandarana, Sharad Madow, William G Renno, James G, Jr Warren, William E Greville, Thomas N E Wasserman, Arthur G Chen, Concordia C Mamelak, Joseph S Renz, Peter L Grubb, Gerd Wehausen, John V Chihara, Theodore S Richards, John F Gurney, Margaret Martin, Hayden Garth Weibel, John L Chinn, William G Rickart, Charles E Mascioni, Vania D Ritter, William E Weill, Lawrence R Clark, Kenneth D Haber, Seymour Mayor, John R Roberts, Richard C Whitney, D Ransom Cleveland, Richard A Halberstam, Heini McAdam, Stephen J Roessler, E B Wilkins, J Ernest, Jr Clifford, Alfred H Hale, David R McClish, Wayne E Rooker, John C, II Williams, Frederick C Cobb, Philip A Harrell, Evans M Meadows, Lawrence F Rosenstein, Williams, Gregory P Cohen, Jack D Harvey, George G Mears, Florence M George M, Jr Wong, Yim~Ming Cohen, Steve Michael Hassinger, Bill, Jr Ryan, Donald E Woythaler, Meisner, Morris J Joseph William Coleman, A John Herrera, R B Herzog, Emil R Mellender, James W Wu, T C Colson, Henry D Salas, Saturnine L Hobbs, Arthur M Meyer, Herman Sarafyan, Diran Comenetz, Daniel Hofer, Robert D Meyer, Jean-Pierre G Sarason, Donald E Yachter, Morris Crane, George E Holberton, John V Mielke, Paul T Sasaki, Shigeo Yamada, Toshihiko Cravetz, Allan E Miller, William David Schlesinger, Ernest C Yasuhara, Ann Critchlow, Florence G Iha, Franklin T Mills, William H Schraut, K C Yasuhara, Mitsuru Cunningham, A B Isaacs, Godfrey L Moller, Raymond W Scott-Thomas, John F Yokoi, Hideo Yoshino, Takashi Mooney, John J Sedgewick, Charles H W Young, Paul M Morris, Chester R GENERAL CONTRIBUTION Sell, George R IN MEMORY OF Moy, Allen Sharp, Henry, Jr Zalcman, Lawrence A ALEXANDER D. WALLACE Mrowka, Stanislaw G Shayib, Mohammed A Zilmer, Delbert E Alfred H. Clifford Murakami, Shingo Shenitz, Charles Muse, Archie Henry Sidney, Stuart J Anonymous (52)

522 AMS RESEARCH FELLOWSHIP FUND

Abbott, James H Brown, Johnny E Ealy, Clifton E, Jr Gurney, Margaret Kahn, Jeff N A'Campo, Norbert Brown, Lawrence G Earle, Clifford J, Jr Gustafson, Robert A Kamp, William P Adachi, Steven H Brown, Scott W Eberlein, Karle, Jerome Adams, William W Brunner, H Patrick Barry Haber, Seymour Karlovitz, Lea A Aissen, Michael I Brunschwig, Ecklund, Earl F, Jr Hackler, Donald B Kelingos, John A Akeh-Ugah, Mildred C Ehrlich, Paul Ewing Halberstam, Heini Kemperman, J H B Eisenbud, David Ufumaka U Burns, Daniel M, Jr Hall, Glen Richard Kent, George A Alexander, John R, Jr Eisner, Elmer Burr, Stefan A Hammer, Carl Kesten, Harry Alexander, Elia, Michele Hammond, William F Khalimsky, Stephanie B Byrnes, James S Elliott, David L Hamstrom, Efim Davidovich Alif, Metod Elliott, George A Khan, Donald W Al-Medhadi, Nora Seif Cain, Bryan E Engle, Jessie Ann Mary-Elizabeth Kibbey, DE Alperin, Calderer, M Carme Ensey, Ronald J Hano, Jun-Ichi Kiefer, James E Roger Charles Camacho, James, Jr Epstein, D I Harbater, David King, Donald R Alspach, Dale E Erdman, John M Harrington, Walter J Campbell, L Andrew King, James R Alvis, Dean L Evens, Leonard Harris, David Carter Caricato, Gaetano Kishi, Masanori Anacker, Steven E Carlson, Bengt G Harris, Steven Guy Ancel, Fredric Davis Hastings, Harold M Kittaneh, Fuad A do Carma, Manfredo P Faith, Carl Anderson, Douglas R Feferman, Solomon Hausmann, Klass, Michael J Carter, Thomas J Anderson, Michael T Fegan, H D Jean-Claude Klatman, David S Cartlidge, James M Andrews, John J Fein, Burton I Haver, William E Klein, Edwin M Castore, Glen M Anabel, Michael Feit, Walter Helgason, Sigurdur Kleinschmidt, Peter deCesare, Kenneth M Archer, Myla M Fenchel, Werner Hempel, John P Knoerr, Alan P Chaffee, James W Arenstorf, Richard F Fendrich, John W Henderson, Kober, Wolfgang Aribaud, F Chen, Goong Ferguson, Helaman Francis McVey Kahn, Robert V Arisaka, Nakaaki Chernoff, Paul R Ferrari, Carlo Herbst, Albert F Kolaitis, Phocion G Arkowitz, Martin A Chiang, Peter H S Ferris, Ian M Hermann, Robert KOlbig, Kurt Siegfried Arms, Judith M Finch, John V Chihara, Theodore S Hernandez, Horacia Kolchin, E R Arnold, Douglas N Finkelstein, Leib Choksi, Jal R Herrera, R B Kolesar, R J Asano, Shigemoto Christensen, Chris Fitzpatrick, Ben, Jr Kolodner, Ignace I Askey, Richard A Fitzsimmons, Herrero, Domingo A Church, Philip T Komm, Horace Atkinson, Frederick V Patrick J Heuer, Gerald A Churchill, Richard C Korman, PhiliP L Flahive, Hilbert, Stephen R Au, C HE Koschorke, Ulrich Azoff, Edward A Clark, Kenneth D Mary Elizabeth Hilton, Peter J Kostenbauder, Clarke, Bob W Floyd, Edwin E Hirano, Tetsutaro Adnah G Bachman, George Clifford, Alfred H Foias, Ciprian Ilie Hironaka, Heisuke Kra, Irwin Bacon, Harold M Cobb, Philip A Folk, Thomas F Hobbs, Arthur M Kramer, Earl S Bade, William G Cohen, Frederick R Folland, Gerald B Hochschild, Gerhard P Kramer, Thomas R Bailyn, Paul M Cohen, Jack D Fang, Che-Kao Hochster, Melvin Krause, Ralph M Baker, Charles R Cohen, Marshall M Fossum, Robert M Hodgson, Kravitsky, Naftali Baker, Edward D Cole, Charles A Freeman, Robert S Jonathan P E Kriete, Thomas L, III Ball, Joseph A Cole, Paul Dana Frohardt, Daniel E Hogesteeger, F W Frohlich, Jurg M Krinik, Alan Cary Balser, A rienne S Collins, Gayle M Holmes, Philip John Bandle, Catherine Fuchs, Wolfgang H Krumpholz, Gary R Conn, Jack Frederick Hosseini, Naser Banda, Shigetoshi Fuchssteiner, Ben no Kumagai, Donna J Conrad, Bruce P Howe, Roger E Baouendi", M Salah Fulton, William Kunen, Kenneth Huard, James G Bari, Ruth A Cootz, Thomas A Kurata, Yoshiki Hudson, D L Barker, Cordero, Luis A Gabber, Ofer Kuroda, S T Hudson, Richard H William Hamblin, II Cowling, Michael G Gacs, Peter Kurshan, Robert P Huerta, Carlos Cuevas Barnet, Frank Critchlow, Florence G Gaier, Dieter Kuras, Herbert Hughes, Anthony Baron, John J Cross, James J Ganelius, Tord H Kurtzke, John F, Jr Hughes, Rhonda J Barth, Karl F Cross, Ronald W Gangolli, Ramesh A Kyner, Walter T Baruch, Herbert M, Jr Currier, Albert W Gantos, Richard L Hughes, Ruth L Huh, Won Barwise, Kenneth J Curtis, Charles W Gautschi, Walter Lam, Tsit-Yuen Bass, Robert James Glloghegan, Ross Huibregtse, Mark E Lanford, Oscar E, III Batt, Jiirgen 0 George, John Clay Hulkower, Neal D Dade, Everett C Langenhop, Carl E Batterson, Steven L Gerber, Mar lies Humphreys, James E Damkohler, Wilhelm L Langston, Baum, Paul F Gerisch, Wolfgang Huneke, Craig L Damon, Michael David Baxter, James A Gill, Tepper L Huneke, John Philip James Norman Lanzano, Paolo Bazer, Jack Gillespie, Robert G Hunt, Walker E D'Angelo, John P Larson, ..lean Beals, R Michael Gilman, Jane P Hutchinson, George A Beals, Richard W D 'Aristotle, Gilman, Robert Larson, Richard G Becker, Howard S Anthony J Glass, AM W LaSalle, Margaret M Iarrobino, Becker, Leigh Carl Davidoff, Giuliana P Gleason, Andrew M Lashof, Richard K Anthony A, Jr Beckmann, William H Davidson, Glenn, Paul G Lawson, H Blaine, Jr lei, Yutaka Bellamy, David P William Robert Goddyn, Luis A Lax, Peter D Iha, Franklin T Bencsath, Katalin A Day, Mahlon M Goebel, K Lazard, Michel P Isaacs, I Martin Benkart, Georgia M Dediu, Mihai Michael Goldhaber, J K Lazarsfeld, Robert K Iwanaga, Yasuo Bercovici, Hari DeFacio, Brian Qoldman, Jerry Le Cam, L M Berger, Alan E Goldman, Dekker, Jacob C K Leger, George F Bergstrand, Deborah J William Mark Jacquet, Herve M Delange, Hubert Lepowsky, James I Bers, Lipman Goldstein, Jerome A Jenkins, Deleanu, Aristide LeVeque, Randall J Bertin, Emile M Goldstein, Nor man J Theodore Allen Levin, Richard G Deprima, Charles R Betsch, Gerhard Gomi, Kunia Jensen, Carey M Levine, Harold I DeTurck, Dennis Bierstedt, Klaus D Goodman, Roe W Jerison, David Levine, Jerome P Deutsch, Daniel H Blackmore, Denis L Goodrich, Jerison, Meyer Li, Li-Ge Blake, Louis H Dickerson, Charles E Suzanne Dinga J eros low, R G Libgober, Anatoly S Blattner, Robert J Dilworth, R P Gray, Robert E Jockusch, Carl G, Jr Lind, David W Blum, Lenore Green, John W Divis-Poracka, Zita M Johnson, Charles N Liu, Jong-Diing de Boor, Carl Greenwood, Robert E Douglis, Avron Johnson, Liulevicius, Arunas L Boos, William Griffin, E L, Jr Dow, Alan D Randolph, Jr Booth, George W Grinberg, Vladimir Lopez, Amparo Dowers, J F Johnson, Joseph F Bowman, Leslie W Gross, Benedict H Louhivaara, Ilppo S Downey, Rodney G Bressoud, David M Gross, Louis J Johnson, Roy A Louton, Tom Dubson, Alberto S Brooks, Robert Grossman, Marvin W Jones, Gordon C Luecking, Daniel H Brothers, John E Duchamp, Thomas E Grosswald, Emil Jones, Harold T Lutterodt, Broughton, S A Duren, Peter L Gualtierotti, A F Jones, Jeffrey P Clement Henry Browder, William Duskin, John W, Jr Guillot, J C Jongen, Hubertus Th Luxemburg, Brown, Anne E Dydak, Jerzy Gulden, S L Jonsson, Bjarni Wilhelmus A J Brown, Edgar H, Jr Dyson, Verena H Gulliver, Robert D, II Joris, Henri E Lyons, Russell D

523 AMS Research Fellowship Fund (Continued}

MacGillivray, Nevai, Paul G Riley, Srinivasan, Bhama Wadhwa, Bhushan Lal Archibald D Nirenberg, Louis Geoffrey William Stackelberg, Olaf P Wagoner, John B Maddux, Roger D Nishiura, Togo Robbins, Leon C, Jr Stafney, James D Wagreich, Philip D Marlow, William G Nobel, John A Rod, David L Stallings, John R Walsh, John J Magnus, Wilhelm Roessler, EB Stanley, Richard P Walters, Eleanor B Malkevitch, Joseph Obi, Wilson C Rooker, John C, II Staples, Edmund Wang, Derming Mann, Benjamin M O'Callaghan, Liam J Roozemond, Leen Beauclerc, III Ware, Buck Marathe, Kishore B Odlyzko, Andrew M Rose, NJ Steele, Alfred T Ware, Roger P Marcus, Michael B Oehmke, Robert H Rosenbaum, Robert A Stein, Sherman K Warne, Ronson J Margolis, Stuart W Oguchi, Kunia Rosenberg, Steinberg, Robert Warren, William E Margulies, William Oldroyd, L Andrew Jonathan M Stern, Ronald J Waterhouse, Marle, Ombe, Hitoshi Rosenthal, Haskell P Stoll, Wilhelm F WilliamC Waterman, Daniel Charles Michael Oprea, John F Ross, Kenneth A Stortz, Clarence B Weaver, John C Marshall, Donald E Orellana, Rossi, Hugo Strauss, Walter A Martelli, Chacin Mauricio Rothaus, Oscar S Sucheston, Louis Wechsler, Martin T Mario Umberto Orlik, Peter P Rothman, Neal J Sullivan, Richard W Weill, Lawrence R Weiss, \Villiam A R Martin, Hayden Garth O'Shea, Donal B Rubin, Herman Suzuki, Haruo Wells, David M Mate, Attila Otermat, Scott C Rubin, JeanE Swokowski, Earl W Matsumura, Hideyuki Ruess, w Szczarba, Robert H Wente, Henry C May, J Peter Wermer, John Paciorek, Joseph W Ruiz Morillo, Margaret W Wessel, W Roy Maziarz, EA Palled, Shivappa V Alejandro J Taft, McBrien, Shigeo West, Donald C Vincent 0 Palmer, Theodore W Taken aka, McCaughan, Dennis J Takesaki, Masamichi White, Dennis E Pan, Ting K Sagan, Bruce E McCoy, Thomas L Sahab, Salem M Tan, Henry K Whitman, Philip M Park, Jae Keol McCready, Robert R Samelson, Hans Tanabe, Hiraki Whitney, Hassler Parrott, David McCullough, Darryl Sarhan, Mahmoud Tartakoff, DavidS Whitney, Stephen Paschke, William L McGibbon, Charles A Sastri, Chelluri C A Taylor, Wicke, Howard H Patton, Charles M McLeod, Robert M Scalzo, Richard C Gladys Gillman Wickerhauser, Peck, N Tenney McLinden, Lynn Schafer, Alice T Taylor, JeanE Mladen Victor Pennell, Stephen A McMillan, Schafer, Richard D Taylor, Laurence R Wilkins, J Ernest, Jr Perlman, Sanford Daniel R, Jr Schauer, Richard L Terao, Hiroaki Williams, Robert F Peterson, Charles L Merriell, David M Schlesinger, Ernest C Thiele, Ernst-Jochen Wirszup, Izaak Pinsky, Mark A Michel, Rene Schmid, Wilfried Thoma, Elmar H Witte, David S Piranian, George Mielke, Paul T Schmidt, Henry J, Jr Thomas, Lawrence E Wolfe, Michael David Pop-Stojanovic, MilitiC, Dragan Schmidt, Kenneth C Thomas, P Emery Wong, Sherman K Zoran R Miller, John Boris Schreiber, Bert M Thorpe, James C Wong, Yim-Ming Porta, Harry J Wood, John W Miller, William David Schreiner, Erik A Tits, Jacques L Pour-El, Marian B Woolf, William B Mishra, Sudhakara Schwarz, Gerald W Tjiok, Mouw-Ching Wright, David J Mitchell, Josephine M Protter, Murray H Scissors, Richard H Todorov, Wu, T C Mityagin, Boris Protter, Philip E Scott, Leonard L, Jr Pavel Georgiev Wyman, Bostwick F Miyadera, Isao Przybylski, Thomas L Scott-Thomas, John F Toll, Kathryn B Miyoshi, Takeo Seebeck, Charles L, III Tomiyama, Jun Yachter, Morris Mochizuki, Horace Y Rabinowitz, Paul H Sehgal, Virindra M Traina, Charles R Yamaguti, Kiyosi Mohamed, Saad Ralston, Elizabeth W Seligman, George B Trucano, Yang, Kung-Wei Monahan, Maurice L Ralston, James V Semmes, Timothy Guy Yang, Paul C Montgomery, M Susan Ramanujan, Stephen William Tsukui, Yasuyuki Yang, Wei-Shih Mooney, John J Melapalayam S Sesay, Mohamed \V I Tsutsumi, Massayoshi Yanowitch, Michael Moore, Richard A Ramsay, Arlan B Shapiro, Joel H Tugue, Tosiyuki Yaqub, Jill S Morris, Robert A Range, R Michael Shields, Allen L Turgeon, Jean M Morrison, John A Rankin, Turner, Philip H Yasuhara, Ann Shin, Yong-Tae Yasuhara, Mitsuru Morse, Burt J Samuel Murray Shiraiwa, Kenichi Yood, Bertram Moschovakis, Raskind, Wayne Mark Uchiyama, Akihito Shorter, William Ivan Yui, Noriko Joan Rand Raymond, Frank A Silberger, Allan J Mostow, George D Reade, Maxwell 0 van der Vaart, Sim, Soon Kiang Zalcman, Lawrence A Mrowka, Stanislaw G Reber, Douglas C H Robert Simen, David C Zaslavsky, Thomas Muhly, PaulS Redheffer, Vamanamurthy, Slater, John B Zimmer, J W Mullin, Albert A Raymond M Mavina K Slodowy, Peter J Zizi, Khelifa Munkholm, Hans J Reed, George Michael Smith, FR Zuckerberg, Hyam L Muse, Archie Henry Rees, PK Van Casteren, Smith, Marianne Zuckerman, Gregg J Myers, Earle F Reid, Freundlich Johannes A Christopher Ervin VanOsdol, Donovan H Smith, Michael G Anonymous (55) Nagami, Keio Reid, James D Smith, Robert Colman VanTuyl, AH Nagle, R Kent Reinoehl, John H Smith, Wilbur L Van Vleck, Fred S Nakano, Kazumi Rejto, Peter A Smaller, Joel A Vance, James T, Jr Narcowich, Francis J Restrepo, Guillermo Sonnesso, Geraldine Vasconcelos, Nathanson, Melvyn B Reynolds, Eugene Sons, Linda R Wolmer V Nelligan, John D Rice, John R Spears, Otis S Veech, William A Nelson, Edward Richards, Pamela C Specht, Edward J Vogan, David A, Jr Nesbitt, Cecil J Richmond, Spencer, Voigt, PaulS Neumann, Bernhard H Lawrence B Thomas Crawford Voytuk, James A

524 ICM-86

A 'Campo, Norbert Albright, Hugh N Anderson, Assmann, Susan F Baker, Edward D Abadie, Jean M Alcorn, David P Robert Joseph Assmus, Edward F Jr Baker, Edward G Abbott, H L Alder, H L Anderson, Robert M Astbury, Kenneth A Baker, Frances E Abbott, James C Alder, Hermann C Anderson, Stuart N Astey, Luis Baker, George A Sr Abbott, James H Aldous, David J Anderson, Atallah, Kamal Baker, Kirby A Abdel-Megied, Alessandrini, Giovanni Theodore W Athanassoulis, Baker, Peter Fritz Mohamed Alexander, Elaine E Andersson, Lars Gerassimos Baksa, Dennis John Abdelaziz, Nazar H Alexander, Herbert J Andersson, Stig I Atherton, Charles Bakshi, Jagjit A bdelkader, Alexander, James C Andima, Susan Atick, Jiana J Bal, Kenan Y Mostafa A Alexander, John R Jr Ando, Shiro Atkin, Joy H Balakrishnan, R Abdullah, Masood B Alexander, Ando, Tsuyoshi Atkinson, Frederick V Balakrishnan, V K Abe, Eiichi Stephanie B Andre, Peter P Atkinson, H R Balanis, George N Abeles, Francine F Alexanderson, Andreadakis, Stylianos Atkinson, Mary L Balbes, Raymond A bellanas, Pedro Gerald L Andreoli, Dorothy W Atkinson, Stuart L Bales, John W Abels, Herbert Alfeld, Peter Andrew, Alfred D Atsuji, Masahiko Balibrea, Francisco Abernathy, Robert 0 Alfonso, Fermin Jorge Andrews, George E Au, C HE Ball, Joseph A Abikoff, William Alford, Neill H III Andrews, H Robert Au-Yeung, Yik-Hoi Ball, N Hansen Abrahamson, Brian Ali, Hydar Andrews, John J Auchmuty, J F G Ball, Richard Neal Abram, Ali, M Kursheed Andru's, Jan F Auer, Jan W Ball, Richard W Thomas Joseph Alif, Metod Angad-Gaur, Aulaskari, Ballantine, Charles S Abramovici, Flavian Aliprantis, Hendrik W K Rauno Pertti Ballard, William R Abramson, Lee R Charalambos D Angelos, James Roy Ault, J C Ballou, Donald H Abrham, Jaromir V Allan, James A Angeniol, Bernard Auslander, Bernice L Balmaceda, Abu-Khuzam, Allday, Christopher J Angus, John Edward Auslander, Louis Jose Maria P Hazar Muhammed Allegretto, Walter Anick, David Jay Auslander, Maurice Baloglou, George C Abudiak, Fawaz M Allen, Harry P Annavedder, Edwin Aust, Catherine C Balser, Arienne S A car, Robert Allen, Howard Anselone, Philip M Austin, Bill R Baltus, Christopher Accola, Robert D M Allen, J Thomas Anshel, Michael Austin, Charles W Banach, Antoni S Acker, Andrew F III Allen, Joseph N Anton, Howard Avelsgaard, Roger A Banchoff, Thomas F Aczel, JAnos D Allender, Eric Antoniadis, Anestis A Aversa, Vincenzo L Bandle, Catherine Adachi, Masahisa Alling, Norman L Apostol, Tom M Avioli, John J Banda, Shigetoshi Adachi, Steven H Allison, Bruce N Appel, Kenneth I Avram, Florin Banerjee, Utpal Appell, Jurgen Adams, David D Alltop, William 0 Avrunin, George S Bang, Seung Jin Adams, David R Almgren, Applebaum, Charles H Axelsson, Owe Bang, Thoger S V Adams, Elliott T Frederick J J r Applebaum, Joseph A Axler, Sheldon Banks, Dallas 0 Appleson, Robert R Ay~rs, Adams, J Frank Alonso, James Kathleen L Bankston, Paul J Adams, Loyce Appling, William D L Alperin, Jonathan L Ayoub, Christine W Bannai, Eiichi Adams, Malcolm R Apter, Arthur William Bansal, Naveen K Alperin, Ayoub, Raymond G Arad, Zvi Bantegnie, Robert Adams, Rebecca S Roger Charles Aytch, Annas Adams, Robert M Araki, Shoro Bao, David D W Alpern, Steven Azarnia, Nozar Addington, Susan L Arapostathis, Aziz, Abdul K Baouendi, M Salah Alpert, Debra Ann Adem, Jose Aristotle Azoff, Edward A Baragar, Francis A Alps, Robert A Adler, Alfred Arason, Jon Kr Azpeitia, Alfonso G Baranow, Michael D Alsholm, Preben Adler, Irving Archbold, Robert J Azumaya, Goro Barbeau, Edward J Jr Alsina, Claudi Adler, Robert J Archer, Myla M Azzaro, Ali Barber, Fredrick W Alspach, Dale E Adler, Roy L Archibald, Ralph G Barber, SF Altergott, Dennis A Adluri, Indrasena Arena, Orazio Baayen, Pieter C Barbosa, J Lucas M Althoen, Steven C Adney, Joseph E Jr Arens, Richard F Babai, Laszlo Barcelo, Helene Altman, Allen B Aeppli, Alfred Arenson, Donald L Babbitt, Albert E Jr Barckett, Joe A Altman, Mieczyslaw AfU.wap~, A Uyi Argemi, Jose Babbitt, Donald G Bardaro, Carlo Altomare, Francesco Agard, Egbert Aribaud, F Babcock, William W Bareiss, Erwin H Agmon, Shmuel Altshuler, Amos Arii, Dayna L Babikir, M A Barendregt, Henk P A gaston, Max K Alvarez, Jairo Ivan Arima, Satoshi Babu§ka, I vo Barety, Julio Edgardo Agronsky, Steve Alvis, Dean L Arisaka, N akaaki Bachelis, Gregory F Barge, Marcy Alzubaidy, Kahtan Aguad~. Jaume Arkin, Joseph Bachman, George Bari, Ruth A Aguilera, Eleanor 0 Ambos-Spies, Klaus Arkowitz, Martin A Bachman, R Barker, Aharoni, Israel Amick, Charles James Armacost, William L Bachmann, Franz William Hamblin II Aharonov, D Amillo-Gil, Jose Armbrust, Manfred K Back, Allen H Barker, William Henry Ahlbrandt, Gisela Amin, Muhammad Armentrout, Steve Backus, George E Barkley, Linda Ahlfors, LV Amir, Dan Armitage, John V Backus, John Barksdale, James B Jr Ahmad, Seema Amir-Moez, Ali R Arms, Judith M Bacon, Harold M Barnard, Roger W Ahmed, N U Arnirkhanian Jolfai, Armstrong, Gerald M Bade, William G Barner, Martin Ahner, John F Vrege Armstrong, Kenneth H Baderian, Barnes, Bruce H Aikele, M Holbrook Amitsur, Shimshon A Armstrong, Philip N Armen Robert Barnes, Charles Ward Aimar, Hugo Am mar, Gregory 8 Armstrong, Thomas E Badrikian, Albert Barnet, Frank Ainsworth, Oscar R Amrine, Brian Arnberg, Robert L Baernstein, Albert II Barnhart, Richard Dee Aissen, Michael I Amsbury, Wayne P Arnold, Douglas N Bagby, Richard J Barnwell, Beth G Aizawa, Sachiko Anacker, Steven E Arnold, Edward M Bagby, S Thomas Baron, John J Akasaki, Takeo Anantharam, Arnold; Leslie K Bagchi, Atish Baron, Simson Akemann, Charles A Venkatachalam Aroian, Leo A Baggett, Lawrence W Barr, Betty Jane Akiba, Tadatoshi Anantharaman, Rajan Aronsson, Gunnar Baginski, Frank E Barr, William J Akin, Ethan J Anastasio, Salvatore Arsenis, Baglivo, Barrar, Richard B Akis, Vladimir N Andersen, E Sparre Spyros Panagis Jenny Antoinette Barreca, J R Akyildiz, Ersan Andersen, Kenneth F Arthur, James G Bahamonde, Antonio Barrera-Sanchez, Al-Amiri, Hassoon S Andersen, Kurt Munk Artin, Michael Baica, Malvina Florica Pablo Al-Bassam, Anderson, Bertil Artola, Michel Baigent, Edith Anna Barrett, Mohammed Ali A Anderson, Chris R Artola, Paul C Baik, Yang Bai Andrew Victor Al-Dhahir, Anderson, Donald W Arveson, William B Bailey, Donald F Barrett, Lid a K Alaaddin A J A'nderson, Douglas R Arz berger, Peter W Bailey, Evelyn C Barron, Beth A Al-Dhahir, M W Anderson, Edmund H Arzt, Shalom Bailey, R A Barros-Neto, Jose Al-Khafaji, Mahmoud Anderson, Eugene R Asadulla, Syed Bailey, Robert E Barshay, Jacob Al-Medhadi, Nora Seif Anderson, Florence R Asano, Shigemoto Baille, Philippe P Barsky, Daniel AI-Rashed, Anderson, Frank W Ascenzi, Maria-Grazia Baily, Walter L Jr Bart, H Abdallah M Anderson, James R Aschbacher, Michael Bailyn, Paul M Barth, Karl F Al-Salam, Waleed A Anderson, Joel H Ash, Avner D Bain, Osias Bartick, Philip R Alavi, Yousef Anderson, John T Ashbaugh, Mark S Baird, Bridget Bartle, Robert G Albers, Donald J Anderson, Joyce E Asher, Donald E Bajusair, Mohamed S Bartnik, Robert A Albert, John P Anderson, Larry R Askey, Richard A Baker, Alan de Bartolomeis, Paolo Albertson, Michael 0 Anderson, Mark D Aslaksen, Helmer Baker, Andrew James Barton, Eric M Albrecht, Felix R Anderson, Michael T Asmussen, Marjorie A Baker, Charles R Barton, Thomas J

525 ICM-86 (Continued)

Baruch, Herbert M Jr Bell, James M Beachler, Edwin F Bloom, Bowers, Philip Lee Baruch, Marjory Jean Bell, 'John B Bessenrodt, Christine Steven Herman Bowman, Leslie W Barzilai, Jonathan Bell, Murray Gordon Besson, Gerard Bloom, Thomas Bowman, Robert H Basor, Estelle Bellamy, David P Betancourt, Carlos M Bloom, Walter R Bowman, Thomas T Bass, Hyman Belley, J M Betounes, David E Blowers, James V Boxer, Laurence R Bass, Robert James Bellissard, Jean V Betsch, Gerhard Blows, T R Boyack, Stephen W Bassett, Edward M Belpaire, Gerald Betz, Ebon E Blum, Dorothee Jane Boyce, Elsa K Bateman, Felice D Ben-Israel, Adi Beukers, F Blum, Howard S Boyce, William E Bateman, Paul T Benardete, Diego Beyer, Arthur T Blum, Ilya Eugene Boyce, William M Batker, Kenneth E Bencsath, Katalin A Beyer, William A Blum, Joseph Boyd, David W Batie Nicolau, Nadal Benda, Miroslav Beylkin, Gregory J Blum, Lenore Boyd, Ernest James Bator, Elizabeth Mary Bendat, Julius S Bhaskaran, M Blumenthal, Robert A Boyer, Steven Patrick Batt, Jiirgen 0 Bender, John Bhattacharya, Prabir Blyth Rentzsch, Boyle, Jeffrey Batten, Benedicty, Mario Bhattacharya, Rabi N Barbara J Boyle, McBlaine M Lynn Margaret Benes, Vaclav E Bhattacharya, Blyukher, Grigory Bozeman, Sylvia T Bauer, Carole Ann Benitez, Carlos Ramendra K Boardman, John M Bozonis, Petros Bauer, Douglas Benkart, Georgia M Bialostocki, Arie Boas, Harold P Braaksma, B L J Bauer, Heinz Benke, George Bickart, Theodore A Bobisud, Larry E Brace, John W Baugh, James R Benkoski, Stanley J Bickel, Marilyn S Bobonis, Augusto Brachman, Malcolm K Baum, John D Bennett, David G Bickel, Thomas F Bochnak, Jacek Brackx, Freddy F Baum, Paul F Bennett, Fredricka T Bidwell, Leonard N Bodfish, Scott E Braden, Eddie Jr Bauman, Patricia E Bennett, John Bruce Bielawski, Jan Bodnar, Andrea C Braden, Lucian J Baumann, Volker H Bennhold, Cornelius Bierstedt, Klaus D Bodnar, Istvan Bradford, Baumeister, Johann Benson, Dave Bierstone, Edward Boduch, David J Giles Edward III Baumgartner, James E Benson, Bilaniuk, Stefan Bodvarsson, Gunnar Bradley, John S Baumslag, Gilbert David Bernard Biles, Charles M Boe, BrianD Bradley, Richard C Bausum, David R Benson, Dean C Billera, Louis J Boehme, Thomas K Bradley, Robert E Bautista, Maurino P Benson, Billik, Martin Boehmer, Klaus W A Bradlow, Steven B Baxendale, Peter H Frederick Challoner Bilodeau, Gerald G Boenecke, Ernst Brady, Walter F Baxley, John V Benson, Jacob Binding, Paul A Bo:ffi, Giandomenico Brady, Wray G Baxter, James A Benson, Steven R Bindschadler, David E Bogar, Gary A Braga-Filho, Baxter, Willard E Benyamini, Yoav Bing, R H Bogart, Kenneth P Luiz Juliao Bayer, Margaret M Berard, Pierre H Bingen, Franz Bogdan, Bragg, Arthur E Bayer, Pilar Berbee, Henry C P Bini, Daria Victor Michael Bragg, Louis R Bayer-Fluckiger, Bercov, Ronald D Binkley, Dean H Boghossian, Artin Brahana, Thomas R Eva M Bercovici, Hari Biot, Maurice A Bogus, Rolf Brahm, Matthew V Bohannon, Barbara Bayo, Enrique Berens, Hubert Bircher, John J Brakke, Kenneth A Bohn, S Elwood Bralver, Peter Jeffrey Bayod, Jose M Berenstein, Carlos A Biriuk, George Bohnke, Bazer, Jack Berent, Paul Alan Birman, Graciela S G C Bramble, Charles C Bazinet, Jacques Beretta, E F Boivin, Andre Bramble, James H Birman, Joan S Beachy, John A Beretta, Giordano B Bojanic, Ranko Bramsen, John D Birnbaum, Eric S Beadle, Allen Jay Berezina, M Bokor, Imre Bramsman, Paul R Birnbaum, Sidney Beale, J Thomas Berg, Gene A Boley, David B Branch, Steve A Birnbaum, Z William Beals, R Michael Berg, Gunnar Bolin, Barbara Brand, Neal E Birnir, Bjorn Beals, Richard W Berg, Michael C Balker, Ethan D Branda!, Willy Birtel, Frank T Bean, Ralph J Berge, Douglas Bollman, D A Brandenburg, Harald Bishop, Alan A Beard, Jacob T B Jr Berger, Alan E Boman, Jan Brandon, Robert L Bishop, Wayne Beasley, Craig J Berger, Marcel Bomba!, Fernando Brandstrom, Bisson, Terrence Paul Beasley, LeRoy B Berger, Marsha J Bona, Jerry L Hugo F K Bisztriczky, Tibor Beaudoin, Robert E Berger, Melvyn S Bond, Robert Joseph Brandt, Dennis E Bivens, Irl C Beauzamy, Bernard M Berger, Neil E Bonet, Jose de Branges, Louis Bix, Michael C Bechtel, Robert D Berger, Robert W Bonnice, William E Branovan, Leo Bjorklund, Peter B Beck, Robert E Berger, Ruth I Bonsall, Frank F Branson, Beck, William A Berger, Thomas R BjOrner, Anders Boo, Per-Anders Thomas Patrick Beckenstein, Edward Bergman, Clifford H Black, Richard H Book, Ronald V Brant, Larry J Becker, David S Bergman, George M Blackall, Clair J de Boor, Carl Branton, Richard G Becker, Eberhard Bergstrand, Deborah J Blackmore, Denis L Boos, William BrattstrOm, Gudrun B Becker, Howard S Bergstrom, Roy C Blackwood, Philip D Booth, George W Braude, Eric J Becker, Leigh Carl Bergum, Gerald E Blad, Michael C Boothby, William M Brauer, Fred Becker, Michael Beringer, Alfred Blair, Charles E Borchers, Hans-Jurgen Brauer, George U Becker, Ronald I Berkey, Catherine Blair, David E Borel, Armand Braun, Hel Becker, Stanley J Susan Grooms Blair, Robert L Borelli, Mario Brawley, Joel V Jr Becker, Thomas Berkey, Dennis D Blair, William D Borgman, Brawley, Lori J Beckman, Frank S Berkowitz, Harry W Blais, JAR William M Jr Breaux, Gary R Beckmann, William H Berkson, Earl R Blake, Louis H Borisewich, John Brechner, Beverly L Beckner, William Berlekamp, Elwyn R Blakemore, Carroll F Borkowski, Mark S Breckenridge, John C Bedard, Robert Berman, Abraham Blanch, Gertrude Borrego, Joseph T Breen, Marilyn Beder, Jay H Berman, David R Blanche, Ernest E Borrelli, Robert L Breen, Stephen Bedford, Eric D Berman, Gerald Bland, John S Borwein, David Breidbart, Seth I Bedient, Berman, J Michael Bland, Paul E Bosch, Carlos Giral Bremermann, Hans J Richard Eugene Berman, Lawrence Blank, Brian Evan Bosch, William W Brender, Melvin Beebe, Alfred S Berman, Robert David Blankenship, Gilmer L Boschan, Paul Brennan, Joseph P Beechler, Barbara J Berman, Stephen Blanton, John D Bose, N K Brenneman, Beekman, John A Bermond, J -C Blasius, Don M Boswell, Rupert D Jr Franklin S Beekmann, Wolfgang BermUdez, Alfredo Blaskie, Marc Bouchard, Pierre Brenner, Susanne C Beem, John Kelly Bernardi, Salvatore D Blatter, Christian Bouchet, Andre Brent, Richard P Beenken, May M Bernau, Simon J Blatter, Jorg Boudourides, Moses Brenti, Francesco Beer, Gerald A Berndt, Bruce C Blattner, Robert J Boultbee, Nan Bressler, David W Beesack, Paul R Berner, Paul David Blaufox, Norman H Bourgeat, Alain P Bressoud, David M Beesley, E Maurice Bernhardt, Chris Bledsoe, Bourgeois, Brian A Bressoud, Thomas C Behnke, K Bernhardt, Robert L Kathryn Phillips Bourgin, Richard D Breuer, Shlomo Behrens, Ernst-August Berning, John A Jr Bleicher, Michael N Bourion, Georges Brewer, James W Beidler, John A Bernis, Francisco Bleiler, Steven A Bousfield, Aldridge K Brewster, Beig, Robert Berresford, Geoffrey C Blij, Frederik Van Der Boutang, Charles Stephen Thomas Bekken, Otto Borger Berrick, A J Blitch, Patricia M Boutin, Pierre Brickman, Louis Belanger, Alain Berriozcibal, Manuel P Bloch, Ethan D Boutot, Jean Francois Brickman, Pamela M Belfi, Victor A Bers, Lipman Bloch, Spencer J Bouwsma, Ward D Bridges, Douglas S Belilgne, Atlaw Berterman, John E Blockus, Marilyn J Bove, Antonio Bridgland, Michael F Bell, Allen D Berthiaume, Pierre Bloom, David S Bower, Julia W Brierly, Bell, Harold Bertin, Emile M Bloom, Melvin Bower, Oliver K Joseph Edward Bell, Howard E Berton, John A Bloom, Stephen L Bowers, Julie M Brieske, Thomas J

526 ICM-86 (Continued)

Brigham, Robert C Brunswick, Byrnes, Carroll, Edward M Chandarana, Sharad Brin, Matthew G Natascha A Raymond Albert Jr Carroll, Jeffrey 8 Chandler, Richard G Brink, Daryl M Bryan, Charles A Byun, Hyeja Carroll, Joel Chandra, Jagdish Brinkmann, HW Bryant, Billy F Carroll, Michael P Chaney, Robin W Brinkmann, Bryant, John G Cable, Charles A Carroll, Robert W Chang, Bomshik Hans-Berndt Bryant, John L Cadogan, Charles C Carroll, Timothy B Chang, Derek Kong Brisson, Erik Bryant, Robert L de Caen, D Carson, John M Sr Chang, Gerard J Britton, John L Bryce, Gerald M Caffarelli, Luis A Cartan, Henri Chang, 1-Lok Broad win, Judith E Brydges, David C Cafl.isch, Russel Carter, David W Chang, Joo Sup Brobeck, Brylinski, Jean-Luc Cagnac, Francis Carter, Karen Chang, Kun Soo Barbara Caldwell Bryn, Milo F Cahill, Pamela M Carter, Chang, Li-Fung Brode, John Brzezinski, Juliusz Cahill, Rotraut G Patricia Hawks Chang, Long F Brodie, Robert Bruce Bube, Kenneth P Cahlon, Baruch Carter, Thomas J Chang, Ping-Tung Brodskii, MS Bucci~o, Alphonse Cai, Haiyan Cartlidge, James M Chang, Shao-Chien Broline, Duane M Buchner, Michael A Cain, Bryan E Carton-Lebrun, c Chang, Shih-Hung Bromberg, Eleazer Buck, Marshall W Cain, George L Jr Casal, Alfonso Chang, Shu-Jung Bromley, Darwin P Buck, Robert E Caine, Bruce R Casanova, Toussaint Chang, Weita Brons, Hans Bucki, Andrzej J Caines, Peter E Casazza, Peter G Chantip, Suchat Brooks, Burrow P Buckley, Joseph T Cairoli, R Case, Bettye Anne Chaperon, Marc Brooks, James 0 Buckley, Stephen M Calcaterra, Robert Case, James H Chapin, Edward W Brooks, Peter S Bucy, RichardS Calderer, M Carme Case, James R Chapin, Steven Arthur Brooks, Robert Buddrus, Lee Edward Caldwell, Casey, Robert Francis Chapman, Robert M Brooks, Robert M Budnik, Paul Jr Roderick PC Casey, Stephen D Chapman, Brosamler, Gunnar A Buechler, Steven Allen Caldwell, Thomas E Casler, Burtis G Scott Thomas Brosowski, Bruno H Bueckner, Hans F Caldwell, William V Cassels, J w s Charles, Josette Brothers, John E Buehler, Royce E Calica, Arnold B Cassidy, Phyllis J Charlton, Harvey J Broue, Michel Buekenhout, F Calkin, Neil J Castagnoli, Erio A Charney, Ruth M Broughan, Kevin A Buell, Duncan A Callahan, James J Castellet, Manuel Chartrand, Gary Broughton, SA Buhler, Joe P Calloway, Jean M Caster, David M Chastkofsky, Leonard Browder, Felix E Buianouckas, Calvert, James E del Castillo, Beatriz M Chatterji, Srishti D Browder, William Francis R Camacho, James Jr Castillo, Jose E Chatters, Brown, Anne E Buis, PaulE Camarena, Vicente Castillo, Oreste N Arthur Walter Brown, Arthur A Bulena, Richard Cameron, Douglas E Castilla-Chavez, Chatzidakis, Zoe Brown, Beverley M Bulirsch, Camillo, Victor P Carlos Chauvin, Andre Brown, C Sue Roland Zdenek Camina, AR Castore, Glen M Chavel, Isaac Brown, Dean R Bull, Everett L Jr Campanino, Massimo Castro, Antonio de Chawla, Lal M Brown, Dennison R Bullen, Peter Campbell, Daniel D Castro, Peter E Chee, Pak Soong Brown, Edgar H Jr Bulman-Flerning, Campbell, Dave A Catanese, Cheeger, Jeff Brown, Edward M Sydney D Campbell, Harold E A Fabrizio ME Cheema, Mohindar S Brown, Gary I Bum by, RichardT Campbell, Hugh G Cateforis, Vasily C Cheer, A Brown, George B Bunce, Gary R Campbell, James T Cathey, Frederick W Chein, Orin N Brown, Gerald D Bunce, John W Campbell, Catlin, Donald E Chen, Concordia C Brown, Gerald L Bunge, Marta C Jean McCroskey Cattabriga, L Chen, Daniel K C Brown, Gordon E Buntinas, MartinG Campbell, L Andrew Cattani, Eduardo Chen, Dayue Brown, Jack B Buoni, John J Campbell, L Lorne Causey, William M Chen, Etang Brown, Jeffrey L Burchard, Hermann G Campbell, Robert Ives Cavanaugh, Joseph M Chen, Goong Brown, Johnny E Burchard, Paul Campillo, Antonio Cavaretta, Alfred S Jr Chen, Haiwen Brown, Kenneth S Burckel, Robert B Campos, LMBC Cavender, James A Chen, Jau-Die W Brown, Lawrence G Burdick, GR Canary, Richard D Caveny, Deanna Marie Chen, Kuang-Ho Brown, Leon Burgess, C Edmund Canavan, Robert I Caviar, Stephen R Chen, Kuo-Tsai Brown, Morton Burgess, Canfield, E Rodney Cawley, Robert Chen, Kwan-Wei Brown, Richard Dennis Canjar, Mike Cay co, Bern Chen, Louis H Y Richard Arthur Burgess, Walter D Cannon, James W Cebollero, Blanca S Chen, M Brown, Richard H Burke, James E Cannon, John R Cecil, Thomas E Chen, MatthewS Brown, Richard K Burkhart, Richard H Cannonito, Frank B Cegrell, Urban Chen, Pengyuan Brown, Richard L Burkholder, Donald L Cantor, David G Celik, Hasan A Chen, W W L Brown, Robert Bruce Burkholder, Cantrell, James C Cenkl, Bohumil Chen, Wen Brown, Robert Dillon Douglas G Cantwell, John C Cerda, Joan Chen, YW Brown, Burkill, John C Cap, Clemens Heinrich Cerlin, Zvonko Chen, Yah-Wei A Robert Freeman Burns, Anne M Capasso, Vincenzo Cernuschi-Frias, Chen, Yongzhuo Brown, Ron Burns, Daniel M Jr Capozzi, Alberto Bruno Cheney, EW Brown, Ronald Burris, Stanley N Capozzoli, Anthony B Cerruti, Umberto Cheng, Jih-Hsin Brown, Russell M Burstall, Francis E Caranti, AE deCesare, Kenneth M Cheng, Kai-Nah Brown, Scott W Burt, Philip B Cardenas, Cesari, Lamberto Cheng, Kuo-Shung Brown, Sharan Inez Burton, Charles R Juan Donoso Cha, Hyung Koo Cheng, Raymond S Brown, Thomas C Burton, Leonard P Carey, Michael C Chabert, J L Cheng, Yungchen Brown, William G Burton, Theodore A Cargal, James M Chachere, Gerald R Cherenack, Paul F Brownawell, W Dale Busemann, Herbert Cargo, Gerald T Chaffee, James W Cherlin, George Y BrowneR F, Henrique Busenberg, Stavros N Cad~cci, Franco Chaiken, Seth D Cherlin, Gregory L Browne, H Nelson Jr Buser, Peter J Carleson, Lennart A E Chakerian, Gulbank D Cherniavsky, John C Browne, Patrick J Bush, George C Carlip, Walter Chakravarti, RS Chernick, Michael R Brownell, Frank H III Bushnell, Colin J Carlisle, Ronald L Chakravarty, Subinoy Chernoff, Herman Browning, William J Buss, Samuel R Carlson, Bengt G Chalk, John H Chernoff, Paul R Brownstein, Beth Butchart, John H Carlson, Bille C Chalkley, Roger Cherny, Julius Brualdi, Richard A Butcher, John C Carlson, David H Chalmers, Bruce L Cherowitzo, William E Bruck, Ronald E Jr Butcher, Carlson, Dean A Chalmers, Graham D Cheung, BKS Bruckner, Andrew M Ralph Stevens Carlson, James A Chamberlain, Chevalier, Michel Bruen, Aiden A Butler, A Fred Carlson, John W Robert Mathieu Jr Chi, Dong Pyo Bruggeman, Roelof W Butler, James William Carlson, Jon F Chamberlin, Chia, Gek-Ling Brumer, Armand Butler, John B Jr Carlson, Kermit H Richard E Chiang, Peter H S Brumfiel, Gregory W Butson, Alton T Carlson, Robert C Chambers, Chiarenza, Filippo Bruna, Joequim Butts, Thomas R Carlson, Timothy J Daniel Warren Chidambaraswamy, Bruner, Robert R Buys, Mutiara Carlsson, Renate Chambers, Jayanthi Bruni, Anthony J Buzano, Ernesto Carlton, Eloise H Florence D Rozniak Chihara, Theodore S BrUning, Jochen Byerly, Robert E Carmichael, Richard D Chambers, Chikwendu, Sunday C Brunk, Hugh D By kat, Alexander do Carmo, Manfredo P Graham A Jr Childs, Lindsay N Bruno, Vincent J Byleen, KarlE Carone, Patricia L Champagne, Carol Z Chillag, David Bruns, Winfried Byrd, Kenneth A Carothers, NealL Chan, Chiu Yeung Chillak, Edward W Brunschwig, Byrne, Catriona M Carpenter, Chan, Jack-Kang Chinburg, Ted C Mildred C Byrne, Charles L Gail Alexandra Chan, Tony F Ching, Wai-Sin Brunson, Barry W Byrnes, James S Carr, Ralph W Chan, YH Chinn, William G

