MSRI After Three Years

Irving Kaplansky

Prelude In the early 1960s I was at a meeting of an advisory committee at the National Science F o u n d a t i o n . Physics, chemistry, and astronomy were represented (in addition to mathematics). We were asked the following question: What, in your discipline, could make for a quantum leap in achievement? When my turn came, I u n h e s i t a t i n g l y replied: "For mathematics, found two new Institutes for Advanced S t u d y - - o n e in the Midwest, and one on the west coast." The next day the Director of NSF joined us and commented on our suggestions. When it was the turn of mathematics he addressed me by name (slightly surprising, since we had never met) saying, "Dr. Kaplansky, I note with interest your suggestion of two new institutes." Twenty years later the dream came true! I have no idea whether there was any connection, however remote, b e t w e e n the incident at NSF a n d w h a t ultimately happened. In particular, nobody ever asked me again. (However, on m y own initiative I did push the institute idea on every possible occasion.) I find it interesting that the prototype NSF institute--the Institute for Theoretical Physics at Santa B a r b a r a - preceded the two mathematics institutes. However, at that committee meeting the physics representative did not mention research institutes (I forget what he did suggest). Where did my enthusiasm for institutes originate? From the granddaddy of institutes, of course: the venerable Institute for Advanced Study in Princeton. I first went there in the fall of 1946. In advance, I had a picture of the place in my mind. I visualized a serene shrine w h e r e one's mathematical adrenalin would almost automatically start flowing. A phrase in Infeld's autobiography Quest comes to mind: a place where one could virtually reach out and grab a theorem anywhere (actually, Infeld was describing Fine Hall, the mathematics building of Princeton University). It lived up to expectations. I spent some time tidying up some projects I already had underway, but in restrospect I can see that in the fall of 1946 the seeds were planted for forthcoming work on noncommutative rings, C*-algebras, AW*-algebras, and a few other 48

things. In other words, I got a mathematical push that lasted a decade. I am not alone in this feeling. I recommend to the reader the Intelligencer interview with Atiyah [1], where there is a warm tribute to the Institute for Advanced Study on page 14. I think I am only exaggerating slightly in saying that, for my generation, "the" Institute was a paradise we had to enter somehow, some time, at all costs. I mean that last phrase literally: some financial sacrifice was common. By the 1960s the feeling was m o u n t i n g that the mathematical community had outgrown the Institute for Advanced Study. One can of course ask, " W h y not just enlarge it?" In point of fact, it was enlarged more than once. But there comes some breaking point. A research institute with a h u n d r e d members can pres u m a b l y be quite agreeable; one with a t h o u s a n d , probably not. Then there is the matter of geography in this large country. In 1932, when the Institute for Advanced Study was founded, American mathematics for the most part was huddled in the Northeast. There was a sturdy outpost at Chicago (if I may be forgiven for saying so myself), but west of the Mississippi mathematical research had at best a precarious foothold. By the 1960s this had changed. Strong departments of mathematics dotted the landscape from coast to coast and from border to border. Twenty years later still these departments were even stronger and new ones had joined the ranks. That the time had come for three institutes to span the continent seemed entirely persuasive to me. Not everyone agreed. In the bibliography I have assembled some of the documentation of the debate that started up about 1978 ([2], [3], and [4]). Items [5] and [6] might be described as the birth certificates of the two new institutes.

1982--84 The Intelligencer article by my colleague Calvin Moore [7] sets forth with fairly complete detail the way MSRI is formally organized and how the scientific aspects are arranged. I shall try to minimize duplicating the material in this article. But let me add something that