527 ICM-86 (Continued)

Chisala, Busiso P Cohen, Leon W Corduneanu, Cuoco, Albert A David, Karl Chivukula, R Rao Cohen, Marshall M Constantin Cupillari, Antonella David,· Nancy A Cho, Choong Yun Cohen,. Martin J Corliss, George F Curjel, Caspar R Davidoff, Giuliana P Cho, Dall-Hoon Cohen, Maurice Cornelius, E Frank Curlovic, Donald L Davidoff, Kathleen S Cho, Sung Je Cohen, Michael P Cornet, H Curran, Peter M Davidson, James R Choe, Young Han Cohen, Moses E Cornette, James L CurraS-Bosch, Carlos Davidson, Kenneth R Choi, Jeen-Kap Cohen, Ralph Correas, J M Currier, Albert W Davidson, Choi, Man-Duen Cohen, Steve Michael Corrigan, Currier, Robert J William Robert Choi, Un Haing Cohn, Harry Thomas Carney Curry, James H Davies, Morton J Choksi, Jal R Cohn, ~arvey Corsi Tani, Gabriella Curry, John D Davis, Burgess J Chollet, John Cohn, Leslie Cortzen, Barbara Curtis, Charles W Davis, Chandler Chong, Chi Tat Cohn, Paul M Cosner, George C Curtis, Doug W Davis, James F Chong, Tjee Hung Cohn, Costa, David G Curtis, Herbert J Davis, Lincoln Ross Chao, Koo-Guan Stephen Edward Cote, Louis J Curtis, Morton L Davis, Paul L Choquet, Gustave Cohn, William S Cotlar, Mischa Curtis, Philip C Jr Davis, Paul W Chou, Col bourn, Cotsaftis, Michel Curto, Raul E Davis, Reid R Wei-Chung Arthur Charles Joseph Cottle, Richard W Cushing, Jim M Davis, Richard L Chouteau, Charles C Colby, Robert R Cotton, Robert M Cusick, David A Davis, Ruth M Chover, Joshua Cole, Brian J Couch, Alva L Cusick, Larry William Davis, William E Chow, Pao-Liu Cole, Charles A Coulter, Malcolm A Cusick, Thomas W Davis, William J Chow, Shui-Nee Cole, Nancy Countryman, Cuthill, Elizabeth H Davison, J Leslie Chow, Yuan S Cole, Paul Dana William Mark Cutler, Doyle 0 Davison, Walter F Choy, Stephen T L Coleman, A John Courter, Richard C Cutrer, M Wayne Davitt, Harold H Chrestenson, Hubert E Coleman, Courtney S Coury, Nicholas J Cygan, Jacek Dawes, AM Christensen, Chris Coleman, William Covey, David N Dawson, Donald A Christilles, William E Colley, Susan Jane Cowan, Jack D D'Agostino, Santo Dawson, John W Jr Christy, Joe Collier, Jarrell D Cowen, Carl C Jr D' Alotto, Louis A Day, B J Chu, Chinku Collins, Carol E Cowens, J Wayne D'Ambrosio, Ubiratan Day, George W Chuang, Pi-Chun Collins, Donald J Cowles, John R D'Angelo, John P Day, Jane M Chuaqui, Rolando B Collins, Gayle M Cowling, Michael G D'Aprile, Margherita Day, K Michael Chung, Fan R K Collins, George E Cox, David A D' Aristotle, Day, Mahlon M Chung, Kai Lai Collins, Heron S Cox, Dennis D Anthony J Day, Peter W Chung, Lung Ock Collins, Jesse H Cox, Gregory V D'Atri, Joseph E Daybell, Dorothy A Church, Alonzo Colquitt, Landon A Cox, Milton D D' Attorre, Leonardo Daykin, Donald R Church, Philip T Colson, Henry D Cox, Raymond H D'Entremont, Dayton, Barry H Churchill, Edmund Colvin, Michael Roy Cox, S H Jr Edward Joseph De Angelis, Valerio Churchill, Richard C Colwell, Peter Coxeter, D'Orazio, Silvana De Bouvere, Karel L Chute, Cheryl Ann Comenetz, Daniel H S MacDonald DaCosta, De Bruin, Marcel G Chyung, Dong Hak Comer, Stephen D Coxson, Pamela G Newton C A De Cecco, Giuseppe Ciarlet, Philippe G Comfort, Steven H Coyne, Steven Thomas Daboussi, Hedi De Iongh, Johan J Cifuentes, Patricio Comfort, W Wistar Cozzens, Dabrowski, Andre De Koning, Jan Cima, Joseph A Compton, Esther A Margaret Barry Dabrowski, Romuald De La Rica, Ciment, Melvyn Compton, Kevin J Crabtree, J B Dade, Everett C Mariano Hermida Cincotta, Comstock, Dale R Craggs, Robert F Dadok, Jiri De Land, Paul N Dominick Anthony Conlon, Joseph G Craig, Cecil Jr Daepp, Ulrich De Lorenzo, Helene Cipra, Barry A Conlon, Lawrence W Craig, Dafermos, De Los Reyes, Cirina, Marco Conn, Jack Frederick Jessica Marguerite Constantine M Josefina P Claasen, H L Connelly, Robert Jr Craig, Walter L Dahari, ASh De Marr, Ralph E Clapp, Michael H Conner, Luther T Jr Craig, William Dahlin, Rolf B De Meersman, Roger Clapp, M6nica Conner, Pierre E Jr Crandall, Michael G Dai, Taen-Yu De Pascale, Espedito Clare, Loren Paul Connolly, Mary V Crandall, Stephen H Daigneault, Aubert De Rezende, Ketty Clark, Cathy Ann Connors, Edward A Crane, George E Dajczer, Marcos De Souza, Vera H G Clark, David M Conrad, Bruce P Crane, Jane P Dale, Knut T De Villiers, Raoul A Clark, Kenneth D Constantinescu, Crapo, Henry H Dale, Richard H De Vries, Hendrik Clark, Robert A Cornelio Crauder, Bruce C Dales, H Garth DeBlassie, Clarke, A Bruce Contessa, Maria Craven, Bruce D Daly, John F Richard Dante Clarke, Douglas A Conti, Roberto Cravetz, Allan E Daly, John T DeFacio, Brian Clarke, Conway, John B Crawford, John David Damiano, David B DeKoninck, Graham Thomas Cook, Clarence H Crawford, Pamela Damkohler, Wilhelm L Jean Marie Clary, Stuart Cook, David Edwin Creekmore, Joseph W Damon, DeLong, R Peter Claus, Elly Cook, Joseph M Creese, Franklyn G James Norman DeShalit, Ehud Cleaver, Charles E Cook, Larry T Creese, Thomas M Dana, Martin P DeTurck, Dennis Clement, Mary Dean Cook, Roger J Creighton, J H C Daney, Charles G DeVore, Ronald A Clench, Ralfe J Jr Cooke, Kenneth L Criscenti, Danforth, Katrine Deal, Albert L III Cleveland, Richard A Cooke, Roger Lee Jacqueline P Dang, X-Hong Dealba-Guerra, Clifford, Alfred H Cooley, Clifford R Criss, Steven P Dangello, Frank Ralph Luz Maria Clifford, Jane E Coombes, Kevin R Critchlow, Florence G Daniel, Timothy Deanin, Alice Clifton, Joseph M Coonen, Jerome T Crittenden, Richard J Daniels, M Peg M Deblasi, Francesco S Cline, Edward T Cooney, Crockett, Diane W Dannan, Fozi Mustafa Debnath, Lokenath Clover, William J Jr Sister Miriam P Cronheim, Arno Dantzig, George B Debrunner, Hans E Cobb, John Cooper, ED Croom, Frederick H Dar, Karamat Hussain Dechene, Lucy I Cobb, Philip A Cooper, Hilda L Cross, James J Darafsheh, Deckert, Kenneth L Cobham, Ian P Cooper, J B Cross, John S Mohammad-Reza Dediu, Mihai Michael Cochran, James A Cooper, Jeffery M Crossman, Howard L Dare, Adrienne M Deeb, Waleed M Cockcroft, Wilfred H Cooper, Mark A Crothers, Derrick S F Dark, R S Deeter, Charles R Coddington, Earl A Cootz, Thomas A Crow, Edwin L Darken, Joanne S Defrancesco, Henry F Cody, William F Jr Cope, Davis K Crumb, Cyndi A Darling, R W R Dehesa, Jesus Sanchez Coen, Salvatore Copeland, Arthur H Jr Cruse, Allan B Darmody, Laurie M Deheuvels, Rene Coffee, Terence Coppel, Cruthirds, John E Darnel, Michael R Deiermann, Paul J Coffey, John William Andrew Csorgo, Miklos Darst, Richard B Deift, Percy Alec Coffin, Tina B Coppersmith, Don Cuculiere, Roger Oas, Anadijiban Dejter, Italo J Cogdell, James Wesley Coppotelli, Fred Cude, Joe E Das, Lovejoy S Dekker, Jacob C E Cogswell, Richard L Coram, Donald S Cudia, Dennis F Dashiell, Fred K Jr Dekleine, Herbert A Cohen, Arjeh M Coray, Daniel Cullinane, Steven Dassios, George Dekster, B V Cohen, Daniel I A Corbacho, Eusebio T Cummings, Jeffrey T Datskovsky, Boris A Del Riego de Del Cohen, David M Corbett, John V Cunningham, A B Datskovsky, Joseph A Castillo, L Cohen, Donald S Cordero, Luis A Cunningham, Datta, Dilip K Delahunty, Michael D Cohen, Frederick R Cordes, Craig M Frederic Jr Dautray, Robert I Delange, Hubert Cohen, Herman J Cordes, Heinz 0 Cunningham, Davey, Brian A Delchamps, David F Cohen, Jack D Cordova, Jeffrey Q Richard J David, Daniel Deleanu, Aristide Cohen, Joel M Cordovil, Raul Cuntz, Joachim David, Florin R Delges, H Wolfgang

528 ICM-86 (Continued)

Dell' Antonio, Disney, Ralph L Drobot, Vladimir Edwards, C M Engler, Hans P Gianfausto Diute, Patricia J Drucker, Daniel S Edwards, Charles H Jr Engquist, Bjorn Della Riccia, Giacomo Divis-Poracka, Zita M Drucker, Thomas L Edwards, Engquist, Michael L Dellwo, David R Dixon, John D Drzyzgula, Robert P David Albert Engstrom, Philip G Delmer, Thomas N Dixon, Martyn Russell Dshalalov, Jewgine Edwards, Harold M Enns, Theodore C Delporte, Jean Dixon, Michael J Du, Ding-Zhu Edwards, Kathleen R Enochs, Edgar E Deluca, Louis J Dixt.n, Peter Grant Du, S W Edwards, Robert D Enomoto, Kazuyuki Delver, Robert Dixon, William 0 Du bins, Lester E Eells, James Ensey, Ronald J Demana, Franklin D Djamboulian, Ara M Dubinsky, Ed Eenigenburg, Paul J Entringer, Roger C Demetrius, Lloyd A DjokoviC, Dragomir Z Dubisch, Roy Egle, David L Epp, Helmut P Demeyer, Frank R Djomehri, M Jahed Dubreil, Paul J Ehlers, F Edward Epp, Susanna S Deming, Robert W Dlab, Vlastimil Dubson, Alberto S Ehrlich, Karel Epperson, James F Denbow, Carl H Dobbins, Robert R Duchamp, Thomas E Ehrlich, Louis W Epstein, Benjamin Deneen, Linda L Dobbs, David E Dudek, Zbigniew C Ehrlich, Paul Ewing Epstein, Bernard Denig, William A Dodds, Peter G Dudley, Richard M Eid, Yousef R Epstein, Charles L Denlinger, Charles G Dodds, Theresa K Dudley, Underwood Eifrig, Bernd Epstein, D I Denman, Dodson, B Dudziak, James J Eigel, Edwin G Jr Epstein, Richard L Richard Thomas Dodson, Duffin, Richard J Eigen, Stanley J Erban, Daniel Dennis, Roger Keith Christopher T J Duffus, Dwight A Eilenberg, Samuel Erbe, Lynn H Dennis, William A Dodziuk, Jozef Duffy, Michael A Jr Ein, Erber, Friedrich M Denny, William F II Doerfler, Willibald Duggal, Krishan Lal Lawrence Man Han Erbland, John P Dent, Elliod Doering, Bora Duhamel, Tammy Lee Einstein-Matthews, Ercolano, Joseph L Denzel, Gene Dolan, James M Duke, Richard A Stanley M Erdelyi, Ivan N Deodhar, Vinay Vithal Dolbeault, Pierre E Dukelow, James S J r Einthoven, Isabella H Erdman, John M Deprima, Charles R Dold, Albrecht E Dumas, H Scott Eisele, Carolyn Erdmann, Michael A Depue, Bill Earl Dolgachev, Igor Duncan, Cecil E Eisenberger, Isidore Erdos. Paul Deretsky, Tatiana Doll, Nelly E Duncan, Hilda Eisenbud, David Erickson, Roger P Deriggi, Dennis F Dollard, John D Duncan, John Eisenstadt, B J Erkiletian, Derr, Leroy J Dolph, Charles L Duncan, Richard D Eisman, Sylvan H Dickran H Jr Deruaz, Marcel A F Dolzycki, Stanislaw J Duncan, Robert L Eisner, Elmer Erlandsson, Thomas Derwent, John E Dombroski, Michael Dunn, D W Eisa, Joseph J Erie, Dieter H nesarmenien, Jacques Dombrowski, Dunn, Stanley M Ekeland, lvar Errico, Thomas G Deskins, WE Joanne M Dunwoody, Eklof, Paul C Ervynck, Desmond, James Eddy Dominguez, J M Martin John Eklund, Anthony D Gontran J 0 H Desoer, Charles A Dominijanni, Roberto Dupont, Johan L Ekman, Kenneth E Eschenbach, Dieter Desolneux-Moulis, Domino, Laurence E Dupre, Maurice J Ekong, Samuel Escobales, Nicole Domokos, Gabor Duran, Wilfred 0 El Kacimi Alaoui, Richard H Jr Dethlefsen, Douglas G Donahue, Durand, Loyal Aziz Eskin, Gregory Detlefsen, Michael E Sister Gabriel Mary Durbin, John R El Kohen, Eslinger, Robert C Dettman, John W Donaly, James P Duren, Peter L Abdelouaha.b Espina, Carlos L Deumens, Erik Donoho, David Leigh Duren, William L Jr El-Helaly, Sherif Tab a Essawy, Abdelrahman Deutsch, Daniel H Donovan, Peter Durfee, Alan H El-Shobaky, Entisarat Hussein Devaney, Robert L Doob, Joseph L Durfee, William H Eldridge, Klaus E Essen, Matts R Deveaux, Raymond C Dooley, A H Durgun, Kanat Elhadad, Jimmy H Esser, Deveney, James K Doplicher, Sergio Durkin, Marilyn B Elhay, Sylvan Anne-Marie Simon Dewallens, Jean H Doran, Robert S Durnota, Bohdan P Elia, Michele Estrada, Luis Dezern, David H Dordal, Peter L Durrett, Richard T Eliason, Stanley B Etheridge, William L Dhaliwal, Ranjit S Doree, Suzanne Durumeric, Oguz C Elich, Joe Etter, Daniel 0 Jr Di Cenzo, Alan Dorfmeister, Josef F Duskin, John W Jr Elkins, Bryce L Evans, Michael Jon Di Franco, Roland B Dorner, Bryan C Duszynski, Martin Jr Elko, Constance B Evans, Trevor Di Maio, Giuseppe Dorsey, Ely A Dutt, J E Ellers, Erich W Evens, Leonard Di Martino, Lino Dorwart, Harold L Dutta, Sankar Prasad Ellingham, Mark N Evyatar, Asriel Di Paola, Jane W Dos Santos, Antonio F Dutton, Elliott, David L Ewen, Ira Di Stefano, Charles J Dotzel, Ronald M Charles Everett Elliott, George A Ewing, George M DiGiallonardo, Dougherty, Robert P Dwork, Bernard M Elliott, Herbert E Jr Ewing, Richard E Joseph A Douglas, Craig C Dwork, Julius S Elliott, P D T A Ewy, Daniel J DiPietro, Alphonso J Douglas, Jim J r Dydak, Jerzy Elliott. Sheldon E Ezeilo, James 0 C Diamond, Jack Douglas, Lloyd E Dye, Henry A Ellis. Alan J Dias Da Silva, J A Douglas, Ronald G Dym, Harry Ellis, James W Fabec, Raymond C Diaz, Gregorio Doust, Ian R Dymacek, Ellis, Kathryn P Fabes, Eugene B Diaz-Miranda, Dovermann, Wayne Marshall Ellis, Robert Fabrykowski, Jacek W Antonio Karl Heinz Dynin, Alexander Ellis, Fagan, James T Dibner, Steve Dovidio, Dynkin, Eugene B Sharon Theleman Fagg, Charles Rocke Dickerson, Charles E Catherine Slanta Dyson, Louise Ellner, Stephen P Faierman, Melvin Dickson, Robert J Dow, Alan Dyson. Verena H Elman, Richard S Fair, Wyman G Diederich, Klas Dow, Stephen J Eloe, Paul W Faires, J Douglas Diehlmann, Richard K Dowd, Robert E Each•_ts, J J Elson, Constance Fairgrieve, Thomas F Diener, Karl-Heinz Dowers, J F Ealy, Clifton E Jr McMillan Faith, Carl Dietz, David Dowlen, Mary M Eames, William P Elton, John H Falbo, Clement E Dietz, Klaus Dowling, ThOmas A Earle, Clifford J Jr Elzanowski, Marek Falconer, Etta Z Dieudonne, Jean A Downey, Rodney G Earle, Frederick 0 Emamzadeh, Falkner, Neil F Digernes, Trond Downing, J Scott Earles, Robert W Abolqasem Fall, Thomas C Dijksma, Aalt Downing, Reginald H Earnest, Andrew G Embry-Wardrop, Faltings, Kai Dikshit, G D Doyen, Jean Ebanks, Bruce R Mary R Famoye, Kiya Felix Dilao, Rui Doyle, Peter G Eberlein, Patricia J Emch, Gerard G Farahat, H K Dilcher, Karl H Dragan, lrinel Eberlein, Emerson, Marion P Faran, James J Dilley, Gary Alan Drager, Lance D Patrick Barry Emery, Faris, William G Dillon, Richard T Dragt, Alex J Eberly, David H Carol J Reinhard Farley, Belmont G Dilts, Gary A Drake, Frank R Ebey, Sherwood F Emmer, Michele Farley, Mark E Dilworth, R P Draper, James A Eby, Edward S Enderton, Herbert B Farmer, Thomas A Diminnie, Charles R Draper, Patricia J Eck, David J Endler, Otto Farmer, Dimitracopoulos, Draper, Richard N Eckhardt, William Endo, Youichi William Michael Constantinos Drazin, Michael P Ecklund, Earl F Jr Endsley, Neil H Farnsteiner, Rolf Dimsdale, Bernard Dreben, Burton S Edelson, Allan L Engber, Michael Farran, Hani Reda Dineen, Sean Drelles, Paul G Edet, Silas B Engel, Lars N Farrand, Scott M Dinitz, Jeffrey Howard Dreseler, Bernd Edgar, Gerald A Engeler, Erwin Farrell, Edward J Dinneen, Gerald P Dressel, Francis G Edmonds, Allan L Engenes, Hans Farrell, Paul A Dian, Gloria S Drew, Gerald C Edmonds, George F Engesser, Hermann Farrell, Roger H Dionne, Benoit Driscoll, Mark J Edmondson, Don E Engl, Heinz W Farrer, Brian H Dionne, Philippe A Driscoll, Richard J Edmundson, H P England, James W Farris, Frank A Diperna, Ronald J Driver, Rodney D Ed rei, Albert Engle, Jessie Ann Farris, Mark K

529 ICM-86 (Continued)

Fass, Arnold L Fishback, William T Frangos, Fwu, Chihchy Gauchman, Hillel Fathi, Albert Fishburn, Nicholas Efstr Gaudefroy, Alain Faudree, Ralph J Jr Peter Clingerman Frank, George N Gaal, Lisl Novak Gaudry, Garth I Faulkner, Frank D Fisher, Edward R Franke, John E Gabai, David Gauthier, Paul M Faulkner, Gary Doyle Fisher, Joe W Frankel, Theodore T Gabay, Jacques Gautschi, Walter Fauntleroy, Amass a C Fisher, Stephen D Franklin, Joel N Gabber, Ofer Gavin, Lloyd A Faurre, Pierre L Fishman, Louis Frantz, Deborah A Gabriel, Richard F Gazik, Ray J Fearn, Dean H Fisk, Steve Frantz, Henry W III Gackstatter, Fritz Gechter, Jerry Fearnley-Sander, Fitch, Frederic B Franzblau, Deborah S Gadbois, Gehring, Frederick W Desmond P Fitz-Gerald, Gary F Franzen, Alan Steven Charles Geissinger, Lad nor D Feder, Moshe Fitzgerald, Eugenia E Frappier, Clement Gadia, Shashi Kumar Geitz, Robert F Feeman, George F Fitzpatrick, Ben Jr Frawley, William J Gage, Michael E Gelbart, Abe Feeman, Timothy G Fitzpatrick, Patrick M Frayne, Thomas E Gaglione, Anthony M Gelbart, Stephen S Feferman, Solomon Fitzsimmons, Frazier, Michael W Gagola, Gloria Potter Gelfand, Hosna Fegan, H D Patrick J Frech, Bruce Gagrat, Mani S Geller, Daryl Neil Fegreus, John F Fixman, Uri Frederickson, Elvy L Gaier, Dieter Geluk, Jaap L Fehlmann, Richard Flahive, Frederickson, Paul 0 Gail, Harry R Jr Geman, Stuart Alan Feibel, Werner M Mary Elizabeth Fredricksen, Harold M Gainer, Dianne M Gemeda, Demissu Feibleman, James K Flanders, Harley Fredriksson, Einar Gaines, Robert Earl Genensky, Samuel M Feichtinger, Flanigan, Francis J Freedman, Allen R Galbraith, A S Genet, Jean Raymond Hans Georg Flapan, Erica L Freedman, Michael H Gallegos-Jarpa, Gentili, Graziano Fein, Burton I Flaspohler, John C Freeland, Mark S Gricelda E Geoghegan, Ross Feinberg, Flath, Dan Freeman, James H Gallier, Jean H Georgakis, Robert Benjamin Fleckner, Oscar L Jr Freeman, Mark E Galovich, Steve Constantine E Feinerman, Robert P Fleming, Charles G Freeman, Michael B Galperin, Efim A George, John Clay Feingold, Alex J Fleming, Daniel J Frei, Armin Galuten, Aaron Georgiou, D A Feinsilver, Philip Fleming, M Brendan Freier, Jerome B Galvin, Fred Geramita, Anthony V Feit, Walter Fleming, Richard J Freire Nistal, J L Gamelin, Theodore W Geramita, Joan M Fejer, Peter Andrew Fleming, Wendell H Freitag, Herta T Games, Richard A Geremia., Feldman, Arnold D Fletcher, Evan Martin Freskos, Mark Allen Gamet, Andreas Raymond S Jr Feldman, Edgar A Fletcher, G Yates Frey, Regina F Gamet, Jens Gerhard, J Arthur Feldman, Jacob Fliess, Michel Freyd, Peter J Ganelius, Tord H Gerhardt, C Feldman, Louis A Florence, John B Fricker, Francois Ganesan, Subramanian Gerig, Stephen R Feldman, Norman Flores, Jorge Gilberta Fridman, Burna L Gangolli, Ramesh A Gerisch, Wolfgang Fell, James M G Floret, Klaus Fridy, John A Gani, Joseph M Gerlach, Fellers, Norman V Jr Flosser, H 0 Fried, David Ganser, Carl C Eberhard G P Fellman, Johan Floyd, Edwin E Friedberg, Solomon Gantos, Richard L Gerlach, Jacob H Fellows, Flynn, David E Friedland, Shmuel Gao, L Zhiyong Gerlach, Jiirgen Germain, Paul Michael Ralph Flynn, Lawrence E Friedlander, Eric M Gapaillard, Jacques Geronimo, Jeffrey S Fenchel, Werner Flynn, Robert Friedlander, John B Gaposchkin, Peter J A Gershenson, Hillel H Fendel, Daniel M Flytzanis, E Friedman, Avner Garb, Regina H Fendrich, John W Gerstein, Larry J Foata, Dominique Friedman, Garcia, Mariano Jr Feng, Qi Garcia~ Gerstenhaber, Murray Foehl, Henry Clay Barbara Bernstein Lafuente, Fenley, John C Gerszonowicz, Jorge A Foglio Susana, F L de Friedman, Joyce B Jose M Fenley, Sergio Roberto Friedman, Nathaniel A Garcia-Loygorri, Gerth, Frank E III Foias, Ciprian Ilie Fennell, James G Friedman, Sy D Cristobal Gervais, Jean Jacques Foley, Am by M Fenner, Murray B H Friesen, Donald K Garcia-Planas, Isabel Gerver, Joseph L Foley, Daniel M Jr Fennig, Joseph N Frink, Orrin Garcia-Rodeja F, Gessaman, Margaret P Folk, Thomas F Fenske, Christian Frisch, Norman J Eduardo Gessel, Ira M Folland, Gerald B Fenstad, Jens E Fritsch, Rudolf Gard, Thomas C Getu, Seyoum Fang, Che-Kao Fenton, William E Fritsche, Richard T Gardiner, Geue, Andrew S Ferguson, Helaman Fang, Kirby W Froese, Richard G Christopher J Gewand, Marlene E Ferguson, Pamela A Fang, Frohardt, Daniel E Gardiner, Cyril F Gewirtz, Stephen J Ferguson, Thomas S Pamela Yuk-Ching Frohlich, Jurg M Gardiner, Frederick P Ghatage, Pratibha Fernandez, Roberto Fang, Paul Frohliger, John A Garding, Lars Ghent, K S Ferrar, Joseph Fonseca, Irene Frucht, Robert W Gardner, Clifford S Ghisa, D Ferrari, Carlo Fontaine, Jean-Marc Fry, Cleota G Gardner, Maria F Ghosh, Bhaskar K Ferreira, Fernando J Fontaine, Marc F Fry, DavidS Gardner, Robert A Ghosh, Bijoy K Ferri, Massimo Fontana, Marco Fry, Thornton C Gardner, Robert B Ghosh, Parimal K Ferris, Ian M Fontenot, Robert A Fryer, S Harvey Garfinkle, Devra Giannini, Paola Ferro, Ruggero Foote, Robert L Fu, Joseph H G Garfunkel, Solomon A Gibson, Peter M Feuer, Albert Foran, James Fuchs, Laszlo Garg, Krishna M Gieseker, David Feuerbacher, Gary A Forbes, Donald Fuchs, Wolfgang H Garland, Howard Giever, John B Fickett, James W Ford, Charles Edwin Fuchssteiner, Ben no Garland, Stephen J Giga, Yoshikazu Fidler, K Steven Ford, David A Fuglede, Bent Garner, Lynn E Gilbarg, David Fieldhouse, D Ford, George P Fuglister, Frederick J Garnett, John B Gilbert, Freeman Fields, David E Forelli, Frank Fuhrken, E Gebhard Garraway, Gilbert, Fields, Jerry L Foreman, Matthew D Fuji-Hara, Ryoh Annie Marie George Thomas Fields, K L Forkeotes, Ann A Fujimagari, Tetsuo Garrett, John W Gilbert, Michael D Fig A- Talamanca, Forman, John W Fuj isaki, Rieko Garrison, Betty B Gilbert, Richard C Alessandro Formanek, Edward W Fujiwara, Daisuke Garrison, David B Gilbert, William J Filgo, Holland C Jr Forster, Otto F Fujiwara, Masahiko Garsia, Adriano M Gildenhuys, Dian Filippenko, Vladimir I Forte, Bruno Fukuda, Komei Garst, Peter Giles, James Edward Finch, David V Foster, A L Fukui, Seiichi Gartenberg, Philip A Giles, John R Finch, John V Foster, Andrew Fukushima, Masatoshi Gartland, Eugene C Jr Gilewicz, Jacek Finch, Steven Richard Foulis, David J Fukushima, Mitsuo Garver, Michael Gilkey, Peter B Fingerman, Joel Foulser, David A Fuller, Kent R Garvin, Gill, Stephen P Finkel, Allan J Fournelle, Thomas A Fuller, Leonard E Sister M Cleophas Gill, Tepper L Finkel, David Fournier, John J F Fuller, Luther B Garza, Gene G Gillard, Roland D Finkelstein, Leib Fowler, Robert S Fuller, Robert Arthur Garzon, Maximiliano Gillespie, Robert G Finkelstein, Louis D Fox, John Michael Fuller, William R Gasca, Mariano Gillette, Richard M Finlayson, Henry C Fox, Ralph Fulton, Curtis M Gaschi.itz, Wolfgang Gilligan, Bruce C Final, Carlos E Fraga, Robert J Fulton, James Paul Gaskill, Herbert S Gillilan, James H Finston, David R Frame, J Sutherland Fulton, William Gaskill, Irving E Gillman, Daniel W Fintushel, Ronald A Frame, Michael Lee Fuqua, Jeffry B Gaskill, Jerry W Gillman, David S Firey, William Francaviglia, Mauro Furuta, Takayuki Gasqui, Jacques H Gilman, Jane P Fischer, David R France!, Margaret Ann Furutani, Kenro Gass, Michael D Gilman, Robert Fischer, Francis, Bruce A Fusaro, B A Gass, Noreen Herefeld Gilmer, Gloria Gerd Otto-Walter Francis, Eugene A Fusco, Giorgio Gastl, George C Gilmer, Patrick M Fischer, H Francsics, Gabor Futaki, Akita Gatterdam, Ronald W Gilmer, Robert Fischer, Hans R Frandsen, Henry Futia, Carl A Gatteschi, Luigi Gilmour, Grant James

530 ICM-86 (Continued)

Gilsdorf, Thomas E Goodearl, Kenneth R Gravesen, Jens Gualtierotti, A F Hajj a, Gine-Masdeu, Evarist Goodman, A W Gray, Denise Guaraldo, Rosalind J Mowaffaq Abdulla Gingell, Robert A Jr Goodman, Gray, Jack D Guay, Merle D Halberstam, Heini Gingrich, Ross B Frederick M Gray, James M Gudder, Stanley P Halchin, Judy D Gioia, Anthony A Goodman, Jacob Eli Gray, Jeremy John Gudimetla, V S Rao Hale, David R Giovannitti, Goodman, Gray, John W Guenard, Francois Hale, Jack K Anthony J James Thomas Gray, Robert E Guerin, Esther E Hale, Margie A Giraud, Daniel Goodman, Nicolas D Gray, Guest, Martin A Hales, Alfred W Girolo, Jack E Goodman, Roe W Samuel Hutchison Guetti, Joan F Hales, R Stanton Jr Gittings, Raymond F Goodner, DB Grayson, Giiichal, Edgardo N Halkin, Hubert Givant, Steven R Goodson, Daniel Richard Guillemin, Victor W Hall, Glen Richard Gladue, Earl C Geoffrey Ross Greaves, G Guillot, J C Hall, L M Glass, AM W Goolsby, Ronnie C Grecos, Alkis Gulbenkian, Garabed Hall, Mark Edwin Glassey, Robert T Goon, Jonathan L Greechie, Richard J Gulden, S L Hall, Peter Glauberman, George Gootman, Elliot C Greek, John W Gulick, Denny Hall, Robert B Glauz, Robert D Gopal, Mangalam R Green, David Jr Gulliver, Robert D II Halperin, S Glaz, Sarah Gordh, George R Jr Green, John W Gundersen, Gary G Halpern, Fred Glazebrook, James F Gordon, B Brent Green, Leon W Gundy, Richard F Halpern, Herbert Glazer, Steven Gordon, Carolyn S Green, William L Gunn, Charles G Halton, John H Gleason, Andrew M Gordon, Derek Green, Willie H Gunn, Michael E Halverson, Kimberly J Glenn, Paul G Gordon, Hugh Greenberg, Peter A Gupta, Hansraj Hamadani-Zadeh, Glier, Germana I Gordon, Robert Greenberg, Philip J Gupta, Rajiv Javad Glimm, James G Gordon, S Robert Greenberg, Ralph Gupta, Ram Niwas Hamana, Masamichi Glover, David W Gordon, Yehoram Greene, John Robert Gupta, Shiv Kumar Hamann, Eloise Ann Glover, Henry H Gorenflo, Rudolf Greene, Peter H Guralnick, Robert M Hamill, William A Glover, Joseph Gorenstein, Daniel Greenfield, Gary R Gurel, Okan Hamilton, Jon W Gluchoff, Alan Daniel Goresky, R Mark Greenhalgh, Mark Gurganus, Kenneth R Hamm, Rose C Goad, Robert E Gori, Franco Greenhall, Charles A Gurney, Margaret Hammer, Carl Goblirsch, Richard P Gorkin, Pamela B Greenleaf, Allan T Curtin, Morton E Hammer, F David Godbole, Anant P GOrlich, Ernest Greenleaf, Frederick P Gustafson, Richard F Hammer, Peter L Goddard, IV Gorman, William J III Greenspan, Bernard Gustafson, Robert A Hammond, William F Goddard, J D Gormley, Michael J Greenspan, Bernard D Gustafson, Sven-Ake Hampton, Charles R Goddyn, Luis A Goroff, Daniel L Greenspan, Donald Gustafson, William H Hamrick, Gary C Godfrey, Colin Gorostiza, Luis G Greenstein, David S Gutek, Andrzej S Hamstrom, Godin, Paul J Gorton, Robert Bruce Greenwood, Robert E Guterman, Martin M Mary-Elizabeth Goebel, K Gosselin, John A Greever, John Guthery, Scott B Han, Jin Kyu Handel, David Goethals, J M E Gosselin, Richard P Gregorac, Robert J Gutierrez, Cristian E Handelman, George H Goetz, Abraham Gossett, Eric James Gregory, David A Gutknecht, Martin H Golan, Jonathan S Hanes, Kit Goth, John A Gregory, John Gutowitz, Ranges, Golber, David L Goto, Hideo Gregory, Howard Andrew Gold, Karen F Thomas Bradford Nicholas William Goto, Midori S Guttalu, Ramesh S Gold, Steven Greif, Anthony David Hanlon, Philip J Gottlieb, Allan J Gutzman, Wayne W Gold, Sydell P Greif, Stanley J Hanna, Charles C Gottlieb, Daniel H Guy, Richard K Goldbeck, Ben T Jr Gould, Henry W Greim, P Hanna, Peter Guy, Goldberg, David M Greiner, Peter C Hannan, James F Gould, Lawrence D Wynne Alexander Goldberg, Mark Kopel Gould, Lillian V Gresser, John T Hano, Jun-lchi Guyker, James Goldberg, Richard Gould, Sydney H Greville, Thomas N E Hansell, Roger W Goldberg, Richard R Goutziers, Constant J Griego, Richard J Hansen, ldar Ha, Kwang Chul Goldberg, Samuel Gouvea, Griffin, E L Jr Hansen, R 0 Ha, Young-Hwa Goldberg, Seymour Fernando Quadros Griffin, Philip Stanley Hansen, Haagensen, Kenneth E Goldberg, Vladislav V Govaerts, Willy J F Griffin, Vagn Lundsgaard Haario, Heikki Goldblatt, Robert I Gover, Eugene H William Gerald Hanson, Stanley C Haarsma, Loren D Golden, Kenneth M Gowda, Muddappa Griffing, Gary R Harada, Koichiro Haas, Felix Goldfeld, Dorian Seetharama Griffiths, Phillip A Harary, Frank Habegger, Nathan B Goldhaber, J K Goze, Michel Griggs, T S Harbater, David Haber, Seymour Goldman, Jay R Graber, James S Grigis, Alain A Harborth, Heiko Goldman, Jerry Grabiel, Federico Grillakis, Manoussos Habetha, Klaus Harbourne, Brian Goldman, Lawrence Grabiner, Sandy Grimmer, Ronald C Habibullah, G M Hardie, Keith A Goldman, Grabner, Gary Clem Crimson, W Eric L Hackborn, William W Hardin, R I William Mark Grace, Dennis R Grinberg, Vladimir Hackenbroch, Hardt, Robert M Goldner, Siegfried R Grace, Edward E Groemer, Helmut Wolfgang Hardy, Dare! W Goldstein, Gerald Grad, Harold Grogan, Eleanor M Hackler, Donald B Hardy, John T Jr Goldstein, Jerome A Graef, John R Grognard, Rene J M Haddad, George F B Hare, Gordon B Goldstein, Larry J Graf, Siegfried Gromov, Mikhael Haddad, Hadi M Hare, William R Goldstein, Norman J Grafakos, Loukas Grone, Bob Haddad, Labib S Hargrove, E E Goldstein, Richard Z Gragg, Karen E Gropp, William D Haddock, A Glen Harinath, K S Goldstein, Sheldon Graham, Aly Gross, Benedict H Haeberly, Harine, Goldston, Daniel A Graham, C Robin Gross, Kepneth I Jean-Pierre A Katherine Jane Goldstone, Richard J Graham, Colin C Gross, Leonard Haebich, William Raring-Smith, Golightly, George Graham, George Gross, Louis J Haefliger, Andre Robert H Golightly, Graham, George P Gross, Mark Thomas Haemmerlin, Guenther Harker, Patrick William L Jr Graham, Ronald L Grossman, Edward H Hafner, James L Harkleroad, Edie Golubitsky, Martin Graham, Sidney W Grossman, Robert Hag, Per Harkleroad, Leon Gomes, Gracinda Graham, Victor W Grosswald, Emil Hagelgans, Nancy L Harman, Jonathan W Gomez Pardo, Grainger, Arthur D Grove, Edward A Hager, Anthony W Harmand, Peter Jose Luis Gramsch, Bernhard Grove, K Hager, Harold W Harmelin, Reuven Gomez, Gerard Crannell, Grove, Larry C Hagis, Peter Jr Harmon, Dennis R Gomez, Jose A Michael John Grover, Carole L Hagler, James N Harnad, John P Gomez-Ruiz, Francisco Grant, . Caroline Galt Groves, J R J Hagood, John W Harper, James D Gomi, Kunia Grasman, William B Grubb, Daniel J Hahn, Gena Harper, John R Gomory, Ralph E Grasser, Grudin, Arnold Hahn, Kyong T Harper, Lawrence H Gompf, Robert E Horst-Siegfried von Grudzinski, Olaf Hahn, Neale M Harras, Barbara M Gondard-Cozette, Grassi, Peter Matthew Gruenberg, Karl W Hahne, Gerhard E Harrell, Evans M Daniell J Gratiaa, Marceline C Gruenhage, Gary F Hailat, Mohammad Q Harrell, Ronald E Gonshor, Harry Gratzer, George A Grundlnd, Haile, Darrell E Harrington, Walter J Gonzalez-Guzman, Gravel, Pierre Alfred Michel Haiman, Mark Harris, Bernard Jorge Graver, Jack E Gruter, Michael Raimo, Deborah T Harris, Bruno Gonzalo, Jesus Graves, Alan S Gryllakis, Hain, Richard Martin Harris, David Gooch, B Dean Graves, Glenn W Constantinos Haji-Ghassem, Harris, David Carter Good, Anton Graves, Larry K Guacaneme, Kamran Harris, Gary A Good, Irving John Graves, Robert L Julio Enrique Hajian, Gerald Harris, John C

531 ICM-86 (Continued)

Harris, Lawrence A Heitsch, J am~s L Hewitt, Gloria C Hofmann, Howe, Margaret R Harris, Morton E Hejtmanek, Hewitt, P R Charles E III Howe, Roger E Harris, Steven Guy Johann Hans Heyneman, Robert G Hofmann, Karl H Howe, Vernon W Harris, Theodore E Heldermann, Norbert Hiai, Fumio Hofmann, Steve C Howell, Joseph 0 Harrison, Monica L Helgason, Sigurdur Hickernell, Fred J HOft, Hartmut Howell, Kenneth Harrold, Orville G Jr Hellberg, H Stefan Hicks, Darrell Lee Hoft, Margret H Howie, John M Harrop, Fred F Heller, Alex Hida, Takeyuki Hogan, Daniel A Hoye, Laurence G Harrop, Ronald Heller, Barbara Ruth Higdon, Robert L Hogan, Guy T Hoyt, Gaylord M Hart, Garry D Heller, Dorothy M Higgins, John A Hogbe-Nlend, Henri Hoyt, Kenneth Hart, Lawrence P Heller, Robert Higgins, John C Hogesteeger, F W Hratz, Joseph A Hart, Neal Hellerman, Leo Higgins, Philip J HOgniis, GOran Hrbacek, Karel Harte, Robin E Hellerstein, Simon Higgins, William J Holberton, John V Hsia, J S Hartl, Johann Helmer, Gary A Hight, Or ian L Holder, L I Hsiang, Wu-Chung Hartman, James L Helson, Henry Hightower, James K Holenweg, Werner Hsiang, Wu-Teh Hartmanis, Juris Helton, Burrell W Hightower, William L Holladay, Kenneth W Hsieh, Po-Fang Hartmann, Helton, Frances J Higman, Donald G Holland, Daniel L Hsieh, Simon C Frederick W Helton, J William Higuchi, Yasunari Holland, Finbarr Hsu, Nai-Chao Hartnett, William E Helton, Jon C Hijab, Omar B Holland, Samuel S Jr Hsu, Pao-Sheng Hartshorne, Robin Helton, Robert Arthur Hilbert, Stephen R Holland, W Charles Hu, Chih-Chen Hartzler, Stanley J Helwig, Karl-Heinz Hildebrand, Adolf Holley, Frieda K Hu, S T Haruki, Hiroshi Helzer, Garry A Hildebrandt, Holley, Hu, Shou-Jen Haruki, Shigeru Hemker, Pieter W Theodore W Richard Andrew Hua, Yang Harvey, Charles A Hemmeter, Joseph S Hildebrant, John A Hollingsworth, B J Huang, Abigail S Harvey, Charles M Hemminger, Robert L Hilden, Hugh M Hollins, Clinton G Huang, Tsang-Chi Harvey, George G Hemmingsen, Erik Hilditch, A Stephen Holm, Kip D Huard, James G Harvey, William J Hempel, John P Hilgers, John W Holman, Huber, Alfred 0 Harzheim, Egbert Hemstead, Robert J Hill, David G B Wayne James III Huber, Heinz Martin Hasegawa, Minoru Henderson, Hill, E Stamford II Holmann, Harald R A Huber, Jamil R Bengt R Huber, Peter J Hashimi, Francis McVey Hill, James Holmberg, Ki-ichiro Charles S Huber, William A Hashimoto, Henderson, James P Hill, Ned W Jr Holmes, Haskell, Peter E Dave R Huch, Donna L Hendricks, Walter J Hill, Robert N Holmes, Haslach, Henry W Jr Holmes, Ernest J Huckaba, James A Henkin, Leon A Hill, Sarah Lynn Hasselbach, Eric R Holmes, John P Huckemann, Henn, Hans-Werner Hill, Shirley A Hassinger, Bill Jr Paul-Louis Holmes, Philip John Friedrich C Hennequin, Hillman, Hasumi, Morisuke Holmes, Richard B Hudson, Charles C Jr Hennessy, Patricia A Jonathan Arthur Hatada, Kazuyuki Guy M Holten, Robert P Hudson, D L Henniart, Hilton, Peter J Hathaway, Dale K Henniger, J P Haltom, Carl Hudson, Richard H Himmelberg, Hattori, Akio Christopher J Holton, Derek A Huerta, Carlos Cuevas Henrich, Charles J Ill Haughton, Henrici, Peter Holubec, Alex Huet, Denise Hindley, James Roger Dominique M A Henriksen, Melvin Holvoet, Roger Huff, Gerald B Hindman, Neil B Hauptman, Henry, Myron S Holzer, Joseph C Huff, Lawrence R Hinman, Peter G Herbert Aaron Hensel, Holzman, Robert E Huff, Melvyn E Hinojosa-Berrones, Hausmann, Robin Ann Morgan Homer, Roger H Huff, W N J A Jean-Claude Hensley, Douglas A Homer, William D Huffer, Ralph C Hinrichsen, D Hausner, Melvin Hensley, Elmer L Homsy, Elham Huffman, Ed W Hinson, Edward K Havens, Jim Henstock, Ralph Honda, Kin-Ya Huffman, William C Haver, William E Herbst, Albert F Hintz, Honda, Koji Hufford, George A Elizabeth Mansfield Hawkins, Herda, Hong, Geck Chan Hugger, Jens Hintzman, William R \Villiam Grant Hans-Heinrich W Hong, Yiming Hughes, Anne Tetsutaro Hay, George E Herfort, Hirano, Hongo, Eishi Hughes, Cephas D Eriko Hay, Louise Wolfgang Norbert Hironaka, Hoobler, Raymond T Hughes, Joseph A Hayashi, Elmer K Hergenrother, Hironaka, Heisuke Hood, Rodney T Hughes, Kimball W Hayashi, Eric Steven L Hirsch, Morris Hook, Julian L Hughes, Ruth L Hayashi, Kazumichi Herink, Curtis D Hirschfelder, Hooker, John W Hughston, L P Hayashi, Mikihiro Herman, Louis M Elizabeth S Hooper, Robert Clark Huh, Won Hayden, John L Herman, Michael Hirschfelder, John J Hoover, D N Hui, Kin Ming Hayes, David F Herman, Richard H Hirschfelder, Hoover, John Craig Huibregtse, Mark E Hayne, Roger M Hermann, Robert Rosemary Hoover, Thomas B Huilgol, Raja R Haynes, Robert C Hernandez, Diego B Hirzebruch, F E P Hopkins, Kevin W Hulbert, Guy William Hayslip, Iris C Hernandez, Eugenio Hitchin, Nigel J Hoppensteadt, Hulkower, Neal D Hazlewood, Carol Hernandez, Horacia Hitotsuyanagi, Nobuo Frank C Hull, Richard B Hazlewood, Donald G Hernandez, Omar A Hively, Gregory A Horblit, Brian D Hummel, James A Head, Richard R Hernandez, Rafael Hjorth, Poul G Horn, Alfred Hummel, Kenneth E Head, Thomas J Herod, James V Hlavac, Paul P Horn, Jean Macgregor H ummelgens, Peter Headley, Vel mer B Herrera, R B Hlavka, James L Horne, J G Jr Humphreys, James E Healey, Timothy J Herrero, Domingo A Ho, Grace P Horner, T S Humphreys, John F Heard, Melvin L Herreshoff, James B Hobbs, Arthur M Hornor, William E Huneke, Craig L Heath, Lenwood S Herrmann, Robert A Hobby, Charles R Horowitz, Maurice Huneke, John Philip Heath-Brown, Roger Herron, David A Hochschild, Gerhard P Horrigan, Timothy J Hung, David C Heatherly, Henry E Herron, Wm Kent Hochstadt, Harry Horrobin, Hung, Henry Hin-Lai Hebert, Michel Herschorn, Michael Hochster, Melvin Elizabeth Ann Hung, Trung Nguyen Hecht, Henryk Hersh, Reuben Hochwald, Scott H Horrocks, Geoffrey Hunkins, Dalton R Hecht, James E Herstein. Israel N Hockett, Kevin G Horsfield, Hunsaker, Worthen N Hecht-Nielsen, Robert Herve, Michel Hocking, J G Christopher H Hunsberger, L Gerald Hecker, David A Herwitz, Paul S Hodel, Richard E Horvath, Hunt, Louis Roberts Hedberg, Lars I Hery, \Villiam J Hodges, Lucille M Charles Daniel Hunt, Robert W Hedlund, Gustav A Herz, Carl S Hodges, Wilfrid A Horvath, Jozsef Hunt, Walker E Hedlund, James H Herzog, Emil R Hodgson, Hosack. John M Hunte, Beryl E Hedstrom, Gerald W Herzog, John 0 Jonathan P E Hosoi. Tsutomu Hunter, Colin B Hefez, Abramo Hesla, Todd Hoeffding, Wassily Hosseini, Naser Hunter, Heifetz, Daniel B Hess, Jacob J Hofer, Robert D Hoste, Jim E Norma Gerkens Heiman, Barbara Hess, Peter Hoffman, Hotta, Ryoshi Hunter, Roger H Heineken, Hermann Hesselink, Wim H Jerome William Hou, Jiashi Huotari, Heinicke, Allan G Hestenes. Marshall D Hoffman, Kenneth M Householder, James E Robert Heikki Heinig, Hans P Hethcote, Herbert W Hoffman, Kenneth R Houston, Evan G Huq, Syed A Heinold, Robert D Hetherington, Hoffman, Michael E Houston, Raymond S Hurd, Spencer Peyton Heinrich, Katherine Richard G Hoffman, Michael J Howard, Edgar J Hurder, Steven E Heins, Albert E Heuer, Gerald A Hoffman, William C Howard, Fredric T Hurley, Donal J Heins, Maurice H Heumos, Michael J Hoffmann, Banesh Howard, Louis N Hurley, James F Heinze, Joachim Heuser, Harro G Hoffmann-Jorgensen, Howard, Paul E Hurley, Michael G Heinzer, Martin !\ Heuvers. Konrad John Jorgen Howard, William A Hurley, Susan E

532 ICM-86 (Continued)

Hurst, C A Jablow, Eric Robert Johnsen, Eugene C Jungreis, Irwin L Kartsatos, Hurvich, Clifford M Jabon, David C Johnsen, Ronald L Jurca, Dan Athanassios G Hurwitz, R Daniel Jackins, Harvey Johnson, Charles N Jurzysta, Sonja H Karube, Takashi Hurwitz, Solomon Jackson, Howard L Johnson, Justice, James H Karush, William Husain, Taqdir Jackson, Lloyd K D Randolph Jr Justus, Charles H Kascic, Michael J Jr Husch, Lawrence S Jackson, Stanley B Johnson, Dale Martin Kasriel, Robert H Hussey, Patricia Ann Jackson, William D Johnson, Daniel P Kaashoek, Marinus A Kato, Goro C Kato, Masakimi Hutchinson, George A Jaco, William H Johnson, Kabak, Bertram S Kato, Shin-ichi Hutchinson, Joan P Jacob, Henry G Jr David Copeland KAdas, Jacob, John Phillip Johnson, David H Zsuzsanna Margit Kato, Takao Hutchinson, John J Jacob, Niels Johnson, David L Kadlecek, David Kato, Tosio Hutson, Holmes L Jacobowitz, Howard J Johnson, David Lewis Kahan, William M Katok, Anatole Hutton, Edward L Jacobs, Felix J Johnson, Diane Louise Kahane, Jean-Pierre Katok, Svetlana R Huynh, Quyen Q Jacobs, Harold Johnson, Diane M Kahaner, Katz, Andrew Hveberg, Jacobs, Jonathan M Johnson, Donald G David Kenneth Katz, Victor J Steinar Magne Jacobs, Marc Q Johnson, Eugene W Kahn, Daniel S Katzen, Dahn Hvidsten, Michael D Jacobsen, Martin Johnson, Kahn, Jeff N Katznelson, Y Hwang, Jang C Jacobson, David Gene Douglas Kahn, Peter J Kauffman, Louis H Hwang, Jacobson, Marc Erik Johnson, Gerald W Kahn, Steven M Kauffman, Robert M John Jew-Chen Jacquet, Herve M Johnson, Guy Jr Kailath, Thomas Kaufman, Allan N Hyers, Donald H Jaffa, Robert E Johnson, James E Kajiwara, Joji Kaufman, Arthur Hyllengren, Anders R Jaffe, Arthur M Johnson, Jean T Kakihara, Yuichiro Kaufman, Edwin H Jr Hyvonen, Jaakko P Jaffe, Martha A Johnson, Jon L Kakiichi, Yoshiaki Kaufman, William E Jaffe, William J Johnson, Joseph F Kalai, Ehud Kaufmann-Bubier, Iarrobino, Jager, H Johnson, Joseph L Jr Kalantari, !raj Walter Anthony A Jr Jahren, Bjorn Johnson, Keith P Kalia, R N Kaup, Burchard Kaup, Wilhelm Ibele, Charles E Jain, M K Johnson, Kenneth D Kalisch, Gerhard K Kawai, Takahiro Ibrahim, A Shawky Jain, Naresh C Johnson, Kall, Peter Jakobsen, Kenneth Walter Kawamata, Yujiro lei, Yutaka Kalliongis, John E Hans Plesner Johnson, P B Kay, David C Igarashi, Akira Kallman, Ralph A Jambois, Thomas F Johnson, Peter M Kallstrom, Anders Kay, Leslie D Igarashi, Masao James, David M Johnson, R Warren Kazamaki, Norihiko Igusa, Jun-Ichi Kalton, N J James, Donald G Johnson, Kaman, Charles H Kazdan, Jerry L Iha, Franklin T Kearfott, R Baker James, Ralph L Robert Shepard Kamejima, Kohji !bringer, Thomas Kearn, Vickie H James, Robert C Johnson, Roberta F Kamel, Hyman Iino, Riichi Kearney, Robert D Janik, Jacqueline C Johnson, Roy A Kamishima, Iio, Tutomu Kearton, Cherry Jankins, Mark Johnson, Russell A Yoshinobou Ijjasz Vasquez, Ede J Johnson, Samuel B Keating, Kevin P Jane, James P Kamiya, Shigeyasu Jansen, Willem G Johnson, William B Keating, ME Ikebe, Teruo Kammler, David W Janson, Barbara J Keating, Maria C Ikeda, Kazumasa Johnston, Ernest R Kamowitz, Herbert M Janssen, Robert D Johnstone, lain M Keaty, John Michael Ikenaga, Bruce M Kamp, William P Jantosciak, James S Johnstone, P T Keeler, Stephen P Illman, SOren A Kamran, Niky Janusz, Gerald J Joichi, James T Keenan, ImHof, Kan, Charn-Huen Janwa, Heera Lal Jonah, David W Daniel Macrae Hans Christoph Kanamori, Akihiro Jara Martinez, Jones, Alfredo R Keeping, Anthony J Im, Chang Ku Kanda, Mamoru Pascual Jones, Carl Hilton Kegel, Otto H Imai, Hideo Kaneda, Masaharu Jarchow, Hans Jones, Eleanor Green Kehayopulu, Niovi Imam, Ibrahim N Kaneko, Akira Jarden, Moshe Jones, F Burton Keiding, Hans Imrich, Wilfried Kaneko, Hideaki Jardine, Jones, Gary D Keisler, James E Kanemitsu, Shigeru Inaba, Hiroshi John Frederick Jones, Gordon C Keisler, Michael Kanenobu, Taizo Indlekofer, K H Jarvi, Pentti K Jones, Harold T Keldys, Ljudmila Kaneyuki, Soji Infante, Ettore Jarvinen, Richard D Jones, James P Kelingos, John A Kang, Ming-Chang Infante, Ronald P Jarvis, Trevor M Jones, Jeffrey Charles Kellar, Margaret E Kani, Ernst J Ingraham, Edward C Jategaonkar, Arun V Jones, Jeffrey P Keller, Arthur M Kaniuth, Eberhard Ingram, John Michael Javor, James A Jones, John Jr Keller, Hans H Kann, Edgar D Ingram, Nancy Jane Jayne, John W Jones, Peter R Keller, Herbert B Ingram, Steven K Jea, Hae-Pyng Jones, Peter Wilcox Kannan, Ravindran Keller, Joseph B Kanner, Richard A Innami, Nobuhiro Jea, Kang C Jones, Theodore W Keller, Wilfrid Kanno, Tsuneo Insel, Arnold J Jean, Michel Jones, Vaughan F R Kelley, AI Kana, Takeshi Ioakimidis, N I Jeanquartier, Pierre Jones, William A Jr Kelley, Walter Gill Kanold, Hans-Joachim Iorio, Rafael J Jr Jech, Thomas J Jones, William R Kellogg, Bruce Kantorovitz, Shmuel Kellogg, Charles N Ipina, Lynne Kamstra Jeffrey, Alan Jones, William R Kao, Calvin C Kelly, G Maxwell Irving, Ronald S Jeggle, Hansgeorg Jongen, Hubertus Th Jeltsch, Rolf Jongmans, Francois Kaper, Hans G Kelly, John B Irwin, Robert C Jenkins, Joe Wiley Jonker, Leo B Kaplan, Alan Kelly, John Clive Isaac, Richard E Jenkins, Jonsdottir, Kaplan, Harold M Kemeny, John G Isaacs, Godfrey L Theodore Allen Kristin Halla Kaplan, Stanley Kemp, Isaacs, I Martin Jenness, Jeffrey W Jonsson, Bjarni Kaplan, Wilfred Daniel Campbell Isaacson, Eli Jennings, George A Jonsson, Thordur Kaplansky, Irving Kemp, Marilyn J Isaacson, Eugene Jensen, Arne Jordan, G Samuel Kappe, Kempachiro, Ohashi Isard, Stephen D Jensen, Carey M Jordan; Kirk E Luise-Charlotte Kemperman, J H B Isbell, John R Jensen, Gary R Jordan, Sister Mary Kappe, Wolfgang P Kempf, George R Ishibashi, Makoto Jenson, Richard Alan Jorgensen, Murray A Karal, Frank C Jr Kendall, Richard P Ishihara, Tadashige Jepsen, Charles H Jorgensen, P E T Karamatsu, Yoshikazu Kengmana, Thongchai Ishii, Jyun Jerison, David Jorgensen, Roy A Jr Karamzadeh, 0 A S Kenig, Carlos E Ishikawa, Hirofumi Jerison, Meyer Joris, Henri E Karch, Paul T Kennedy, Edward S Ishimoto, Hiroyasu Jerome, Carlos Joseph, James E Karel, Martin Lewis Kennedy, Judy Anita Iskander, A wad A Jerome, Joseph W Josephson, Karle, Jerome Kennedy, Julia P Karlof, John K Ismail, Mourad Jeroslow, R G William Allen Kenner, Paul M Karlovitz, Les A El-Houssieny Jerrard, Richard P Josephy, R Michael Kenney, Charles S Karnik, Kennison, John F Ito, Noboru J'erri, Abdul Jabbar Joshi, Padmini T Subhash Moreshwar Kenschaft, Patricia C ItO, SeizO Jessen, Borge Jou, S S Jha, V Joyal, Andre Karoubi, Max Kent, Clement F Itzkowitz, Gerald L Jiang, Boju Joyner, David W Karpilovsky, G Kent, George A Iverson, I B Jimenez, Jose Alfredo Juberg, Richard K Karpinski, Marek Kent, J T Iwanaga, Yasuo Jockusch, Carl G Jr Jumper, Gregory S Karr, Alan F Kent, Joseph F Iwasawa, Kenkichi Jodeit, Max A Jr Jun, Byung Moon Karras, Ulrich Kent, Mark D Iwata, Koichi Johanson, Arnold A Jun, Kil Woung Karreman, Herman F Kenyon, Hewitt Ize, Jorge A Johansson, Bo Ingvar J ungck, Gerald F Karren berg, Keown, Nancy A Izuchi, Keiji John, David J Junghenn, Hugo D Frederick G Kerckhove, Izumi, Shin-Ichi John, Fritz Jungnickel, Dieter Karrer, Guido Michael George

533 ICM-86 (Continued)

Kerman, R A Kirch, Allan M Kodama, Yuji Kramer, Robert J Kustin, Kernast, John C Kirchbraber, Urs Koditschek, Kramer, Thomas R Andrew Richard Kerr, Anton G Kirchherr, Walter W Daniel Eliezar Kranjc, Marko Kusunoki, Yukio Kerr, Sandria N Kirk, David B Koehler, Anne B Krantz, Steven George Kuttler, Kersting, Gary Steven Kirk, Joe E Jr Koehler, T L Kranzer, Herbert C Kenneth Latimer Kervaire, Michel A Kirk, William A Koellner, William G Krasner, Marc Kutzko, Philip C Kerzman, Norberta L Kirmayer, Greg A Koethe, Gottfried M Krasny, Robert Kuusalo, Tapani J Kesten, Harry Kirmser, P G Koether, Robb T Kraus, Jon E Kuyk, Willem Kestenband, Barbu C Kirsch, Werner Kogelman, Stanley Krause, Ralph M Kwak, Edward K Kettner, James E Kirwan, Frances C Koh, Jee Heub Krause, Ulrich Kwak, Hyo Chul Key, Eric Stephen Kirwan, William E II Koh, Kwangil Kraut, Gertrud L Kwak, Nosup Keyfitz, Barbara L Kiselman, Christer 0 Kohler, Peter Kravvaritis, D Kwon, Do-Ha Keys, Charles David Kiser, Terry Lee Kohls, Carl W Krebs, Neil E Kwun, Kyung W Keys, Robert Gene Kishi, Masanori Kohn, Joseph J Kreider, Donald L Kydoniefs, Anastasios Khadr, Amr S Kishore, Masao Kahn, Robert V Krein, Mark G Kye, Seung-Hyeok Khajah, H G Kister, James KOhnen, Walter Krein, Kyner, Walter T Khalifah, M F Kister, Jane Koizumi, Shoji Selim Grigorjevitch Kyrouz, Thomas J Khalimsky, Kita, Hiroo Kojima, Sadayoahi Kreith, Kurt Kythe, Prem K Efim Davidovich Kitada, Hitoshi Kokoska, Stephen M Krener, Arthur J Kyuno, Shoji Khan, Donald W Kitano, Koh-Ichi Kolaitis, Phocion G Krengel, Ulrich Khan, Liaqat Ali Kitchen, Joseph W Jr KOlbig, Kurt Siegfried Kress, Michael E LaBorde, Hasell T Khan, Mizan R Kittaneh, Fuad A Kolchin, E R Kreyszig, Erwin 0 LaBudde, Christian D Khan, Kittappa, R Kit Kolesar, R J Krieger, Henry A LaSalle, Margaret M Mohammad Ahmad Kittel, Phyllis M Kolesnik, Grigori Kriegman, David A Labbe, Marcel Adrien Kharaghani, Hadi Kiyek, K Kolk, Johan A C Kriegsman, Helen F Labelle, Jacques Khavinson, Dima Kiyohara, Mineo Kollar, Janos Krinik, Alan Cary Labrousse, Khazal, Klaasen, Gene A Koller, Herbert Krishnamoorthy, Jean-Philippe Reya.dh Rashed Klainerman, Sergiu Kolman, Bernard Mukkai S Laca, Marcelo E Khor, Hoe Peng Klambauer, Gabriel Kalmus, Peter A Krishnamurthy, Lacampagne, Carole B Khosraviyani, Firooz Klamkin, Murray S Kolodner, Ignace I Visvanatha Lachance, Kibbey, DE Klass, Michael J Koltracht, I Kristensen, Leif Michael Anthony Kichenassamy, Klassen, Vyron M Komanska, Kristiansen, Lacher, R C Satyanad Klatman, David S Henryka Krystyna Gundorph K Lachlan, Alistair H Kidwaii, Hariss Klawe, Komatsu, Hikosaburo Krogstad, Harald E Lacomba, Ernesto A Kidwell, Mark E Maria Margaret Komatu, Yusaku Kronheimer, Erwin H Lada, Thomas J Kiefer, James E Klee, Victor L Jr Komiya, Yurin Kronstein, Karl M Ladde, Gangaram S Kielczewski, Zbigniew Kleiman, Steven L Komm, Horace Kropp, Karl V Laderman, Kiernan, Bryce M Klein, Abraham Komornicki, Wojciech Krumpholz, Gary R Julian David Kieval, Harry S Klein, Benjamin G Laduke, Jeanne KOmura, Takaka Kruskal, Martin D Kifer, Yuri Klein, David M Ladwig, John A Kon, Mark Andrew Krystock, Kilgour, D Marc Klein, Edwin M Kong, Man Cheong Robert Lawrence Laffey, Thomas J Kilhefner, Dale Z Kleiner, Lafontane, Jacques Konhauser, Ku, Hsu-Tung Killeen, John Alexander F J r Lagarias, Jeffrey C Joseph DE Kubelka, Richard P Killgrove, Raymond B Kleiner, Mark Konig, Heinz J Lagnese, John E Kubert, Daniel S Kilpi, Yrjo 0 Kleinerman, Samuel Konvisser, Marc W Laha, Radha G Kubo, Fumio Kiltinen, John 0 Kleinfeld, Erwin Koornwinder, Tom H Lahr, Charles Dwight Kubo, Tadao Kim, Chong-Rae Kleinschmidt, Peter Kopacek, Hal W Laidacker, Michael A Kubota, Yoto Kim, Chao W Kleisli, H Kopell, Nancy J Laird, Cameron B Kuczumow, Tadeusz Kim, Hayon Kleitman, Daniel J Kopperman, Ralph D Lakshmanan, N eel a Kudla, Stephen S Kim, Hong Jong Kleppner, Adam Koni.nyi, Adam Lakshmikantham, V Kudo, Tatsuji Kim, Hyuk Kletzing, Dennis R Korenblum, Boris Lam, Siu-Por Kuebler, Roy R Jr Kim, In Su Kline, Robert M Korevaar, Jacob Lam, Tsit-Yuen Kueker, David W Kim, Jin Bai Klingen, Helmut Korevaar, Lamb, Charles W Kuelbs, James D Kim, Jong-Chul Klingsberg, Paul R Nicholas Jacob Lambarth, David A Kuga, Michio Kim, Jongsik Kliorys, Koriyama, Akira Lambek, Joachim Kugler, Lawrence D Kim, Joong Ho Constantine K Korsmo, Eric Paul Lambert, Harvey W Kuhlmann, Norbert Kim, Kyongsoo Kloeppel, Peter K Kortanek, Kenneth 0 Lambert, Joseph M Kim, Kyung Hwa Klopp, Howard D Korzeniowski, Andrzej Kuhn, Nicholas J Lambert, Richard W Kim, Sang Og Klopsch, H Peter Kosarew, Siegmund Kuhn, Robert M Lambert, Robert J Kim, Soon-Kyu Knapp, Anthony W Kosinski, Antoni A Kuiken, Kathryn Lamken, Esther R Kim, Sung-Ock Knapp, Wolfgang D Kosier, Karl Andrew Kuiper, Hendrik J Lampe, William A Kim, Wan Se Knaust, Helmut Koslowski, Juergen Kuiper, Nicolaas H Lampone, Leo W Kim, Woo Jong Kneece, Roland R Jr Koss, Walter E Kuku, A 0 Lancaster, Peter Kim, Young Joan Knight, Frank B Kossack, C R Kulp, Daniel Teibi Landau, Susan Kim, Young-Kook Knight, Julia Kossack, Carl F Kumagai, Donna J Landauer, Christopher Kimber, John E Jr Knight, Lyman C Jr Kossowski, Marek B Kumahara, Keisaku Landers, Billy L Kime, Katherine A Knill, Ronald J Kostenbauder, Kumar, A Anil Landin, Joseph Kimme, Ernest G Knoebel, Arthur Adnah G Kumasawa, Nin Landman, Kimmel, John Edward Knoerr, Alan P Kotiuga, P Robert Kumm, Elizabeth A Bruce Michael Kimura, Kyoko Knopf, Peter M Kottman, Clifford A Kummer, Dawn M Landrock, Peter Kimura, Tatsuo Knopp, Marvin I Kottwitz, Kummer, Martin P Landweber, Peter S Kincaid, Wilfred M Knorr, K R K Robert Edward Kunen, Kenneth Lane, Norman D Kinderlehrer, David S Knowles, J David Katzen, Marshall J Kunle, Heinz Lane, Richard N Kindler, Philippe Knowles, R J Kountz, Linnea M Kunz, Ernst A Lanford, Oscar E III King, Alan J Knox, Nathaniel Kourouniotis, Christos Kunze, Ray A Lang, George E Jr King, Amy C Knuth, Donald E Kovacs, Anton Amos Kiinzle, Hans Peter Lang, Patrick M King, Donald R Knutson, Donald I Kovc\tfk, Zdislav V Kuo, Tzee Char Lang, Serge King, Henry C Ko, Hai-Ping Kovesi-Domokos, S Kupferwasser, Marcelo Lang, William E King, James A Ko, Ki Hyoung Kowtha, Vijayanand C Kupka, Ivan Langebartel, Ray G King, James R Ko, Ohseun Park Kozin, Frank Kuranishi, Masatake Langenhop, Carl E King, L Richardson Ko, Tsz Mei Kra, Irwin Kurata, Yoshiki Langer, Robert W King, Randall R Kobayashi, Shoshichi Krabbe, Gregers L Kurowski, Gary J Langsam, Joseph A Kingston, Albert W Kobayashi, Yuji Kracht, Manfred Kurshan, Robert P Langstaff, John E Kinoshita, Shin 'ichi Kober, Wolfgang Krafft, Olaf W Kuras, Herbert Langston, Kinser, Kocan, Daniel Kraft, Hanspeter Kurtz, Douglas S Michael David Hubert Blevins Koch, Charles F Kraines, David P Kurtz, Stuart A Laning, J H Kioustelidis, John Koch, Richard M Krakowski, Fred Kurtz, Thomas G Lankford, Dallas S Kiraly, Emery Kochendorfer, Anne Krall, Harry L Kurtzke, John F Jr Lanski, Charles P Kirby, A R Kochman, Kramer, Earl S Kurtzman, Lanteri, Antonio Kirby, Mary D Stanley Oscar Kramer, Stephen Joseph Lantz, David C Kirby, Robion C Kodama, Tetsuo Raymond F Jr Kushner, Harvey Lantz, Vicky L

534 ICM-86 (Continued)

Lanza de Cristoforis, Lee, Jyh-Hao Levit, Robert J Liporace, Louis A Love, Thomas Robert M Lee, Ke-Seung Levitt, Nor man J Lipp, D W Loveland, L Duane Lanzano, Paolo Lee, Kotik K Levitt, Tod S Lipps, Lo·vell, John R Lapidot, Eitan Lee, Kyung Bai Levitz, Hilbert Frederick Weissner Lovett, JaneT Lapidus, Arnold Lee, Myrna Pike Levko, John J III Lipschutz, Seymour Lovric, Miroslav Lapidus, Michel L Lee, Peng-Yee Levy, Azriel Lipshitz, Stanley P Lowe, Peter G Lappan, Peter A Jr Lee, Richard T Levy, Gene Lipsich, H David Lowengrub, Morton Lardy, Lawrence J Lee, Ronnie Levy, Joel Lipton, R J Loxton, John Harold Larkin, Martha A Lee, Sa-Ge Levy, Lawrence S List, Stephen E Lu, Chin-Pi Larney, Violet H Lee, Shyling Levy, Paul Lister, William G Lu, Kau-Un Larsen, Kenneth M Lee, Tack-Wang Lewan, Douglas Litchfield, Kay P Lu, Yung Chen Larsen, Soren Lee, Tai-Chung Lewin, Jacques Litherland, Rick Lubin, Clarence I Larson, Edward H Lee, Vincent H Lewin, Mordechai Little, Charles H C Lucas, Billy J Larson, Jean Lee, WillY Lewis, Charles J Little, John B Lucas, Thomas R Larson, Lee M Lee, Yuh-Jia Lewis, Daniel Ralph Littman, Walter Lucas, William F Larson, Loren C Leehey, Patrick Lewis, David W Liu, Chyan-Jyue Luce, R Duncan Larson, Richard G Leemans, H W Lewis, George M Liu, Hongde Luchins, Edith H Larson, Roland E Leep, David B Lewis, Gilbert N Liu, Hsi-Ching Lucht, Lutz Gerhard Larson, Leger, George F Lewis, Houston I Liu, Jong-Diing Lucier, Bradley J Suzanne Lynne Legg, David A Lewis, Hugh L Liu, Tai-Ping Ludden, Gerald D Larue, David M Leggett, Anne M Lewis, Ira Wayne Liu, Tsai-Sheng Ludington, Anne L Lascarides, Leggett, Robert N Jr Lewis, Jeff E Liu, Wei-Nung Ludwig, Donald A Constantine G Legorreta, Leonardo Lewis, John A Liu, Zhishun Ludwig, Garry Lasher, Sim Lehman, R Sherman Lewis, John Howard Liu, Zhong-Dong Lue, Huei-Shyong Lashof, Richard K Lehmer, Derrick H Lewis, L Gaunce Jr Liulevicius, Arunas L Luecking, Daniel H Laska, Michael Lehner, Joseph Lewis, Robert H Livingston, Albert E Luehr, Charles P Laskar, Renu Lehnigk, Siegfried H Lewis, William James Livingston, Charles Lueneburg, E Lassettre, Edwin R Lehr, Marguerite Lewittes, Joseph Livingston, Luersen, Barbara LaszlO, Gabor Lehrer-Ilamed, Y Lewkowicz, Ellen Schmitz Luft, Erhard Latch, Dana May Lehtinen, Matti T Raymond E Livingston, Luh, Jiang Lattin, Christopher Lehto, OlliE Lhee, Byung Su Marilynn L Luk, Hing-Sun Lau, Anthony T Lehto, Pertti Li, Bingxi Livsic, Moshe S Luke, Jonathan Hwa Lau, Chi-Ping Lehtonen, Li, Li-Ge Llave, Rafael de La Chung Lau, William G Ari Taavi Einari Li, Luen-Chau Llewellyn, Mark A Lumer, Gunter Lauchli, Hans Leibowitz, Rochelle Li, Peter W K Llibre Salo, Jaume Luna, Eduardo A Laudenbach, Francois Leichtweiss, Kurt Li, Tien-Yien Llovet, Juan Lundell, Albert T Lauer. Robert M Leighly, Joseph Li, Wen-Ching Winnie Lloyd, Noel G Lundgren, J R Laufer, Henry B Leighton, Liang, David Shien Lloyd, Stuart P Luo, Feng Lauria, Francesco E Frank Thomson Liang, Jaung Lo, Chi Yeung Luskin, Mitchell B Lauritzen, Steffen L Leipholz, Horst Liang, Joseph J Lo, Libo Lustman, Liviu R Laursen, Kjeld B Hermann Eduard Liao, Luby Lo, Tzee-Nan Kuo Lusztig, George Lausch, Hans Leipnik, Roy B Liardet, Pierre Loane, Edward P Luther, C F Laush, George Leith, Cecil E Jr Libby, Richard Allan Loatman, R Bruce Luther, Norman Y Lavalle, Irving H Leitzel, James R C Libera, Richard J Lochs, Gustav E Lutoborski, Adam Lavallee, Lorraine D Leitzel, Joan P Liberman, Zvie Lockhart, Lutterodt, Lavine, David Leitzell, James D Libgober, Anatoly S Deborah Frank Clement Henry Lavine, Richard B Leja, Stanislaw Lichnerowicz, Andre Lockhart, Jody Meyer Lutzer, David J Law, Peter R Lelek, Andrzej Lichtenbaum, Stephen Loday, Jean-Louis Luvisetto, Maria Luisa Lawler, Gregory F Lelong, P J Lichtin, Benjamin L Lodha, Suresh K Luxemburg, Lawniczak, Anna T Lemaire, Luc Lichtman, Alexander Loeb, Henry L Wilhelmus A J Lawrence, Lemay, William H Lick, Dale W Loeb, Peter A Lyche, Tom J Anthony Lee Lemei, Han Lick, Don R Lofgren, Lars Lynch, Robert E Lawrence, John W Lenard, Andrew Lid!, Rudolf Lohuis, David J Lynch, William C Lawson, David F Lengyel, Bela A Lieb, Elliott H Lok, Walter L Lyndon, Roger C Lawson, H Blaine Jr Lenhart, Lieberman, David I Lombardo, Charles J Lyon, Lawson, Walter R Suzanne Marie Lieberman, Gary M Laminae, Thomas C Her bert Chancellor Lawvere, F William Lenihan, Stanley R Lien, Magnhild Lemont, John S Lyons, Carter G Lax, Anne!i Lenstra, H W Jr Liggett, Thomas M Londen, Stig Olof Lyons, Russell D Lax, Peter D Leon, Steven J Light, Christina London, David Lysko, Janusz M Lax, Robert F Leonard, Henry S Jr Light, F W Jr Lonergan, Francis D Lytle, Raymond A Laywine, Charles F Leonard, Isaac E Lighterman, Mark S Long, David Darren Lyubeznik, Lazar, Aldo J Leonard, P Scott Lightner, Michael R Long, Ding-Gwo Gennady M Lazarov, Connor Leonard, Philip A Lim, Chong-Keang Long, John J Jr Lazarsfeld, Robert K Leonardi, Sharon Ann Lima, Asvald Long, John M Ma, Tsoy-Wo Lazarus, Andrew J Leong, Yu Kiang Lin, Charlotte Long, Robert L Ma, Xiaoyun Lazerson, Earl E Lepisto, Timo Valter Lin, Jindra Longo, Roberto Mac Lane, Saunders Le Cam, L M Lepowsky, James I Lin, Pei-Kee Longobardi, Patrizia MacCaull, Wendy A Le, Diing Tr8.ng Lerat, Alain Lin, Qing Longstaff, W E MacCluer, Barbara D LeBrun, Claude R Jr Leray, Jean Lin, Shen Longyear, Judith Q MacDonald, LeMesurier, Lerche, Kenneth D Lin, Wen-Hsiung Loomis, Irene H James Hugh Brenton John Leschen, John G Linamegi, Edmonts G Lopez, Amparo MacDonnell, John J LeVeque, Randall J Leslie, Arthur S Lind, David W Lopez-Carmona, MacDonnell, Joseph F LeVeque, William J Lesmes, Jaime Lind, Douglas A Antonio MacGillivray, Lea, James D Lesniak, Linda Lindberg, John A Jr L6pez-Escobar, Archibald D Leach, Ernest B Leu, Hsiao-Tung Lindemann, Willi Edgar G K MacGregor, Thomas H Leader, Solomon Leung, Anthony W Linden, Carl N Lopp, Kenneth W MacKenzie, Robert E Leahey, William J Leung, Denny Ho-Hon Lindenstrauss, Joram Lord, James M MacLaren, M Donald Leahy, John V Leung, Dominic S P Lindgren, John E Lord, Nicholas J MacNaughton, JohnS Lebedin, Neal D Leung, Kar-Koi Lindquist, W Brent Lorell, Jack MacNeil, Robert L Lebel, Jean E Leung, Tat-Wing Lindsay, John W Lorenz, Dan H MacQueen, David B Lebow, Arnold Levatin, JoAnne Lee Lindstrom, Mikael Lorimer, Macbeath, A M Lebowitz, Joel L Levenberg, Norman Lindstrom, Wendell D Joseph Wilson Macdonald, Ian D Lee, Amos T Levi, Mark Ling, Lott, Fred W Jr Madar lane, Lee, Carl W Levin, Jacob J William Halstead Lou, Shih-Pe Andrew Ian Lee, Cheng-Ming Levin, Richard G Linnell, P A Loud, Warren S Macgibbon, Brenda Lee, Daniel Tai Lik Levine, Daniel S Linsenmeyer, Louhivaara, Ilppo S Mach, Lenka Lee, D6-Won Levine, Eugene Edward C Lounesto, Pertti Machado, Armando Lee, Hung Hwan Levine, Harold I Lions, J L Loustau, John A Machover, Maurice Lee, Jeh Gwon Levine, Howard A Lipkin, Leonard J Louton, Tom Macfas, Roberto A Lee, John M Levine, Jerome P Lipman, Joseph Love, Mack, John E Lee, Joseph Han-Tack Levinger, Bernard W Lipman, Myra A R Sister M Genevieve Mack, John Michael

535 ICM-86 (Continued)

Mackenzie, Kirill Marcus, Steven I Matsumura, Hideyuki McGehee, 0 Carruth Messer, Karen Mackey, Michael C Marde!HC, Sibe Matsunaga, Teru McGehee, Richard P Metakides, George Maclachlan, Colin Margolis, Stuart W Matthews, Jane McGibbon, Charles A Metas, Nick Macphail, Moray S Ma.rgosian, Zaven Matthies, Karl H McGovern, Richard J Metzler, Richard C Macpherson, Robert D Margulies, William Matti, Joseph T McGovern, William M Metzner, Ellen L Madan, Manohar L Maria, May H Mattingly, McGuigan, Robert A Meuser, Diane Mae Madan, Ved P Marin, Francisco L Robert Bruce McGuinness, Mewing, Robert Maddux, Roger D Marin, Irwin C Mattson, Leroy T Francis Oisin Meyer, Burnett C Mader, Adolf Marion, Jean Albert Mattuck, Arthur P Mcintosh, Harold V Meyer, Donald V Madison, Bernard L Markanda, Raj K Mau, Mary Sue Mcintosh, William D Meyer, Gerard G L Madow, William G Marker, David E Maurer, Donald E McKay, James H Meyer, Herman Madsen, Gregory W Markert, Matthew J Maurer, Peter M McKay, Thomas G Meyer, Jean-Pierre G Madsen, Ib H Markin, Jack T Maurer, Stephen B McKean, Henry P Jr Meyer, Kenneth R Maecher, John H P Markoff, Nicholas G Mawhin, Jean L McKee, Timothy W Meyer, Norman C Jr Maeda, Fumi-Yuki Markot, Robert P Mawyer, Farley McKelvey, Robert W Meyer, Paul R Maeda, Michie Markowsky, George Maxfield, John E McKenzie, Lionel W Meyer, Victor Arthur Magee, Gordon R Markus, Lawrence Maxson, Carlton J McKenzie, Ralph N Meyer, W Weston Magenes, Enrico Marie, Maxwell, James W McKeon, James P Meyer, Wolfgang T Magid, Martin A Charles Michael May, Coy L McKibben, William P Meyerhoff, Robert Magill, Kenneth D Jr Marley, Anthony A J May, E Lee Jr McKinney, Richard L Meyerowitz, Aaron D Magnanini, Rolando Marley, Gerald C May, J Peter McKinney, Wilbur T Meyers, Charles Frank Magnus, Arne Marley, Thomas J May, Michael K McLaughlin, Donald E Mez, Hans-Christian Magnus, Wilhelm Marlowe, Thomas J Jr May, Warren L McLaughlin, Harry W Mezzino, Michael J Jr Magnuson, Marotto, Frederick R Mayans, Robert A McLaughlin, Jack E Miao, Lung-Chi Alan William Marques, Carlos A Mayberry, John P McLaughlin, Renate Michael, Ernest A Mah, Peter F Marr, Robert B Mayer, Dieter H McLaughlin, T G Michael, Mark Mahdavianary, Marrone, Lydia B Mayer, John Clyde McLeod, Edward B Jr Michael, William A Seyed Kazem Marrone, Pedro A Mayer, Karl Heinz McLeod, Robert M Michaels, Brenda Maher, Mary Anne Marsden, Jerrold E Mayer, Meinhard E McMahon, Douglas C Michel, Rene Mahler, Kurt Marsh, Marcus M Mayer, Raymond A J r McMahon, Edith Mary Michel, Russell J Mahoney, Anne Marshall, Bernard P Mayor, John R McMillan, Audrey W de-Michele, Leonede Mahowald, Mark Marshall, David Imler Mazhar, Syed M McMillan, Brockway Michiaki, Watanabe Maiellaro, Michele Marshall, Maziarz, E A McMillan, Daniel R Jr Michler, G Maier, Robert S Murray Angus Mazumdiir, Tiipas. McMillan, Evelyn R Michon, Gerard Ph Mair, Bernard A Marstrand, John M Mazur, Barry McMillen, Mickens, Ronald E Mairhuber, John C Martelli, Mazzeo, Rafe R William Robert Middleton, David Maj, Mercede Mario Umberto Mazziotto, Gerald McMinn, Trevor J Middleton, Majda, George J Martens, Henrik H McAdam, Stephen J McMullen, P Jeffrey Blake Majeed, Abdul Marti. Jurg T McArthur, C W McMurtrie, Kenneth A Miel, George Majumdar, Samir R Martin, E Dale McAuley, Louis F McNab, Robert Mielke, Marvin V Makar, Boshra H Martin, George E McBrien, Vincent 0 McNeill, Robert B Mielke, Paul T Makar-Limanov, L G Martin, Hayden Garth McBroome, Greg McNulty, George F Mientka, Walter E Makepeace, Coline M Martin, James E McCall, James D Jr McNulty, Miesner, Walter Maki, Daniel P Martin, John R McCallum, William G Nieves Austria Mignot, Fulbert Makowsky, Martin, Joseph E McCarthy, Charles A McNulty, Sallie Keller Mihalik, Michael L Johann Andreas Martin, Margaret A McCarthy, Donald J McOwen, Robert C Miklos, Dezso Malafronte, Thomas A Martin, Nathaniel F G McCarthy, Edward D McRae, D George MikoliC-Torreira, Igor Malcolmson, Peter Martin, Nigel McCarthy, John McShane, Edward J Mikusinski. P Maleki, Amir Martin, Norman M McCarthy, John David McWilliams, Ralph D Milcetich, John G Malitz, Jerome I Martin, Peter E McCarthy, Paul J Meacham, Robert C Miles, E P Jr Malkevitch, Joseph Martin, Robert H Jr McCarty, Carl P Mead, David G Miles, Joseph B Mallach, Lawrence E Martin, William T McCarty, George S Meadows, Miles, Philip E Mallet-Paret, John J Martindale, McCarty, Kevin P Douglas Shelby Miles, Thomas J Malliaris, Tassos G Wallace S III McCasland, Roy Lynn Meadows, Lawrence F Milgram, Martin Malon, David M Martinez, Jorge McCaughan, Dennis J Meagher, Jack R MilifiC, Dragan Malone, J J Martinez, Rafael A McCleary, John H Meakin, John C Millar. Robert F Maloof, Giles W Martins, Joyce A McClenahan, Mears, Florence M Millar, T S Maltese, George J Martonak, Stephen H Maurice D Meder, Albert E Jr Miller, Dale A Malvern, Lawrence E Marty, Roger H McClish, Wayne E Medhin, N egash G Miller, Douglas E Mamelak, Joseph S Marumoto, Yoshihiko McConnell, John C Meeker, Loren David Miller, Forrest R Man, Chi-Sing Maruyama, Toru McConnell, Tanya D Megginson, Miller, Haynes R Mand, Lawrence R Marvin, John W McConnell, Terry R Robert Eugene Miller, Hugh D B Mandelbrot, Benoit B Marzuq, Maher M H McConnell, Thomas Mehlman, Marc H Miller, Jack M Manfredi, Juan J Masani, Pesi R McCormick, Ferris E Mehrmann, Volker Miller, James Curtis Manis, Merle E Mascioni, Vania D McCoy, Daisy C Meilan, Gerard Miller, John C Manjarrez, Victor M Masha, Edward V McCoy, Robert A Meisner, Morris J Miller, John G Mann, Benjamin M Mashburn, Joe D McCoy, Thomas L Meister, Erhard V Miller, Kenneth S Mann, Henry B Maskit, Bernard McCray, Patrick D Meixner, Th Miller, Matthew Mann, W Robert Maslanka, McCready, Robert R Mejia, Raymond Miller, Richard Alan Mann-N achbar, David Joseph McCrory, Clint G Mejlbo, Lars C Miller, Richard K Pauline Maslennikov, M V McCulloh, Leon R Melin, Anders E Miller, Richard R Manogue, Joseph F Mason, Ian McCullough, Darryl Mellender, James W Miller, Stephen S Manougian, Edward Massa, Silvio McCurley, Kevin Snow Melrose, Richard B Miller, Victor S Mansfield, Maynard J Mass am, Helene McDaniel, Andrew Melzi, Giovanni Miller, W F Manske, Brian K Massell, Paul B McDivett, Robert D Memory, Margaret C Miller, Willard Jr Mantel, Francis Masser, D W McDonald, Bernard R Menconi, Mario Miller, William David Mantel, Harold John Massey, Fred A McDonald, Gerard Mendelson, Elliott Miller, William V Mantel, Jeffrey Hunt Masten, Jeffrey Paul McDonald, John N Mendelssohn, Roy Miller, Zevi Mantini, Lisa Masters, Harold U McDonald, Mendoza, Millett, Kenneth C Mapa, Felina G Masur, Howard A Katherine J Charles Gerard Millman, Richard S Marathe, Kishore B Matchett, Andrew J McDonough, J M Mercier, Armel N Mills, Harlan D Marble, Robert P Mate, Attila McDowell, Kenneth P Meredith, David B Mills, Janet E Marchand, Margaret 0 Mather, John Norman McDuff, Dusa Merkel, Rudolph B Mills, Stuart E Marchetti, Federico Matheson, John D McEliece, Robert J Merriell, David M Mills, William H Marchiafava, Stefano Mathis, Darrell L McElroy, Betty K Merrien, Jean Milman, Mario Marcou, R J Mathison, Neil T McFaden, Harold H Merrill, Milman, V D Marcum, Howard J Mathsen, Ronald M McFarland, Thomas L Kathy Donovan Milne, Stephen C Marcus, David J Matijevic, Jacob R McGaffey. Thomas N Merryfield, Kent G Milner, Eric C Marcus, Leo G Matsuda, Shigeo McGarry, Dorothy Mertens, John R J r Milner, Fabio A Marcus, Morel M Matsui, Kiyoshi McGavran, Dennis R Meserve, Bruce E Miloslavsky, G C Marcus, Philip S Matsumoto. Yukio McGee, Robert I Mesirov, Jill P MilutinoviC, Uroi!l