THE MATHEMATICAL INTELLIGENCER VOL. 7, NO. 4 9 1985 Springer-Verlag New York

a reader of [7] might not adequately realize: founding a research institute from scratch involves a myriad of problems that have to be solved. I can only say that it all looked like a miracle to me when I first saw MSRI in February 1983 (I was there for a meeting of the program committee for the 1983-84 program on infinitedimensional Lie algebras). MSRI was a going concern; the place was humming; and the programs in statistics and nonlinear partial differential equations were flourishing. Three months later I was astounded to receive a phone call from H y m a n Bass (Chairman of the MSRI Board of Trustees) asking if I might be interested in the Directorship of MSRI. (The founding Director, Shiing-Shen Chern was planning to step down in the fall of 1984, despite universal pleas--in which I instantly joined--to reconsider.) Well, my answer was "yes". I arrived for good in March 1984, on leave from the University of Chicago to participate in the Lie algebra program (as had been planned before the possibility of the directorship arose). The other 1983-84 program was on ergodic theory and dynamical systems. Since I am going to concentrate in this article on the 198485 programs that are current as this is being written, I shall just record one minor personal incident of 198384. It illustrates, in a very modest way, how bringing together people in two different fields can have unpredictable consequences. Mike Boyle was one of the participants in the ergodic theory program. (An amusing sidelight is that he is one of my academic descendants: K ~ Ornstein Lind ~ Boyle; both Douglas Lind and Donald Ornstein were also involved in the ergodic theory program.) In his work he ran into a matrix question that was right up m y alley, and fortunately he called it to my attention. Question: Do there exist integral matrices A and B which are not similar (over the integers) but do become similar w h e n blown up to (O

o) and(O

I)

?

I managed to find such matrices. Actually, what I did was not quite good enough. One needed in addition that A and B should have d e t e r m i n a n t _+1; Mike promptly remedied that. His matrices are

(i i) '~

(i i)

A matrix implementing similarity of the enlarged matrices is (i

112501 ! ) 12 3

"

Program Committee for Operator Algebras: (I to r) Alain Connes, Ronald Douglas, Masamichi Takesaki.

Incidentally, this example clarifies a mis-statement in the literature (I shall skip the technical details). See page 374 of [8]. 1984--85

The three programs for 1984-85 were chosen long before I came on board. So I trust I will not be thought self-serving if I say that the selection was marvelous. I would like to convey to the reader a feeling for what 1984-85 has been like at MSRI. It got off to an early start with an intensive week-long workshop entitled " G e o m e t r y and Operator Algebras", running from August 20 to 25, 1984. Each of the six mornings was devoted to a single presentation; the three committee members (Connes, Douglas, and Takesaki) took three of the mornings, and the other three featured speakers were Bott, Quillen, and Singer. During the afternoons there were a number of shorter talks. The old Lecture Hall, with a capacity of 80, spilled over into the lobby, where an additional thirty or so people followed the proceedings on TV. A closing conference with the same title was planned for June 5-12, 1985, as a kind of mirror image. The low-dimensional topologists planned a monotone increasing sequence of workshops with dimension moving from 2 (October 11-16, 1984) to 3 (January 9-19, 1985) and then to 4 (May 20-25, 1985). Differential geometry featured a single workshop (May 28June 1, 1985). By late October 1984 things had simmered down and a pattern emerged. Let me take as a sample the week of October 22-26, 1984. There were ten seminars at MSRI that week (see the displayed box). As I shall shortly explain, the lines are not easily drawn this year, but I will declare six of them to be on operator algebras, three on topology, and one on differential geometry. Some of the talks reported on the speaker's recent research. Two of the operator THE MATHEMATICAL INTELLIGENCER VOL. 7, NO. 4, 1985