536 ICM-86 {Continued}

Mimura, Mamoru Montgomery, Philip R Mozzochi, C J Nance, Dana W Nichols, John C Min, Kang·Joo Montgomery, Mrowka, Stanislaw G Nandakumar, N R Nicholson, Victor A Min-oo, Maung W Duane Mrowka, Tomasz S Nanzetta, Philip Nicholson, W Keith Minagawa, Takizo Month, Leslie A Muckenhoupt, Napolitano, Nickel, James A Mine, Henryk Monticone, Leone C Benjamin Margaret M Nickerson, Helen K Minda, C David Moody, Robert V Muder, Douglas J Narasimhan, Revathi Nico, William R Minehart, Moon, Kyung-Ho Mueller, Bruno J W Narayanaswami, P P Nicol, Sherrie Jean Deborah Fairley Mooney, Charles 0 Mueller, Carl Douglas Nariboli, Gundo A Nicolas, Jean-Louis Minemura, Katsuhiro Mooney, John J Mueller, James R N arushima, Hiroshi Nicolas-Carrizosa, Mines, Ray Moons, Theodoor M Mueller, Julia Nash, David Alfredo Miniowitz, Ruth Moore, Alan L Mugele, Raymond A Nash, Stephen G Niederhauaen, Minkus, Jerome B Moore, Hal G Muhly, PaulS Nash-Williams, Heinrich Minsker, Steven Moore, John D Mulase, Motohico Crispin St J A Niedzwecki, G P Mintel, Ernest J Moore, John G Mulay, Shashikant B Nastold, Nielsen, Peter A Minton, Roland B Moore, Muldoon, Mark Hans-Joachim Nievergelt, Yves Miranda, Rick Judy Hennessey Mulflur, James H Nath, Radha G Niino, Kiyoshi Mirchandaney, Moore, Lawrence C Jr Muller, George M Nathanson, Melvyn B Nijenhuis, Albert Arjan Sobhraj Moore, Marion E Muller, Gert H Nation, James B Nirenberg, Louis Mirsky, NormanD Moore, Marvin G Mullikin, Thomas W Naugle, Nishida, Takaaki Mishoe, Luna I Moore, Polly Mullin, Albert A Norman Wakefield Nishiura, Eizo Mishra, Sudhakara Moore, Richard A Mullins, Sarah E Navarro, Francisco J Nishiura, Togo Mislin, Guido Moore, Robert H Mullis, James W Jr Nayak, M K Nissim, Leonard J Misra, A K Moore, Robert T Mullis, Nayar, Brij M Nitecki, Zbigniew H Misra, Prabudh Ram Moore, W Keith Robert Harvey Jr Naylor, Lawrence W Niven, B S Mital, J N Moore, William E Mulrow, Edward J Nduka, A Niven, Ivan Mitchell, Alan R Moori, Jamshid Mulry-Liggan, Mary H Nebres, Bienvenido F Njastad, Olav Mitchell, Emerson C Mora, Xavier Mumford, David Needham, C Sam NObauer, Wilfried H Mitchell, George E Morales, Pedro A Munoz, Edgar Neff, John D Noble, Andrewa R Mitchell, James C Moran, Charles W Munoz, Miguel-C Neff, Mary M Noble, William Mitchell, John W Moran, William Murakami, Shingo Neggers, Jospeh Nobusawa, Nobuo Mitchell, Josephine M Morawetz, Cathleen S Murali, V Negro, Angelo L Noel, Guy G J Mitchell, Roger W Mordeson, John N Murasugi, Kunio Negron, Pablo Victor Noguchi, Hiroshi Mitchell, Moreau, Jean Jacques Murata, Hiroshi Neher, Erhard Noguchi, Junjiro Stephen Ames Moreland, Robert A Murayama, John T Nebs, Robert M N ohel, John A Mitchell, Stephen G Moretz, Marjorie A Murphy, Brian B Neidhardt, Arnold L Nolan, John P Mitchem, John A Morey, Philip S Jr Murphy, Catherine M Neidhardt, Wayne L Nold, Annett Mitrovic, Dragisa Morgan, Christopher L Murphy, Donald P Nel, Louis D Noll, Landon Curt Mittal, D M Morgan, Frank Murphy, Grattan P Nelimarkka, Esa Nomizu, Katsumi Mittenthal, Lothrop Morgan, Raymond Jr Murphy, Nelligan, John D Noonburg, Virginia A Morgantini, Edmondo Mityagin, Boris Katherine Adele Nelms, Dorothy N oor, Khalida Inayat Miura, Robert M Morimoto, Akihiko Murphy, Paul F Nelson, Edward Nordgren, Eric A Miyadera, Isao Morley, Larry J Murphy, R B Nelson, Edward 0 Norfleet, Jesse Peter Miyahara, Yoshio Morrel, Bernard B Murphy, Thomas N Nelson, George C Norman, Miyazaki, Ken-Ichi Morrel, Judith H Murray, Alan G Nelson, James J r Francis Alexander Miyoshi, Takeo Morrill, John E Murray, G Graham Nelson, Larry D Norman, Peter Mizel, Victor J Morrie, Alun 0 Murray, Nelson, Roger B N .rman, R Daniel Mizumoto, Hisao Morris, Chester R Margaret AM Nelson, Stuart A Normand, Gerard L Mizuno, Hirobumi Morris, Joseph R Mur•e, Jacob P Nelson, William Norris, Douglas A Mlitz, Rainer Morris, L E Muse, Archie Henry Nemethi, Csaba Northcutt, Robert A Moak, Daniel Stuart Morris, Peter C Muto, Hideo Nesbitt, Cecil J Norton, V Alan Mochizuki, Horace Y Morris, Robert A Myers, Dale W Nesin, Ali H Mochizuki, Leslie Y Morrisette, Thomas M Myers, Earle F Neubauer, Gerhard J Nottrot, Roel Modica, Luciano Morrison, David R Myers, John R Neuberger, John W N ovacky, George A Moeller, Helmut Karl Morrison, John A Myers, Russell W Neugebauer, Nowlin, Benjamin Moen, Courtney H Morrison, JohQ. F Myerson, Gerald Christoph J N owosad, Pedro Moet, H J K Morrison, Kent E Myszewski, Mathew Neumann, Bernhard H Nulton, James D Mohamed, I J Morrison, Michael A Myung, Hyo Chul Neumann, Nummela, Eric C Mohamed, Saad Morrison, Sally D Walter David Nunez, Mohapeloa, Morrison, Tim H Naatanen, Marjatta Neustadter, Pascual Ballesta Khomo T S Morrow, James R Nachbin, Leopoldo Siegfried F Nunke, Ronald J Moise, Edwin E Morse, Burt J Nachman, Louis J Neuwirth, Jerome H Nussbaum, A Edward Moite, Sally M Morse, Jack Neil Nadel, Mark E Neuwirth, Lee P Nussbaum, Roger D Molino, Pierre Morton, Lynda S Nadell, Paul T Nevai, Paul G Nusse, Helena E Moller, Raymond W Moschovakis, Nadler, Edmond J Neveu, Jacques Nutt, Michael D Molloy, C T Joan Rand Naeem, Rana Khalid Newberger, Stuart M Nyman, Philip D Molnar, Joseph R Moschovakis, N agahara, Takasi Newcomb, Lynn T Nymann, Dewayne S Molnar, Richard K Yiannis N Nagai, Osamu Newell, David C Molnar, Moscovici, Henri Nagami, Keio Newhouse, Sheldon E O'Callaghan, Liam J Suzanne Miller Moser, Joseph G Nagarkatte, Umesh P Newlander, August Jr O'Connell, James R Jr Monahan, Maurice L Moser, Jiirgen K N agasaka, Kenj i Newman, Morris O'Connor, Anthony J Mond, Bertram Moser, Lucy I Nagata, 'K Wayne Newton, MervinE O'Donnell, Daniel F Mong, Kai Cheong Moser, William 0 J Nagel, Alexander J Ney, Peter E O'Farrell, Mongelli, Dominic Mosher, James R Nagel, Bengt C H Ng Wong Hing, John Michael Monheit, Lilly A Mosier, Ronald G Nagisetty, Rao V John Dilwyn O'Halloran, Joyce Monk, J Donald Moskowitz, Nagle,- R Kent Ng, Che Tat O'Keefe, David F Monleon-Pradas, Daniel Nat han Ndmpally, Ng, O'Leary, Daniel J Manuel Moskowitz, Martin A Somashekhar A Eugene Kwan Sang O'Meara, 0 Timothy Monma, Clyd~ L Moss, Kenneth N Nairn, Dean C Ng, Ho Kuen O'Neil, Kevin A Monna, Antonie F Moss, Lawrence Stuart Najar, Rudolph M Ng, Kok Lip O'Neil, Philip J Monrad, Ditlev Moss, William F Ill Najarian, John P Ng, Kung-Fu O'Neill, Anne F Monro, Gordon P Mostpw, George D Nakagawa, Kiyokazu Ng, Sze-Kui O'Neill, Bernard V Jr Monsky, Paul H Motamedi, Mansoor Nakagawa, Yoko Ng, Tze-Beng O'Neill, Edward J Monson, Stephen R Motte, David L Nakamura, Masahiro Nguifl'o, B Michel O'Neill, Ronald C Montague, Motto, Michael H Nakamura, Riichiro Nguyen, Hung Thac O'Rourke, Joseph Patricia Tucker de Mottoni, Piero L Nakanishi, Yasutaka Nguyen, Thanh Van O'Shea, Donal B Montaldi, James A Moulden, Trevor H Nakano, Kazumi Nguyen, True T O'Sullivan, John J Montesinos, Jose M Mourant, Wanda J Nakashima, Martin M Nguyen, Van Hien Oates-Williams, Sheila Montgomery, Deane Movahedi-Lankarani, Nakata, Mie Nguyen-Huu-Bong Obaid, Samih A Montgomery, Hugh L Hossein Namba, Kanji Nicholls, Peter J Obata, Morio Montgomery, M Susan Moy, Allen Nambooripad, S K S Nichols, Edward Obeid, Mustafa A Montgomery, Peter L Moyle, Benjamin N Namioka, Isaac Nichols, Eugene D Oberg, Edwin N

537 ICM-86 (Continued)

Oberguggenberger, Ortmey.er, Park, Jong Geun Peligrad, Magda Piccinni, Paolo Michael B William Albright Park, Samuel Pelikan, Stephan Pickel, Paul Frederick Oberlin, Daniel M Ortner, Gene M Park, Sehie Pelin, Alex Pickrell, Douglas M Oberschelp, Walter Osborn, Howard A Park, Young Sao Pell, William H Piech, MAnn Obi, Wilson C Osborn, J Marshall Parker, Elwood G Pelletier, Donald H Piegl, Leslie de las Obras, Carmen Osborn, John E Parker, Francis D Peltier, Charles F Piene, Ragni Obrock, Arthur E Osborn, T A Parker, George D Pence, Dennis D Pier, Jean-Paul Ochoa, Juan C Osborne, Mason S Parker, Phillip E Pence, Sallie E Pierce, John Festus Ochoa-Lions, Osborne, Parker, Thomas H Penenberg, Pierce, Rebecca L Jorge Daniel Michael Robert Parker, Willard A Darryl Neil Pierce, Richard S Ocneanu, Adrian Osborne, Richard P Parks, Harold R Pengelley, David John Piger, Jean Ocone, Daniel L Oslon, Steve G Parlett, Beresford N Penn, Howard Lewis Pignedoli, A Oda, Tadao Osner, H J Parr, James T Pennell, Stephen A Pigno, Louis Odlyzko, Andrew M Osofsky, Barbara L Parrish, Daniel M Penner, Robert Clark Pigozzi, Don L Oehmke, Robert H Osondu, Kevin E Parrish, Herbert C Penney, Carol W Pilgrim, Donald H Oertel, Ulrich Ossa, Erich Parrott, David Penney, David E Pillay, Anand Oertly, Alda F Osserman, Robert Parrott, Mary E Pennisten, John W Pilz, Gunter Oertly, Frederick Jay Ostaszewski, Parry, R Bryce Pepper, Paul M Pinch, Richard Ogawa, Hajimu Krzysztof M Parry, Walter R Percus, Jerome K Gilmour Eric Ogden, Nancy C Osteen, Robert Ernest Parry, William Pereira da Silva, C Pinchen, Didier Ogg, Andrew P Osterburg, James Parsons, Lee Perel, William M Pinkham, Henry C Ogg, Frank C Jr Oster by, Ole Parsons, Torrence D Perez-Gonzalez, Pinkus, Allan M Oguchi, Kunia Osterwalder, Konrad Parter, Seymour V Fernando Pinney, Karen R Oh, Yong-Geun Ostrogorski, Tatjana Parzen, Emanuel Perez de Vargas, Pinsky, Mark A Ohkuma, Tadashi Ota, Clem Z Pas, Michael Francis Alberto Pinsky, Ross George Ohm, Jack E Cta, Sch6ichi Paschke, William L Perkel, Manley Pinter, Joseph Kalman Ohme, Paul A Otermat, Scott C Pasini, Antonio Perkins, Peter Piotrowski, Zbigniew Ohta, Haruto Otter, Richard R Pass, Ralph P III Perko, Lawrence M Pippert, Raymond E Ohtsuka, Makoto Ottmann, Thomas A Passell, Nicholas Perlis, Robert V Piranian, George Ohwaki, Shin-Ichi Oursler, C C Passerini, Henry J Perlman, Sanford Pitcairn, Joel Oikawa, KOtara Ouyang, Chung-Hsing Passman, Donald S Perry, William J Pitcher, Everett Okada, Masami Overdeck, John M Passty, Gregory B Perry, William L Pitt, Loren D Okada, Satio Owa, Shigeyoshi Pastor, David E Persek, Stephen C Pittenger, A 0 Okada, Susumu Owen, David R Pastro, Pier-Ivan Perseus, Jan Pitteri, Mario G Okamoto, Kiyosato Owens, Owen G Pasztor, Ana Persson, Ulf Pittnauer. Franz Okamoto, Sigeru Owings, James C Jr Patera, J Pertica, Horacia N Pitts, Andrew M Okayasu, Takateru Oxley, Theron D Jr Patnaik, Pervin, William J Pitts, Jon T Okoronkwo, Oxtoby, John C Surendra-Nath Pesotan, Hoshang Pixley, Alden F Emmanuel 0 Oyabu, Takashi Patrick, E Vincent Pestana, Dinis Duarte Pixley, Emily C Oksendal, Bernt K Ozols, Vilnis Patrizio, Giorgio Petermann, G E Pixley, Henry H Okuyama, Akihiro Ozsvath, Istvan Patten, Donald L Peters, James Vincent Pixton, Dennis G Oldham, Frank Ernest Patten, Paul R Peters, Klaus Pizer, Arnold K Oldroyd, L Andrew Pacella, Filomena Patton, Charles M Peters, Meinhard H Piziak, Robert Oler, Norman Paciorek, Joseph W Patton, L Raphael Peters, Pizzo, Joseph T Olesen, Packer, Judith A Patty, C Wayne Thomas Joseph Plachky, Detlef Dorte Marianne Pade, Offer Patula, William T Petersen, Bent E Plank, Donald L Oliger, Joseph E Padma, Paul, DeDe Petersen, Gordon M Plano, Richard A Oliker, Vladimir I Narasimhachari Paul, Jerome L Petersen, Karl E Plante, Joseph F Olin, Robert F Pagano, Miriam E Paulling, John M B Petersen, Peter Plaut, Conrad P Oliveira, Page, David A Paulowich, David G Petersen, Troels Plavchak, Tom Joseph Stephen Pai, Chikaung Paulsen, Vern I Petersen, Wesley P Pless, Vera S Oliver, Fatma Paige, Eugene C Jr Paur, Sandra 0 Peterson, Chariya P Plotkin, Neal J Olivier, Reinhard M Paige, Lowell J Pavel, Monique L Peterson, Charles L Plotnick, Daniel H Olmstead, P S Painter, George W Pavlakos, Panaiotis Peterson, Franklin P Plummer, Michael D Olsen, Martin Paisner, Helen L Pavlicek, Glenn H Peterson, Ronald J Plymen, Roger J Olshen, A C Pak, Jingyal Pawelke, Siegfried H R Pethe, S Podio-Guidugli, Paolo Olson, Dwight M Palais, Eleanor G Payne, Fred R Petricka, Russell J Poenaru, Valentin A Olson, Loren D Palais, Richard S Payne, Lawrence E Petro, John Pohl, William F Olson, Pearl Palenz, Diana G P Payne, Stanley E Petro, John W Pohle, Olum, Paul Paley, Hiram Pazy, Am non J Petroni, Ann T Frederick Valentine Olver, Frank W J Palka, Bruce P Pearce, Kent Petryshyn, W V Pokrass, David J Olver, Peter J Palled, Shivappa V Pearcy, Carl M Pettitt, Richard Bruce Polachek, Harry Oman, John Arthur Palm, Andrew C Pears, Alan R Petty, Clinton M Polemarchakis, H Omar, A A H Palmer, Edgar M Pearson, Bennie J Pew, Thomas Vincent Polking, John C Ombe, Hitoshi Palmer, John N Pearson, Kenneth R Pfeffer, Washek F Pollack, Richard Omori, Hideki Palmer, Kenneth J Pearson, Robert W Pfeiffer, Paul E Pollak, Barth Onat, Emin T Palmer, Theodore W Peccati, Lorenzo Pflaum, C W Pollatsek, Harriet S Ong, Sing-Cheong Pan, Ting K Peck, Emily Mann Pflug, Peter Pollingher, Adolf Onishi, Hironori Panchal, Champak D Peck, N Tenney Phadke, Bhalchandra Polly, Mary R Onneweer, Cornelia W Panchapagesan, T V Peck, PaulS Balvant Palombo, Albert Ono, Takashi Pandian, Peck, Pham, Tuan Dinh Polonsky, Ivan P Onose, Hiroshi Ramaiah Devadoss Vincent Christopher Phanord, Dieudonne D Pomerance, Carl Oodaira, Hiroshi Pandit, Onkar N ath Pedersen, Edward A Pheidas, Athanasios C Ponce, Ooms, Alfons I Pantoja-Macari, Pedersen, Flemming P Phelps, Robert R Gustavo Alberto Opfer, Gerhard H Jose Eduardo Pedersen, Franklin D Phili, Christine Melle Pons, Montserrat Opalka, Hans Panzone, Rafael Pedersen, Henrik Philipp, Walter V Pool, James C T Oppenheim, Joseph H Papangelou, Fredos Pedersen, Jean J Phillips, Anthony V Poon, Yat Sun Oprea, John F Papantonopoulou, Pedersen, John F Phillips, Poor, Walter A Ordman, Edward T Aigli Helen Pedersen, Katherine L Christopher K Pop-Stojanovic, Orellana, Pappas, Peter C Pedersen, Phillips, Daniel Zoran R Chacin Mauricio Pappas, Richard C Niels Vigand Phillips, John Gilbert Poritz. Alan B Orihuela, Jose Paradise, Michael E Pedersen, Stig Andur Phillips, Keith L Port, Sidney C Orland, George H Parchment, Osborne C Peek, James M Phillips, Ralph S Porta, Harry J Orlik, Peter P Pardee, Otway O'M Peetre, Jaak Phillips, Veril Porte, Jean Orloff, Daniel Pareek, Pefkaros, Kenneth Phillips, William J Porter, Gerald J Orloff, Leo N Chandra Mohan Peglar, George W Piacentini, Porter, James F Ornstein, Avraham J Pareja-Heredia, Diego Pego, Robert L Giulia Maria Porter, Lawrence D Ornstein, Donald S Parikh, Rohit J Peitgen, Heinz 0 Piacenza, Robert John Posey, Eldon E Ornstein, Wilhelm Paris, Alan Terry Peixoto, Mauricio M Piatetski-Shapiro, I Pass, Richard L Orated, Bent Park, Chull Peleg, Bezalel Picard, Rainer H Post, Karel A Ortiz-Franco, Luis Park, Jae Keol Peletier, Lambertus A Piccinini, Renzo A Postema, Dale

538 ICM-86 (Continued)

Pothoven, Kenneth L Rabinowitz, Paul H Reese, Matthias F Richmond, Glayvera Rochberg, Richard H Potts, Donald H Rabinowitz, Stanley Reese, Richard H Richmond, Jean B Roche, Claude Poulin, J Rabung, John R Reese, Sylvester Richmond, Rockland, Charles Poulsen, Ebbe T Racine, Michel L Reeves, Roy F Lawrence B Rockwell, Richard D Pour-El, Marian B Radin, Lon Berk Reeves-Hicks, Richter, Gary Hardin Rod, David L Pourahmadi, Mohsen Radjavi, Heydar Ernestine Richter, Guenther E Rodin, Burton Pourciau, Radzikowski, Pawel Regan, Francis Richter, Michael M Rodino, Luigi Bruce Hunter Raggio, Guido A Rehder, Wulf Richtmyer, Robert D Rodman, Leiba Pouzet, Maurice Rahe, Maurice H Rehm, Allan S Rickart, Charles E Rodrigue, Garry H Powell, Charles H Rahman, Mizanur Rehmann, Ulf Rickert, Neil W Rodrigues Filho, Joao Powell, David E Raimi, Ralph A Reichard, Rosalind Rickey, V Frederick Rodriguez, Gilmar Powell, James H Rains, M A Reichaw, Meir Rickman, B Roeder, David W Powers, Morrie Walton Rainville, Russell Reid, Riddle, Lawrence H Roels, Jacques Albert Powers, Robert T Rajca, John F Christopher Ervin Riddle, Malcolm Lee Roessler, E B Pawls, Keith W Rajnak, Stanley L Reid, James D Ridenhour, Jerry R Roettinger~Kaplan, Prabhu, Rakesh Reid, Kenneth B Rider, Daniel G Ida N arahari U manath Ralevic, Nebojsa V Reid, R Clay Ridolfo, Anthony S Rogawski, Pradines, Jean Rail, Louis B Reid, Walter M Riedel, Herbert H J Jonathan David Prakash, Jagdish Ralston, Elizabeth W Reid, William H Rieffel, Marc A Roger, Claude Prakash, Nirmala Ralston, James V Reilly, Rieger, Georg J Rogers, Edwin Prange, John D Ramachandran, Norman Raymond Riehemann, Robert M Rogers, Leland E Pranger, Walter A Doraiswamy Reineck, James F Riehm, Carl R Rogers, Thomas D Preissendoerfer, Ramakrishnan, Reiner, Irma M Riemen, William P Roggenkamp, Klaus W Manfred W Dinakar Reiner, Irving Riemenschneider, Rohatgi, Vijay K Prest, Michael Y Ramalho, Margarita Reinhardt, Elsi A Oswald Rohl, Frank D Preston, Gordon B Ramanujan, Reinhart, Bruce L Ries, John F X III Rohrbach, Hans Price, David T Melapalayam S Reinoehl, John H Riesel, Hans 1 Rohrl, Helmut Price, G Baley Rambavilas, C D Reissner, Eric Riess, Karlem Rohrlich, David E Price, John F Ramey, Wade C Reiter, Edna E Riffelmacher, David A Roitman, Judith Price, Justin J Ramfrez de A, Enrique Reiter, Stanley Rigdon, Robert D Rojo, Oscar Price, Ray H Ramirez, Reitwiesner, George W Riihentaus, Rokni, Kamran A Price, Roderick A Arturo Flores Rejali, Ali Leo Juhani Romanelli, S Price, Thomas E Jr Ramsay, Arlan B Rejto, Peter A Rijnks, H Ronga, Felice L L Priddy, Stewart B Ran, Abselom Rem, Martin Riley, Bruce V Rood, John J Priest, Cynthia J Ran, Andre C M Rempel, Richard S Riley, Robert F Rooker, John C II Priestley, William M Randall, A Duane Remus, Dieter Riney, John S Rooney, Prill, David D Randall, John D Renardy, Michael Ringel, Gerhard Elaine Kasimatis Prime, Mirko Randol, Burton S Renault, Jean N Ringrose, John R Rooney, Gerald E Jr Prisco, Mary K Randolph, John F Renegar, James M Rio, Sheldon T Rooney, Paul G Pritchard, Frank Leon Rangaswamy, K M Reneke, James A Rippon, Philip J Roonguthai, Warut Proctor, C Wayne Range, R Michael Renner, Lex E Ripy, Sara L Roos, Cornelia Proctor, Robert A Rankin, R A Renno, James G Jr Rishel, Thomas W Roos, Guy Promel, Hans Jurgen Rankin, Renz, Peter L Risnes, Martin Roos, Jan~Erik I Propes, J C Samuel Murray Resch, Richard I Ritt, Robert K Roos, Paul Protas, David S Ransom, Adelbert W Restrepo, Guillermo Ritter, David L Roosenraad, Cris T Protter, Philip E Rao, Retzloff, David G Ritter, William E Root, William L Proulx, Vidhyanath Kajana Reuterdahl, J Thomas Rival, Ivan Roozemond, Leen Viera Krflanov8. Rapp, P E Reyna, Luis Guillermo Rivet, Roger Rosa, Alexander Provencher, Normand Rascle, Michel Maria Rna, Jung-Sik Rosales, Rodolfo Pruitt, William E Raskind, Wayne Mark Reynolds, David W Roach, G F Rosales, Silvia Huerta Prussner, Laird Drew Rasoulian, Amid Reynolds, Donald F Roach, Michael D Rosalsky, Andrew Przebinda, Tomasz Rassias, George M Reynolds, Eugene Robacker, John T Rosati, Mario Przybylski, Thomas L Rassias, Themistocles Reynolds, Jim D Robbie, Desmond A Rosato-Furia, Pu, Hwang-Wen Michael Reynolds, Robert R Robbins, Steven Frances M Pucci, Carlo Ratcliff, Gail D L Reynolds, William F Robbins, Wade B Rose, Donald J Pucci, Patrizia Ratiu, Tudor Stefan Reza, Fazlollah Robbins, Walter M Rose, N J Puckette, Stephen E Ratliff, Louis J Jr Reznick, Bruce A Robel, Gregory F Roselle, David P Pudaite, Paul R Ratschek, Helmut Rhai, Tong-Shieng Robert, Alain Roseman, Puel, Jean Pierre RB.tz, Ji.irg J Rheinboldt, Werner C Roberts, Dorothy K Dennis Martin Puerta, Fernando S Raugh, Michael R Rhemtulla, Roberts, Roseman, Joseph J Puig, Lluis Rausen, John Akbar Hussein Florence Ellen Rosen, Julie A Puigbo, Jaume Vila Rawsthorne, Daniel A Rhie, Young H Roberts, George A Rosen, Kenneth H Pukanszky, Lajos Ray, Nigel Rhoades, Clark P Roberts, Howard M Rosen, Michael I Pumplun, Dieter Rayburn, Marlon C Jr Rhodes, Burgess H Roberts, Jean E Rosen, Ned Ira Pun, Ping-Sham Raymond, Frank A Rhodes, Frank Roberts, Joel L Rosenbaum, Robert A Puppe, Dieter Rayna, Gerhard Riaza, Roman Roberts, Joseph B Rosenberg, Iva G Puri, Pratap Read, Dwight Ribeiro, Hugo B Roberts, Leslie G Rosenberg, Purisch, Steven D Read, Thomas T Ribenboim, Paulo Roberts, Richard C Jonathan M Pursell, Lyle E Reade, Maxwell 0 Ribes,, Luis Roberts, Stephen G Rosenberg, Steven Putcha, Mohan Sastri Reardon, Philip C Ribet, Kenneth A Robertson, Rosenberger, Gerhard Putnam, Alfred L Reber, Douglas C Ricci, Fulvia Alexander P Rosenblatt, David Putnam, Calvin R Rechard, Ottis W Ricci, Michael G Robertson, Malcolm S Rosenblatt, Pyle, Ernest Recillas, Felix J Ricci, Stephen J Robertson, Neil Joseph Max Pyle, H Randolph Rector, David L Rice, Barbara Slyder Robinson, Clark Rosenblum, Marvin Pym, JohnS Reddy, William L Rice, Henry G Robinson, Daniel A Rosenfeld, Gerald Redfield, Robert H Rice, John R Robinson, Derek J S Rosenfeld, Norman S Quackenbush, Redheffer, Rice, John William Robinson, G de B Rosenfeld, Ronald L Robert Willis Raymond M Rice, Richard E Robinson, Rosenlicht, Maxwell A Quade, Edward S Redlin, Lothar H Richard, B K Helen Drummond Rosenstein, Quartararo, Philip Jr Redmond, Don Richards, Donald St P Robinson, Ivor George M Jr Quigg, John C Redner, Richard A Richards, John F Robinson, Rosenthal, Arthur J Quigley, Frank D Reed, Bruce E Richards, Pamela C Joanna Ryten Rosenthal, David S Quine, J R Reed, David L Richardson, Henry R Robinson, Karl H Rosenthal, Eric S Quinn, Frank S III Reed, Irving S Richardson, Robinson, Michael L Rosenthal, Erik J Quint, Richard A Reed, James H Raymond E Robinson, Raphael M Rosenthal, Haskell P Quint, Simon R Reed, Myra Jean Richardson, Robinson, Thomas J Rosier, Ronald C Quintas, Louis V Reed, Robert E Roger W Jr Rabkin, Eugene E Ross, Kenneth A Quinto, Eric Todd Reedy, Christopher L Richert, Arthur Robson, Robert 0 Ross, Paul Quiring, Donald K Rees, Charles S Richert, Hans-Egan Rocha, Alva.ny Ross, Sharon C Quirk, Paul B Rees, MinaS Richert, Norman J Rocha, Carlos V Rossa, Robert F Quiroga, Rafael P Rees, P K Richman, Daniel J Rocha, Luiz F C Rossi, Guido A

539 ICM-86 (Continued)

Rossi, Hugo Saaty, Thomas L Sattinger, David H Schmidt, Roland Seaman, Douglas D RoSsi, John F Saccoman, John J Sauer, Timothy D Schmidt, Wolfgang M Searcy, Charles J Rossmann, Wulf Sachs, David Saul, Alan Bruce Schneebeli, H R Searl, James E Rosson, John Sacks, Paul E Saunders, Frank W Schneider, David I Sedgewick, Rota, Gian-Carlo Sacksteder, Richard C Saunders, William G Schneider, Dennis M Charles H W Roth, Ben G Sadosky, Cora S Saut, Jean Claude Schneider, Manfred F Sedory, Stephen A Roth, Klaus F Saeki, Sadahiro Savage, James R Schneider, Victor P Seebach, J Arthur Roth, Richard L Saff, Edward B Savage, Richard P Jr Schnitzer, Franz J Seebach, Linda Roth, Rodney J Sagan, Bruce E Savariraj, G A Schober, Anton G Seebeck, Charles L Ill Rothaus, Oscar S Sah, Chih-Han Sawka, John M Schochet, Claude L Seeley, Robert T Rothberger, Fritz Sahab, Salem M Saworotnow, Schochet, Steven H Seenithamby, Rothe, Erich H Saibel, Edward A Parfeny P Schoen, Richard M Sivasundaram Rothman, Neal J Saint Mary, Donald F Sawtell, Roy G Jr Schoenberg, Isaac J Segal, David M Rothman, Sheldon N Saito, Akira Sawyer, Emerson L Schoenenberger-Deuel, Segal, Irving E Rothschild, Bruce L Saito, Kazuyuki Sawyer, Eric T Johanna Segers, Richard G Rothschild, Joseph G Saito, Shiroshi Saxton, Ralph Arthur Schoenfeld, Lowell Sehgal, Surinder Rothschild, Saito, Yoshimi Sayasov, Yuri S Schoenwaelder, Sehnert, James Ellis Linda Preiaa Saitoh, Saburou Sayeki, Hidemitsu Ulrich F Seibert, Peter Rothwell, Franklin G Sakai, Katauro Sayre, John E Scholz, E Seidel, Johan J Sakai, Shoichiro Seiden, Esther Rottenberg, Reuven R Sbordone, Carlo Schon, Rolf Sakuma, Motoyoshi Scalzo, Richard C Schonbek, Maria Elena Seidenberg, Abraham Roush, Fred Salane, Douglas E Scanlon, Jane Cronin Schoneborn, Heinz Seidman, Thomas I Rousseau, Cecil C Salas, Saturnine L Scaramuzzi, Roberto Schoonover, Melvin R Seifert, George Rousseau, Thom&a H Saleem, Mohammad Scariano, Stephen Schori, Richard M Seifert, Ralph L Rouaaou, Athina Salehi, Ebrahim Scarpi, Giambattista Schot, Steven H Seifter, Norbert Rouya, Robert L Salinas, Luis C Scaraini, Marco Schottenloher, Seitelman, Leon H Rovnyak, James L Salinas, Norberto Scavo, Thomas R R Martin Sejeeni, Fowzi A Rovnyak, Virginia G Salisbury, Thomas S Seedrov, Andre Schramm, Ruben Sekimoto, Toshihiko Rowan, William H Sallee, G Thomas Schaal, Werner G H Schrauf, Geza H Sekino, Kaoru Rowe, Niles W Ballman, Jeffrey E Schacher, Murray M Schraut, K C Selby, Alan M Rowell, William W Sally, Judith D Schaefer, Donald J Schreiber, Bert M Selden, Rowen, Louis Halle Sally, Paul J Jr Schaefer, Paul T Schreiber, Morris Annie Alexander Roxin, Emilio 0 Salmon, Jessie Blake Schaeffer, David G Schreiner, Erik A Selden, John J r Royden, Halsey X.. Saltiel, Leon Schaer, Jonathan Schrijver, Alexander Seldin, Jonathan P Royer, James Samuel Saltz, Daniel Schafer, Alice T Schroder, M Belgrade, James F Royster, Wimberly C Saltzman, Schafer, Richard D Schroeck, Seligman, George B Ruben, Jeffrey Stewart Schiffer, Juan Jorge Franklin E Jr Seligman, Mark L Flores Espinoza Salvadori, Anna Schaffer, Matthew J Schroeder, Johann Seligson, Stuart A Rubenfeld, Lester A Salwach, Chester J Schaller, Kay-Uwe Schruben, Johanna Sell, George R Ruberman, Daniel Salzberg, Betty Schaps, Schryer, Nor man L Sellers, Peter H Rubin, Arthur L Salzberg, Pablo M Maika Elisheva Schubert, Cedric F Selmer, Ernst S Rubin, Herman Samayoa, Rodolfo Scharlau, Rudolf M E Schulman, Seltzer, Stanley E Rubin, Jean E Samelson, Scharlemann, Michael David Semmes, Rubinstein, Joachim H Christopher L MartinG Schulte, Felix Stephen William Rubio de Francia, Samelson, Hans Schattschneider, Schulte, Thomas R Sempi, Carlo E Jose L Samolis, Doris W Schultz, Phillip Sen, Dipak K Rudin, Mary E Ruth Anastasia Schatz, James R Schultz, Reinhard E Senderovizh, Rivka Rudnick, Karl Sampson, Gary Schatz, Joseph A Schumacher, Carol S Senechal, Lester J Ruedy, Reto A Samuel, Pierre Schatzman, Schumacher, Georg B Benge, George Ruelle, David P Sanchez, Antonio James Carl Schumacher, Scott Senger 1 Steven 0 Ruess, W Sanchez, David A Schauer, Richard L Schumaker, John A Seppala, Ruh, Ernst A Sanchez, Juan Miguel Schaumberger, Schumer, Peter David Mika Kristian Ruijsenaara, Sanchez-Diaz, Rafael Norman Schumitzky, Alan Serapioni, Raul Simon Norbert Sanchez-Pedreno, Schay, Geza Schurle, Arlo W Serbyn, William D Ruiz Morillo, Salvador Schechter, Samuel Schuster, Seymour Serfling, Robert J Alejandro J Sanchez-Vazquez, Schecter, Stephen Schutz, John W Serfozo, Richard F Ruiz, Ceferino Gonzalo Scheffer, Carel L Schuur, Susan E Serizawa, Shozo Rumely, Robert S Sancho San Roman, Scheffer, Vladimir Schwa bauer, Robert J Serre, Jean-Pierre Rumford, Fred K Juan Scheier, M A Schwandt, Ernst Serrell, Robert Rumsey, Howard C Jr Sanders, Georgia E Scheim, David E Schwanz, CheJ;"yl L Servedio, Frank J Runck, Paul Otto Sanders, Jan A Scheinberg, Stephen Schwartz, Ira B Sesay, Mohamed W I Runckel, Sanders, Robert W Schellhorn, Schwartz, Jacob T Seshu, Lily H Hans-Joachim Sandomierski, Frank L Jean-Pierre Schwartz, Laurent Sethuraman, Bharath Rund, Hanno Sands, Bill Scherkoske, Schwartz, Randy Ken Sethuraman, J Rung, Donald C Sands, Kenneth D Schwartz, William Seubert, Steven M Ruoff, Dieter Jonathan Winslow Schibell, Stephen T Schwarz, Binyamin Sever, Michael Rush, David Eugene Sands, Suzanne E Schiffer, Max M Schwarz, Friedheim Sevilla, Alicia N Sangren, Ward C Sevrain, Christophe Rusinak, William A Schiller, Ilya Schwarz, Gerald W Sangwine-Yager, Schindler, Guenter M Schwarz, Thomas Jean-Paul Ruokai, M Beth Jane Rosamund Schweiger, Fritz Sexauer, Norman E Russell, David Martin Schirmer, Helga H Santa Pietro, John J Schlagbauer, Hartmut Schweitzer, Eric Seydel, Robert E Jr Russell, Gary L Santalo, LA Schlesinger, Ernest C Schwerdtfeger, Sgro, Joseph A Russell, Thomas F Santos Filho, Schlichenmaier, Hans WE Shabazz, Abdul Alim Russo, Bernard Jose Ruidival S Martin Schwermer, Joachim Shabell, Barbara J Russo, Dominick A Santos, Rafael Schmeelk, John F Schwietzer, Michael J Shader, Bryan L Russo, John A Sanz-Serna, J M Schmerl, James H Scissors, Richard H Shadwick, Rutherford, Saphar, Pierre David Schmetterer, Scoppola, Carlo M William Francis Ralph Gordon Sapozhnikova, Vera D Leopold K Scott, Dana S Shafer, Douglas S Rutland, Leon W Jr Saracino, Daniel H Schmid, Josef Scott, Frederick J Shaffer, Dorothy B Rutledge, Dorothy S Sarafyan, Diran Schmid, Jurg Scott, Leonard L J r Shahabi, M A Ruttimann, Sarason, Donald E Schmid, Scott, Peter Shahidi, Freydoon Gottfried T Sarason, Leonard Norma Pauline Scott, Ridgway Shahin, Ryan, Donald E Sarhan, Mahmoud Schmid, Peter P Scott, Warner Henry Mazen Mohammed Ryan, Francis D Sarian, Edward Schmid, Rudolf Harvey III Shahshahani, Ryan, Patrick J Sario, Leo Schmid, Wilfried Scott, William R Mehrdad M Ryan, Peter M Sarrat, Charles F Schmidlin, Dean J Scott-Thomas, John F Shakil, Mohammed Ryan, Robert D Sastri, Chelluri C A Schmidt, Darrell Scourfield, Eira J Shallit, Jeffrey Outlaw Rydeen, Clarence F Satake, Ichiro Schmidt, Dieter S Scowcroft, Philip H Shand, Sherman M Ryeburn, David Sathaye, A vi nash Schmidt, Harvey J Jr Scriba, Christoph J Shannon, Christine A Rykken, Charles John Sato, Hiroki Schmidt, Henry J Jr Seal, Charles Edwin Shannon, David L Rzedowski, Martha Sato, Ken-iti Schmidt, Kenneth C Sealy, Robert P Shannon, Susan

540 ICM-86 (Continued)

Shapiro, Daniel B Short, Donald R Jr Skau, Christian F Snyder, Stakgold, Ivar Shapiro, GeorgeS Shorter, William Ivan Skeath, James E William Merritt Jr Staley, Walter L Shapiro, Harold N Shortt, Rae Michael Sklar, Abe Soardi, Paolo Maurizio Stallings, John R Shapiro, Helene Shreve, Darrell R Sklar, David Soare, Robert I Stallmann, Shapiro, JackS Shreve, David C Skora, Richard Kevin Sobel, Bernard A Friedemann W Shapiro, Jay Shreve, Warren E Skoug, David L Sobolev, Serguei L Stamey, WilliamL Shapiro, Joel H Shrikhande, Skowronski, Socolovsky, Stamm, Emil Shapiro, Louis W Mohan Sharad Janislaw M Eduardo Adolfo Stammbach, Urs Shapiro, Michael Shu, Wun-Item Slack, Stephen Sohmer, Bernard Stampfli, Joseph G Shapiro, Victor L Shu be, Beatrice Slade, Gordon D Sohrab, Houshang H Stanaitis, Otonas E Shar, Albert 0 Shucker, DavidS Slaminka, Edward E Soldate, Albert M Jr Stancl, Donald L Sharkey, Michael Shulberg, Gary W Slater, John B Solheid, Ernie S Stand, Mildred L Sharma, PL Shultz, Harris 8 Slater, Michael Solian, Alexandru Stanczak, Romuald Sharp, Henry Jr Shutt, Elsie G Slattery, Michael Solin, Robert N Stanley, Lee James Sharp, Jack Carroll Jr Sibille, E Sleeman, BrianD Salmon, Donald C Stanley, Richard P Sharp, Rodney Y Sibley, Thomas Q Slifker, James F Solomon, Louis Stanley, Walter L Sharpe, Michael J Sibner, Lesley M Sloane, Neil J A Solomon, Ronald M Stanojevi~, Caslav V Sharpe, Richard W Sibner, Robert J Slodowy, Peter J Somer, Lawrence E Stanton, Charles M Shaughnessy, Sibuya, Yasutaka Slook, Thomas H Somes, John D Stanton, Charles S J Michael Bichler, J Sloss, James M Sommer, Richard D Stanton, Nancy K Shaw, Guy B Siddiqi, Rafat N Slowikowski, Wojciech Somolinos, Alfredo S Staples, Edmund Shaw, Mei-Chi Sidney, Stuart J Sluchin, Benny Soni, Kusum K Beauclerc III Shaw, Sen-Yen Siebenmann, Slud, Eric V Soni, Raj Pal Staples, John C Shayib, Mohammed A Laurence C Slye, John M Sonn, Jack Starbird, Thomas W Shearer, Sieg, Wilfried Smale, Stephen Sonneborn, Lee M Stark, Christopher W James William Siegel, Alan Small, Charles Sonner, Johann Stark, Eberhard L Shearer, Michael Siegel, Eli A Small, Kenneth A Sonnesso, Geraldine Stark, Harold M Sheffer, Iaador M Siegel, Jerrold N Small, Lance W Sons, Linda R Stark, Sheldon H Jr Sheil-Small, Siegel, Paul H Small, Lynne Barnes Sontag, Starr, Edith N Terence Brian Siegrist, Kyle T Smart, John R Alexia Henderson Stasheff, James D Shekoury, Raymond Siersma, Dirk Smiley, Leonard Sontag, State, EmileY Shell, Niel Sigrist, Francois Smillie, John Eduardo Daniel Staton, William A III Shelly, Maynard W Sigurdsson, Gunnar Smith, Bryan A Sontz, Stephen Bruce Stavroudis, Orestes N Shelton, Chrispian E Silberger, Allan J Smith, Soong, Roland Stavru, Hristos Shelton, Richard M Silberstein, Esther Constance Flahive Sorensen, Niels T Stecher, Michael J Shelupsky, David I Silbert, Gary Allen Smith, David A Sorenson, John R Steel, Thomas B Jr Shemanske, Thomas R Siler, Jack G Smith, Donald R Sormani, Michael J Steele, Alfred T Shen, CY Silva, Alessandro Smith, Edgar C Jr Sorvali, Tuomae Steele, Charles A Jr Shen, John Silva, Maria Welleda Smith, FR Souganidis, Steelman, John Henry Panagiotis E Shen, Mei-Chang Silvennoinen, Smith, Frank A Steen, FH Shen, Nien-Tsu Risto Pekka Smith, Frank D Jr Sourour, Steen, Lynn A Herb Ahmed Ramzi Stefani, Stefano T Shen, Yuan-Yuan Silverman, Smith, Fredrick L Shepherd, Raymond T Joseph H Southgate, Theodore J Stef,naaon, Silverman, Smith, Gene Ward J6n R Shepherdson, Southworth, Steflik, Theresa M John C Silverman, Smith, Raymond W Sher, D Richard Allan Harold Robert Jr Stegeman, Jan D Sherk, FA Silverman, Spanier, Jerome Stegenga, David A Smith, Harvey A Sherling, Robert David Spatz, Ian N Steger, Arthur Smith, Herman J Jr Dorothy Davia Evelyn Marie Spears, Otis S Steger, Tim Joshua Silvia, Smith, James Lester Sherlock, Sim, Soon Kiang Specht, Edward J Steggles, Irving Smith, John Howard Philip Thomas Simen, David C Spector, Rene Stegun, Irene A Smith, Sherman, Clayton C Simmons, Harold Speed, Terence Paul Stein, Alan H John Huntington Sherman, Gary J Simmons, Speh, Birgit Stein, Elias M Sherman, Malcolm J John Wiseman II Smith, John M Spellman, Dennis Stein, Howard Smith, Sherr, JosephS Simms, David J Spence, Lawrence E Stein, Ivie Jr Kenneth Woodward Sherrill, M David Sim6, Carles Spencer, DominaE Stein, Marjorie L Sherwood, Howard SimOes-Pereira, JMS Smith, Kirby C Spencer, Guilford L II Stein, Michael Smith, LoisE B Shestakov, Simon, Imre Spencer, Joel H Stein, Sherman K Aleksei Ilyich Simon, Jonathan K Smith, Marianne Spencer, Steinberg, Arthur Sheu, Simon, Robert Freundlich Thomas Crawford Steinberg, Leon Albert Jeu-Liang Simons, Smith, Mark A Spencer, W Dean Steinberg, Maria W Shiba, Masakazu Gordon Edward Smith, Matthew D Sperry, Marvin G Steinberg, Paraskevi Shibata, Masanori K Simons, Roger A Smith, Mi-Soo Bae Spiegel, w Steinberg, Robert Shibata, Yukio Simons, Stephen Smith, Spielberg, Jack Steinberg, Stanley Lee Shibuya, Yhuji Simpson, James E Michael Ainsworth Spielberg, Stephen E Steinberger, Mark Shick, PaulL Sims, Benjamin T Smith, Michael G Spieler, Barry S Steinberger, Meir Shiffman, Bernard Sims, Charles C Smith, Perry B Spikes, Steiner, Anne K Shiffman, Max Sinclair, Allan M Smith, Raymond F Dolores Richard Steiner, Donald D Shifrin, Theodore Sinclair, Smith, Robert Colman Spitzer, FrankL Steiner, Ray Phillip Shiga, Koji George Edward Smith, Roland F Sprecher, David A Steinhardt, Fritz Shih, Kung-Sing Singer, David Allen Smith, Ronald K Spring, David H Steinborn, Charles I Shih, Weishu Singer, Michael Smith, Scott Springer, Tonny A Steketee, Jakob A Shilepsky, Arnold C Singer, Phyllis E Smith, Step)len B Spruck, Joel Stell, George R Shim, Jae Ung Singer, Seymour Smith, Stephen B Spurr, Michael J Stelzer, Joerg M Shimada, Nobuo Singer, WilliamM Smith, Stephen D Squires, William A Stemmer, Gaspar H Shimamoto, Don H Singerman, David Smith, Tom C Srebro, Uri Stenger, Frank Shimizuike, Yuzi Singh, Bhagat Smith, Tr-uett Lee Sribney, WilliamM Stenger, William Shin, Kyo II Singh, Kanhaya L Smith, Wayne Stewart Srinivasan, Bhama Stenson, David L Shin, Yong-Tae Singh, Sankatha P Smith, Wilbur L Srinivasan, VK Stenson, Ellen M Shiohama, Katauhiro Singh, Surjeet Smith, WilliamK Sritharan, Sivaguru Stephens, Clarence F Shiohara, Hiroshi Bingman, David H Smithson, Raymond E Srivaatav, Anupam Stephenson, Kenneth Shipp, R Dale Sinkiewicz, Gerald P Smoke, WilliamH Srivastav, RamP Stephenson, Shiraiwa, Kenichi Sinkov, Abraham Smolarski, Dennis C Srivastava, Krishna J Robert M Jr Shiratani, Kataumi Sinnott, Warren Mann Smoller, Joel A St Vincent, Michael Stepp, James Wilson Shiu, Elias 8 Siatar, Carolyn Denise Smyth, WilliamF Stacey, Peter John Sterba-Boatwright, Shivaji, Ratnasingham Sitia, Candido Snaith, Victor P Stackelberg, Olaf P Blair D Shively, Barbara S Siu, Man-Keung Snell, J Laurie Staffeldt, RossE Sterling, Theodor D Shoaff, WilliamD Siu, Yum Tong Snover, James E Stafford, Jonathan Stern, Alan Spector Sholander, Marlow C Sivera, Rafael Snow, Dennis M Stafney, James D Stern, Mark A Shor, S W Williston Siwiec, FrankE Snow, Joanne R Stagnaro, Ezio Stern, Ronald J Shore, Richard A Sizer, WalterS Snyder, David F Stahl, Saul Sternberg, David Shore, Terence R Skaff, MichaelS Snyder, Richard D Stair, Lynda F Sterrett, Andrew Jr