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Program Committee for Low-Dimensional Topology: (l to r) Robert Edwards, John Morgan, William Thurston, Robion Kirby. algebra seminars were frankly instructional and each ran for a bone-grinding two hours. Wednesday afternoon saw an orgy of low dimensional topology. Friday morning belonged to Vaughan Jones and his disciples (more about this later). A selection of Evans Hall seminars are regularly listed in the MSRI weekly bulletin (Evans Hall is the mathematics building). Especially important in 198485 were two run by Bill Arveson and Marc Rieffel: function analysis on Tuesday and operator algebras on Wednesday. (I recall with special pleasure Gert Pedersen's talk on Tuesday October 23: "AW*-algebras: The theory that refuses to die".) On Mondays and Thursdays there is a general trek from MSRI to Evans. When MSRI started, Chern inaugurated the "Director's Seminar", a w a y of showcasing MSRI talent at Evans Hall. Under a new name it continues to thrive under the new regime, and I hope it will continue forever. On Monday, October 22 Paul Baum spoke in this seminar on "Atiyah-Singer for beginners". Thursday is the department's regular colloquium day; on October 25 Vaughan Jones spoke on "Braids and knots, Hecke and yon N e u m a n n algebras". Cal Moore and I have been interested in C*-algebras for a long time; my attention strayed but he remains active in the field. But how the subject has changed! Sitting in a seminar one day I looked around with wonder at the sea of young faces, all born long after the appearance of Rings of Operators I in 1936. Even the senior members of the group (Paul Baum, Ronald Douglas, and Peter Fillmore) were at most babies then. But not everything has changed. Well behaved C*-algebras (call them GCR, Type I, postliminal, or what you will) still pop up n o w and then. Led by the indefatigable Alain Connes, C*-algebras are being used by an army of investigators in an invasion of large chunks of mathematics. That there is a subject appropriately called noncommutative measure theory was firmly established long ago. This has now 50

THE MATHEMATICAL INTELLIGENCER VOL. 7, NO. 4, 1985

been joined by noncommutative topology and nonc o m m u t a t i v e differential g e o m e t r y . Foliations are being studied via C*-algebras. Atiyah-Singer index theory fits in. In short, C*-algebras have burst out of the narrow boundaries that formerly confined them and are spilling over in all sorts of directions. Perhaps most exciting of all has been a totally unexpected connection between knots and operator algebras. This was the discovery of Vaughan Jones and was announced at the opening workshop of the operator algebra year. As a result, the two major programs of 1984-85 have come into contact in a resounding way. At a typical seminar it is hard to decide what is going on: C*-algebras or low dimensional topology; at any rate, you cannot tell for sure just by listening to the speaker or looking at the audience. The starting point was Jones's paper [12]. Let A be a factor of type II1, and B a subfactor. There is a natural way to define a relative index; the unqiue traces that exist on each of A and B play a key role in the definition. A priori, the index looks like it is entitled to be any real number 21. But here comes a surprise: while every number ->4 is indeed eligible, only a scattering of values b e l o w 4 are permissible. The w a y this is proved is perhaps even more intriguing than the result. Jones constructs a sequence e i of projections (selfadjoint idempotents) satisfying, among other things, the equations. eiei_+lei = rei, eiej = ejej for li-jl 2 2, where r is the reciprocal of the index. From these equations a self-contained analysis leads to the restrictions on the index mentioned above. The above equations are reminiscent of Artin's presentation of the braid group: SiSi+lS i =

Si+lSiSi+l,

sisj = sjsi for [i-j[ >_- 2. This is not just a resemblance: for a suitable scalar t, the elements t 2 (e i + tei_l) satisfy the braid relations. A close analysis of this situation led Jones to the discovery of a new polynomial invariant for knots, and of course everyone is calling it the "Jones polynomial". It differs from the classical Alexander polynomial; perhaps most striking is the fact that the Jones polynomial can distinguish a knot (such as the trefoil) from its mirror image. See the announcement [13] for details. Promptly there was another advance. Several mathematicians (Adrian Ocneanu, Kenneth Millett, Ray Lickorish, Peter Freyd, David Yetter and Jim Hoste) independently discovered a two-variable polynomial invariant that combines the t w o polynomials and generalizes them. There is as yet no name for this polynomial. This work no longer makes any explicit reference to operator algebras. But the fact remains that

the new idea came from operator algebras. Within the world of low dimensional topology itself excitement reigns. In the 1950s and 1960s the topology of manifolds of dimension _->5 attained a high degree of polish, with many of the main problems completely solved. (A leader in these advances was Steve Smale, w h o will be active in MSRI in 1985-86 in two of its programs: Computational complexity and mathematical economics). Since then it has become increasingly clear that the lower dimensions are different. This is most visible in dimension 3, where the Poincar4 conjecture continues to hold out and shows no sign of yielding. But, modulo the Poincar~ conjecture, 3-manifolds are no longer a complete mystery. Thurston and his school are restoring order. Tools from a wide variety of disciplines are playing a role: differential geometry, algebraic geometry, representation theory, group theory, and n o w C*-algebras have joined in. I shall mention one contribution to Thurston's pro-