541 ICM-86 (Continued)

Stetkaer, Henrik Stuart, H W Jr Szalajka, Walter S Teleman, N icolae Tinsley, Frederick C Steuerwalt, Michael H Stuckwisch, Stephen E Szambien, Horst H Telg&rsky, Rastislav Tippett, James M Stevens, Stummel, Friedrich Szankowski, Andrzej Temam, Roger Tischler. David C Glenn Howard Sturgeon, Michael L Szarek, Stanislaw J Ternes, Lorne Titani, Satoko Stevens, John G Sturley, Eric A Szczarba, Robert H Temple, John Blake Tits, Jacques L Stevens, Randy S Sturnfield, James F Szczepanski, Susan Temte, Mark C Toby, Ellen H Stevens, Richard S Styer, Robert A Szeto, Mabel Tena, Juan Toczycki, Wladyslawa Stevens, T Christine Stynes, Martin Sznitman, Alain·Sol Tench, Douglas Toda, Nobushige Stewart, Cameron L Su, Yuh·Ling Szucs, Joseph M Tenenblat, Keti Todd, Aaron R Stewart, G W Suarez, Pedro A Szulkin, Andrzej Teply, Mark L Todd, John Stewart, Robert C Suchanek, Ana M Szwec, R J Terao, Hiroaki Todd, Peter M Stiel, Edsel F Sucheaton, Louis Szyld, Terkelsen, Frode Todor, Fabian Stiles, Joseph L Sudderth, William D Daniel Benjamin Terng, Chuu·Lian Todorov, Still, Charles Herbert Sueiro·Bal, Juan Terrae, Riho Pavel Georgiev Stinson, Kevin Sueyoshi, Yutaka Tabak, Barbara L Terrell, Steven W Togo, Shigeaki Stob, Michael Sugarman, Paul J Tadic, Marko Terwilliger, Paul M Toledo, Domingo Stock, John R Sugawara, Masahiro Tadmor, Eitan Terzuoli, Andrew J Toll, Charles H Stocker, Sugden, Dorothy A Taft, Earl J Tesei, Alberto Toll, Kathryn B Harold Edward Sugiyama, Linda E Taft, Margaret W Tews, Melvin C Tolle, Louis F Stockmeyer, Paul K Suita, Nobuyuki Tagawa, Mas a· Yoshi Textorius, Bjorn 0 B Tolosa, Juan Stocks, Douglas R Jr Sulanke, Robert A Taggart, Marcia D Thakur, Tolsted, Elmer Stoddard, James H Suleman, Hermono Tahani, Hossein Umesh Sarwottam Tolstoy, Alexandra I Stokes, H Christine B Sullins, Kenneth W Taiani, Geraldine Thaler, Alvin I Tomasovic, Joseph Stokes, Marion Jeffries Sullivan, Dennis P Taibleson, Mitchell H Thedy, Armin Tomber, Marvin L Stokes, Russell A Sullivan, Joseph A Taira, Kazuaki Thelen, Brian J Tomczak-Jaegermann, Stokes, William G Sullivan, Richard W Tait, Robert James Theodorescu, Radu N Stolaraky, Kenneth B Sullivan, Wayne G Tait, William W Theohari·Apostolidi, Tomei, Carlos Stolberg, Harold J Sulski, Leonard C Takacs, Lajos F Theodora Tomer, Eugene R Stoll, Wilhelm F Summerhill, Takahashi, Reiji Theusch, Colleen J Tominaga, Hisao Stoller, Gerald S R Richard Takahashi, Shuichi Thickstun, Thomas L Tomiuk, Daniel B J Stoltenberg-Hansen, Summers, John W Takeda, Ziro Thiele, Ernst-Jochen Tomiyama, Jun Viggo Summers, Stephen J Takemoto, Hideo Thieler, Peter Tompson, R N Stoltzfus, Neal W Summers, Timothy W Takenaka, Shigeo Thierrin, Gabriel Tomsky, Jack Stone, Alexander P Summerville, Takesaki, Masamichi Thoele, Edward C Tomter, Per Stone, Arthur H Richard M Takeuchi, Maearu Tholen, Walter P Tong, Mary Powderly Stone, Arthur L Sun, Hugo Sui-Hwan Takeuti, Gaisi Thorn, Rene Tonne, Philip C Stone, Charles J Sun, Twan Takizawa, Kiyoshi Thoma, Elmar H Topsoe, Flemming Stone, David A Sun, Tze.Chien Takizawa, Seizi Thomas, Alan T Toptsis, Anestis A Stone, Eugene A Sundararaman, D Talaga, Paul C Thomas, Clayton J Torelli, Paul A Stone, Lawrence D Sundberg, Carl Talenti, Giorgio G Thomas, Erik G F Torrejon, Ricardo M Stone, Michael G Sunderland, Talge, Gordon J III Thomas, Garth H M Torres, Euclides Stong, Robert E Benjamin B Jr Taliaferro, Steven D Thomas, Janet D Torres, Waldo A Store, Lise Sung, Cheng Chih Talmadge, Richard B Thomas, Jay F Torrico, James 0 Storey, Albert J Sunley, Judith S Tam, Bit·Shun Thomas, Lawrence E Toshi, Iida Stormes, James M Sunouchi, Gen·ichiro Tam, Kwok-Wai Thomas, Marc Phillip Tosquea, Mario Storrer, Hans H Sunouchi, Haruo Tamari, Dov Thomas, Mark A Totaro, Burt J Stortz, Clarence B Sunseri, Mary V Tan, Henry K Thomas, P Emery Toth, Michael Richard Story, Donald D Suri, Manil Tan, Sin·Leng Thomas, Tousley, Horace W Story, Helen F Suryanarayan, E R Tanabe, Hiraki Richard Kenneth Town, George G Stout, Edgar Lee Sussman, Myron M Tanaka, Hisao Thomason, Steven K Townsend, Stout, Lawrence N eft' Sussmann, Hector J Tanaka, Jun-Ichi Thomee, Vidar C Douglas Wayne Stout, Quentin F Suter, Bruce Tanaka, Toshie Thomeier, Siegfried Townsend, James T Stowe, Dennis C Sutherland, Scott Tancredi, Alessandro Thompson, Harold B Traina, Charles R Straffin, Philip D Jr Sutner, Klaus Tang, Alfred S Thompson, Jon H Tran, Lanh Tat Stralka, Albert R Sutton, Charles S Tang, Betty M Thompson, Lee D Tranah, David A Strang, W Gilbert Suttorp, L G Tang, Chi-Ming Thompson, Mary E Transue, William R Strassberg, Helen A Suziedelia, Eligijus S Tang, Francis Chi-Yu Thompson, Robert C Transue, William R R Strassen, Volker Suzuki, Haruo Tang, Hwa Tsang Thompson, Trapp, G E Jr Straube, Emil J Suzuki, Michio Tang, Wai-Shing Robert James Trautman, David A Straubing, Howard Suzuki, Yoshiya Tangeman, Richard L Thompson, Russell C Trautmann, G Straughan, Brian Svendsen, Tangora, Martin C Thomson, Brian S Traynor, TimE Straume, Eldar J Edward Charles Tanimoto, Tatree T Thomson, James E Trebels, Walter Strauss, Leon Svensson, Erik J Tania, Elliot A Thorbergsaon, Treibergs, Andrejs E Strauss, Monty J Sverdlove, Ronald Tannenbaum, Peter Gudlaugur Treiman, Jay S Strauss, Walter A Swaminathan, Tanny, David Thorne, Trench, William F Streater, David Neil Srinivasa Tao, Yoko John Farley III Trenholme, Alice Ruth Strebel, Kurt Swamy, Srikanta M N TaqQu, Murad S Thornton, Melvin C Tretkoff, Carol Strebel, Ralph Swan, Richard G Targonski, Gyorgy I Thorpe, James C Treves, Francois Stredulinaky, Swann, Howard S G Targonski, Jolan M Thorpe, John A Triantafillou, Georgia Edward W Swanson, David L Tarski, Jan Thrall, Robert M Trias Pair6, Joan Street, Ross H Swanson, Ellen E Tartakoff, David 8 Throop, T Tribbey, William P Streit, Franz Sward, Gilbert L Tashiro, Yoshihiro Thurber, Tricerri, Franco Streitman, Max Sward, Marcia P Tashjian, Gloria J Edward Gerrish Trigiante, Donato Strikwerda, John C Swardson, Mary Anne Tatevossian, Leon H Tiberio, Ronald S Trimble, Selden Y Stroeker, R J Swartwout, Donald E Taussky, 0 lga Tichy, Robert F Trinh, Sinh Han Stroessner, Heinz Swartz, Blair K Tavantzis, John Tierney, Ann R Trione, Susana Elena Strohl, G Ralph Jr Swartz, Charles W Taylor, Angus E Tierney, John A Tripp, John C Strom, Arne Swartz, Edward B Taylor, B A Tietz, Horst Trager, Hans Strom, Bernee D Swartz, Joseph H Taylor, Donald E Tijdeman, Robert Tromba, A J Stromberg, J 0 Swearingen, Daniel B Taylor, Tijo, Stef H Trombi, Peter C Stromberg, Karl Sweeney, Edward T Gladys Gillman Tilidetzke, Tronel, Gerard Stromquiat, Walter R Sweeney, Orval L Taylor, Howard M Robert James Troas, Ralph G Stroock, Daniel W Sweeney, William J Taylor, Jean E TiHmann, Heinz G Trotter, Hale F Stroth, Gernot H W Sweet, Melvin M Taylor, Laurence R Timmer, Kathleen M Trotter, P G Stroud, Eric G Swick, Kenneth E Taylor, Michael D Timmerscheidt, Klaus Troue, Jacques Strout, John E Swift, Clare Taylor, Samuel J Timoney, Richard M Trout, John D Stroyan, K D Swokowski, Earl W Taylor, Thomas J Timourian, James G Troutman, John L Struik, Dirk J Swong, Khyson Taylor, William W Tin, Saw Troyanov, Marc Struik, Ruth Rebekka Synowiec, John A Tee, Garry J Tinberg, N alsey B Troyer, Stephanie F Struppa, Daniele C Sysikaaki, Pirjo Teitelbaum, Jeremy T Tindle, Graham L Trudinger, Neil S Strzelecki, Emmanuel Szabo, Manfred Egon Teitelbaum, Pinchas Tinkelman, Robert P Trueman, DC

542 ICM-86 (Continued)

Truglio, Marilyn Ann van der Vaart, Vernon, Michael Walker, James 8 Watkins, Trytten, Deborah A H Robert Veron, Laurent Walker, James W Robert James Trytten, George N Vachuska, Colleen A Verona, Andrei Walker, Julia A Watkinson, Anthony Tsai, Chester E Vainio, Reino Johan Verrios, Walker, Larry R Watnick, Richard M Tsai, Long-Item Vajravelu, Katherine G D Walker, LeRoy H Watro, Ronald J Tsai, Yen-Shung Kuppalapalle Veratraelen, Leopold Walker, Marshall Watson, James D Tsai, Yuh-Dong Valdivia-Gutierrez, Vervaat, Wim Walker, Michael F Wataon, Martha F Tsangaris, Panayiotis Oscar Vesentini, Edoardo Walker, Philip W Watson, Saleem Tsao, Anna Valk, Henry S Vesley, Richard E Walker, Russell Bruce Watson, Stephen Tsao, Chia K Valle, Antonio Vetter, Udo Wall, Charles T C Watt, Jeffrey Xavier Tsao, Liang-Chi Vamanamurthy, Vietsch, W Karel Wall, Curtiss E Watt, Stephen B John J Tschantz, Steven T Mavina K Vigneras, Wall, GE Wavrik, Keith Tseng, Shiojenn Van Bendegem, Marie France Wallace, David A R Wayland, K Tsucbikura, Tamotsu Jean Paul Vila-Freyer, Wallace, David Robert Wayman, Arthur Edward C Tsui, Sze-Kai Jack Van Casteren, Ricardo Francisco Wallen, C J Waymire, Alan Tsui, Waikin Johannes A Villa-Salvador, Wallen, Lawrence J Wayne, George E Tsuji, Kaza Van De Water, Gabriel D Wallis, Walter D Weaver, Weaver, James R Tsuji, Mikio Arthur M Jr Villamil, Hilda P Walsh, Dennis Patrick Weaver, John C Tsukada, Makoto Van Den Ban, Erik P Vilma, Jaak Walsh, John J Weaver, Tsukiyama, Kouzou Van Der Hoek, John Vinacua, Alvaro Walsh, Mary L Robert Wooddell Tsukui, Yasuyuki Van Der Kallen, Vincent, Alan W Walster, G William. Webb, Cary H Tsutsumi, Massayoshi Wilberd Vinograd, Robert E Waltcher, Azelle B Webb, William A Tu, Yih-0 Van Der Maas, Vinti, Calogero Gilbert G Walter, Webber, Robert P Tubaro, Luciano Gerard J Viola, Carlo John H Walter, Weber, Claude A Tucci, Ralph P Van Der Poorten, Virsik, George MartinE Walter, Weber, Waldemar C Tucker, Alan C Alfred J Viterbo, Claude Walter, Norman R Tucker, Don H Van Engelen, F Vitter, Albert L III Webster, Sidney M Walter, Rolf W Wechsler, Martin T Tucker, Howard G Van Horne, John W Vitulli, Marie A Walter, Wolfgang L H Wedel, Arnold M Tucker, John R Van Lint, Jacobus Vives, Luis A Walters, Eleanor B Tucker, Susan L Van Mulbregt, Paul A Viviente, Jose L Wefelacheid, Heinrich Walters, Peter Wehauaen, John V Tugue, Tosiyuki Van Rootselaar, Bob Voehl, Deanna Lynn Walters, Randall K Marijke Wehlau, David L Tuler, Randy Van Rossum, Voelker, Dietrich H Walters, Robert F C Strien, Wehlen, Joseph A Jr Tulipani, Sauro Van Vogan, David A Jr Waltman, Paul E J Wei, S Walter Tull, Jack P Sebastian Vogel, Kurt Walton, Jay R Shih-Hsiung VanTuyl, A H Thomas I Weibel, Charles A Tung, Vogel, Walton, Lewis F Barry Van Vleck, Fred 8 Weibel, John L Turett, J Vogelius, Wanby, Goran Van Wesenbeeck, Wei!, Clifford E Turiel, F J Michael Steenstrup Wan_g, Ai-Nung Walter F Vogt, Andrew Weiler, Fred W Turisco, Wang, Cecilia Y Joann Stephanie Van Winter, Clasine Vogt, E Weiler, Jerome T Wang, Derming Turkington, Bruce E Van de Wetering, Vogtmann, Karen Weill, Lawrence R Wang, Edward T H Turner, Edward C R Lee Vohra, Rakesh V Weinberg, David A Wang, Hwai-Chiuan Turquette, Atwell R Vance, Elbridge P Voichick, Michael Weinberg, Elliot C Wang, Kwang-Shang Turull, Alexandre Vance, James T Jr Voigt, Paul S Weinberger, Hans F Wang, McKenzie Y Turyn, Lawrence Vanden Boss, Voigt, Robert G Weiner, Jerome H Wang, Ming-Chang Turyn, Richard J Eugene L Vojta, Paul A Weinert, Wang, Paul K C Tuttle, Keith Allan Vanderbauwhede, A L Volkmann, Bodo Hanna Joachim Shicheng Tuvell, Walter E Vandevelde, J Richard Voreadou, Rodiani Wang, Weinfeld, Joel M Tuzar, J aroslav Vanhecke, L Vougiouklis, Wang, Weinitachke, Stuart Sui-Sheng Tuzhilin, Avgustin Vanstone, James R Thomas N Hubertus J Tseng-Chan Twersky, Victor Varadarajan, Voytuk, James A Wang, Weinkam, James J Ying Hsiung Tyrl, Paul Kalathoor Vrbik, Jan Wang, Weinstein, Alan D Evelyn K Tysk, Johan Varadarajan, V S Vrscay, Edward R Wantland, Weinstein, Jack Tzafriri, Lior Varadhan, Vuilleumier, Monique Wanzong, Robert C Weinstein, Michael I N Tzavaras, Srinivasa S R Vuorinen, Matti Ward, Harold Weinstein, Stephen T Athanassios E Varberg, Dale E Vyas, U D Ward, James E Ill Weintraub, Elaine Y Tzeng, Chun-Hung Varga, Richard S Vybor-ny, Rudolf Ward, James Norman Weintraub, Steven H Vargas, John David Ward, John T Weia, Lutz Ucci, Jack Vasak, Janet Trzcinski Wada, Hidekazu Ward, Joseph D Weiser, Alan Uchiyama, Akihito Vasconceloa, Wada, Junzo Ward, Josephine A Weisner, Louis Uchiyama, Saburo Wolmer V Wade, William R Ware, Buck Weiss, Asia lviC U e belacker, James W Vaseratein, L N Wadhwa, Bhushan Lal Ware, Roger P Weiss, David M Uesugi, Toshitane Vasquez, Alphonse T Waelbroeck, Lucien Warfield, Robert B Jr Weiss, John Uhlenbeck, Karen Vassiliou, Efatathios Wage, Michael L Warga, Jack Weiss, Norman J Uhlmann, Gunther A Vatsala, AS Wagener, Raymond Warne, Ronson J Weiss, Richard M Uhlmann, Werner Vaughan, Herbert E Wagner, Daniel H Warnecke, Gerald Weissler, Frederic B Ullery, William D Vaughan, Jerry E Wagner, David H Warner, Frank WIll Weitkamp, Galen L Ullian, Joseph 8 Vaughan, Wagner, Diane M Warnock, Robert L Welch, Lawrence V Ullman, Joseph L Theresa Phillips Wagner,· Eric G Warren, Bette L Welch, Lloyd R Ulmer, Douglas L Vaughn, James M Jr Wagner, Gretchen B Warren, William E Weiland, Grant V Ulmer, F Vayl, Vladimir Wagner, Neal R Warrinnier, Alfred E Wellner, Jon A Ulrich, Bernd Vazquez, Juan L Wagon, Stan Warshauer, Max Leon Wells, Charles F Ulrich, Stuhler Vega de Prada, WagoD.er, John B Warsi, Shaukat Ali Wells, David M Umble, Ronald N Jose Manuel Wagreich, Philip D Washburn, Wells, Jane F Umeda, Takato Vein, Howard B Wagstaff, Samuel S Jr Robert B Jr Wells, R 0 Jr Umoh, Hanson M Veldman, Willem H Wahba, Grace G Washburn, Sherwood Welters, G E U nai, Yasushi U nai Velleman, Daniel Wahl, John J Washington, Wen, Shih-Liang Unal, Aynur Veneri, James M Wahl, Jonathan M Lawrence C Wendel, James G Underwood, Ventre, Aldo Giuseppe Wahlbin, Lara B Waaow, Wolfgang R Wendland, Wolfgang L Douglas H Ventura, Waid, C Carter Wasserman, Arthur G Wendroff, Burton Ungar, Abraham Belisario Augusto Wainwright, Eric Wasserman, Richard S Wendt, Dieter W F Ungar, Peter Venugopalan, P Wainwright, John Watanabe, Hisao Wene, Gregory P Unger, Ernst R Venzi, P Wajnryb, Bronislaw Watanabe, Sh6ji Wenger, Ronald H Upatisringa, Visutdhi Venzke, Paul B Wakana, Kaoru Watatani, Yaauo Wente, Henry C Updike, John Bernard Vepsalainen, Wake, Bob Alan Waterhouse, Wenzel, John A Upham, Mary H Seppo Juhani Walczak, Hubert R William C Wermer, John Upmeier, Harald Verde Star, Luis Walde, Ralph E Waterman, Daniel Werner, Willes L Upton, John A W Verdera, Joan Wales, David B Waterman, Michael 8 Werschulz, Arthur G Urda, Michael Vergne, Michele Walker, Carol L Wathen, Judy A Wesael, W Roy Uribe Espinosa, Verma, Ghasi R Walker, Elbert A Watkins, David S West, Dennis E Guillermo Vermeer, Johannes Walker, Homer F Watkins, Joseph C Weet, Donald C Usmani, Riaz A Vermes, Robert Walker, James Edward Watkins, Murray S West, James E

543 ICM-86 (Continued)

Westbrook, Edwin P Williams, Dana P Wolkowicz, Gail S K Yamaguchi, Keizo Zaanen, Adriaan Western, D W Williams, David A Wolpert, Scott A Yamaguchi, Ryuji Zabrodsky, Alexander Westman, Joel J Williams, Dorothy J Wolsson, Kenneth Yamaguti, Kiyosi Zacharia, Dan Weston, Jeffrey D Zacharias, Gail Williams, ·E Bruce Wong, Edward T Yamaki, Hiroyoshi Weston, Kenneth W Williams, Frederick C Wong, James S W Zacks, Shelemyahu Yamamoto, Koichi Westphal-Schrnidt, U Williams, Gareth Wong, Kwok Chi Zadeh, Lotfi A Westwood, Derek .J Williams, George K Wong, Yamanoshita, Tsuneyo Zafrany, Samy Wetzel, John E Williams, Gerald W Philip Pit-Wang Yamaoka, Kenya Zagier, Don Bernard Wexler, D Williams, Gregory P Wong, Pit-Mann Yamasaki, Masayuki Zahid, Wheeler, Charles H III Williams, Hugh C Wong, Raymond Y Yanagawa, Minoru Muhammad Ishaq Wheeler, John A Williams, Hugh M Wong, Yang, Chao-Hui Zajac, Edward E Roman Woon-Ching Zaks, Abraham Wheeler, Robert F Williams, James G Yang, Chung-Tao White, Alvin M Williams, Joyce W Wong, Sherman K Zalik, R A Yang, Deane White, Williams, Lawrence R Wong, Warren J Zama, Nobuo Arthur Thomas II Williams, Lynn R Wong, Yim-Ming Yang, Jeong Sheng Zambrini, Jeau-Claude White, Benjamin S Williams, Mark Wong, Yuen-Fat Yang, Kung-Wei Zanca, Clemente A White, Charles Scott Williams, Neil H Wong, Yung-Chow Yang, Paul C Zandi, Ahmad White, Denis A Williams, Peter David Woo, K Y Yang, Shaochen Zanolin, Fabio White, Dennis E Williams, Wood, Geoffrey V Yang, Wei-Shih Zapf, Hermann White, Donald B Richard Wood, John C Zaretzki, Philip M R Jr Yanik, Joe White, George N Jr Williams, Robert F Wood, John W Zaring, Wilson M Yano, Kentaro White, Janis E Williams, Ronald 0 Woodroofe, Michael B Zaslavsky, Thomas White, John T Williams, Ruth J Woodrow, Yanowitch, Michael Zdinak, Edward G White, Neil L Williams, Stephen A Robert Edward Yao, Andrew Chi-Chih Zedek, Mishael White, Stevan Russell Williams, Susan Gayle Woodruff, Edythe P Yap, Leonard Y H Zehnder, Eduard J White, Steven J Williams, Vincent C Woodruff, William M Yaqub, Adil M Zeigler, Fredrick J White, Wm A Williams, William 0 Woods, Alan C Yaqub, Fawzi M Zeleznikow, John Whitehart, F E Woods, Zelinsky, Daniel Williamson, Charles K Dale Yaqub, Jill S Whitehead, George W Williamson, Clifton J Woods, Jerry D Zelinsky, David S Yasue, Kunia Whitehead, Williamson, Francis U Woods, R Grant Zelle, Joseph Kathleen B Williamson, Frank Jr Woodward, Gordon S Yasugi, Mariko Zeller, Karl Whiteman, Albert L Williamson, Jack Woolf, William B Yasuhara, Ann Zemaitis, Vincent R Whitfield, John H M Williamson, Richard E Wooster, Kenneth Yasuhara, Mitsuru Zemmer, Joseph L Jr Whitley, W Thurmon Williamson, Robert E Wooton. William Yau, Stephen S-T Zerzan, Terrance P Whitman, Andrew P Willis, Paul A Worley, Patrick H Yeager, Dorian P Zhu, Jianchao Whitman, Philip M Wills, Jorg M Worsham, Robert D Ziebur, Allen D Yebra, Whitmore, William F Wortman, Dennis H Ziege, Roland W Wilson, Edward N Jose Luis Andres Whitney, D Ransom Wilson, James B Wouk, Arthur Ziegler, Zvi Whitney, Hassler Wilson, John H Woyczyrlski, Wojbor A Yee, George S Ziemer, William P Whitney, James N Wilson, Leslie C Woythaler, Yeh, James J Zierler, Neal Whitney, Ronald F Wilson, Raymond B Joseph William Yeh, Roger C Ziller, Wolfgang Whitney, Stephen Wilson, Robert L Wrench, John W Jr Yen, David H Y Zilmer, Delbert E Whitson, Kim J Wilson, Robert Lee Wrentmore, Anita K Yerian, Katherine Ann Zimering, Shimshon Whitt, Lee B Wilson, Robert R Wright, David G Yetter, David N Zimmer, J W Whittaker, James V Wilson, Ted C Wright, David J Zimmerman 1 Grenith J Yhap, Whitten, Wilbur Wilson, Wright, Jeffrey Allen Zimmerman, Jay J Terence E Ernesto Franklin Whittlesey, Emmet F Wilson, W Stephen Wright, Randy Zink, Robert E Yntema, Wick, Brian D W inarsky, Nor man D Wright, Ron Mary K Zinn, J Wicke, Howard H Windham, Wright, Steve J Yocom, Kenneth L Zipfel, Christian Wickelgren, Wayne A Michael Parks Wright, Yohe, James M Zippo, Siro W ickerhauser, Wine, James D Thomas Perrin .J r Yokoi, Hideo Zipse, Philip W Mladen Victor Winfrey, W R Wrona, Wlodzimierz Yoneda, Kaoru Zirilli, Francesco Zitarelli, Widger, Charles F Wing, G Milton Wu, Hung-Hsi Yoneguchi, Hajimu David E Widlund, Olaf B Wingler, Eric J Wu, Ling-Erl Eileen T Zitron, Norman R Yood, Bertram Widman, Kjell-Ove H Winograd, Shmuel Wu, T C Zitzler, Siham Braidi Wiedemann, Alfred Winslow, Dennis N Wulf, Leo M Yorke, James A Zizi, Khelifa Wiedemann, Winston, Clement Wunderlich, Marvin C Yoshino, Genji Zizler, Vaclav Douglas H Winston, Kenneth Wurfel, Tilmann Yoshino, Takashi Zlatev, Zahari Wiegand, Roger A Winter, Eva P Wurster, Marie A Yoshizawa, Taro Zo, Felipe J Wiegand, Winters, Robert Wyler, Oswald Yosida, Kosaku Zoch, Richmond T Zoercher, Sylvia Margaret Winther, Ragnar Wylie, Clarence R Jr Young, Barry H C Z Wiegmann, Wintrobe, F Wyman, Bostwick F Zomorrodian, Reza Young, David M Jr Klaus Werner Wirszup, lzaak Wyneken, Zondek, Bernd Wielenberg, Norbert J Wirth, Andrew Matthew Fair Young, Donald F Zorn, M A Wiener, Don Wise, Gary Lamar Wyss, Walter Young, Elmer Lorne Zorn, Paul Wiener, Howard W Wisniewski, Helena S Young, Eutiquio C Zoroa, P Wiener, Matthew Withers, Wm Douglas Xia, Daoxing Young, Gail S Zorzitto, Frank A Wightman, Arthur S Witt, Donald M Xia, Jingbo Young, Lael M Zsid6, Lciszl6 Zucker, Steven Wihstutz, Volker Witte, David S Young, Laurence C M Wilcox, Theodore W Witte, Franklin P Yablon, Marvin Zuckerberg, Hyam L Young, Paul M Wildman, Louis R Woan, Wen-Jin Yachter, Morris Zuckerman, Gregg J Wiles, Andrew J Woeppel, James J Yackel, James W Young, Sam Wayne Zuckerman, Paul R Wilke, Frederick W Woess, Wolfgang Yadin, Micha Young, Wo-Sang Zweifel, Paul F Wilken, Donald R Wofsy, Carla Yahya, S M Younger, Daniel H Zwick, D Wilker, John B Woldar, Andrew J Yajima, Yukinobu Yovanovich, Zwier, Paul J Wilker, Peter Wolfe, Carvel S Yaku, Takeo M Michael Zygmund, Antoni Wilkerson, Wolfe, Dorothy W Yale, I Keith Yu, Chien-Chih Anonymous Clarence W Wolfe, Harold E Yale, Paul B Yu, Lucille Chieh (130) Wilkins, J Ernest Jr Wolfe, John E Yamada, Miyuki Yui, Noriko Wilkinson, Steven V Wolfe, Michael David Yamada, Shinichi Willard, Stephen Wolff, Manfred P H Yamada, Toshihiko Yun, Dall-Sun Willey, Richard N Jr Wolfson, Paul R Yamaguchi, Itaru Yung, Tin-Gun Williams, Bennie B Walk, Elliot S Yamaguchi, Jinsei Yus, Nicolas

544 Election Information

The Nominating Committee and the Preferential Ballot The method of counting the ballots for places It is too much to attempt a complete summary on the Nominating Committee (but not for here. The spirit is that every ballot shall count the election of officers) is the one known as somewhere if possible; if it is not needed (as the "single transferable vote." As this was when the candidate is either elected without codified in British law, there were certain it or defeated with it), it shall be transferred ambiguities and elements of chance. These to the next choice. There is a "quota", the have been removed in a subsidiary document, number of votes required to elect a candidate. also followed, known as the "senatorial rules." With N ballots and m places to be filled, the The reader who wishes general background on quota Q is the smallest integer J such that preferential ballots and on the single transfer­ at most m candidates can receive at least J able vote may read Voting in Democracies by votes, that is Q = [1 + N/(m + 1)]. Ballots are Enid Lakeman and James D. Lambert (Faber distributed according to first choice votes, any and Faber, 1955). The reader who wishes to candidate receiving the quota being declared go further and to become familiar with the elected. Then votes are transferred, repeatedly case advanced against preferential ballots may if necessary, first using up excesses over the read books by F. A. Hermens, such as Europe quota, the largest excess first, and then elim­ between Democracy and Anarchy (University inating candidates with insufficient votes, the of Notre Dame, 1951). There is a criticism smallest first. In transferring either an excess of the preferential ballot by Steven J. Brams number of votes or an insufficient number of in Notices, vol. 29, pages 136-138 (February votes, numbers of votes with total equal to the 1982). number to be transferred are assigned to the In a preferential ballot, the voter ranks his remaining candidates, the assigned number for choices and indicates the ranks with consecu­ a candidate being proportional to the number tive integers, beginning with one for the can­ of next choice votes for that candidate. Trans­ didate of his first choice. The voter may order fers follow the order of preference of the voters. any desired number of candidates. However, The process is determinate, but it is too long the voter may reach a point where ignorance to explain completely here. or indifference makes further ordering impos­ Please do not correspond with the writer sible. A ballot becomes not transferable and or the editors about the ambiguities that are thus of no further use (see the next paragraph) unresolved in the above abbreviated descrip­ when an integer is used twice or is skipped tion. Instead, read Appendices IV and V in indicating the order of choice, so that the already referred to, where eleven pages of tellers are not effectively informed. rules, essentially a teller's flow chart, appear to leave no ambiguities. The precise rules by which the votes are counted are stated in Appendices IV and V Everett Pitcher, Secretary of the cited text of Lakeman and Lambert. This is a passage of thirty-three pages of def­ initions, rules, interpretations, and examples.

1986 Election Information The names of candidates for the election of 1986 would appear in this issue of Notices if it were possible to follow the plan of past years. However, the spring meeting of the Council, at which nominations are made, was unusually late this year and occurred after this issue of Notices went to press. Thus names of candidates will not appear until the August issue.

Everett Pitcher, Secretary Bethlehem, Pennsylvania

545 Logan, October 10-11, Utah State University First Announcement of the 828th Meeting

The eight hundred and twenty-eighth meeting of David Gabai, Dennis Garity, Matt Grayson, Jim the American Mathematical Society will be held Henderson, Jim Hoste, Stephen Humphries, Louis at Utah State University in Logan, Utah, on Kaufman, Chris Lacher, Duane Loveland, Jerr Friday and Saturday, October 10-11, 1986. Levine, Vo Thanh Liem, John Luecke, Dale Rolf­ son, T. Benny Rushing, Martin Scharlemann, Jon Invited Addresses Simon, Mike Starbird, Abigail Thompson, Keith By invitation of the Committee to Select Hour Wolcott, and David Wright. Speakers for Far Western Sectional Meetings, Most of the papers to be presented at these there will be two invited one-hour addresses. The special sessions will be by invitation. However, speakers and the titles of their talks are: anyone submitting an abstract for the meeting PETER LI, University of Utah, Harmonic who feels that his or her paper would be particu­ functions on complete manifolds. larly appropriate for one of these special sessions KENNETH C. MILLETT, University of Cali­ should indicate this clearly on the abstract form fornia, Santa Barbara, New combinatorial methods and submit it by July 28, 1986, three weeks before in three dimensional topology. the deadline for contributed papers, in order that it may be considered for inclusion. Participants Special Sessions are reminded that a charge of $16 is imposed for By invitation of the same committee, there will retyping abstracts that are not in camera-ready be five special sessions of selected papers. The form. topics of these sessions, the names and affiliations of the organizers, and partial lists of the speakers, Contributed Papers are as follows: There will also be sessions for contributed ten­ Applications and computational aspects of minute papers. Abstracts should be prepared numerical continuation methods, EUGENE ALL­ on the standard AMS form available from the GOWER, Colorado State University. The tentative AMS office in Providence or in Departments of speakers are Eusebius Doedel, Kurt Georg, Her­ Mathematics. Abstracts should be sent to the bert Keller, Hans Mittelmann, T.-Y. Li, Alexan­ Editorial Department, American Mathematical der Morgan, Aubrey Poore, Chris Tiahrt, and Society, Post Office Box 6248, Providence, Rhode Layne Watson. Island 02940, so as to arrive before the August Analysis on manifolds, S. Y. CHENG, Uni­ 18, 1986, abstract deadline. Participants are versity of California, Los Angeles. Some of the reminded that a charge of $16 is imposed for speakers are Michael Anderson, Robert Brooks, retyping abstracts that are not in camera-ready Hui-Dong Cao, Robert Greene, Douglas Moore, form. Late papers will not be accommodated. V. Oliker, Luen-Fai Tam, Andrejs Treibergs, and Johan Tysk. Registration Representations of reductive groups, DAVID The meeting registration desk will be located on H. COLLINGWOOD, University of Oregon. Some the main floor of the Eccles Conference Center. of the speakers are Brian Boe, J. Chang, Tom The desk will be open from 10:00 a.m. to 4:30 Enright, Devra Garfinkle, Henryk Hecht, R. Irv­ p.m. on Friday, and from 8:00 a.m. until noon ing, J. Johnson, Dragan Milicic, Ivan Mirkovic, on Saturday. The registration fees are $10 for Hugo Rossi, R. Scaramuzzi, Brad Shelton, Peter members of the AMS, $16 for nonmembers, and Trombi, and Joseph Wolf. $5 for students or unemployed' mathematicians. Random fields, random measures and ap­ plications, ED WAYMIRE, Oregon State Univer­ Petition Table sity. The speakers are Kenneth S. Alexander, A petition table will be set up in the registration Florin Avram, Rabi Bhattacharya, Michael D. area. Additional information can be found in a Brennan, Peter Brockwell, Robert M. Burton, box in the New Orleans meeting announcement Colleen Cutler, Richard Darling, Stuart Ethier, on page 72 of the January 1986 issue of Notices. Peter Gaenssler, Evarist Gine, Thomas G. Kurtz, Charles M. Newman, Mina Ossiander, Richard Accommodations F. Serfozo, Winfried Stute, Murad Taqqu, and A block of rooms is being held for conference Stanley C. Williams. participants at the University Inn. This is a Geometric topology, DAVID WRIGHT and full-service hotel complex located on campus. J. W. CANNON, Brigham Young University. Other motels are located within 1.5 to 3 miles of The speakers are Colin Adams, Ric Ancel, campus in downtown Logan. Participants should Mladen Bestvina, Steven Bleiler, John Bryant, Ed make their own reservations directly with the Burgess, J. W. Cannon, Bill Eaton, Bill Floyd, University Inn or other motel of their choice.

546 Rates are subject to possible change and do not Persons driving from Salt Lake City to Logan include local taxes. should take 1-15 north to Brigham City, exit at University Inn 363 and take US highway 89-91 to Logan. In Utah State University, Logan, UT 84322 Logan, turn east off Main Street to 400 North Telephone: 801-750-1153 and then turn left off 400 North to 700 East to Single $28 Double $32 the University Campus. Baugh Motel Parking is available in the visitor parking ter­ 153 South Main Street, Logan, UT 84321 race on 700 North, adjacent to the University Inn, Telephone: 801-752-5220 and to the east of campus at 1200 East. Terrace Single $26 and up Double $30 and up parking will be validated for conference partici­ pants and the University Inn guests. Parking is Best Western Weston's Lamplighter permitted virtually anywhere on weekends. 250 North Main Street, Logan, UT 84321 Regularly scheduled limousine service be­ Telephone: 801-752-5700 tween Salt Lake City and Logan is available Single $28 Double $32 through Key North Limousine. Current one-way Holiday House prices are $36.40 for one person and $25 each for 447 North Main Street, Logan, UT 84321 two or more persons. Up to six departure times Telephone: 801-752-9141 per day are offered weekdays, and four times Single $26 and $28 Double $31 and $33 per day on weekends, with additional limousines added as needed. Reservations must be made Food Services no later than 24 hours in advance by calling Key The cafeteria in the Taggart Student Center North Limousine, 801-394-7743. will be open during both days of the meeting. Logan is at an elevation of 4000 feet. Autumn In addition, there are numerous restaurants along days are cool and frost is likely at night. Rain or Main Street, and a restaurant list will be available snow is possible, but unlikely. at the registration desk. Travel and Local Information Hugo Rossi Salt Lake City, Utah Associate Secretary Logan is located approximately 90 miles north of Salt Lake City. Salt Lake City is served by Amtrak, Greyhound and major airlines. Most car rental agencies have counters at the airport terminal.

547 Charlotte, October 17-18, Adam's Mark Hotel First Announcement of the 829th Meeting

The eight hundred and twenty-ninth meeting of Most of the papers to be presented at these the American Mathematical Society will be held special sessions will be by invitation. However, at the Adam's Mark Hotel in downtown Charlotte, anyone submitting an abstract for the meeting North Carolina, on Friday and Saturday, October who feels that his or her paper would be particu­ 17-18, 1986. This meeting will be hosted by the larly appropriate for one of these special sessions University of North Carolina at Charlotte. should indicate this clearly on the abstract form and submit it by July 30, 1986, three weeks before Invited Addresses the deadline for contributed papers, in order that By invitation of the Committee to Select Hour it may be considered for inclusion. Participants Speakers for Southeastern Sectional Meetings, are reminded that a charge of $16 is imposed for there will be three invited one-hour addresses. retyping abstracts that are not in camera-ready The speakers are as follows: form. PATRICK EBERLEIN, University of North Carolina, Chapel Hill, Structure of manifolds of Contributed Papers non-positive curvature. There will also be sessions for contributed ten­ STEVEN C. FERRY, University of Kentucky, minute papers. Abstracts should be prepared title to be announced. on the standard AMS form available from the FRANK RAYMOND, University of Michigan, AMS office in Providence or in Departments of Ann Arbor, title to be announced. Mathematics. Abstracts should be sent to the Editorial Department, American Mathematical Special Sessions Society, Post Office Box 6248, Providence, Rhode By invitation of the same committee, there will Island 02940, so as to arrive before the August be four special sessions of selected twenty-minute 20, 1986, abstract deadline. Participants are papers. The topics of these sessions and the reminded that a charge of $16 is imposed for names and affiliations of the organizers, are as retyping abstracts that are not in camera-ready follows: form. Singularities and algebraic geometry, GARY Information concerning registration, accom­ KENNEDY, Duke University. modations, food service, etc., will be available in Geometric topology, JOHN MAYER, Univer­ the August issue of Notices. sity of Alabama, Birmingham. Group actions on manifolds, DAVID ROYS­ Frank T. Birtel TER, University of North Carolina, Charlotte. New Orleans, Louisiana Associate Secretary Twistor theory and four-dimensional geome­ try, ALBERT L. VITTER III, Tulane University.

548 Denton, October 31-November 1, North Texas State University First Announcement of the 830th Meeting

The eight hundred and thirtieth meeting of the Geometric structures on manifolds and Klein­ American Mathematical Society will be held at ian groups, RAVI S . KULKARNI. North Texas State University in Denton, Texas, Invariant theory, V. LAKSHMIBAI, Texas on Friday, October 31 and Saturday, November 1, A & M University. 1986. Most of the papers to be presented at these special sessions will be by invitation. However, Invited Addresses anyone submitting an abstract for the meeting By invitation of the Committee to Select Hour who feels that his or her paper would be particu­ Speakers for Central Sectional Meetings, there larly appropriate for one of these special sessions will be four invited one-hour addresses. The should indicate this clearly on the abstract form speakers are as follows: and submit it by August 4, 1986, three weeks PAUL FONG, University of Illinois at Chicago, before the deadline for contributed papers, in Modular representations of finite groups. order that it may be considered for inclusion. RAVI S. KULKARNI, Indiana University, Participants are reminded that a charge of $16 Bloomington, title to be announced. is imposed for retyping abstracts that are not in ALEXANDER NAGEL, University of Wiscon­ camera-ready form. sin, Madison, title to be announced. MARY WHEELER, Rice University, title to Contributed Papers be announced. There will also be sessions for contributed ten­ minute papers. Abstracts should be prepared Special Sessions on the standard AMS form available from the By invitation of the same committee, there will be AMS office in Providence or in Departments of several special sessions of selected twenty-minute Mathematics. Abstracts should be sent to the papers. The topics of the sessions, and the names Editorial Department, American Mathematical and affiliations of the organizers, are as follows: Society, Post Office Box 6248, Providence, Rhode Banach spaces and related topics, ELIZABETH Island 02940, so as to arrive before the August BATOR, RUSSELL BILYEU and PAUL LEWIS, 25, 1986, abstract deadline. Participants are North Texas State University. reminded that a charge of $16 is imposed for Connections between combinatorics, algebra retyping abstracts that are not in camera-ready and geometry, NEAL BRAND and JOSEPH KUNG, form. It appears unlikely that late papers can be North Texas State University. accommodated. Differential equations, ALFONSO CASTRO, Information concerning registration, accom­ Southwest Texas State University, and JOHN modations, food service, etc. will be published in NEUBERGER, North Texas State University. the August issue of Notices. Mathematics for large scale computing, J. C. DIAZ, University of Oklahoma. Robert M. Fossum Computational mathematics, WARREN FER­ Urbana, Illinois Associate Secretary GUSON, JR., Southern Methodist University.