gram made during 1984-85 at MSRI; it is delightfully easy to state (if not to prove). The theorem is due to Joel Hass, a postdoctoral fellow. Let M be a compact 3-manifold with 71 (M) infinite and ~r2 (M) = 0. Then the covering space of M is Euclidean 3-space. Four-manifolds continue to be dominated by the spectacular achievements of Michael Freedman and Simon Donaldson (both in residence at MSRI for parts of 1985). Freedman [10] showed that simply connected four-manifolds are topologically as well-behaved as can be hoped, being essentially determined by a unim o d u l a r integral quadratic form; this includes the Poincar6 conjecture in d i m e n s i o n four. Moreover every unimodular integral quadratic form is eligible. But s m o o t h four-manifolds inhabit an entirely different world. For instance, if the quadratic form of a smooth four-manifold is positive definite it must be simply xl 2 + . . . + xn2. Donaldson's proof of this [9] was based on ideas from mathematical physics, such

THE MATHEMATICAL INTELLIGENCER VOL. 7, NO. 4, 1985 5 1

formation the metric on a compact Riemannian manifold can be transformed to one of constant scalar curvature. This was the first part of Yamabe's program to prove the Poincar6 conjecture by differential geometry, for a compact simply connected 3-manifold will be diffeomorphic to the 3-sphere if it can be given a metric of constant Ricci curvature. Shiing-Shen Chern has been Director Emeritus since September 1, 1984 but he remains very active; in particular, he is the chairman of the program committee for Differential Geometry in 1984-85.

Future Programs Program Committee for Differential Geometry: (l to r) Blaine Lawson, S.-S. Chern, I. M. Singer. as the Yang-Mills equations. Thus there are hordes of four-manifolds that do not admit a smooth structure: just take any positive definite unimodular integral quadratic form which is not a sum of squares. The determination of just which 4-manifolds admit a smooth structure is a problem awaiting a complete solution. There is a conjecture that the only ones are the obvious ones. (This conjecture, of course, is hard to defeat since the definition of "obvious" is subject to change.) Ron Stern of Utah, visiting MSRI for parts of 1984-85, has advanced beyond Donaldson and made progress toward proving the conjecture. His Intelligencer article [15] is a very readable account of these developments. On the question of the uniqueness of the smooth structure when there is one (i.e., the existence of exotic structures) there is unfinished business and a surprise. The unfinished business is the differentiable Poincar6 conjecture in dimension 4: Is there an exotic structure on the 4-sphere? The surprise is the existence of exotic structures on E4, ordinary Euclidean 4-space. (In no other dimension does this happen!) This was observed by Freedman and is seen by adroitly combining his work with that of Donaldson; see Kolata's popular account [14] of these developments. Subsequently, R.E. Gompf [11] found three such structures. In later work done at MSRI, w h e r e he is an NSF Postdoctoral Fellow, he extended this to obtain infinitely many exotic structures on E4. Still later, Clifford Taubes found uncountably many. One effect of Donaldson's work has been to bring about a significant rapproachement between differential geometry and topology. That these are two of the three fields being pursued in 1984-85 at MSRI is fortunate, to say the least. The timing of the differential geometry workshop, soon after the workshop on 4manifolds is another happy fact. Within the world of differential geometry itself there is solid progress. Notable is Schoen's proof of the Yamabe conjecture, which says that by a conformal trans52

THE MATHEMATICAL INTELLIGENCER VOL. 7, NO. 4, 1985

The heart of MSRI lies in its annual programs. As a corollary, the work of the Scientific Advisory Council and the program committees is absolutely vital. The six member Scientific Advisory Council picks the programs; the program committees conduct them scientifically. If both do their work well, success is virtually certain; if they falter, nothing else is going to help. The role of the program committees is so important that I thought it appropriate, in this article devoted largely to 1984-85, to display all their photographs. The box displays the programs and program committees from the beginning through the year 1987-88. (I should add that the programs for 1987-88 are tentative, for that is the first year of the five year renewal which is pending as this is being written.) I shall let this box speak for itself in giving the reader an overall view of mathematics at MSRI, but let me record here the workshops planned for 1985-86, to the extent that they have been determined at this writing.