549 Invited Speakers October 1986 Meeting in Charlotte Southeastern Section and Special Sessions Deadline for arganizers: Expired Deadline for consideration: July 90, 1986 Invited Speakers at AMS Meetings Gary Kennedy, Singularities and algebraic geom- The individuals listed below have accepted invi­ etry tations to address the Society at the times and John Mayer, Geometric topology places indicated. For some meetings, the list of David Royster, Group actions on manifolds speakers is incomplete. Albert L. Vitter, III, Twistor theory and four- dimensional geometry

Logan, October 1986 October 1986 Meeting in Denton Peter Li Kenneth C. Millett Central Section Deadline for organizers: Expired Charlotte, October 1986 Deadline for consideration: August 4, 1986 Elizabeth Bator, Russell Bilyeu, and Paul Lewis, Patrick Eberlein Frank Raymond Banach spaces and related topics Steven C. Ferry Neal Brand and Joseph Kung, Connections be­ tween combinatorics, algebra and geometry Denton, October 1986 Alfonso Castro and John W. Neuberger, Differ­ ential equations Paul Fong Alexander Nagel J. C. Diaz, Mathematics for large scale computing Ravi S. Kulkarni Mary Wheeler Warr~n Ferguson, Jr., Computational mathemat­ ~cs Ravi S. Kulkarni, Geometric structures on mani­ San Antonio, January 1987 folds and Kleinian groups Marc Culler David J. Saltman V. Lakshmibai, Invariant theory Peter D. Lax Lesley M. Sibner (Colloquium Lecturer) Thomas Spencer Fall1986 Meeting Robert J. McEliece (Gibbs Lecturer) Eastern Section No meeting will be held Newark, April 1987 Robert V. Kohn Birgit Speh January 1987 Meeting in San Antonio Rodolfo Rosales Lars S. Wahlbin Associate Secretary: Robert M. Fossum Deadline far arganizers: Expired Organizers and Topics Deadline for consideration: September 2.4, 1986 Ronald J. DiPerna, Nonlinear partial differential of Special Sessions equations The list below contains all the information about Burton Fein, David J. Saltman, and Murray Special Sessions at meetings of the Society avail­ Schacher, Brauer groups and Galois theory able at the time this issue of the Notices went Michael Gage and Edwin Lutwek, Geometric to the printer. The section below entitled Infor­ inequalities mation for Organizers describes the timetable for Philip J. Hanlon, Combinatorics and group repre- announcing the existence of Special Sessions. sentations Paul D. Humke, Complex analysis Lesley M. Sibner, Gauge theory October 1986 Meeting in Logan Lynn McLinden and Jay S. Treiman, Theoretical Far Western Section optimization Deadline far arganizers: Expired Gregory P. Wene, Mathematical physics Deadline for consideration: July 28, 1986 Eugene L. Allgower, Applications and compu­ Spring 1987 Meeting tational aspects of numerical continuation methods Eastern Section S. Y. Cheng, Analysis on manifolds Deadline for organizers: October 15, 1986 Deadline for consideration: To be announced David Collingwood, Representations of reductive groups Ed Waymire, Random fields, random measures Spring 1987 Meeting and applications Central Section David Wright and J. W. Cannon, Geometric Deadline far organizers: October 15, 1986 topology Deadl''ne for consideration: To be announced

550 Spring 1987 Meeting Information for Speakers Far Western Section A great many of the papers presented in Special Deadline for organizers: October 15, 1986 Sessions at meetings of the Society are invited papers, Deadline for c0118ideration: To be announced 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 Spring 1987 Meeting deadline announced above and in the announcements of the meeting at which the Special Session has been Southeastern Section scheduled. Contributors should know that there is Deadline for organizers: October 15, 1986 a limitation in size of a single special session, so Deadline for c0118ideration: To be announced that it is sometimes true that all places are filled by invitation. Papers not accepted for a Special Session are considered as ten-minute contributed papers. Abstracts of papers submitted for considera­ Information for Organizers tion for presentation at a Special Session must be received by the Providence office {Editorial Depart­ Special Sessions at Annual and Summer meetings are ment, American Mathematical Society, Post Office held under the general supervision of the Program Box 6248, Providence, RI 02940) by the special dead­ Committee. They are administered by the Associate line for Special Sessions, which is usually three weeks Secretary in charge of the meeting with staff assistance earlier than the deadline for contributed papers for from the Society office in Providence. the same meeting. The Council has decreed that no Some Special Sessions arise from an invitation paper, whether invited or contributed, may be listed to a proposed organizer issued through the Associate in the program of a meeting of the Society unless an Secretary. Others are spontaneously proposed by abstract of the paper has been received in Providence interested organizers or participants. Such proposals prior to the deadline. are welcomed by the Associate Secretaries. The number of Special Sessions at a Summer or Annual Meeting is limited to twelve. Proposals, Send Proposals for Special Sessions to the invited or offered, which are received at least nine months prior to the meeting, are screened for suitabil­ Associate Secretaries ity of the topic and of the proposed list of speakers, The programs of sectional meetings are arranged by the and for possible overlap or conflict with other pro­ Associate Secretary for the section in question: posals (specific deadlines for requesting approval for Far Western Section (Pacific and Mountain) Special Sessions at national meetings are given above). Hugo Rossi, Associate Secretary If necessary, the numerical limitation is enforced. Department of Mathematics Proposals for Special Sessions should be submit­ University of Utah ted directly to the Associate Secretary in charge of the Salt Lake City, UT 84112 meeting (at the address given in the accompanying (Telephone 801-581-8159) box). If such proposals are sent to the Providence Central Section office, addressed to Notices, or directed to anyone Robert M. Fossum, Associate Secretary other than the Associate Secretary, they will have Department of Mathematics to be forwarded and may not be received before the University of Illinois quota is filled. 1409 West Green Street In accordance with an action of the Executive Urbana, IL 61801 Committee of the Council, no Special Session may (Telephone 217-333-3975) be arranged so late that it may not be announced Eastern Section in Notices early enough to allow any member of the W. Wistar Comfort, Associate Secretary Society, who wishes to do so, to submit an abstract for Department of Mathematics consideration for presentation in the Special Session Wesleyan University Middletown, CT 06457 before the deadline for such consideration. (Telephone 203-347-9411) Special Sessions are effective at Sectional Meet­ Southeastern Section ings and can usually be accommodated. They are Frank T. Birtel, Associate Secretary arranged by the Associate Secretary under the super­ Department of Mathematics vision of the Committee to Select Hour Speakers for Tulane University the section. The limitation on the number of sessions New Orleans, LA 70118 depends on the space and time available. The same (Telephone 504-865-5646) restriction as for national meetings applies to the As a general rule, members who anticipate orga­ deadline for announcing Special Sessions at sectional nizing Special Sessions at AMS meetings are advised to meetings: no Special Session may be approved too seek approval at least nine months prior to the sched­ late for its announcement to appear in time to allow a uled date of the meeting. No Special Sessions can be reasonable interval for members to prepare and sub­ approved too late to provide adequate advance notice to mit their abstracts prior to the special early deadline members who wish to participate. set for consideration of papers for Special Sessions. The Society reserves the right of first refusal for the publication of proceedings of any special session. These proceedings appear in the book series Contemporary Mathematics.

551 Call For Topics For 1988 Conferences

Suggestions are invited from mathematicians, ei­ ther singly or in groups, for topics of the various conferences that will be organized by the Soci­ Topics of Current and ety in 1988. The deadlines for receipt of these Recent Conferences suggestions, as well as some relevant information AMS-SIAM Symposium in about each of the conferences, are outlined below. An application form to be used when submitting Applied Mathematics suggested topic(s) for any of these conferences 1980- Mathematical psychology and psychophysiology, (except the Short Course Series) may be obtained organized by STEPHEN GROSSBERG of Boston Uni­ by writing to the Meetings Department, American versity. Mathematical Society, P.O. Box 6248, Providence, 1983- Inverse problems, organized by D. W. Rhode Island 02940, or telephoning 401-272-9500, MCLAUGHLIN of the University of Arizona. extension 296. AMS Summer Institute Individuals willing to serve as organizers should be aware that the professional meeting 1983- Nonlinear functional analysis and applications, staff in the Society's Providence office will provide organized by FEI,IX BROWDER of the University full support and assistance before, during, and of Chicago. after each of these conferences. Organizers should 1984- Geometric measure theory and the calculus of also note that for all conferences, except Summer variations, organized by WILLIAM K. ALLARD of Research Conferences, it is required that the Duke University and FREDERICK J. ALMGREN, proceedings be published by the Society, and that JR. of Princeton University. proceedings of Summer Research Conferences are 1985 -Algebraic geometry, organized by DAVID frequently published. A member of the Organizing EISENBUD of Brandeis University. Committee must be willing to serve as editor of 1986- Representations of finite groups and related top­ the proceedings. ics, organized by JON ATHAN L. ALPERIN of the All suggestions must include (1) the names University of Chicago. and affiliations of proposed members and chair­ AMS-SIAM Symposium on Some man of the Organizing Committee; (2) a two- or Mathematical Questions in Biology three-page detailed outline of the subject(s) to be 1983- Muscle physiology, organized by ROBERT M. covered, including the importance, timeliness of MIURA of the University of British Columbia. the topic, and estimated attendance; (3) a list 1984-DNA sequence analysis, organized by of the recent conferences in the same or closely ROBERT M. MIURA of the University of British related areas; (4) a tentative list of names and Columbia. affiliations of the proposed principal speakers; (5) 1985- Plant biology, organized by ROBERT M. a list of likely candidates who would be invited MIURA of the University of British Columbia. to participate and their current affiliations; and 1986-Modeling circadian rhythms, organized by (6) any other observations which may affect the GAIL A. CARPENTER of Northeastern Univer­ size of the conference and the amount of support sity. required. Any suggestions as to sites and dates 1987-Models in population biology, organized by should be made as early as possible in order to ALAN HASTINGS of the University of California, allow adequate time for planning. By action of Davis. the AMS Board of Trustees, the Meetings De­ partment of the Society is responsible for the final AMS-SIAM Summer Seminar selection of the site for each conference and for 1983-Large scale computations in fluid mechanics, all negotiations with the host institution. Indi­ organized by RICHARD C. J. SOMERVILLE, viduals submitting suggestions for the conferences Scripps Institution of Oceanography. listed below are requested to recommend sites or 1984-Nonlinear systems of P DE in applied mathe­ geographic areas which would assist the Meetings matics, organized by BASIL NICOLAENKO of Los Department in their search for an appropriate Alamos National Laboratories. site. In the case of Joint Summer Research 1985- Reacting flows: Combustion and chemical re­ Conferences in the Mathematical Sciences, a one-, actors, organized by G. S. S. LUDFORD of Cornell two-, or three-week conference may be proposed. University. Refer to the accompanying box titled Topics of Current and Recent Conferences for lists of topics.

552 1988 Symposium In 1988 Joint AMS-IMS-SIAM Summer Pure Mathematics Research Conferences in the This four-day symposium in pure mathemat­ Mathematical Sciences ics usually takes place in even-numbered years, These conferences are similar in structure to those sometimes in conjunction with a spring sectional held at Oberwolfach, and represent diverse areas meeting. Since none is scheduled in 1986, how­ of mathematical activity, with emphasis on areas ever, the next such symposium is due to be currently especially active. Careful attention held in the spring of 1987 at a location to be is paid to subjects in which there is important announced. Subject to final approval by the interdisciplinary activity at present. Topics for the AMS Board of Thustees, the topic selected by fifth series of one-week conferences, being held in the Committee on Summer Institutes and Special 1986, are Mathematics in general relativity, Large Symposia is The mathematical heritage of Her­ scale data analysis via computer graphics, Time man W eyl. Other topics in recent years have been reversal of Markov processes and potential theory, Relations between combinatorics and other parts Artin's braid group, Discrete and computational of mathematics (1978); The mathematical heritage geometry, and Representation theory of Lie groups. of Henri Poincare (1980); Several complex vari­ If proceedings are published by the Society, they ables (1982); and Pseudodifferential operators and will appear as volumes in the series Contemporary Fourier integral operators with applications to par­ Mathematics. tial differential equations (1984). Proceedings are published by the Society as volumes in the series Deadline For Suggestions: February 1, 1987 Proceedings of Symposia in Pure Mathematics. Deadline For Suggestions: August 15, 1986 Call for Topics for 1988 AMS Short Course Series 1988 AMS Summer Institute The AMS Short Courses consist of a series of intro­ Summer institutes are intended to provide an ductory survey lectures and discussions ordinarily understandable presentation of the state of the art extending over a period of one and one-half days in an active field of research in pure mathematics, immediately prior to the Joint Mathematics Meet­ and usually extend over a three-week period. ings held in January and August each year. Each of Dates for a summer institute must not overlap the courses is devoted to a specific area of applied those of the Society's summer meeting, which in mathematics or to areas of mathematics used in 1988 is the AMS Centennial Celebration from the study of a specific subject or collection of prob­ August 8-12, and there should be a period of at lems in one of the physical, biological, or social sci­ least one week between them. Proceedings are ences. Topics in recent years have been Approxima­ published by the Society as volumes in the series tion Theory (January 1986), Actuarial Mathematics Proceedings of Symposia in Pure Mathematics. (August 1985), Fair Allocation (January 1985), En­ vironmental and Natural Resource Mathematics (Au­ Deadline For Suggestions: August 15, 1986 gust 1984), Mathematics of Information Processing (January 1984). Proceedings are published by the 1988 AMS-SIAM Summer Seminar Society as volumes in the series Proceedings of Sym­ posia in Applied Mathematics, with the approval of The goal of the summer seminar is to provide an the Editorial Committee. environment and program in applied mathematics Deadline for Suggestions: July 1, 1986 for Jan­ in which experts can exchange the latest ideas and uary 1988 course, and December 1, 1986 for newcomers can learn about the field. Proceedings August 1988 course. are published by the Society as volumes in the Submit suggestions to: Professor Stefan A. Burr, series Lectures in Applied Mathematics. Chairman, AMS Short Course Subcommittee, De­ Deadline For Suggestions: August 15, 1986 partment of Computer Sciences, CUNY, City Col­ lege, New York, New York 10031.

553 Special Meetings

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

1985-1986. Academic Year Devoted to Nonlinear Dif­ 2--6. Greco Calcul Forme!, Marseille, France. ferential Equations, Mittag-Leffler Institute, Djursholm, Information: A. Zeller-Meirer, CIRM, Luminy Case 916, Sweden. (January 1985, p. 89) Route Leon-Lachamp 70, F-13288 Marseille Cedex 9, 1985-1986. Special Year in Complex Analysis, Univer­ France. sity of Maryland, College Park, Maryland. (August 1985, 2--6. Mathematical Modeling, Salisbury State College, p. 522) Salisbury, Maryland. (January 1986, p. 132) 1985-1986. Special Year in Differential Geometry, Uni­ 2--6. NSF-CBMS Regional Conference on Hierarchical and versity of Illinois, Urbana-Champaign, Illinois. (October Empirical Bayes, Bowling Green State University, Bowling 1985, p. 674) Green, Ohio. 1985-1986. Special Year in Operator Theory, Indiana Information: J. H. Albert, Department of Mathematics, University, Bloomington, Indiana. (August 1985, p. 522) Bowling Green State University, Bowling Green, Ohio 43403. (Note change from March 1986, p. 366) 1985-1986. Special Year in Singularities and Algebraic Geometry, University of North Carolina, Chapel Hill, 2--6. NSF-CBMS Regional Conference on Mathematical North Carolina. (June 1985, p. 397, note change in years Modelling in the Energy and Environmental Sciences, West since previous listings.) Virginia University, Morgantown, West Virginia. 1985-1986. Year in Mathematical Logic, University of Informati.on: C. L. Irwin or I. Christie, Department of Notre Dame, Notre Dame, Indiana. (October 1985, p. 674) Mathematics, West Virginia University, Morgantown, West Virginia 26506. (Note change from March 1986, August 19, 1985-July 31, 1986. Program on Stochastic p. 366) Differential Equations and Their Applications, University of Minnesota, Minneapolis, Minnesota. (Note date change 2-6. Seventeenth Yugoslav Congress of Theoretical and from June 1985, p. 397) Applied Mechanics, Zadar, Yugoslavia. (January 1986, p. 132) October 1985--0ctober 1986. Material Instabilities in Continuum Mechanics, Heriot-Watt University, Edinburgh, 2-11. Twenty-third Symposium on Functional Equations, Scotland. (June 1985, p. 397) Gargnano, Italy. Information: C. Pagani, Dipartimento Matematica, Politec- 1986-1987. Academic Year Devoted to Algebraic Geome­ nico, 9 via Bonardi, I-20133 Milano, Italy. try, Mittag-Leffler Institute, Djursholm, Sweden. (January 1986, p. 130) 2-12. Eleventh Annual International Conference on Op­ erator Theory, Bucharest, Romania. (March 1986, p. 366) JUNE1986 3--6. IMACS International Symposium on Modeling and Simnlation of Lumped and Distributed Control, Lille, 1-5. First Japan Conference on Graph Theory and its France. (January 1986, p. 132) Applications, Tokai University, Hiratsuka, Japan. (October 4--6. Japan-United States Joint Seminar on Discrete 1985, p. 677) Algorithms and Complexity Theory, Kyoto, Japan. (March 1-8. Ninth International Conference on Analytic Func­ 1986, p. 366) tions, Maria Curie-Sklodowska University, Lublin, Poland. 4--6. 1986 National Educational Computing Conference, (January 1986, p. 132) Town and Country Hotel, San Diego, California. (October 2-4. The Second Symposium on Computational Geometry, 1985, p. 677) IBM T. J. Watson Research Center, Yorktown Heights, 5-12. International Conference on Mathematical Problems New York. (October 1985, p. 677) from the Physics of Fluids, Roma, Italy. 2-4. Workshop on Differentiability Properties of Real­ Information: G. Benfatto, Dipartimento Matematica, Uni­ Valued Functions, Universite de Montreal, Quebec, Canada. versita di Tor Vergata, Via 0. Raimondo, I-00173 Roma, (March 1986, p. 366) Italy.

554 8-12. Internationaler Workshop: Geometrische und Quan­ 16-18. Forefronts '86, Rensselaer Polytechnic Institute, titative Komplexe Analysis, Wuppertal, Federal Republic Troy, New York. of Germany. Program: Invited speakers will explore the common infras­ lnfarmatian: Workshop Komplexe Analysis, D. Lindner, tructure of computing technology upon which applica­ FB Mathematik, Univ.-GHS Wuppertal, Gau6str. 20, tions of large-scale computation are based. D-5600 Wuppertal 1, Federal Republic of Germany. lnfarmatian: H. Raveche, Dean of the School of Science, 8-13. Alpine--United States Seminar on Inverse and Ill- Rensselaer Polytechnic Institute, Troy, New York 12180- posed Problems, Saint Wolfgang, Austria. (March 1986, 3590, 518-266-6305. p. 366) 16-19. 1986 National Computer Conference, Las Vegas, 8-14. International Symposium on Topological and Geo­ Nevada. (March 1986, p. 366) metric Methods in Field Theory, Espoo, Finland. 16--20. Colloque International du CNRS en L'Honneur de lnfarmatian: J. Mickelsson, Res. Inst. theor. Phys., 20-C A. Lichnerowicz "Geometrie et Physique", Paris, France. Siltavuorenpenger, SF-00170 Helsinki, Finland. Infarmatian: M. Stefan, IRMA, 7 rue Rene Descartes, F- 67084 Strasbourg Cedex, France. 9--13. Discrete Mathematics, Salisbury State College, Salisbury, Maryland. (January 1986, p. 132) 16--20. Fifteenth International Symposium on Rarefied Gas Dynamics, Grado, Gorizia, Italy. (October 1985, 9--13. 1986 Gordon Research Conference on Theoretical p. 677) Biology and Biomathematics, Tilton School, Tilton, New Hampshire. (March 1986, p. 366) 16-20. Septieme Colloque International "Simulation d'Ecoulements par Elements Finis", Antibes, France. 9--14. Applications Harmoniques, Marseille, France. Infarmatian: INRIA, Serv. des Relations exterieures, Roc- Infarmatian: A. Zeller-Meier, CIRM, Luminy Case 916, quencourt, B.P. 105, F-78153 Le Chesnay Cedex, France. Route Leon-Lachamp 70, F-13288 Marseille Cedex 9, France. 16--20. Tenth United States National Congress of Applied Mathematics, University of Texas at Austin, Texas. (March 9-14. Symposium on Ill-Posed and Inverse Problems, 1986, p. 366) Lake St. Wolfgang, Austria. Infarmatian: H. Eng!, Kepler-Univ., A-4040 Linz, Austria. 16--20. The 1986 Johns Hopkins Mathematical Sciences Lecture Series, The Johns Hopkins University, Baltimore, 9-14. Workshop on Universal Algebra and Lattice Theory, Maryland. (March 1986, p. 366) Berkeley, California. (March 1986, p. 366) 16--21. Theorie des Nombres, Marseille, France. 9--19. Stochastic Differential Systems with Applications Infarmatian: A. Zeller-Meier, CIRM, Luminy Case 916, to Control Theory, Electrical/Computer Engineering, and Route Leon-Lachamp 70, F-13288 Marseille Cedex 9, Operations Research, University of Minnesota, Minneapo­ France. lis, Minnesota. (June 1985, p. 402) 16--4 July. Dynamique des Structures, Centre d'Etudes 9--20. First China-U.S.A. International Conference on du Breau-Sans-Nappe, Ablis, France. Graph Theory and its Applications, Jinan, Shandong Infarmatian: Secretariat des Ecoles d 'Ete, Electricite de Province, China. (March 1986, p. 366) France, 1, Avenue du General de Gaulle, 92140 Clamart, 9--August 22. Gordon Research Conferences, New Hamp­ France. Telephone: (1) 47 65 36 44. shire. 17-21. Analysis Conference, National University of Singa­ Program: Meetings on a variety of topics at various locations pore, Republic of Singapore. (October 1985, p. 677) in New Hampshire will extend the Frontiers of Science 18-20. Colloque sur les Proprietes Structurelles des by fostering a free and informal exchange of ideas Systemes Lineaires Multivariables, Paris, France. among persons actively interested in the subject under Infarmatian: M. Malabre, LAN ENSM, 1 rue de Ia Noe, discussion. For a description of the conferences and for F-44072 Nantes Cedex, France. applications to the conferences, see below. lnfarmatian: Before June 8: A. Cruickshank, University of 19--21. Second Meeting on the Mathematics of Fuzzy Rhode Island, Kingston, Rhode Island 02881-0801, 401- Systems, Napoli, Italy. 783-4011 or 401-783-3372. After June 8: A. Cruickshank, Infarmatian: A. Di Nola, Institute Matematiques, Faculte Director, Gordon Research Conferences, Colby-Sawyer d'Architecture, 3 Via Monteoliveto, I-80134 Napoli, Italy. College, New London, New Hampshire 03257, 603-526- 2()-.July 5. Third Workshop on Nonlinear Evolution 2870. Equations and Dynamical Systems, Baia Verde, Gallipoli, 12-14. Institute of Mathematical Statistics Central Re­ Italy. gional Meeting, Purdue University, West Lafayette, Indi­ Infarmatian: M. Boiti, Dip. Fis., Universita, via Arnesano, ana. (March 1986, p. 366) I-73100 Leece, Italy. 12-14. 1986 Annual Summer Meeting of the Canadian 22-29. Society of Women Engineers 1986 National Con­ Mathematical Society, Memorial University, Saint John's, vention, Sheraton Hartford and Parkview Hilton, Hartford, Newfoundland. (March 1986, p. 366) Connecticut. (October 1985, p. 678) 12-16. Workshop on Analysis, National University of 23-27. NSF-CBMS Regional Conference on Stochastic Singapore, Singapore, Republic of Singapore. Processes in the Neuro Sciences, North Carolina State Program: Lectures will be given on Interpolation of Linear University, Raleigh, North Carolina. Operators on Product Measure Spaces, Singular Inte­ Infarmatian: C. E. Smith, Department of Statistics, North gral Operators and Pseudo-Differential Operators, and Carolina State University, Raleigh, North Carolina 27695. Probabilistic and Volume Methods in the Geometry of (Note change from March 1986, p. 367) Banach Spaces by S. Igari, A. Miyachi, and G. Pisier 23-28. Colloque International de Theorie des Graphes et respectively. Combinatoire, Luminy, France. lnfarmatian: Organizing Committee, Analysis Workshop, Infarmatian: Colloque Theorie des Graphes, ER 175, c/o Department of Mathematics, National University of Sin­ M. Ulrich, C.A.M.S.-M.S.H., 54 boul. Raspail, F-75270 gapore, Singapore 0511, Republic of Singapore. Paris Cedex 06, France. 15-20. Fourth International Symposium on Statistical 23-28. La Combinatoire, Marseille, France. (January 1986, Decision Theory and Related Topics, Purdue University, p. 133) West Lafayette, Indiana. (January 1986, p. 133) 24-27. Seventh International Conference on Analysis and 15-21. Function Spaces and Applications, Lund, Sweden. Optimization of Systems, Antibes, France. (October 1985, (March 1986, p. 366) p. 678)

555 25-July 3. Secondo Corso del CIME: Mathematical Eco­ 6-13. Deuxieme Universite d'Ete sur l'Histoire des nomics, La Querceta, Montecatini Terme, Italy. Mathematiques, Toulouse, France. lriforrnJJti.on: CIME, c/o Ist. Mat. U. Dini, 67-A viale Infarmation: M. Guillemot, IREM de Toulouse, Univ. P. Morgagni, I-50134 Firenze, Italy. Sabatier, 118 route de Narbonne, F-31062 Toulouse France. 29-July 4. Ninth Dundee Conference on the Theory of Cedex, Ordinary Partial Differential Equations, Dundee, Scotland. 7-9. The Fourth IMA Conference on the Mathematical (March 1986, p. 367) Theory of the Dynamics of Biological Systems, University of Oxford, Oxford, England. (October 1985, p. 678) 3Q-July 2. Seventh European Workshop on Applications and Theory of Petri Nets, Oxford, Great Britain. 7-9. The Mathematical Theory of the Dynamics of Bio­ InforrnJJti.on: G. Cutts, Dept. Comp. Studies, Sheffield City logical Systems, Oxford, England. Polyt., Pond Str., GB-Sheffield S11WB, England. Infarmation: The Deputy Secretary, The Institute of Math­ ematics and its Applications, Maitland House, Warrior on The 3Q-July 4. NSF-CBMS Regional Conference Square, Southend-on-Sea, Essex SS1 2JY, England. Formation Space of Algebraic and Geometric Structures, University of California, Los Angeles, California. 7-11. Nonstandard Analysis and its Applications, Univer- lnforrnJJti.on: J. J. Millson, Department of Mathematics, sity of Hull, England. (January 1986, p. 133) University of California, Los Angeles, California 90024. 7-11. The Fourth International Conference on Bound­ (Note change from March 1986, p. 367) ary and Interior Layers-computational and Asymptotic 3Q-July 4. Tenth Prague Conference on Information The­ Methods, Institute of Mathematics, Siberian Branch of ory, Statistical Decision Functions and Random Processes, the U.S.S.R. Academy of Sciences, Novosibirsk, U.S.S.R. Institute of Information Theory and Automation of the (October 1985, p. 678) Czechoslovak Academy of Sciences, Prague, Czechoslo­ 7-12. Fifth Brazilian Conference on Topology, Universi­ vakia. (October 1985, p. 678) dade de Sao Paulo, Siio Carlos, Brasil. 3Q-July 6. The First KIT Mathematics Workshop, Korea Infarmation: P.F.S. Porto, Universidade de Siio Paulo, Caixa Institute of Technology (KIT), Taejon, Korea. Postal 668, 13560 Siio Carlos, SP-Brasil. Telephone: Program: The workshop will consist of series of lectures by (0162)71-2238. invited speakers and one hour special sessions by some 7-19. Abidjan Symposium of Mathematics, University of of the participants. Abidjan, Abidjan, Ivory Coast. Invited Speakers: W. Baily, D. Buchsbaum, T. Ono, K. H. 'I'Mme: Methods of Group Theory and their Applications. Dovermann, T. Petrie, C. N. Lee, and R. Lashof. Program: The morning sessions will be devoted to lecture InforrnJJti.on: J. K. Koo, Mathematics Workshop, Korea courses and the afternoon sessions to discussions and Institute of Technology, 400 Kusong-dong, Chung-gu, presentation of contributed scientific papers by partici­ Taejon, Korea 300-31. pants. S. Toure, Institute de Recherches Mathema- 3Q-July 12. Quatrleme Ecole d'Ete de Didactique des InforrnJJti.on: 08 Post Office Box 2030 Abidjan 08, Ivory Coast. Mathematlques, Orleans, France. tiques, lnforrnJJti.on: R. Douady, IREM Univ. Paris -7, 2 place 7-25. Seminaire de Mathematiques Superieures-NATO Jussieu, F-75251 Paris Cedex 05, France. Advanced Study Institute-Variational Methods in Nonlin­ ear Problems, Universite de Montreal, Montreal, Canada. 26. Ecole d'Ete du CIMPA: Lea Mathematiques 3Q-July (January 1986, p. 133) Appliquees a Ia Mechanique, Monastir, France. InforrnJJti.on: Centre intern. Math. Pures et Appl., 1 av. 7-25. Tendances dans les Architectures Normalisees de Edith Cavell, F -06000 Nice, France. Reseaux, Centre d'Etudes du Breau-Sans-Nappe, Ablis, France. InforrnJJti.on: Secretariat des Ecoles d'Ete, Electricite de JULY 1986 France, 1, Avenue du General de Gaulle, 92140 Clamart, France. Telephone: (1) 47 65 36 44. 1-2. Mathematics in Major Accident Risk Assessment, Oxford, England. (January 1986, p. 133) 8-11. Practical Bayesian Statistics, Cambridge, United Kingdom. (March 1986, p. 367) 1-3. Sixteenth Symposium on Fault-Tolerant Computing, Wien, Austria. 9-19. A Workshop on Stochastic Analysis, Silivri Center, lnforrnJJti.on: H. Kopetz, Interkonvention Hofburg, P.O. Box Silivri, Turkey. 80, A-1107 Wien, Austria. Program: First week will be devoted to lectures on Stochas­ tic Partial Differential Equations, Stochastic Calculus 1-3. Structural Failure, Product Liability and Technical of Variations (Malliavin Calculus) and an introductory Insurance, Vienna, Austria. (January 1986, p. 133) lecture on Brownian Motion and Diffusions. The second 1-4. A European Turbulence Conference, Lyon, France. week will be devoted to presentation of contributed pa­ (January 1986, p. 133) pers on topics related to the above subjects and their applications. Nonlinear Waves in Solids, Toledo, 1-4. Euromech 212: H. Korezlioglu, Ecole Normale Superieure des Spain. Infarmation: Telecommunications, 46 rue Barrault, 75634 Paris Cedex InforrnJJti.on: E. Alarcon, Univ. Politecn., E.T.S. Ingen. 13, France. industr., 2 J. Gutierrez Abascal, E-28006 Madrid, Spain. 14-17. First International Conference on Numerical Grid Differential Equations in Banach 2-4. Conference on Generation in Computational Fluid Dynamics, Landshut, Bologna, Italy. Spaces, Federal Republic of Germany. Porta IriforrnJJti.on: G. Dore, Universite de Bologna, 5 Piazza lnforrnJJti.on: J. Hauser, Dept. mech. and electr. Eng., Col­ Italy. S. Donato, I-40127 Bologna, lege, 32-34 Stetheimerstr., D-8300 Landshut, Federal 2-12. LMS Symposium on Non-classical Continuum Me­ Republic of Germany. chanics: Abstract Techniques and Applications, University 14-18. Chaotic Motion in Nonlinear Dynamic Systems, of Durham, Durham, England. (November 1985, p. 812) Udine, Italy. 3-5. Fourth Geometry Symposium at Siegen on Discrete IriforrnJJti.on: Segr. CISM, Palazzo del Torso, 18 Piazza and Combinatorial Geometry, Universitat Siegen, Siegen, Garibaldi, I-33100 Udine, Italy. Federal Republic of Germany. (March 1986, p. 367) 14-18. Introduction to Working Statistics, Ohio State 5-£. Incontro di Analisi Funzionale, Firenze, Italy. University, Columbus, Ohio. InforrnJJti.on: C. Franchetti, Ist. Mat. appl., 3 viaS. Marta, Program: This short course presents the basic concepts I-50139 Firenze, Italy. underlying statistical data analysis including necessary

556 probability concepts, effective methods of collecting data, 27-August 1. International Conference on Algebraic and simple descriptive techniques for initial interpreta­ Topology, Humboldt State University, Arcata, California. tion of data. (March 1986, p. 367) Inforrnat:ian: L. B. Larew, Ohio State University, Depart­ 27-August 1. Thirteenth International Biometric Confer­ ment of Conferences and Institutes, Office of Continuing ence, Seattle, Washington. (June 1985, p. 402) Education, Fawcett Center for Tomorrow, 2400 Olen­ tangy River Road, Columbus, Ohio 43210, 614-422-8571. 27-August 2. Eugene Strens Memorial Conference on Intuitive and Recreational Mathematics and its History, 14-18. NSF-CBMS Regional Conference on Harmonic The University of Calgary, Calgary, Alberta, Canada. Analysis and Probability, Depaul University, Chicago, (March 1986, p. 367) Illinois. Inforrnat:ian: R. L. Jones, Department of Mathematics, 28-August 1. Applied Discrete Data Analysis, Ohio State De Paul University, Chicago, Illinois 60614. (Note change University, Columbus, Ohio. from March 1986, p. 367) Program: This short course introduces various statistical 14-19. Logic Colloquium 1986, University of Hull, Hull, methods for analyzing data organized in tables of count. England. (March 1986, p. 367) InfOT1JULtion: L. B. Larew, Ohio State University, Depart­ ment of Conferences and Institutes, Office of Continuing 14-25. Partial Differential Equations, Eighth Latin Amer­ Education, Fawcett Center for Tomorrow, 2400 Olen­ ican School of Mathematics, Instituto de Matematica Pur a tangy River Road, Columbus, Ohio 43210, 614-422-8571. e Aplicada, Rio de Janeiro, Brazil. (October 1985, p. 678) 28-August 1. NSF-CBMS Regional Conference on Prob­ An Interna­ 14-26. Oscillation, Bifurcation and Chaos: abilistic Methods in Combinatorics, Fort Lewis College, Conference in Differential Equations, The University tional Durango, Colorado. Canada. (November 1985, p. 812) of Toronto, Toronto, Inforrnat:ian: R. A. Gibbs, Department of Mathematics, Fort 15-17. Symposium on Recent Advances in Simulation of Lewis College, Durango, Colorado 81301. (Note change Complex Systems, Tokyo, Japan. (January 1986, p. 133) from March 1986, p. 368) 15-19. Treizieme Colloque Automates, Langages et Pro­ 28-August 1. Conference on Singularities, University of grammation, Rennes, France. Iowa, Iowa City, Iowa. (January 1986, p. 133) lnforrnat:ian: E. Lebret, IRISA-INRIA, Campus de Beaulieu, F -35042 Rennes Cedex, France. 28-August 1. International Conference on Harmonic Mea­ sure and Potential Theory in Euclidean Spaces, University 16-25. Eighth International Congress on Mathematical of Toledo, Toledo, Ohio. (March 1986, p. 368) Physics, Marseille, France. (March 1986, p. 367) 28-August 1. Operator Algebras Conference, University Introductory Biostatistics, Ohio State University, 21-25. of California at Santa Barbara, Santa Barbara, California. Ohio. Columbus, (March 1986, p. 368) Program: This short course provides an introduction to basic concepts and techniques in statistics applied to life 28-August 1. The Seventh International Conference on sciences. Nonlinear Analysis and Applications, The University of InjOT1JULtion: L. B. Larew, Ohio State University, Depart­ Texas, Arlington, Texas. (October 1985, p. 678) ment of Conferences and Institutes, Office of Continuing 28-August 2. Extremal Families of Sets, Designs and Fi­ Education, Fawcett Center for Tomorrow, 2400 Olen­ nite Geometries, Universite de Montreal, Quebec, Canada. tangy River Road, Columbus, Ohio 43210, 614-422-8571. (March 1986, p. 368) 21-25. Ninth Escola de Algebra, University of Brasilia, 29-August 1. Conference on Continuous Time, Fractional Brasilia, Brasil. (March 1986, p. 367) and Multiobjective Programming, St. Lawrence University, 21-25. Third International Conference on Logic Program­ Canton, New York. (November 1985, p. 813) ming, London, England. 30-August 1. Computers and Mathematics, Stanford lnforrnat:ian: K. Clark, Imp. Coli. of Sc. and Techn., London, University, Stanford, California. (March 1986, p. 368) England. 21-26. International Congress on Computational and Ap­ 30-August 1. Conference on Universal Algebra and Lattice plied Mathematics, University of Leuven, Belgium. (Octo­ Theory, National Institutes of Health, Bethesda, Maryland. ber 1985, p. 678) (October 1985, p. 678) 21-31. Workshop on Functional Integration with Empha­ 30-August 2. Computers in Mathematics, Stanford Uni­ sis on the Feynman Integral, Universite de Sherbrooke, versity, Palo Alto, California. (November 1985, p. 813) Sherbrooke, Quebec, Canada. Program: Two expository lectures of six hours each, ten AUGUST 1986 lectures of one hour, and contributed papers of thirty minutes. The proceedings should appear in a special issue 2. United States Commission on Mathematical Instruction of the Rendiconti del Circola Matematico di Palermo. Pre-Congress Series of Invited Survey Talks, New York P. Morales and J. Dubois, Departement de Inforrnat:ian: City Technical College, Brooklyn, New York. (March 1986, Mathematiques et d'Informatique, Universite de Sher­ p. 368) brooke, Sherbrooke, Quebec, Canada J1K 2Rl. Tele- · phone: 819-821-7035 and 819-821-7032 respectively. 3-11. International Congress of Mathematicians, Berkeley, 22-26. Conference on Constructive Function Theory, Uni­ California. (February 1984, p. 159) versity of Alberta, Edmonton, Alberta, Canada. (August 4-8. Workshop on Micromechanical Aspects of Creep 1985, p. 525) Fracture in Metals and Polymers, Mathematical Sciences 23-31. 1986 Corvallis Conference: Quadratic Forms and Institute, Cornell University, Ithaca, New York. Real Algebraic Geometry, LeSell Stuart Center, Ore­ lnfOT1JULtion: Program Coordinator (Hui or Phoenix), Math- gon State University, Corvallis, Oregon. (November 1985, ematical Sciences Institute, Cornell University, Caldwell p. 812) Hall, Ithaca, New York 14853. 26-31. Fourteenth Canadian Symposium on Operators 6-9. First IMACS Symposium on Computational Acous­ and Operator Algebras, University of Victoria, Victoria, tics, Yale University, New Haven, Connecticut. (January British Columbia. (March 1986, p. 367) 1986, p. 134) 27-30. Tenth Summer Symposium on Real Analysis, Uni­ 11-16. Second International Conference on Teaching versity of British Columbia, Vancouver, British Columbia, Statistics, University of Victoria, Victoria, British Colum­ Canada. (November 1985, p. 812) bia, Canada. (January 1985, p. 93)

557 12-14. SIAM Conference on Linear Algebra in Signals, SEPTEMBER 1986 Systems and Control, Park Plaza Hotel, Boston, Mas­ sachusetts. (March 1986, p. 368) 1-3. Elft Symposium iiber Operations Research, Darm­ 12-16. Structure, Coherence and Chaos in Dynamical stadt, Federal Republic of Germany. Systems, The Technical University of Denmark, Lyngby, Inf0'f'171.0i:ion: Elft SOR, FB Mathematik, TH Darmstadt, Denmark. (March 1986, p. 368) SchloBgartenstr. 7, D-6100 Darmstadt, Federal Republic of Germany. 13-16. Second International Conference on Fibonacci 1-5. Analyse Harmonique, Marseille, France. Numbers and their Applications, San Jose State University, Inf0'f'171.0i:ion: A. Zeller-Meier, CIRM, Luminy Case 916, San Jose, California. Route Leon-Lachamp 70, F-13288 Marseille Cedex 9, Program: Talks on branches of mathematics and science re­ France. lated to the Fibonacci numbers and their generalizations will be given. 1-5. COMPSTAT 1986 Seventh Symposium on Compu­ In/O'f'17I.Oi:ion: C. Long, Department of Mathematics, Wash­ tational Statistics, Universita "La Sapienza," Rome, Italy. ington State University, Pullman, Washington 99163 (March 1986, p. 369) or G. Bergum, The Fibonacci Quarterly, Department 1-5. Representations des Groupes et Analyse Complexe of Mathematics, South Dakota State University, Post (Journees S.M.F.), Marseille, France. Office Box 2220, Brookings, South Dakota 57007-1297. InJO'f'17I.Oi:ion: P. Torasso et M. Rais, Dept. de Matbematiques, 13-29. Workshop on Jordan Structures, University of Univ. de Poitiers, 40 av. du Recteur Pineau, F-86022 Ottawa, Ottawa, Ontario, Canada. Poitiers Cedex, France. Program: Informal lectures and discussion sessions. 1-5. Tenth World Computer Congress, Trinity College, lnf0'f'171.0i:ion: E. Neher or M. Racine, Department of Math- Dublin, Ireland. (March 1986, p. 369) ematics, University of Ottawa, Ottawa, Ontario K1N 9B4, Canada. 1-5. Theorie des Nombres, Marseille, France. (March 1986, p. 369) 18-22. Bifurcation, Analysis Algorithms Applications, 2-3. Mathematical Modelling of Non-Destructive Evalua­ Universitat Dortmund, Dortmund, Federal Republic of Germany. (October 1985, p. 679) tion, Cambridge, Great Britain. InJO'f'17I.Oi:ion: The Secretary and Registrar, IMA, Maitland 18-29. Opening Workshop on Basic Methods of Numerical House, Warrior Square, Southend-on-Sea, GB- Essex Analysis and Introduction to State of the Art Research, SS1 2JY, England. University of Minnesota, Minneapolis, Minnesota. (March 3-5. Symposium on Control Theory, Somerville Col­ 1986, p. 368) lege, University of Oxford, Oxford, England. (March 1986, 19--24. Third International Conference on Applied Systems p. 369) Research, Information, and Cybernetics and International 3-6. Workshop on Sign-Pattern Analysis of Linear Symposium on Mathematics and Logie, Baden-Baden, and Nonlinear Systems, Universite de Montreal, Quebec, Federal Republic of Germany. (March 1986, p. 369) Canada. (March 1986, p. 369) 2Q-September 6. Seizieme Ecole d'Ete de Caleul des 4-7. Polish Symposium on Interval and Fuzzy Mathe­ Probabilites, Saint-Flour, France. matics, Technical University of Poznan, Poznan, Poland. Inf0'f'171.0i:ion: P. L. Henniquin, B.P. 45, F-63170 Aubiere, (October 1985, p. 679) France. 7-9. The Mathematics of Surfaces, University College, 23-25. Optimisation and Simulation of Large Seale Sys­ Cardiff, United Kingdom. (January 1986, p. 134) tems, Reading, United Kingdom. (January 1986, p. 134) 8-13. Algebra-Tagung Halle 1986, Martin Luther Uni­ 23-26. International Symposium on Probability and versitat, German Democratic Republic. (October 1985, Bayesian Statistics, Innsbruck, Austria. (January 1986, p. 679) p. 134) 8-14. First World Congress of the Bernoulli Society 25-29. Congres Franco Chilien de Matbematiques Ap­ for Mathematical Statistics and Probability, Tashkent, pliquee&, Santiago, Chile. U.S.S.R. (October 1985, p. 679) Inf0'f'171.0i:ion: C. Carasso, Faculty of Sciences, 23 rue du 11-13. Spinors in Physics and Geometry, International Docteur Paul Michelon, 42023 Saint-Etienne cedex 2, Centre for Theoretical Physics, Trieste, Italy. France or R. Correa, Universidad de Chile, Departa­ InJO'f'17I.Oi:ion: J. Eells, International Centre for Theoretical mento Matematicas, Casilla 170 Correo 3, Santiago, Physics, Post Office Box 586, Miramare, Strada Costiera Chile. 11, 34100 Trieste, Italy. Telephone: 2240-1. 25-29. Sixth Prague Topological Symposium, Prague, 12-17. International Conference on Graphs, Steiner Sys­ Czechoslovakia. (June 1985, p. 402) tems and their Applications, Santa Tecla, Catania, Italy. 25-30. Analyse Harmonique, Marseille, France. (March Program: The program will consist of invited lectures and 1986, p. 369) contributed papers (talks). Topics include graph theory, block-designs theory, their applications. 25-September 13. Microprogram on Nonlinear Diffusion InJO'f'17I.Oi:ion: Professor M. Gionfriddo, Dipartimento di Equations and their Equilibrium States, Mathematical Matematica, Universita, viale A. Doria numero 6, 95125 Sciences Research Institute, Berkeley, California. (October Catania, Italy. 1985, p. 679) 14-20. Nonlinear Random Vibration, Oberwolfach, Fed­ 25-September 20. Ecole d'Automne 1986 du CIMPA: eral Republic of Germany. "L' Analyse Fonetionelle et Applications", Nice, France. Infarmation: M. Barner, Institutsdirektor, Geschaftsstelle, lnf0'f'171.0i:ion: CIMPA, 1 av. Edith Cavell, F-06000 Nice, Alberst.rasse 24, D-7800 Freiburg, Federal Republic of France. Germany. 31-September 6. International Meeting on Ring Theory, 14-20. Seehst Pannonisehes Symposium iiber Mathema­ Grenada, Spain. tisehe Statistik, Bad Tatzmannsdorf, Austria. Inf0'f'171.0i:ion: B. Torrecillas, dept. de Algebra y Funda­ Infarmation: W. Wertz, Inst. fiir Statistik u. Wahrscheinlich­ mentos, Fac. de Ciencias, Univ. de Grenada, E-18071 keitsth., TU Wien, Wiedner Hauptstr. 8-10, A-1040 Grenada, Spain. Wien, Austria.

558 15-19. International Conference on Stochastic Program­ 29-0ctober 2. Wirtschaftsmathematik in Beruf und Aus­ ming, Charles University, Prague, Czechoslovakia. bildung, Klagenfurt, Austria. Topics: Stochastic programming including related parts of lnfamwtian: Fiinft Kiirtner Symposium fiir Didaktik der mathematical statistics, probability theory and math­ Mathematik, lnst. fiir Mathematik, Univ. fiir Bildung­ ematical programming; applications of stochastic pro­ swiss., Universitatsstr. 65--67, A-9022 Klagenfurt, Aus­ grams and experience with their numerical solution. tria. lnfamwtian: T. Cipra, Department of Statistics, Charles University, Sokolovska 83, 186 00 Prague, Czechoslo­ OCTOBER 1986 vakia. 1--4. Concurrence Imparfaite et Modeles de Marche, 15-19. Workshop on Global Differential Geometry, Inter­ Luminy, France. national Centre for Theoretical Physics, Trieste, Italy. lnfamwtian: A. Zeller-Meier, CIRM, Luminy Case 916, lnfamwtian: J. Eells, International Centre for Theoretical Route Leon-Lachamp 70, F-13288 Marseille Cedex 9, Physics, Post Office Box 586, Miramare, Strada Costiera France. 11, 34100 Trieste, Italy. Telephone: 2240-1. 3-4. The Fourteenth Annual Mathematics and Statistics 15-20. Journees de Probability, Marseille, France. Conference, Miami University, Oxford, Ohio. lnfamwtian: A. Zeller Meier, CIRM, Luminy Case 916, Theme: Discrete Mathematics. Route Leon-Lachamp 70, F-13288 Marseille Cedex 9, Speakers: R. Graham, A. Tucker, and H. Wilf. France. lnf=tion: G. Gilbert, Department of Mathematics and 15-26. Workshop on Computational Fluid Dynamics Statistics, Miami University, Oxford, Ohio 45056, 513- and Reacting Gas Flows, Institute for Mathematics and 529-5818. its Applications, University of Minnesota, Minneapolis, 5-9. International Symposium on Information Theory, Minnesota. (March 1986, p. 369} University of Michigan, Ann Arbor, Michigan. (March 1986, p. 363} 17-19. The Impact of Mathematical Analysis on the Solution of Engineering Problems, University of Maryland, 9-11. Interdisciplinary Conference on Inference, 0 hio College Park, Maryland. (March 1986, p. 369} University, Athens, Ohio. Program: The program will include tutorials and con­ 18-19. Second Catalan International Symposium on tributed papers. The topics should deal with inference, Statistics, Barcelona, Spain. as it applies to computer science, linguistics, mathemat­ lnfamwtian: Seer. Symp. Cons. Inform., 1 Doc. de Catalunya, ics, philosophy, and psychology. 187 Urgell, E-08036 Barcelona, Spain. Infamwtian: M. Swardson, Department of Mathematics, 21-0ctober 4. Mathematics and Computer Science in Ohio University, Athens, Ohio 45701, 614-594-6603. Medical Imaging, II Ciocco, Italy. 13-November 29. Topology and Number Theory, Univer­ ln/amwtian: M. A. Viergever, Delft Univ. of Techn., Dept. site de Montreal, Quebec, Canada. (March 1986, p. 369} of Mathematics & Informatics, P.O. Box 356, NL-2600 15-18. Convegno su Ipergruppi, Altre Strutture Multi­ AJ Delft, Netherlands. voche e Applicazioni, Udine, Italy. 22-26. Euromech 216: Integrable Systems in Nonlinear lnf=tion: P. L. Corsini, 1st. Mat., 3 via Mantrice, I-33100 Analytical Mechanics, Leeds, England. Udine, Italy. lnfamwtian: A. P. Fordy, Centre for nonlinear studies, Univ. 17-18. Fifteenth Annual Midwest DiftTerential Equations of Leeds, GB- Leeds L52 9JT, England. Conference, Marquette University, Milwaukee, Wisconsin. 22-26. IFAC-IMACS Symposium on Simulation of Control (March 1986, p. 369} Systems, Vienna, Austria. (January 1986, p. 134} 2Q-24. Les Sondages, Marseille, France. 22-26. Theorie Analytique des Nombres, Luminy, France. lnfamwtian: A. Zeller Meier, CIRM, Luminy Case 916, lnfamwtian: A. Zeller-Meier, CIRM, Luminy Case 916, Route Leon-Lachamp 70, F-13288 Marseille Cedex 9, Route Leon-Lachamp 70, F-13288 Marseille Cedex 9, France. France. 24-25. Second Eastern Small College Computing Confer­ 22-27. Orthogonal Polynomials and Their Applications, ence, University of Scranton and the Hilton at Lackawanna Segovia, Spain. (November 1985, p. 813} Station, Scranton, Pennsylvania. (March 1986, p. 369} 23-25. Optimisation and Simulation of Large Scale Sys­ 24-25. The Sixth Southeastern-Atlantic Regional Con­ tems, University of Reading, Reading, Berkshire, United ference on Differential Equations, Clemson University, Clemson, South Carolina. Kingdom. (January 1986, p. 134} Program: Lectures will be given by J. Burns, G. Fix, and 23-26. International Symposium on Probability and J. Herod. In addition, there will be sessions for twenty Bayesian Statistics, Innsbruck, Austria. minute contributed talks. lnfamwtian: Prof. R. Viertl, Inst. fiir Statistik und Wahr· lnfamwtian: T. Proctor, Mathematical Sciences Depart­ scheinlichkeitsth., TU Wien, A-1040 Wien, Austria. ment, Clemson University, Clemson, South Carolina 26-28. Congress on Variational Methods in Differential 29634-1907, 803-656-3434. Problems, Trieste, Italy. lnjO'T"TTUltion: E. Mitidieri, 1st. Mat., Univ., 1 piazzale Eu- NOVEMBER 1986 ropa, 1-34127 Trieste, Italy. 2--6. Fall Joint Computer Conference, Dallas, Texas. 29-30. Computers in Mathematical Research, University (March 1986, p. 370} College, Cardiff, United Kingdom. (January 1986, p. 134} 3-7. Workshop on Numerical Algorithms for Modern Par­ 29-0ctober 1. Fifth Meeting on Mathematical Physics, allel Computer Architectures, Institute for Mathematics University of Coimbra, Coimbra, Portugal. and its Applications, University of Minnesota, Minneapolis, Topics: Differential geometry, variational methods, dif­ Minnesota. (March 1986, p. 370} ferential equations in mathematical physics, analytic 9-15. Austrian Symposium on History of Mathematics, mechanics, and field theories. Neuhofen an der Ybbs, Austria. Call for Summaries: Those interested in contributing a Topics: "Mathematics-stimulated or stimulating? and "On short talk should submit a summary to the Organizing the interaction between 'pure' and 'applied' mathematics Committee by June 30, 1986. Send summary to the through the ages." address below. lnf=tion: Ch. Binder, Inst. fiir techn. Mathematik, Tech­ lnfamwtian: Grupo de Ffsica-Matematica, Departamento nische Universitat Wien, Wiedner Hauptstrasse 6-10, de Matematica, Apartado 3008, 3000 Coimbra, Portugal. A-1040 Wien, Austria.

559 1G-15. Workshop on Artin £-functions and Related 15-July 3. Microprogram on Commutative Algebra, Math­ Topics, Universite de Montreal, Quebec, Canada. (March ematical Sciences Research Institute, Berkeley, California. 1986, p. 370) (October 1985, p. 679) 1G-28. Workshop on Representation Theory of Lie Groups, 23-26. Sixth IMACS International Symposium on Com­ International Centre for Theoretical Physics, Trieste, Italy. puter Methods for PDE's, Lehigh University, Bethlehem, Inforrootion: J. Eells, International Centre for Theoretical Pennsylvania. (January 1986, p. 134) Physics, Post Office Box 586, Miramare, Strada Costiera 23-27. International Conference on Generalized Func­ 11, 34100 Trieste, Italy. Teleph~ne: 2240-1. tions, Convergence Structures and Their Applications, 18-22. Didactique des Mathernatiques, Marseille, France. Dubrovnik, Yugoslavia. Inforrootion: A. Zeller Meier, CIRM, Luminy Case 916, Inforrootion: Inst. of Math., GFCA-87, Dr Ilije DjuriCica 4, Route Loon-Lachamp 70, F-13288 Marseille Cedex 9, YU-21000 Novi Sad, Yugoslavia. France. 29--July 4. Joint IMA-GAMM-SIAM-SMAI Conference on First Joint International Conference on Industrial 24-28. Geometrie Symplectique, Marseille, France. and Applied Mathematics, Paris, France. (January 1986, Inforrootion: A. Zeller Meier, CIRM, Luminy Case 916, p. 134) Route Leon-Lachamp 70, F-13288 Marseille Cedex 9, France. JULY 1987 13-17. Fourteenth International Colloquium on Automata, DECEMBER 1986 Languages, and Programming, University of Karlsruhe, Federal Republic of Germany. (March 1986, p. 370) 3-5. Forty Second Annual Conference on Applied Statis­ 19--24. Conference on Potential Theory, Charles Univer­ tics, Atlantic City, New Jersey. {March 1986, p. 370) sity, Prague, Czechoslovakia. 3--6. Theory of Robots Symposium, Vienna, Austria. Program: One-half and one hour lectures will be given (January 1986, p. 134) on various aspects of potential theory including the applications of potential theory in other areas. 15-17. Cryptography and Coding, Cirencester, Glos., Inforrootion: Potential Theory, Faculty of Mathematics and England. Physics, Charles University, Sokolovska 83, 186 00 Praha Inforrootion: The Secretary and Registrar, IMA, Maitland 8, Czechoslovakia. House, Warrior Square, Southend-on-Sea, GB- Essex SS1 2JY, England. AUGUST 1987 ****** 9--15. International Conference on Abelian Groups, Perth, Western Australia. JANUARY 1987 InfamwiJian: P. Schultz, Department of Mathematics, Uni­ versity of Western Australia, Nedlands, Western Aus­ January-May. Nonlinear PDE's, Brigham Young Univer­ tralia 6009, Australia. Telephone: 380 3838. sity, Provo, Utah. 17-20. International Conference on Rings, Modules, and Program: During the winter semester N. Alikakos, G. Cagi­ Radicals, Hobart, Tasmania. nalp, and P. Fife will lead seminars in topics related to Infarmatinn: B. Gardner, Mathematics Department, Uni­ continuum models of phase transistion, singular pertur­ versity of Tasmania, General Post Office Box 252C, bations and reaction-diffusion equations. Hobart, Tasmania 7001, Australia. Inforrootion: P. Bates, Department of Mathematics, 24-28. Conference on Differential Equations "Equadiff Brigham Young University, Provo, Utah 84602. '87", Democritus University of Thrace, Greece. (January 1986, p. 134) FEBRUARY 1987 24-28. International Conference on Web Geometry and Related Fields, Szeged University, Szeged, Hungary. Organizing Committee: S. S. Chern (Honorary Chairman), 13-18. Annual Meeting, American Association for the M.A. Akivis, V. V. Goldberg, P. Nagy, K. Strambach. Advancement of Science, Chicago, Illinois. (March 1986, Infarmatinn: P. Nagy, Bolyai Institute, Szeged University, p. 370) Aradi Vertanuk tere 1, H-6720 Szeged, Hungary. 24-28. Second International Conference on Combinatorial MARCH1987 Mathematics and Computing, Canberra, Australia. (March 1986, p. 370) 22-25. Institute of Mathematical Statistics Central Re­ 24-28. Sixth National Conference on Artificial Intelli­ gional Meeting, Dallas, Texas. {March 1986, p. 370) gence, Seattle, Washington. (March 1986, p. 370) 23-30. NSF-CBMS Conference on Mathematical Statis­ tics, Ohio State University. SEPTEMBER 1987 InfamwiJian: S. Leurgans, Department of Mathematics, 9--12. lnternationale Konferenz iiber Anwendungen und Ohio State University, Columbus, Ohio 43210. Modellbildung im Mathematikunterricht, Kassel, Federal Republic of Germany. InfamwiJian: W. Blum, Univ. GHS Kassel, FB Mathematik, MAY 1987 Heinrich-Plett-Str. 40, D-3500 Kassel, Federal Republic of Germany. 25-29. The Ninth International Symposium on Noise 13-19. Journees Arithmetiques 1987, Ulm, Federal Re­ in Physical Systems, Universite de Montreal, Quebec, Canada. (March 1986, p. 370) public of Germany. Infarmatinn: E. Wirsing, Univ. Ulm, Abt. Mathematik II, Postfach 4066, D-7900 Ulm, Federal Republic of JUNE1987 Germany. 20-26. DMV-Jahrestagung 1987, Berlin, Federal Republic 8-19. Singapore Group Conference Theory, National of Germany. University of Singapore, Republic of Singapore. (March InfamwiJian: D. Sauer, DMV, Albertstr. 24, D-7800 Freiburg, 1986, p. 370) Federal Republic of Germany.

560 NEW AMS PUBLICA liONS

applications of interpolation group theory to rings and C* -algebras via ordered Ko. A discussion of PARTIALLY ORDERED ABELIAN some open problems in interpolation groups and GROUPS WITH dimension groups concludes the book. INTERPOLATION Of interest, of course, to researchers in ordered K. R. Goodearl algebraic structures, the book will also be a valuable (Mathematical Surveys and Monographs, source for researchers seeking a background in Volume 20) interpolation groups and dimension groups for applications to such subjects as rings, operator In the past decade a new branch of ordered algebras, topological Markov chains, positive algebraic structures has grown, motivated by polynomials, compact group actions, or other K-theoretic applications and mainly concerned areas where ordered Grothendieck groups might be with partially ordered abelian groups satisfying the useful. Riesz interpolation property. This monograph is 1980 Mathematics Subject Classifications: the first source in which the algebraic and analytic 06F20. 46A55. 19K14. 19A49. 16A30 16A54. 46L05. 46L80 aspects of these interpolation groups have been ISBN 0-8218-1520-2. LC 86-7876 integrated into a coherent framework for general ISSN 0076-5370 reference. The author provides a solid foundation 358 pages (hardcover). due June 1986 in the structure theory of interpolation groups Individual member 141. List price $68. Institutional member $54 and dimension groups (directed unperforated To order. please specify SURV /20N interpolation groups), with applications to ordered K-theory particularly in mind. Although interpolation groups are defined as purely algebraic structures, their development has been GROUP RINGS, CROSSED strongly influenced by functional analysis. This PRODUCTS AND GALOIS cross-cultural development has left interpolation groups somewhat estranged from both the THEORY algebraists, who may feel intimidated by compact Donald S. Passman convex sets, and the functional analysts, who may (CBMS Regional Conference Series, Number 64 feel handicapped by the lack of scalars. This book, Supported by the National Science Foundation) requiring only standard first-year graduate courses in algebra and functional analysis, aims to make the For readers with a basic graduate level background subject accessible to readers from both disciplines. in algebra, these ten articles provide a readable introduction to three major interrelated subjects High points of the development include the of noncommutative algebra. The theme is the following: characterization of dimension groups interplay between group theory and ring theory, as direct limits of finite products of copies of dealing specifically with group rings, crossed the integers; the double-dual representation of products, and the Galois theory of rings. The an interpolation group with order-unit via affine author has carefully included most definitions, continuous real-valued functions on its state space; to keep the required background minimal. the structure of dimension groups complete with Furthermore, each article contains a selection respect to the order-unit norm, as well as monotone of results on the given topic, a limited number sigma-complete dimension groups and dimension of proofs or sketches, and at least a few open groups with countably infinite interpolation; and problems. an introduction to the problem of classifying extensions of one dimension group by another. The Contents book also includes a development of portions of ll -methods in group rings the theory of compact convex sets and Choquet The Jacobson radical of group rings simplices, and an expository discussion of various Zero divisors in group rings (continued)

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561 Polycyclic group rings The L 1-modification theorem Crossed products of finite groups Approximation by inner functions Crossed products of infinite groups The LSC property of H= Computing the symmetric ring of quotients Max-sets and nonapproximation theorems Galois theory and crossed products Inner maps Galois theory of prime rings A Lusin type theorem for A(B) Rings and fixed rings Continuity on open sets of full measure The closure of in 1980 Mathematics Subject Classifications: A( B) (LHjP(B) 16A27, 16A74 Open problems ISBN 0-8218-0714-5, LC 86-1177 1980 Mathematics Subject Classifications· ISSN 0160-7642 32A07, 32A35, 32A40, 32H99 80 pages (softcover), due May 1986 ISBN 0-8218-0713-7, LC 86-1205 All individuals S7, List price $12 ISSN 0160-7642 To order, please specify CBMS/64N 96 pages (softcover), due May 1986 All individuals SB. List price $13 To order, please specify CBMS/63N

NEW CONSTRUCTIONS OF IN FUNCTIONS HOLOMORPHIC EXTREMAL GRAPH THEORY THE UNIT BALL OF en EMPHASIS ON Walter Rudin WITH (CBMS Regional Conference Series, Number 63 PROBABILISTIC METHODS Supported by the National Science Foundation) Bela Bollobas (CBMS Regional Conference Series, Number 62 The starting point for the research presented Supported by the National Science Foundation) in this book is A. B. Aleksandrov's proof that nonconstant inner functions exist in the unit ball Problems in extremal graph theory have B of en. The construction of such functions has traditionally been tackled by ingenious methods been simplified by using certain homogeneous which made use of the structure of extremal polynomials discovered by Ryll and Wojtaszczyk; graphs. In this book, an update of his 1978 book this yields solutions to a large number of problems. Extremal Graph Theory, the author focuses on the more recent trend towards probabilistic methods. The author has organized the lectures, presented He demonstrates both the direct use of probability at a CBMS Regional Conference held at Michigan theory and, more importantly, the fruitful adoption State University in June 1985, into a body of of a probabilistic frame of mind when tackling main results discovered in the preceding four years in line extremal problems. this field, simplifying some of the proofs and generalizing some results. The book also contains Essentially self-contained, the book does not merely results that were obtained by Monique Hakina, catalog results, but rather includes considerable Nessim Sibony, Erik Lpw and Paula Russo. Some discussion on a few of the deeper results. The of these, surprisingly, are new even in one variable. author addresses pure mathematicians, especially combinatorialists and graduate students taking An appreciation of techniques not previously used graph theory, as well as theoretical computer in the context of several complex variables will scientists. He assumes a mature familiarity with reward the reader who is reasonably familiar with combinatorial methods and an acquaintance with holomorphic functions of one complex variable and basic graph theory. The book is based on the with some functional analysis. NSF-CBMS Regional Conference on Graph Theory Contents held at Emory University in June, 1984. The pathology of inner functions Contents RW-sequences Subdivisions Approximation by £-polynomials Contractions The existence of inner functions Small graphs of large girth Radial limits and singular measures Large graphs of small diameter £-functions in the Smirnov class Cycles in dense graphs Almost semicontinuous functions, and Iu + vfl The evolution of random graphs Approximation in L 1/ 2 (continued)

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562 The size Ramsey number of a path 1980 Mathematics Subject Classifications: 58B10, 58C25, 46E15, 41A65 Weakly saturated graphs ISBN 0-8218-5059-8, LC 86-3510 List colourings ISSN 0271-4132 136 pages (softcover), due June 1986 Mathematics Subject Classifications: 1980 Individual member 111, List price $18, 05C35, 05C25, 05C10, 05C40, 05C65, 60C05 Institutional member $14 ISBN 0-8218-0712-9, LC 85-30670 To order, please specify CONM/54N ISSN 0160-7642 72 pages (softcover), due May 1986 All Individuals 17, List price $12 To order, please specify CBMS/62N THE SELBERG TRACE FORMULA AND RELATED DIFFERENTIAL ANALYSIS IN TOPICS INFINITE DIMENSIONAL Dennis A. Hejhal, Peter Sarnak, SPACES and Audrey Anne Terras, Editors Kondagunta Sundaresan and Srinivasa (Contemporary Mathematics, Volume 53) Swaminathan, Editors volume provides both a good introduction 54) This (Contemporary Mathematics, Volume to current work and a collection of survey articles This volume focuses on developments made in the on the Selberg trace formula and its applications. past two decades in the field of differential analysis As the proceedings of a joint AMS-IMS-SIAM in infinite dimensional spaces. New techniques such Summer Research Conference, these twenty papers as ultraproducts and ultrapowers have illuminated represent a diverse set of attitudes toward the the relationship between the geometric properties trace formula, including those from mathematical of Banach spaces and the existence of differentiable physics, classical analytic and modern adelic functions on the spaces. The wide range of topics number theory, modern geometric representations, covered also includes gauge theories, polar subsets, and classical harmonic analysis. partial approximation theory, group analysis of required varies considerably and actions on The background differential equations, inequalities, from introductory papers to those requiring infinite groups. familiarity with representation theory, the theory of Addressed to both the expert and the advanced automorphic forms. and number theory. graduate student, the book requires a basic Contents knowledge of functional analysis and differential Jeffrey Adams and Joseph F. Johnson, topology. Endoscopic groups and stable packets of certain Contents derived functor modules M. S. Berger, The impact of gauge theories on Dorothy Wallace Andreoli, A preliminary nonlinear infinite dimensional analysis version of the Selberg trace formula for Sean Dineen, Polar subsets in infinite dimensional PSL(3, 1) x PSL(3, R)/S0(3, R) spaces-small sets in large spaces James Arthur, Rebecca A. Herb, and Paul M. P. Heble, Approximation of differentiable S. Sally, Jr., The Fourier transform of weighted functions on a Hilbert space. II orbital integrals on SL(2, R) C. C. A. Sastri, Group analysis of some partial Daniel Bump, Solomon Friedberg, and Dorian differential equations arising in applications Goldfeld, Poincare series and Kloosterman sums Mau-Hsiang Shih and Kok-Keong Tan, Minimax J.-M. Deshouillers and H. lwaniec, The inequalities and applications non-vanishing of Rankin-Selberg zeta-functions at T. N. Subramanian, Slices for actions of infinite special points dimensional groups Jugen Elstrodt, Fritz Grunewald, and Jens K. Sundaresan, Convex functions on Banach Mennicke, Eisenstein series for imaginary lattices quadratic number fields K. Sundaresan and S. Swaminathan, Yuval Z. Flicker, Unitary quasi-lifting: preparations Differential analysis and geometry of Banach spaces-isomorphism theory J. H. M. Whitfield and V. Zizler, A survey of rough norms with applications (continued)

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563 David Fried, Fuschian groups and Reidemeister torsion Paul B. Garrett, Theta-correspondences on THE 8-CONJECTURE: unitary groups CHARACTERIZATION OF Paul Gerardin and Wen-Ch' ing Winnie Li, Establishing correspondences without the trace CHEVALLEY GROUPS formula John H. Walter George T. Gilbert, Multiplicity theorems for GL(2) (Memoirs of the AMS, Number 345) and GL(3) A. Good, The convolution method for Dirichlet This memoir provides a well-organized and efficient series approach to proving the B-conjecture, a theorem Martin C. Gutzwiller, Physics and Selberg's trace which is an essential step in the classification of formula finite simple groups. This conjecture asserts that Ki-ichiro Hashimoto, Representations of the finite the 2-components of the centralizer of an involution symplectic group SP(4, Fp) in the spaces of Siegel in a finite simple group are quasisimple, and is modular forms proven using the signalizer functor method. Dennis A. Hejhal, Roots of quadratic congruences and eigenvalues of the non-Euclidean Laplacian The first part of this publication contains many M. N. Huxley, Exceptional eigenvalues and interesting theorems on the characterization of congruence subgroups group, which are needed for the proof of the Herve Jacquet, An introduction to the trace principal result in Part II. formula from the adelic point of view The memoir is aimed at specialists in the theory N. Mandouvalos, The theory of Eisenstein series of finite simple groups. It presumes a thorough for Kleinian groups knowledge in the arguments and principal results M. Scott Osborne, The continuous spectrum: of classification theory. some explicit formulas Contents M. Scott Osborne, Selberg's trace formula and a character identity for generalized Heinsenberg Part I. Characterization of Chevalley groups and groups some locally c-unbalanced groups J. D. Rogawski, Some remarks on Shalika germs Introduction Peter Sarnak, On cusp forms 1. Preliminary concepts and results Audrey A. Terras, Some simple aspects of the 2. 2-Components of type PSL(2, q) and A7 theory of automorphic forms for GL(n, Z) 3. Characterization of groups of type C which are V. S. Varadarajan, The eigenvalue problem on not Chevalley type negatively curved compact locally symmetric 4. Characterization of Chevalley groups manifolds Part II. The B-Conjecture; signalizer functors Marie-France Vigneras, Correspondances entre Introduction representations automorphes de GL(2) sur une 1. Properties of uneven 2-components extension quadratique de GSp(4) sur D. conjecture 2. Verification of A-oddness locale de Langlands pour GSp(4) 3. Regular elementary 2-groups and the generation Garth Warner, Traceability on the discrete of layers spectrum 4. Construction of proper subgroups by signalizer Rainer Weissauer, Stable modular forms functors Floyd L. Williams, Finite spaces of non-classical 5. Properties of a minimal counterexample Poincare theta series 6. Proof of Theorem I 1980 Mathematics Subject Classifications: References 10020, 10024, 10040, 12A85, 22E40, 22E45, 22E50, 22E55, 30F35, 30M10, 30N10 Index ISBN 0-8218-5058-x, LC 86-3512 Index of notation ISSN 0271-4132 1980 Mathematics Subject Classification: 568 pages (softcover), due June 1986 Individual member 125, List price $42, 20005 Institutional member $34 ISBN 0-8218-2345-0, LC 86-3388 To order, please specify CONM/53N ISSN 0065-9266 204 pages (softcover), 1986 Individual member 111, List price $19, Institutional member $15 To order, please specify MEM0/345N

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564 HOMEOMORPHISMS OF OVERGROUPS OF SYLOW 3-MANIFOLDS WITH SUBGROUPS IN SPORADIC COMPRESSIBLE BOUNDARY GROUPS Darryl McCullough and Andy Miller Michael Aschbacher (Memoirs of the AMS, Number 344) (Memoirs of the AMS, Number 343)

This memoir studies the group G(M) of isotopy This memoir systematically describes the maximal classes of self-homeomorphisms for a P2-irreducible overgroups of noncyclic Sylow subgroups of 3-manifold M with compressible boundary. When the sporadic finite simple groups. The author M is orientable, G(M) is proved to be finitely associates with this collection of overgroups a generated, and the subgroup J(M) generated geometric structure which can be used to study by Dehn twists about discs, annuli, and tori the group and its permutation representations. is shown to have finite index. Moreover, any He includes reasonably self-contained proofs of orientation-preserving self-homeomorphism the existence and uniqueness of the relevant inducing the identity on 7ri{M) is proved to be subgroups. Though the text is aimed primarily at isotopic to a product of Dehn twists about properly simple group theorists, the tables which describe imbedded 2-discs. When M is nonorientable, the the maximal subgroups will be useful to any authors develop a different set of techniques to mathematician seeking information about the obtain appropriate extensions of these results. subgroups, permutation representations, or the geometry of the sporadic groups. and graduate students Aimed at mathematicians Contents with an interest in geometric topology, especially the topology of 3-manifolds, this memoir requires 1. Introduction lemmas a knowledge of the basic "classical" techniques of 2. Preliminary lemma 3-manifold topology. 3. A generation 4. A lemma on the representations of SL2(q) Contents 5. Certain subgroups of symplectic groups Incompressible neighborhoods 6. Groups generated by transvections Standard homeomorphisms of an orientable 7. Some subgroups of alternating groups product-with-handles 8. Large p-subgroups of finite groups The mapping class group of an orientable 9. Centralizers of semisimple elements in groups product-with-handles of Lie type Finite generation and the Johannson subgroup for 10. Centralizers of unipotent elements in groups of mapping class groups of orientable 3-manifolds Lie type The homomorphism H(\I,Xo)---> Aut(7ri{V,x0 ) 11. 3-structure in Sz and Ly The homomorphism H(M,Xo)---> Aut(7r1 (M,x0 ) 12. 2-structure in certain sporadic groups The nonorientable case 13. 3-structure in J3 1980 Mathematics Subject Classifications: 14. 3-structure in F3 57M99, 57R50, 55P10, 55S37 15. Some subgroups of the Monster ISBN 0-8218-2346-9, LC 86-3387 16. Some groups with abelian Sylow groups ISSN 0065-9266 116 pages (softcover), April 1986 17. Groups of GF(p)-type Individual member 18, List price $14, 18. Almost simple groups of GF(p)-type Institutional member Sll 19. Random lemmas on groups of GF(p)-type To order, please specify MEM0/344N 20. The proof of Theorem B 21. Large extraspecial subgroups of width 1 22. Large extraspecial 5-subgroups of width 1 23. Nonlocal overgroups of large symplectic subgroups 24. Small Sylow 3-groups 25. Overgroups of elementary abelian p-groups

(continued)

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565 26. Small overgroups of large symplectic Adaptation of Schoen and Simon's results p-subgroups Behavior near a corner 27. Geometries Results when the boundary is an extreme polygon

28. Parabolics 1980 Mathematics Subject Classification: 29. Overgroups of Sylow p-groups in groups of 49F22 GF(p)-type ISBN 0-8218-2339-6, LC 86-1039 30. Overgroups of large symplectic 2-subgroups ISSN 0065-9266 96 pages (softcover). March 1986 31. The Todd module Individual member 17, List price $12. 32. Geometries for 2 E6 (2) and M(22) Institutional member $10 To order. please specify MEM0/342N 33. A 112-dimensional module GF(2)J4-module References

1980 Mathematics Subject Classifications: 20B25. 20008. 20005 ISBN 0-8218-2344-2. LC 86-1040 ISSN 0065-9266 240 pages (softcover). March 1986 ILL-POSED PROBLEMS OF Individual member U3. List price $21. MATHEMATICAL PHYSICS AND Institutional member $17 To order. please specify MEM0/343N ANALYSIS M. M. Lavrent' ev, V. G. Romanov, and S. P. Shishat-skil (Translations of Mathematical Monographs, EXPLICIT DETERMINATION OF Volume 64) AREA MINIMIZING HYPERSURFACES, II Aimed at specialists in applied mathematics, Harold R. Parks physics, and geophysics, this volume fills some (Memoirs of the AMS, Number 342) of the gaps in the monographic literature on the theory of ill-posed problems. The theory has The author presents a method for explicitly developed intensively in the past two decades, aided determining an oriented area minimizing by the advent of modern computing technology. hypersurface with a given boundary. The method, It is connected with a wide variety of applied which uses the least gradient approach, is presented problems, including interpretation of readings of as quickly as possible after the introduction of the many physical instruments and of geophysical and necessary notation. As an essential hypothesis, the astronomical observations, optimization of control, author assumes that the given boundary lies on the management, and planning, and the synthesis of surface of a bounded convex subset of Rn. This automatic systems. strengthens an earlier result of the author, where the uniqueness of the area minimizing hypersurface The authors begin by presenting a number of is no longer required, and the given boundary is examples which lead to ill-posed problems arising replaced by an appropriate approximation. The with the processing and interpretation of data of goal of the remainder of the book is to determine physical measurements. Basic postulates and some how accurate an approximation ensures that all the results in the general theory of ill-posed problems topological complexity of the minimizing surface follow. The exposition also includes problems of has been expressed in the approximating set. analytic continuation from continua and discrete sets, analogous problems of continuation of Aimed at the research mathematician interested solutions of elliptic and parabolic equations, the in the calculus of variations in general and area main ill-posed boundary value problem for partial minimizing problems in particular, the text requires differential equations, and results on the theory of a working knowledge of basic geometric measure Volterra equations of the first kind. A very broad theory. presentation is given of modern results on the Contents problem of uniqueness in integral geometry and on Introduction inverse problems for partial differential equations. Notation The method Interior topology (continued)

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566 Contents Contents Physical formulations leading to ill-posed problems Introduction Basic concepts of the theory of ill-posed problems Analytic continuation Chapter I. Solutions of Direct and Inverse Problems Boundary value problems for differential equations and Some of Their Relations Volterra equations 1. Convolution formulas Integral geometry 2. Connections between solutions of second-order Multidimensional inverse problems for linear equations of various types differential equations 3. Direct and inverse problems for the heat equation and the method of incomplete 1980 Mathematics Subject Classifications: 35R30, 45D05 separation of variables ISBN 0-8218-4517-9. LC 86-3642 Chapter II. Source Problems ISSN 0065-9282 296 pages (hardcover), May 1986 1. A linearized formulation of the problem of Individual member asg, List price $98, determining q(z) and f(t); the case J1(z) = 1 Institutional member $78 2. The linearized problem. The case of a coefficient To order, please specify MMON0/64N of the leading derivative 3. The problem in the exact formulation; the case J1(z) = 1 ONE-DIMENSIONAL INVERSE Chapter Ill. A One-Dimensional Inverse Problem for the Wave Equation PROBLEMS OF MATHEMATICAL 1. Generalized solutions of boundary value PHYSICS problems for the wave equation (American Mathematical Society Translations, 2. The concept of a solution of an inverse problem Series 2, Volume 130) in the case of information given on a finite segment This monograph deals with the inverse problems 3. "Local" existence of a unique solution of the of determining a variable coefficient and right side inverse problem with a "distributed" source for hyperbolic and parabolic equations on the basis 4. A method of constructing a "global" solution of of known solutions at fixed points of space for the inverse problem with a "distributed" source all times. The problems are one-dimensional in 5. Uniqueness and stability of the solution of the nature since the desired coefficient of the equation inverse problem with a "distributed" source is a function of only one coordinate, while the 6. Uniqueness of the solution of the inverse problem desired right side is a function only of time. The with a source of perturbation concentrated at a authors use methods based on the spectral theory point of ordinary differential operators of second order Appendix and also methods which make it possible to reduce 1. The Laplace transform the investigation of the inverse problems to the 2. The exponential Fourier transform investigation of nonlinear operator equations. The 3. The Fourier cosine transform problems studied have applied importance, since 4. The method of spectral theory of second-order they are models for interpreting data of geophysical ordinary differential operators prospecting by seismic and electric means. 5. Connections between solutions of linear In the first chapter the authors prove the one-to-one differential equations in Banach spaces correspondence between solutions of direct Cauchy Bibliography problems for equations of different types, and they 1980 Mathematics Subject Classifications: present the solution of an inverse problem of heat 35K05, 35L05, 35R30 conduction. In the second chapter they consider ISBN 0-8218-3099-6, LC 86-7917 a second-order hyperbolic equation describing ISSN 0065-9290 80 pages (hardcover), due July 1986 a wave process in three-dimensional half-space. Individual member $22, List price $37, The third chapter investigates formulations of Institutional member $30 one-dimensional inverse problems for the wave To order, please specify TRANS2/130N equation in multidimensional space.

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567 Miscellaneous

Personal Items Deaths James Eells of the University of Warwick, Daniel T. Finkbeiner II, Professor Emeritus Coventry, England, has been appointed head of Kenyon College, Gambier, Ohio, died on March of mathematics at the International Centre for 28, 1986, at the age of 66. He was a member of Theoretical Physics in Trieste, Italy. He will the Society for 38 years. share his time between the University and the Henri G. Garnir of Angleur, Belgium, died Centre. on November 18, 1985, at the age of 58. He was B. K. Ghosh, Professor and former Chairman a member of the Society for 24 years. of the Department of Mathematics at Lehigh Emmert A. Gassman of Mount Carmel, Illi­ University, has been selected by the Alexander nois, died on January 13, 1986, at the age of 80. von Humboldt Foundation to receive a Senior He was a member of the Society for 20 years. He will visit the Institute U.S. Scientist Award. Tadeusz Mackowiak of Wrodaw University, of Mathematical Statistics at the University of Poland, died on February 4, 1986, at the age of during the academic year 1986-1987. Munster 37. He was a member of the Society for 6 years. of Santa Clara University Paul R. Halmos A. A. H. Omar of Cairo, Egypt, died on was awarded the honorary degree of Doctor of December 11, 1985, at the age of 37. He was a Letters by Kalamazoo College, at a Humane member of the Society for 9 years. Scholars' Day Convocation held on March 14, 1986. Wolfram Schwabhiiuser of the University of Stuttgart, Federal Republic of Germany, died on of Fordham University has Robert H. Lewis 27, 1985, at the age of 54. He was a to Associate Professor at that December been promoted of the Society for 17 years. institution. member Kurt Vogel, Professor Emeritus of the Uni­ M. Zelmanowitz, Professor of Math­ Julius versity of Munich, Federal Republic of Germany, at the University of California, Santa ematics died on October 27, 1985, at the age of 97. He Barbara, assumed the post of Associate Vice was a member of the Society for 22 years. Chancellor for Academic Affairs in September, 1985. PARTICLE SYSTEMS, RANDOM MEDIA AND LARGE DEVIATIONS Richard Durrett, Editor "The boolc is an excellent introduction to the exciting recent developments which combine ideas from physics with mathematical techniques from the theory of probability." - Frank L. Spitzer Cornell University

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

568 Visiting Mathematicians

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

American Mathematicians Visiting Abroad Name and Home Country Host Institution Field of S12ecial Interest Period of Visit Babbitt, Donald G. (U.S.A.) Universite de Dijon, France Mathematical Structures Arising 11/86- 6/87 in Quantum Theory Blackwell, John H. Oxford, United Kingdom Heat Conduction in Media with 10/86 (Canada) Change of Phase Buckley, B. (Canada) Institute National de Recherche Optimization 1/87- 7/87 en Informatique et Automatique, France Dennis, S.C.R. (Canada) University of Poi tiers, France; Fluid Mechanics 7/86; 1/87- 9/87 University College, England 8/86; 10/87- 12/87 Gamelin, Theodore W. University of Saarbrucken, West Function Algebras, Analytic 7/86- 6/87 (U.S.A.) Germany Functions Ghosh, B. K. (U.S.A.) University of Munster, West Mathematical Statistics 9/86- 6/87 Germany Groetsch, C. W. (U.S.A.) Australian National University Integral Equations 7/86- 8/86 Hamacher, Horst W. Technical University Graz, Combinatorial Optimization 10/86- 7/87 (U.S.A.) Austria Michelson, Daniel (U.S.A.) Hebrew University, Israel Partial Differential Equations, 7/86- 6/87 Numerical Analysis Nagano, Tadashi (U.S.A.) Sophia University, Japan Differential Geometry 6/86- 7/87 Naylor, Derek (Canada) King's College, England Integral Transforms and 1/87- 12/87 Eigenfunction Expansions Rasmussen, Henning University of Edinburgh, Fluid Mechanics 5/86- 9/86 (Canada) Scotland Sillith, Penny (U.S.A.) Max Plank Institute, West Nonlinear Analysis/Geometry 9/86- 6/87 Germany Stakgold, Ivar (U.S.A.) Ecole Polytechnique Federale Nonlinear Analysis 2/87- 5/87 de Lausanne, Switzerland Talman, James D. (Canada) University of Oslo, Norway Theoretical Atomic Physics 9/86- 6/87

Visiting Foreign Mathematicians Andreatta, Marco (Italy) University of Notre Dame Algebraic Geometry 8/86- 1/87 Andreka, Hajnal (Hungary) Iowa State University Universal Algebra, Logic 8/86- 5/87 Avellaneda, Marco M. Courant Institute, New York Partial Differential Equations, 9/86- 5/87 (Argentina) University Probability Beauville, Arnaud (France) Columbia University Algebraic Geometry 9/86- 12/86 Ben-David, Shai (Israel) University of Toronto Set Theory 7/86- 6/87 Berman, Abraham (Israel) University of California, Santa Linear Algebra 1/87- 6/87 Barbara Brenier, Yann (France) University of California, Los Applied Mathematics 1/87- 3/87 Angeles Canuto, Giuseppe (Italy) Princeton University Algebraic Geometry 9/86- 6/87 Cao, Z. (China) Florida State University Fuzzy Sets Theory 8/86- 8/87 Catanese, Fabrizio (Italy) Columbia University Algebraic Geometry 1/87- 5/87 Ciecielski, Krzysztof Auburn University Set Theory 9/86- 6/87 (Poland) Coen, Salvatore (Italy) Johns Hopkins University Several Complex Variables 7/86- 6/87 Constanda, Christian Iowa State University Analysis 8/86- 5/87 (Scotland) Conway, John H. (England) Princeton University Finite Mathematics 9/86- 6/87 DeConcini, Corrado (Italy) Brandeis University; Algebraic Geometry 9/86- 12/86 Harvard University 1/87- 6/87 Eastham, Michael S. P. University of Western Ontario Differential Equations 9/86 (England) Epstein, David (England) University of Minnesota Geometry 9/86- 12/86

569 Name and Home Country Host Institution Field of S~ecial Interest Period of Visit Escobar, Jose (Colombia) Courant Institute, New York Differential Geometry and Partial 9/86- 6/87 University Differential Equations Floer, Andreas (West Courant Institute, New York Differential Equations 9/86- 5/88 Germany) University Foata, Dominique (France) University of Pennsylvania Combinatorics 7/86- 12/86 Fraenkel, Edward (England) University of Minnesota Analysis 7/86- 12/86 Friedman, Menachem Florida State University Computational Mathematics 8/86- 8/87 (Israel) Frohlich, A. (United University of Western Ontario Algebraic Number Theory 8/86- 10/86 Kingdom) Graham-Eagle, James (New University of Delaware Functional Analysis, Partial 9/86- 5/88 Zealand) Differential Equations Grillakis, Manoussos Courant Institute, New York Partial Differential Equations 9/86- 5/88 (Greece) University Gyiiri, Ervin (Hungary) Vanderbilt University Combinatorics 9/86- 5/87 Gardner, B. J. (Australia} Dalhousie University Ring Theory 8/86- 1/87 Han, Yong Shen (People's Washington University Analysis 9/86- 5/87 Republic of China) Hills, Roger N. (Scotland) University of California, Los Applied Mathematics 9/86- 6/87 Angeles Hilton, Anthony J. W. Auburn University Graph Theory and Combinatorics 9/86- 6/87 (England) Hirano, Yasuyuki (Japan) University of Cincinnati Ring Theory 4/86- 3/87 Hsu, Pei (People's Republic Courant Institute, New York Probability 9/86- 6/87 of China) University Iacob, Andrei (Israel) Brandeis University Symplistic Geometry and 9/86- 5/87 Mechanics Jarosz, Krystof (Poland) University of California, Santa Banach Space Theory, Banach 9/86- 6/87 Barbara Algebras Journe, Jean-Lin (France) Princeton University Analysis 2/87- 6/87 Kasch, Friedrich (West University of California, Santa Ring Theory 10/86- 12/86 Germany) Barbara Kato, Akio (Japan) University of Toronto Set Theoretic Topology 8/86- 7/87 Koslowski, Jiirgen (West Vanderbilt University Category Theory 9/86- 5/87 Germany) Lascar, Bernard (France) University of California, Los Analysis, Partial Differential 9/86- 6/87 Angeles Equations Lin, Fang-Hua (People's Courant Institute, New York Partial Differential Equations 9/86- 5/87 Republic of China) University Little, C.H.C. (New University of Waterloo Graph Theory 9/86- 8/87 Zealand) Liu, Guizhen (People's Simon Fraser University Combinatorics 7/86- 6/87 Republic of China) Looijenga, Eduard Columbia University Algebraic Geometry 1/87- 5/87 (Netherlands) Ma, Daowei (People's Washington University Analysis 9/86- 5/87 Republic of China) Magidor, Menachem (Israel) University of California, Los Logic 9/86- 6/87 Angeles Martin, Paul A. (England) University of Delaware Scattering Theory 9/86- 5/87 Mawhin, Jean (Belgium) University of Colorado, Boulder Differential Equations 9/86- 12/86 Mohar, Bojan (Yugoslavia) Simon Fraser University Combinatorics 9/86- 8/87 Moszkowski, Paul (France) University of Pennsylvania Combinatorics 7/86- 6/87 Motohashi, Yoichi (Japan) University of Colorado, Boulder Number Theory 1/87- 5/87 Nemeti, Istvan (Hungary) Iowa State University Universal Algebra, Logic 8/86- 5/87 Nochetto, Ricardo (Italy) University of Minnesota Numerical Analysis 8/86- 6/87 Ozluk, Ali E. (Turkey) Simon Fraser University Discrete Mathematics 9/86- 8/87 Paivarinta, Lassi J. University of Delaware Scattering Theory 2/87- 5/87 (Finland) Fakes, Anthony G. Colorado State University Probability 1/86- 12/86 (Australia) Papageorgiou, Demetrius Courant Institute, New York Fluid Dynamics 9/86- 5/87 (England) University Paulraja, P. (India) University of Waterloo Cycles in Graphs and Digraphs 9/86- 8/87

570 Name and Home Country Host Institution Field of SQecial Interest Period of Visit Peikert, Ronald University of Minnesota Computer Algebra 7/86- 1/87 (Switzerland) Pisanski, Tomaz Simon Fraser University Combinatorics 5/87- 8/87 (Yugoslavia) Power, Steve (United University of Alabama Operator Algebras, Operator 1/87- 5/87 Kingdom) Theory and Function Theory Przytycki, J ozef (Poland) University of Toronto Algebraic Topology 9/86- 8/87 Rademacher, Hans-Bert University of Pennsylvania Differential Geometry 7/86- 6/87 (West Germany) Ramanan, S. (India) Harvard Uttiversity Algebraic Geometry 9/86- 1/87 Rend!, F. (Austria) University of Waterloo Combinatorial Optimization 9/86- 8/87 Roan, Shi-Shyr (Taiwan) Brandeis University Complex Algebraic Geometry 9/86 - 12/86 and Complex Manifolds Romanowska, Anna Iowa State University Lattice Theory 8/86- 5/87 (Poland) Sakai, Katsuro (Japan) University of California, Santa Geometric Topology 6/86- 12/86 Barbara Scott, Alastair J. (New University of Wisconsin, Statistics 8/86- 1/87 Zealand) Madisoii Shinkai, Kenzo (Japan) University of Minnesota Partial Differential Equations 8/86- 2/87 Sibony, Nessim (France) University of California, Los Analysis 1/87- 6/87 Angeles Smedsaas, Tom (Sweden) University of California, Los Numerical Analysis 9/86 - 12/86 Angeles Stock, Josef (West University of California, Santa Ring Theory 9/86- 6/87 Germany) Barbara Sudo, Masaki (Japan) Johns HopJ<:ins University Number Theory 8/86- 12/86 Szpiro, Lucien (France) Columbia University Number Theory 9/86- 12/86 Tangerman, Folkert M. Courant Institute, New York Dynamical Systems 9/86- 5/88 (Holland) University Teicher, Mina (Israel) Columbia University Algebraic Geometry 1/87- 5/87 Thomann, Enrique Courant Institute, New York Partial Differential Equations 9/86- 6/87 (Argentina) University van der Geer, Gerard Harvard U11iversity Algebraic Geometry 1/87- 5/87 (Netherlands) Viterbo, Claude M. Courant Institute, New York Symplectic Geometry 9/86- 1/87 (France) University Wajnryb, Bronislaw (Israel) University of Toronto; Algebraic Topology 9/86- 12/86 Columbia University Wilson, John (Great University of California, Santa Group Theory 1/87- 3/87 Britain) Barbara Yu, Chen (China) University of Notre Dame Algebra 8/86- 5/87 Zhu, Kehe (People's Washingtoll University Analysis 9/86- 5/87 Republic of China)

571 AMS Reports and Communications

Recent Appointments of the individuals in the above categories based on name identification and that additional informa­ tion on the number Committee members' terms of office on standing of Ph.D.'s granted to women would also be collected using the AMS committees expire on December 31 of the year Annual Survey. Since name given in parentheses following their names, identification was used, the unless otherwise specified. information for some categories necessitated the use of four classifications: Male - names that Joel M. Cohen has been appointed to the were obviously male; Female - names that Committee to Select Hour Speakers for the Eastern were obviously female; Sectional Meetings by President Irving Kaplansky. Unknown - names that could not be identified as clearly Continuing members of the committee are W. male or female (i.e., only initials Wistar Comfort (ex officio), Roger Keith Dennis given); and Foreign - foreign (1987), Clifford J. Earle, Jr., (1986), and Srinivasa names that could not be S. R. Varadhan, chairman (1986). identified as clearly male or female. The following is the first reporting of this in­ A joint AMS-MAA Committee on Arrange­ formation. Updated reports will appear annually ments for the San Antonio Meeting (January in Notices. 22- 25, 1987) has been appointed by Presidents Lynn A. Steen (MAA) and Irving Kaplanksy Members of the AMS Residing in the U.S. (AMS). Committee members are Donald F. Bai­ Male: 12,527 72% ley, Robert M. Fossum (ex officio), William J. Female: 2,200 13% LeVeque (ex officio), Kenneth A. Ross (ex officio), Unknown: 2,157 12% Gregory Wene, chairman, Lawrence R. Williams, Foreign: 497 3% and Bennir A. Zinn. Total checked: 17,381 Kenneth M. Hoffman, Thomas R. Kramer, Invited Hour Address Speakers Louise A. Raphael, T. Christine Stevens, and at AMS Meetings (1975-1985) Marcia P. Sward have been appointed by President Male: 372 85% Irving Kaplansky to be the selection panel for the Female: 10 2% Congressional Fellow for 1986-1987. Professor Unknown: 19 4% Kramer is chairman. Foreign: 36 8% Total checked: 437 Statistics on Women Mathematicians Speakers at Special Sessions Compiled by the AMS at AMS Meetings (1981-1985) At its August 1985 meeting, the Council of the Male: 2,545 77% AMS approved Female: 169 5% a motion to regularly assemble Unknown: and 386 12% report in Notices information on the relative Foreign: 187 6% numbers of men versus women in at least the Total checked: 3,287 following categories: membership in the AMS; invited hour addresses at AMS meetings; speakers Trustees and Council Members at special sessions at AMS meetings; and members 1985 1984 1983 1982 of editorial boards of AMS journals. Total: 71 70 It was subsequently 63 64 decided that this infor­ Male: 61 86% 62 89% 55 87% 58 91% mation would be gathered by determining the sex Female: 10 14% 8 11% 8 13% 6 9%

Members of Editorial Boards of AMS Journals 1985 1984 ____!2§_L ___l2§_L ___!m_ ~ ___l_[ZlL_ ____m§______mz_ ____!_[ffi_ Total: 102 93 90 83 85 82 82 67 59 58 Male: 94 92% 85 91% 84 93% 77 93% 79 93% 77 94% 78 95% 64 96% 56 95% 55 95% Female: 8 8% 8 9% 6 7% 6 7% 6 7% 5 6% 4 5% 3 4% 3 5% 3 5%

Ph.D.'s Granted to U.S. Citizens 1985 1984 1983 1982 1981 1980 1979 1978 1977 1976 Total: 396 433 455 519 567 578 596 634 689 722 Male: 315 80% 346 80% 366 80% 431 83% 465 82% 491 85% 503 84% 545 86% 602 87% 636 88% Female: 81 20% 87 20% 89 20% 88 17% 102 18% 87 15% 93 16% 89 14% 87 13% 86 12%

572 Classified Advertisements

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

POSITIONS AVAILABLE STANFORD UNIVERSITY Departments Northern Michigan University invites applicants for a of Computer Science temporary position in the Department of Mathematics and Mathematics & Computer Science. The person filling this position is Applications are invited for a joint tenure-track pro­ expected to teach undergraduate mathematics or statistics fessor with interest in numerical analysis and scientific or computer science. depending on preparation. and must computing for a newly proposed Applied Mathemat­ have at least a Master's degree in one of these areas. ics/Scientific Computing program. Applicants must have a Applicants should send a resume, transcripts and three record of strong research and an interest and competence letters of recommendation to: in teaching. Candidates should send curriculum vitae with suggested Dr. Terrance L. Seethoff. Head references to Professor Gene H. Golub. Com­ puter Science Department. Stanford University, Department of Mathematics & Computer Science Stanford. CA Northern Michigan University 94305. Further inquiries can be made to the above Marquette. Ml 49855-5340 address or to ARPAnet address: Golub

573 POSITIONS AVAILABLE UNIVERSITY OF CINCINNATI, DEPT. OF MATH. SCIENCES MAIL LOCATION #25, MATHEMATICS FACULTY POSITIONS IN CINCINNATI, OH 45221 FLORIDA INTERNATIONAL, UNIVERSITY The University of Cincinnati expects to rnake several at Florida The Department of Mathematical Sciences tenure-track appointments at the assistant professor level. tenure-track as­ International University announces several Preference will be given to candidates with potential to the associate or sistant professorships, and one opening at strengthen the existing research areas in the Department. Requirements: full professor level beginning August 1986. All candidates must have outstanding potential for re­ demonstrated Ph.D. in Mathematics. research potential. search. scholarship, and teaching. Send vita and direct. demonstrated teaching ability. Senior position requires 3 letters of reference to C. W. Groetsch. Head. Depart­ hours per research record. Teaching load: 15 semester ment of Mathematical Sciences, M.L. #25. University of teaching available. Preferred academic year: summer Cincinnati, Cincinnati. OH 45221. U.C. is an AA/EOE. specialties: harmonic analysis, several complex variables. mathematical logic. algebra. combinatorics: other areas considered. Send resume by January 30. 1986 to: Re­ Tenure-track and visiting positions in mathematics. cruitment Committee, Dept of Mathematical Sciences. applied mathematics and statistics are available beginning Florida International University, Miami. FL 33199. FlU is September 1986. Excellent teaching and a commitment a member of the State University System of Florida and to research are required. Some 3-year instructorships an Equal Opportunity/Affirmative Action Employer. may also be open. The department offers B.S. and M.S. degrees. MTU is a state supported university emphasizing science and engineering. To apply, write: Dr. Martyn invites THE AUSTRALIAN NATIONAL UNIVERSITY R. Smith, Head. Mathematical and Computer Sciences. and men for ap­ applications from suitably qualified women Michigan Technological University, Houghton. Ml 49931. pointment to the following position: POSTDOCTORAL Michigan Technological University is an equal opportu­ FELLOW, DEPARTMENT OF FELLOW/RESEARCH nity educational institution/equal opportunity employer. MATHEMATICS. RESEARCH SCHOOL OF PHYSICAL SCIENCES. The Department of Mathematics (Head: Pro­ fessor D. W. Robinson. FAA} carries out research in TEXAS A&M UNIVERSITY Algebras: Semigroups several areas. namely Operator Applications are invited for tenure-track positions in Partial Differential Equations and Mathematical Physics: Mathematics starting September 1. 1986. Candidates Lie Groups and Algebraic and Geometry: Foundations: must have a Ph.O. in Mathematics and salaries are com­ Equations and Control The­ Groups: Ordinary Differential petitive. We are especially interested in applicants in date: 31 July ory: Global Analysis: Group Theory. Closing algebra. applied mathematics. and non-linear analysis. Salary will be in accordance with 1986. Ref: PS. 25.3.3. There is a possibility of appointments at Professor. Asso­ within the range: Postdoc­ qualifications and experience ciate Professor. and Assistant Professor ranks. Visiting (at a fixed point) $23,473-$26.889 toral Fellow Grade 1 positions are also expected. Send vita and arrange for Research Fellow $27.233-$35.777 p.a. Appoint­ p.a.: three reference letters to be sent to ment. unless otherwise stated. will be: Research Fellow normally up to three years with the possibility of ex­ Dr. H. E. Lacey, Head tension to maximum of five years: Postdoctoral Fellow Department of Mathematics normally two years. with the possibility of extension to Texas A&M University maximum of three years. Grants towards travel. as­ College Station. TX 77843-3368 sistance with housing. superannuation. Maternity leave Texas A&M is an equal opportunity employer. is available. The University reserves the right not to make an appointment or to make an appointment by invitation at any time. Applicants should quote the refer­ The University of Wyoming ence number when submitting applications to R. V. Dubs Head-Department of Mathematics Registrar. GPO Box 4, Canberra. ACT. 2601 Australia. The Department of Mathematics at the University of THE UNIVERSITY IS AN EQUAL OPPORTUNITY Wyoming invites applications for the position of Depart­ EMPLOYER. ment Head. The University is the sole four-year institution of higher education in the state of Wyoming with an en­ rollment of 10,000 students. The mathematics program OTTO SZASZ VISITING PROFESSORSHIP offers degrees in mathematics. applied mathematics. and The Department of Mathematical Sciences of the Uni­ several joint-degree options at the bachelor. master and versity of Cincinnati expects to establish. pending final doctoral levels. The department has a growing major budgetary approval. a visiting professorship named in research component in applied mathematics including a honor of the late Otto Szasz. The professorship will be petroleum research institute with funding from major in­ tenable for a period of one year. Applicants with strong dustrial supporters. Other active research areas in the research accomplishments in an area compatible with one department include numerical analysis. partial differen­ of the existing research groups of the Department may tial equations. functional analysis. optimization theory. address inquiries to C. W. Groetsch. Head. Department dynamical systems. rigidity theory, and combinatorics. of Mathematical Sciences. Mail Location 25. University of Candidates should have a strong research record com­ Cincinnati. Cincinnati. Ohio 45221. patible with department interests and a commitment to The University of Cincinnati is an AA/EOE. excellence in instruction. Applicants should submit a current curriculum vita and the names of at least three suitable references to UNIVERSITY OF CRETE, GREECE Department of Mathematics W. G. Bridges. Chair Search Committee Applications are invited from Greek-speaking mathe­ Mathematics Department maticians. holding the Ph.D. degree. for visiting positions University of Wyoming at all ranks for both semesters of the whole Academic Laramie. WY 82070 Year 1986-7. Send expression of interest. vita and Applications will be considered through January 31. representative research work to Professor Susanna Pa­ 1986. The University of Wyoming is an equal opportunity padopoulou. Chairperson. Mathematics Department. Uni­ employer. versity of Crete. lraklion. Greece.

574 POSITIONS AVAILABLE UNIVERSITY OF BRITISH COLUMBIA-DEPART­ MENT OF MATHEMATICS. Applications are invited for a tenure track position for the academic year commencing 1 MERRIMACK COLLEGE July, 1986. This position is subject to final budgetary ap­ Department of Math/CS proval. Candidates must have a Ph.D. and demonstrated Chairperson and two other positions (one in mathe­ excellent research potential in an area of interdisciplinary matics and one in Computer Science or a combination) applied analysis. preferably related to one or more of the available Fall 1986. For chairperson, a PhD in CS is following fields: geophysical fluid dynamics, non-linear preferred but a Masters accompanied by appropriate expe­ waves, nonlinear diffusion, continuum mechanics. applied rience will be considered. Rank and salary commensurate differential equations, applied optimization, biomathemat­ with qualifications. For the other positions, a PhD in ics. or resource modelling. In addition to carrying out Math and/or a Masters in CS with demonstrated teaching undergraduate and graduate instruction. the successful experience preferred. Tenure slots available. Send resume, applicant will become an active member of the Institute of salary requirement and references to: R. E. Ozimkoski. Applied Mathematics, which offers interdisciplinary M. Sc. Dept. of Math/CS. Merrimack College, North Andover. and Ph.D. degrees. Salary commensurate with experience MA 01845. An Equal Opportunity Employer. and research record. Applicants must include a complete C.V. and arrange for at least three letters of reference to be mailed directly to: MATHEMATICS EDITOR Head. Mathematics Department Cambridge-based international scientific publisher has The University of British Columbia an opening for Mathematics Editor. This job entails Vancouver. Canada V6T 1 Y 4 the development and expansion of an already existing Applications received after May 31. 1986 will be consid­ strong graduate-level program in pure and applied math­ ered only if a position remains unfilled. In accordance with ematics. Applicants are expected to be able to assume Canadian immigration requirements, priority will be given responsibility for the development of such a program. to to Canadian citizens and permanent residents of Canada. work. under initial guidance, independently and to main­ tain good personal and professional ties with members of the mathematical community. Prior publishing experience THE UNIVERSITY OF ALABAMA is desirable but not essential: substantial exposure to DEPARTMENT OF MATHEMATICS graduate-level mathematics is required. Salary commen­ As part of ongoing development in applied mathemat­ surate with educational background/experience. Excellent ics, we have received funding for a new position in applied benefit package. Send resume and salary history to: mathematics. Salary will be negotiable. up to a maxi­ AMS. #1 June Classifieds, P. 0. Box 6248, Providence. mum of $35,000 and rank will depend upon qualifications. Rl 02940. We invite applications from experienced mathematicians in applied mathematics with excellent records in research The University of B. C.-Department of Mathematics: and teaching. We are looking for someone who will help Applications are invited for a tenure track position to fill us develop our research grouP. and our graduate pro­ a vacancy in Geometry /Topology for the academic year grams in applied mathematics. Applications or inquiries commencing 1 July, 1986. This position is subject to final should be sent to: Alan Hopenwasser. Chairman. Depart­ budgetary approval. Candidates must have a Ph.D. and ment of Mathematics. P. 0. Box 1416. University, AL demonstrated excellence in research and ability to teach 35486. THE UNIVERSITY IS AN EQUAL OPPORTU­ undergraduate and graduate courses in the Department. NITY/ AFFIRMATIVE ACTION EMPLOYER. Salary commensurate with experience and research record. Applicants must include a complete C.V. and arrange for UNIVERSITY OF TORONTO at least three letters of reference to be mailed directly to: Department of Mathematics Head, Department of Mathematics Toronto, Ontario MSS LAl The University of British Columbia Canada Vancouver, Canada V6T IY 4 The Department of Mathematics. University of Toronto Applications received after May 31, 1986 will be con­ is looking for strong applicants in pure or applied math­ sidered only if a position remains unfilled. ematics to nominate as candidates for NSERC Research In accordance with Canadian immigration requirements, Fellowships beginning July 1. 1987. These are five year priority will be given to Canadian citizens and permanent research positions (subject to a review in the third year) residents of Canada. with teaching load of at most one course per year. One of the five years may usually be taken as a sabbatical. Successful candidates may in special circumstances be AUSTIN PEAY STATE UNIVERSITY, DEPT. considered directly for a tenure-stream position. OF MATHEMATICS AND COMPUTER Applicants should be mathematicians with a relatively SCIENCE, CLARKSVILLE, TN 37044 recent doctorate and who have demonstrated their ability A tenure-track position in computer science will be avail­ with some substantive post-thesis research accomplish­ able September 1986. A doctorate in computer science or ment. They must be Canadian citizens or landed im­ a doctorate in a related field with substantial experience migrants by November 1, 1986. University of Toronto in computer science is required. All areas of specialization encourages both men and women to apply for these will be considered. The department offers a curricu­ fellowships. lum structured according to ACM guidelines. Computing Applicants should send an up to date curriculum vitae equipment includes a DEC VAX 11/780 dedicated to aca­ and a short description of their research program to demic use. a DEC PDP 11. a microcomputer laboratory, Professor T. Bloom. Chairman, and arrange to have sent and a hardware laboratory. The successful applicant will three letters of reference. This material should arrive be expected to participate in curriculum development and before Monday, September 15. 1986. The Department's academic advisement in addition to teaching responsibili­ choice of candidates will be made in late September, and ties. Send letter of application. resume. transcripts. and the final decision by NSERC is announced (by NSERC) in three letters of recommendation to Dr. William Stokes, the spring. Chairman. APSU is an Equal Opportunity, Affirmative Action Employer.

575 MATH SCI PRESS, 53 Jordan Rd. Brookline MA. POSITIONS AVAILABLE 02146. 617-738-0307. Just published: Proceedings of the Berkeley-Ames Conference on Nonlinear Problems in THE UNIVERSITY OF TEXAS OF THE Control and Fluid Dynamics. L. R. Hunt and C. F. Martin. PERMIAN BASIN (eds). 450 pages. $50. (Lie Groups. Se~ies B: Syste~s Department of Mathematics and Computer Information and Control. vol. II). Toptcs 1n the Geometnc Science Theory of Linear Systems. by Robert Hermann. $50. Odessa, TX 79762-8301 (Interdisciplinary Mathematics. v. 22). The Geometric Opening for Assistant or Associate Professor. starting theory of ordinary Differential Equations and Algebraic in the fall of 1986. The candidate must hold a Ph.D .. Functions. by G. Valiron. $75. (Lie Groups. v. 14). in Mathematics or Computer Science. U.T. Permian Topics in the Geometry Theory of Integrable Systems. Basin is an upper level institution with no Freshmen or $65. by Robert Hermann (Interdisciplinary Mathematics. Sophomore students. The total enrollment is about 2000 v. 23.). Write or call for special prices on back list. with approximately 130 of these majoring in Computer Science or Mathematics. Contact: Dr. James A. Nickel. Foundations of Semiological Theory of Numbers Chairman. The University of Texas of the Permian Basin H. A. Pogorzelski and W. J. Ryan is an AA/EOE. Volume 1 {1982). General Semiology, 597 pp .. $29.95 Volume 2 {1985). Semio. Functions. 695 pp .. $34.95 THE AUSTRALIAN NATIONAL UNIVERSITY (Eight Volumes Projected) UMO Press. Univ. of Maine. Orono. ME 04469 invites applications from suitably qualified men and (Post free within US if check included) women for appointment to the position of POSTDOC­ TORAL FELLOW /RESEARCH FELLOW in the DE­ PARTMENT OF MATHEMATICS. RESEARCH SCHOOL Mathematics library. Request list. Bert Ross. Univer­ OF PHYSICAL SCIENCES. The Department of Mathe­ sity of New Haven. West Haven. CT 06516. matics (Head: Professor D. W. Robinson. FAA) carries out research in several areas. namely Operator Algebras; HELDERMANN VERLAG. Nassauische Str. 26. Semigroups and Mathematical Physics; Partial Differen­ D-1000 Berlin-West 31. Just published: M. Jiinger.Poly­ tial Equations and Geometry; Foundations; Lie Groups hedral combinatorics and the acyclic subdigraph problem. and Algebra Groups; Ordinary Differential Equations and $15. G. Reinelt. The linear ordering problem - algorithms Control Theory; Global Analysis; Group Theory. Two and applications. $15. A. B. Romanowska and J. D. H. positions are available for one year from dates between Smith. Modal theory - an algebraic approach to order. 1 September 1986 and 28 February 1987 and preference geometry. and convexity. $16. G. Mazzola. Grupp~n und will be given to applicants working in Partial Differ­ Kategorien in der Musik. $20. R. Lowen. On the extstence ential Equations. Geometric Measure Theory or related of natural nontopological. fuzzy topological spaces. $14. fields of research in mathematical analysis. Enquiries Ask for full catalogue! may be made to Professor Leon Simon. Department of Mathematics. Closing date: 31 July 1986. Ref: PS. 2.4.1. Salary will be in accordance with qualifications A special textbook by Gilbert Strang and experience within the range; Postdoctoral Fellow Grade 1 (at a fixed point) $23.473-$26.889 p.a.; Research INTRODUCTION TO Fellow $27.233-$35.777 p.a. Grants towards travel. as­ APPLIED MATHEMATICS sistance with housing. superannuation. Maternity leave [750 pages. $36] is available. The University reserves the right not to make an appointment or to make an appointment by Please order directly from Wellesley-Cambridge Press. invitation at any time. Applicants should quote the refer­ Box 157. Wellesley. MA 02181 617-235-9537 ence number when submitting applications to R.V. Dubs Registrar. GPO Box 4. Canberra. ACT. 2601 Australia. THE UNIVERSITY IS AN EQUAL OPPORTUNITY Proceedings of the Nineteenth EMPLOYER. Nordic Congress of Mathematicians, University of Iceland, August 13-17,1984 SHEPHERD COLLEGE Detailed advertisement with list of contents in the Shepherdstown, West VIrginia 25443 January 1986 Notices. p. 181. Send order and pay­ ment ($20) to: Icelandic Mathematical Society. Science MATHEMATICS: FALL. 1986. For tenure-track ap­ Institute. Dunhaga 3. IS-107 Reykjavik. Iceland. pointment. PhD required. or date commitment for degree completion; one-year temporary appointment possible for other candidates. Commitment to quality teaching at a BOOKS WANTED growing. undergraduate. state-supported liberal arts col­ lege with over 3700 enrollment. located on the banks of Required back volumes of journals in mathematics. the Potomac River. 90 minutes drive from Washington. Contact Librarian with the list of journals and terms of DC. Applications reviewed beginning June 2 and received offer. Institute of Mathematical Sciences. Taramani S.O .. until position filled. Send vita and three letters of recom­ Madras-600 113. India. Telegram: Matscience. Madras. mendation. at least one referring to teaching abilities. to Dr. Peter Morris. Shepherd College. Shepherdstown. WV WANTED 25443. AA/EOE English language/mathematical symbols typewriter old inexpensive ok. Send sample script. Peter Bralver.13110 FOR SALE Bloomfield Street. Sherman Oaks. CA 91423. Math/Stats Library: List from R. Chesters. Suite 1900. 2 First Canadian Place. Toronto. Canada M5X 1E3.

576 The American Mathematical Society Transactions of the Moscow Mathematical publishes books and journals at the Society graduate and research levels in pure and Translations of Mathematical Monographs applied mathematics. About 70 new books REVIEWS VOLUMES are published annually and a backlist of Reviews in Graph Theory about 1,300 titles is maintained. Fifteen Reviews in Group Theory journals are published, including primary Reviews in Number Theory research journals, translation journals, and Reviews in Ring Theory Reviews of Papers the prestigious abstracting journal in Algebraic and Differential Topology. Topological Mathematical Reviews, which contains Groups. and Homological Algebra 40,000 reviews annually of mathematics Reviews in K-theory books and research journals from around INDEXES AND DICTIONARIES the world. Indexes of Mathematical Papers Index to Mathematics of Computation Index to Translations Selected by the PROCEEDINGS AND MONOGRAPHS American Mathematical Society Mathematical Reviews Cumulative Author CBMS Regional Conference Series in Indexes and Subject Mathematics Indexes Romanian-English Dictionary Colloquium Publications Russian-English Dictionary of the Conference Proceedings. Canadian Mathematical Sciences Mathematical Society Contemporary Mathematics Lectures in Applied Mathematics Abstracts of Papers Presented Lectures on Mathematics in the Life to the AMS Sciences Bulletin of the American Mathematical Society Mathematical Surveys and Monographs - New Series Current Memoirs of the American Mathematical Mathematical Publications Mathematical Society Reviews Mathematics Proceedings of Symposia in Applied of Computation Mathematics Memoirs of the AMS Notices of the Proceedings of Symposia in Pure AMS Mathematics Proceedings of the AMS Transactions Selected Tables in Mathematical Statistics of the AMS SIAM-AMS Proceedings TRANSLATIONS - BOOK SERIES Mathematics of the USSR - lzvestiya American Mathematical Society Mathematics of the USSR - Sbornik Translations - Series 1 Proceedings of the Steklov Institute of American Mathematical Society Mathematics Translations - Series 2 Soviet Mathematics - Doklady Proceedings of the Steklov Institute of Theory of Probability and Mathematical Mathematics Statistics Selected Translations in Mathematical Transactions of the Moscow Mathematical Statistics and Probability Society 1-~\\EM..th +. C:-t For a complete catalogue of all American Mathematical Society AMS titles. call in the continental PO Box 6248 us 1-800-556-7774. Providence. Rhode Island 02940 USA

577 Universiteit van Amsterdam

The Department of Mathematics of the University of Amsterdam invites applicants for the position of professor of algebra (m/0* Full-time. Vacancynumber N. 0889. Applicants should have extensive knowledge of algebra, an excellent research record and a feeling for the links between algebra and other mathematical disciplines. Good teaching qualities and ability to collaborate with colleagues are requested. Research activity of the applicant must be in algebra, for example • algebraic number theory • algebraic geometry • group theory, e.g. algebraic groups. Applicants must be willing to acquire a working knowledge of the Dutch language within two years after appointment.

Applicants are requested, quoting vacancynumbe~; to send a curriculum vitae, list of publications and the names of three references- before August, 1st. 1986- to the Chairman of the Appointment Committee, Dr. P. van Emde Boas, Mathematisch Instituut, Roetersstraat 15, 1018 WB Amsterdam, The Netherlands. *T#Jmen in particular are invited to apply.

MATH TEXT CARNEGIE MELLON UNIVERSITY Versic;m 1. 85 HEAD, DEPARTMENT OF MATHEMATICS

The easiest, most natural Nominations and applications are invited for technical word processor available. the position of Professor and Head of the De­ Fully what-you-see-is-what-you-get. partment of Mathematics. Candidates should have an international reputation for mathemat­ {" { sin 2n£ } d (f) ical research and be able to provide leadership I0 ef + ln(5) + 12 ~ to a diverse and growing department of applied and pure mathematics with strong programs on Configurable for IBM compatibles, the Z-100, Tandy 2000, and TI-PC. both graduate and undergraduate levels. Many supported printers. Nominations and applications should be $199 (Demo disk: $5) addressed to: Professor Victor J. Mizel NOW COMPLETELY Head of Search Comittee Produced, distributed and Department of Mathematics supported by Carnegie Mellon University HUFF Sc:>ft"lf\Ja.:re Pittsburgh, PA 15213 410 S. Fraser Street (412) 268-2562/2545. State College, PA 16801 Carnegie Mellon University is an equal (814)239-0885 opportunity /affirmative action employer.

578 The Swiss Federal Institute of Technology New at Lausanne, Switzerland SUPERCOMPUTERS Algorithms, Architectures, and invites applications for three newly created Scientific Computation professorships in Edited by F. A. MATSEN and T. TAJIMA Twenty-six articles dealing with high-level lan­ guages, cosmology, biology, seismology, elementary particles, field theory, magnetism, superconductivity, NUMERICAL ANALYSIS AND SIMULATION quantum chemistry, and reservoir engineering-the APPLIED STATISTICS proceedings of an international conference on su­ ARTIFICIAL INTELLIGENCE percomputers conducted at the University of Texas at Austin in March 1985. 488 pages, 102 line drawings, 17 tables, 1 photograph ISBN 0-292-70388-0 30th, 1986 Applications deadline: June $35.00 hardcover Beginning of activity: Fall 1986 or later (512) 471-4032 Further information at the following address: Secretariat general de I'Ecole polytechnique federale de Lausanne, CE-Ecublens, 1015 Lausanne, Suisse ~University of Box 7819 Austin, Texas 78713

JOrMath"' a text-formatter for mathematical papers, books and theses At the Swiss Federal Institute of Technol­ ogy in Zurich a chair of 1 Lim ·~-1(1+1_) _I_]2' = __!_ [{'Jo ~log(l·x) dx -1]-I MATHEMATICS n+OOI .!.:.!L_ _I_ 2 II] I is to be filled. L t 7r2;2 i=l of the new professor include teach­ Duties • matrices, big symbols sized and ing and research in the field of numerical placed automatically and applied mathematics. He will be respon­ • concise command syntax; macros sible for undergraduate and graduate courses further reduce typing bibliographic for students of mathematics, natural sciences • automatic numbering; references, chapters, equations, and engineering. cross-references The applicants should have a Ph.D., success­ • table of contents, indexes word­ fully completed own research projects in the • automatic justification, splitting field of applied mathematics and a proven • automatic numbering and place­ ability to perform and direct high quality ment of footnotes research and development. Willingness to • multiple fonts, sizes, special symbols teach at all university levels and to cooperate • output on CRT, dot-matrix printers, daisywheels, laser printers, with colleagues is expected, photo-typesetters Applications with curriculum vitae and list • use on micros

579 OUTSTANDING JOURNALS IN MATHEMATICS JOURNAL OF JOURNAL OF COMPLEXITY MATHEMATICAL A Unifying Theme ANALYSIS AND of the Sciences APPLICATIONS Editor: J. F. Traub Classical Analysis Columbia University,New York City and Its Applications Journal of Complexity publishes original Editors: Ralph P. Boas, Northwestern research papers that contain substantial University, Evanston, Illinois mathematical results on complexity as George Leitmann, brC?adly conceived. In the area of compu­ University of California, Berkeley tatiOnal complexity, Journal of Complexity Journal of Mathematical Analysis and focuses on problems that are approxi­ Applications presents mathematical mately solved and for which optimal papers that treat classical analysis and its algorithms or lower-bound results are numerous applications. The emphasis is available. The journal also presents on articles devoted to the mathematical p_apers that provide major new algo­ treatment of questions arising in physics, nthms or make important progress on chemistry, biology, and engineering, par­ upper bounds. ticularly those that stress analytical aspects and novel problems and their Volume 2 (1986), 4 issues solutions. ISSN 0885-064X Volumes 113-120 (1986), 16 issues U.S.A. and Canada: $80.00 ISSN 0022-247X All other countries $93.00 U.S.A and Canada: $912.00 U.K.: £69.00 All other countries: $990.00 U.K.: £733.50

~ample copies and personal subscription rates are available on request. For more mformation, please write or call: ACADEMIC PRESS, INC. or ACADEMIC PRESS, INC. Journal Promotion Dept. (LONDON) LTD. 1250 Sixth Avenue Journal Marketing Dept. San Diego, CA 92101 @ 24-28 Oval Road U.S.A. London NW1 7DX, England (619) 230-1840 01-267-4466 Academic Press, Inc. Harcourt Brace Jovanovich, Publishers San Diego Orlando New York Austin Boston London Sydney S6029 Tokyo Toronto l#jl#il#jl#jl#il#jl#jl#jl#jl#jl#ll#jl#jl#ll#jl#jl#ll#ll#jl##wl#li

580 ~,New Volumes Contemporary Mathematicians

With each volume published in the Contemporary Mathematicians series, Birkhauser Boston makes available the collected or selected writings of some of this century's most outstanding mathematicians. The contributions are brought together in organized, annotated volumes, with accompanying comments which provide valuable insight on the history of the papers and their influence on modern mathematics.

KURT OTTO FRIEDRICHS: Collected Mathematical Papers edited by Cathleen Morawetz The vast majority of the collected papers of Friedrichs are in partial differential equations and spectral theory where the author had the greatest impact. He had also made fundamental contributions in the theory of asymptotics: two papers, including Friedrich's Gibbs Lecture are included. The profound contributions Freidrichs made to applied mathematics are represented by some papers in elasticity and magneto-hydrodynamics. Available in June I Cloth Vol. I I 496 pages I 0-81 76-3268-9 I $64.00 tent. Vol. II I 448 pages I 0-81 76-3269-7 I $64.00 tent.

JAKOB NIELSEN: Collected Mathematical Papers edited by Vagn Lundsgaard Hansen and Fleming Topsoe Jakob Nielson was one of the developers of combinatorical group theory and the topology of surfaces. His papers, available for the first time in English translation, will play a fundamental role in recent investigations on low dimensional topology. Contents include Nielsen's thesis containing the seeds to his later research on the topology of surface transformations. His four long memoirs in Acta mathematica from 1927, 1929, 1932, and 1942, with the details of his pioneering work on surface transformations, appear in English from the original German text, along with his famous paper on free groups from 1921 originally published in Danish. Also included is a set of hitherto unpublished notes from two inaugural lectures given by Nielsen in Breslau in 1921, in which he formulated many of the problems on surface transformations to which he devoted so much of his research later in life. Available in June I Cloth Vol. I I 4 72 pages I 0-81 76-3140-2 I $60.00 tent. Vol. II I 432 pages I 0-81 76-3151-8 I $60.00 tent.

FRITZ JOHN; Collected Papers edited by Jurgen Moser The deep and original ideas of Fritz John have had a great influence on the development of various fields in mathematical analysis. John's fundamental papers on convexity, uniqueness, ill posed problems, numerical analysis, quasi-isometry, and blow-up have all started new developments. Each is followed by commentary by a leading expert in the field and worth careful and attentive study. Available I Cloth Volume 1 I 648 pages I 0-8176-3266-2 I $97.50 Volume 2 I 760 pages I 0-8176-3267-0 I $97.50

Forthcoming GARRETT BIRKHOFF: Selected Papers on Algebra and Topology Available in July I Cloth I 0-1876-3114-3 Birkhauser Boston .., 380 Green Street, Cambridge, MA 02139 M

581 See the world of mathematics through the minds of experts

An Introduction to Mathematical An Introduction to Differentiable Logic and Type Theory Manifolds and Riemannian To Truth through Proof Geometry PETER B. ANDREWS Second Edition This book facilitates the study of logic as a vehicle WILLIAM M. BOOTHBY for formal reasoning. It presents issues related to the problem of constructing formal proofs as mod­ In Paperback 1986, 448 pp. els of informal reasoning, and details information $32.95/ISBN: 0-12-116053-X used in computerized systems which involve au­ $59.00/ISBN: 0-12-116052-1- Casebound tomated deduction. In Paperback June 1986, 320 pp. $29.95/ISBN: 0-12-058536-7 Eilenberg-Mac Lane: $55.00/ISBN: 0-12-058535-9- Casebound Collected Works SAMUEL ElLENBERG and Complex Cobordism and Stable SAUNDERS MAC LANE In Paperback Homotopy Groups 1986, 856 pp. of Spheres $39.50/ISBN: 0-12-234021-3 DOUGLAS C. RAVENEL $69.50/ISBN: 0-12-234020-5- Casebound This is a unified account of the application of complex cobordism to the stable homotopy groups of spheres. It presents self-contained, A favorite text-now available in well-documented treatments of the relevant pre­ paperback! requisites in homotopy theory, homological algebra, and formal group law theory. Fourier Series and Integrals In Paperback H. DYM and H. P. McKEAN 1986, 440 pp. $45.00/ISBN: 0-12-583431-4 In Paperback $90.00/ISBN: 0-12-583430-6- Casebound 1972, 295 pp. $31.00/ISBN: 0-12-226451-7 $57.50/ISBN: 0-12-226450-9- Casebound

To Order Call Collect (305) 345-2731 ACADEMIC PRESS Harcourt Brace Jovanovich, Publishers Orlando, Florida 32887 Orlando San Diego New York Austin Boston London Sydney Tokyo Toronto 12066 HR!GD

582 ''If it weren't for AMS Members' Life Insurance, I'd be underinsured ... by thousands of dollars!''

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583 THE LONDON MATHEMATICAL SOCIETY

Burlington House, Piccadilly, London WlV ONL Incorporated by Royal Charter

The Society was established in 1865 for the promotion and extension of mathematical knowledge. Members of the American Mathematical Society are very welcome to join the London Mathematical Society under the Reciprocity Agreement, which enables certain formalities to be dispensed with and a reduced subscription to be applied. For the year 1985/86 the reduced subscription is £3.00 (US$5.00).

Members of the American Mathematical Society may commence or continue their reciprocity membership of the London Mathematical Society during periods of temporary residence in the United Kingdom. They may then take advantage of all the facilities offered to London Mathematical Society members, such as use of the Society's extensive Mathematical Library at University College, London. They are also welcome to attend Society meetings, which take place about seven times a year.

Members of the American Mathematical Society resident outside the United Kingdom may be chiefly interested in subscribing to the periodicals of the L.M.S. These are the Bulletin, the Journal and the Proceedings of the London Mathematical Society. One volume of the Bulletin is published per year, and two volumes each of the Journal and Proceedings. Members of the American Mathematical Society may purchase periodicals at special rates on condition that they are for personal use of members and not for the supply of libraries or similar institutions.

The London Mathematical Society also publishes a series of Monographs, Lecture Notes, Students Texts and the proceedings of its Instructional Conferences. Members may purchase these books at a discount of 25%.

Full particulars of the activities of the L.M.S. and application forms for reciprocity membership may be obtained from the Administrative Assistant of the L.M.S. at the above address.

584 A REVOLUTIONARY TEXT PROCESSOR FOR THE SCIENTIFIC COMMUNITY. TELH/~ r For IBM PC's* and Compatibles. DOS 2.0 or higher. 256K, IBM Color, IBM EGA, or Hercules** Board Required. Registered trademarks- *International Business Machines Corp. and **Hercules Computer Technology Inc.

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585 Personal AT LAST: Mathematical Typesetting 1EX Capability for PC Users! Inc the producers of PC '!EX offer a complete line of software, hardware and fonts for journal quality output. T.EX is a state-of-the-art typesetting program developed by Professor Donald E. Knuth at Stanford University. T.EX is being supported as a standard language for mathematical type­ setting by the American Mathematical Society. The AMS has developed a special package of mathematical typesetting tools for T.EX, called AMS-T.EX, which greatly simplifies the setting of complex mathematical formulas. T.EX inputs a standard ASCII computer file, and generates output which can be directed to print on a variety of devices, from dot matrix printers to laser printers to phototypesetters. This entire ad was typeset using PCT.EX and printed on the Corona LP300 Laser Printer. At the recent AMS Conference in New Orleans, we asked mathematicians to suggest com­ plex formulas which we then typeset using T.EX· The following was submitted by Bernard Harris, Department of Statistics, University of Wisconsin.

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586 MATHEMATICAL SCIENCES RESEARCH INSTITUTE BERKELEY, CALIFORNIA

The Institute solicits applications for memberships in the Institute for the 1987-88 year, which begins in September 1987.

In 1987-88 two programs will be featured.

Classical analysis. The following areas will receive particular attention: singular integrals and pseudo-differential operators, Fourier analysis in Euclidean space, Hardy space theory, applications of probability theory to Fourier analysis, and applications of harmonic analysis to partial differential equations and several complex variables. The program committee consists of C. Fefferman, E. Stein (chairman), and G. Weiss.

Representations of Lie groups. The program will focus on the representations of semisimple Lie groups. Among topics to be emphasized are:

( 1) The systematic use of asymptotics and homological methods,

(2) The development of new geometric methods,

(3) Recent advances in the understanding of the structure of Harish-Chandra modules, (4) The structure of the unitary dual.

The program committee consists of W. Schmid, D. Vogan, and J. Wolf (chairman).

Although these two areas will be emphasized, applications from candidates in all fields will be welcome.

The academic year 1987-88 will conclude with a "microprogram" on the Structure of Banach Spaces, running from June 27 to July 15, 1988. The program committee consists ofJ. Bourgain, W. Johnson, Y. Lindenstrauss, G. Pisier, and H. Rosenthal (chairman). The mathematical public is cordially invited to participate. At a later time there will be a separate poster giving details.

The Institute does not use formal application forms. However, an information sheet for prospective applicants is available on request. Write to the Mathematical Sciences Research Institute, 1000 Centennial Drive, Berkeley, CA 94720. Applicants' files should be completed by December 15, 1986. Please note that this is a half month earlier than the deadline of previous years.

For 1988-89 the Institute plans yearlong programs on Combinatorial Group Theory and Geometry and on Symplectic Geometry and Mechanics. Suggestions for programs for future years are welcome.

The Institute is committed to the principles of Equal Opportunity and Affirmative Action.

587 STATISTICAL SCIENCE a review journal of the institute of mathemadcal stadsdcs

EXECUTIVE EDITOR: Morris H. DeGroot, Carnegie-Mel/on University EDITORS: David R. Brillinger, University ol California, Berkeley J. A. Hartigan, Yale University Ingram Olkin, Stanford University On the occasion of its fiftieth anniversary, the Institute of Mathematical Statistics announces publication of Statistical Science, a new quarterly review journal in statistics and probability. Statistical Science will present the full range of contemporary statistical thought at a modest technical level accessible to the broad community of practitioners, teachers, researchers, and students in statistics and probability. featuring / Volume I Number I February 1986 • discussions of methodological and theoreti­ cal topics of current interest and importance, Freedman & Navidi on MODELS FOR ADJUSTING THE CENSUS • surveys of substantive research areas with Efron & Tibshirani on promising statistical applications, THE BOOTSTRAP Le Cam on • evaluations of research papers in specific THE CENTRAL LIMIT THEOREM areas of statistics and probability, AROUND 1935 Geisser on • discussions of classic articles from the THE COLLECTED WORKS OF literature with commentary on their impact GEORGE E. P. BOX on contemporary thought and practice, Genest & Zidek on COMBINING PROBABILITY • comprehensive book reviews, and DISTRIBUTIONS Interviews with • interviews with distinguished statisticians DAVID BLACKWELL & T. W. ANDERSON and probabilists. / ' / Statistical Science joins the Institute's distinguished journals, The Annals of Statistics and The Annals of Probability, in February 1986 and is included as a privilege of membership. Nonmember subscriptions are available to individuals and organizations. All subscriptions to the Annals include a subscription to Statistical Science for 1986. For additional information on how to receive Statistical Science, please write to the IMS Business Office.

IMS Business Office 3401 Investment Boulevard #7 Hayward, California 94545 (USA)

588 PUBLICATIONS OF L'ENSEIGNEMENT MATHEMATIQUE

MONOGRAPHS SERIES

A. WEll: Essais historiques sur Ia P. ERDOS AND R. L. GRAHAM: Old theorie des nombres. - 55 pages, 28 Swiss and new problems and results in com­ Francs. binatorial number theory. - 128 pages, 43 Swiss Francs. THEORIE ERGODIQUE - Proceedings of a Seminar. - Lectures by P. Arnoux, P.-L. Aubert, P. de La Harpe and K. Jhabvala, M. Misiure­ wicz, C. Series, B. Weiss. 112 pages, 40 Swiss Francs. LOGIC AND ALGORITHMIC- An International Symposium N

IMU LECTURES- Series of survey Lectures given under the sponsorship of the International Mathematical Union

W. M. SCHMIDT: Approximation to D. MUMFORD: Stability of projective algebraic numbers. - 70 pages, 30 Swiss varieties. - 74 pages, 32 Swiss Francs. Francs. A. G. VITUSHKIN: On representation of functions by means of superposi­ F. HIRZEBRUCH: Hilbert modular sur- tions and related topics. - 68 pages, faces.- 103 pages, 37 Swiss Francs. 30 Swiss Francs.

NEW Masaki KASHIWARA: Introduction to microlocal analysis These survey lectures were given at the University of Bern in June 1984, under the sponsorship of the International Mathematical Union. They explain the microlocal point of view (consideration of the cotangent bundle) for the study of linear differential equations. 38 pages, 1986; 25 Swiss Francs.

Georges de RHAM:

20o/o discount available on prepaid orders placed with L'Enseignement Mathematique EM POB 240, CH-1211 Geneva 24, Switzerland

589 Directory of Women in the Mathematical Sciences Questionnaire for Women who Hold the Ph.D. or Have Equivalent Training, and Ph.D. Candidates

(PLEASE TYPE. Handwritten forms may not be used.)

1. A. Name {last name first)------

B. Cross listing of name {if desired) ------

2. Mailing Address {Preferred)

*Current Status D (Employed) D {Retired) D {Graduate Student) D {Unemployed) (*This information is for record keeping and will not be printed in the Directory.)

3. Name of employer or institution------­ (1£ retired, give last institution where employed.)

Address of employer or institution ------

Title of position------

4. Highest academic degree conferred------

Year degree conferred ______Institution------

5. Fields of mathematical interest (Write 1 for primary field, 2 for other. Only two will be printed.) A. __ Algebra H. __ Logic and foundations B. __ Analysis I. __ Number theory C. __ Applied mathematics J. __ Operations research D. __ Combinatorics K. __ Statistics and Probability E. __ Computer science L. __ Topology F. __ Functional analysis M. __ Other G. __ Geometry

6. Publications: Bibliographic information for two most important publications. In addition to the title, give MR number and date only, if available; otherwise for an article, use MR abbreviation for journal, give volume number and date.

i. ------11. ------

Return this form as soon as possible to Directory of Women in the Mathematical Sciences, c/o R. Struik, Math. Dept., Campus Box 426, University of Colorado, Boulder, CO 80309.

590 American Mathematical Society P. 0. Box 6248 Charge by Providence, Rhode Island 02940 Phone (401) 272-9500 c800-556-7774 Ordered by------Mail to (if different)------

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Fasten PAYMENT securely Transform Analysis of Logic, Methodology and Generalized Functions Philosophy of Science VII By O.P. MISRA and J.L. LAVOINE Proceedings of the Seventh International Congress, Salzburg, Austria, 11-16 July 1983 NORTH-HOLLAND MATHEMATICS STUDIES, 119 Edited by A. BARCAN MARCUS, G.J.W. CORN and Notas de Matematica (1 06) Editor: Leopolda Nachbin P. WEINGARTNER

Transform Analysis of Generalized Functions concentrates STUDIES IN LOGIC AND THE FOUNDATIONS OF on finite parts of integrals, generalized functions and MATHEMATICS, 114 distributions. It gives a unified treatment of the distributional setting with transform analysis, i.e. Fourier. Logic, Methodology and Philosophy of Science presents Laplace, Stieltjes, Mellin, Hankel and Bessel Series. the results of recent research into the foundations of science. Included are accounts of applications of the theory of This volume contains 41 invited papers presented at the integral transforms in a distributional setting to the solution Congress, covering the areas of Logic, Mathematics, of problems arising in mathematical physics. Information Linguistics, Natural Sciences and Humanities. Also on distributional solutions of differential, partial differential included is a discussion of current problems in the Ethics equations and integral equations is conveniently collected of Science. here. 1986 xiv + 738 pages. Price: US $125.00/Dfl. 400.00 The volume will serve as introductory and reference ISBN 0-444-87656-1 material for those interested in analysis, applications, physics and engineering.

1986 xiv+ 332 pages. Price: US $48.251Dfl. 140.00 ISBN 0-444-87885-8

Harvey Friedman's Research on the Foundations of By YU.I. MANIN Mathematics NORTH-HOLLAND MATHEMATICAL LIBRARY, 4 In the ten years since this book was first published in Edited by L.A. HARRINGTON, M.D. MORLEY, English, there has been important progress in a number of A. SCEDROV and S.G. SIMPSON related topics. The class of algebraic varieties cloee to the rational ones has crystallized as a natural domain for the STUDIES IN LOGIC AND THE FOUNDATIONS OF methods developed and expounded in this volume. MATHEMATICS, 117 The original text has been left intact, except for a few This volume discusses various aspects of Harvey corrections, and brought up to date by the addition of an Friedman's research on the foundations of mathematics Appendix and recent references. over the past fifteen years. It should appeal to a wide audience of mathematicians, computer scientists, and This Appendix sketches some of the most essential new mathematically oriented philosophers. results, constructions and ideas, including the solutione of the Liiroth and Zariski problems, the theory of the descent Much of Friedman's work enjoys a quality of broad and obstructions to the Hasse principle on rational accessibility and communicability to anyone who is even varieties, and recent applications of K-theory to slightly interested in the foundations of mathematics. arithmetic.

1985 xvi +408 pages. Price: US $55.50/Dfl. 150.00 1986 X+ 326 pages. Price: US $64. 75/Dfl. 175.00 ISBN 0-444-87834-3 ISBN 0-444-87823-8

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