Mathematical Economics (topics and organizers to be announced), Winter 1986 (Friday to Sunday).

Numerical Analysis and Complexity Theory, January 1986.

Holomorphic Functions and Moduli, March 13-19, 1986.

The Theory of General Economic Equilibria, April 1986 (Friday to Sunday).

Complexity Aspects of Parallel and Distributed Computing, May 19-23, 1986. Peter Lax Conference, June 9-12, 1986. From the start not all the available resources were reserved for the major programs. There remained a residue which, for lack of a better name, was called "Area III". Visitors (at any level) fall under Area III if they are unconnected with the main programs or have only a peripheral connection. For the most part Area III w o r k e d well, b u t there w e r e e n o u g h question marks to suggest the desirability of an experiment in 1986-87 (or, more exactly, two experiments). The first experiment is an organized offering of "midcareer" sabbaticals. This is in response to the ten-

dency of MSRI awards to concentrate on senior mathematicians and postdoctoral fellows, with midcareer ones "falling between the cracks". A widely distributed poster and a full-page advertisement in the March 1985 Notices of the American Mathematical Society have publicized the plans. In the second experiment short programs will be tried. They have been nicknamed "microprograms", contrasting with the full programs and miniprograms (typically half size) of 1982-86. The main motivation for their introduction was the feeling that there are branches of mathematics that are important to the overall health of the mathematical enterprise, b u t which may not stand u p - - a t least for a w h i l e - - i n competition with fields where progress is particularly promising and exciting. As exhibited in the box, plans are moving ahead for three microprograms, two in 1986-87 and one in 1987-88. The summer of 1986 will see enhanced activity at MSRI. The International Congress of Mathematicians will meet in Berkeley from August 3 to 11. Special visitors to MSRI will include Beno Eckmann and Friedrich Hirzebruch for most of the summer, and Raoul Bott for several weeks after the Congress.

The New Building Ground was broken for MSRI's new building on Saturday, January 28, 1984. The mild climate of the San Francisco Bay area m a k e s y e a r - r o u n d construction work feasible, and so it was hoped that the building would be occupied within a year. But there were the inevitable delays. At length MSRI shut d o w n for the week beginning March 25, 1985 and reopened on April 1 in its new building. I am going to write about the building in glowing terms. While I sat in on some of the final sessions of the building committee as the structure neared completion, the reader will surely realize that none of the credit is mine. In addition to Shiing-Shen Chern, Cal Moore, Iz Singer, and the professionals involved, let me single out one person whose tireless efforts deserve high praise: MSRI's business manager, Diane Wegner. The box summarizes some of the highlights. I shall try to supplement this by giving the reader some sense of what it has been like to work in the building. Parking is at a level below the building. After a brisk climb up the stairs (or a short walk down a ramp if one takes the bus), one enters from the east through an imposing gateway, surmounted by MATHEMATICAL SCIENCES RESEARCH INSTITUTE in large letters. It was the architect's plan that on entering the lobby one would face the Golden Gate Bridge through a large picture w i n d o w - - a n d it works! (Despite rumors, the Bay area is usually not fogbound.) THE MATHEMATICAL INTELLIGENCER VOL. 7, NO. 4, 1985

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Mathematical Sciences Research Institute, University of California, Berkeley. The lobby conveys a feeling of spaciousness. A large coffee table surrounded by easy chairs invites casual conversation. The entire building has wall-to-wall carpeting which makes for quiet walking and a touch of luxury. Two large blackboards which adorn the lobby promptly got a t h o r o u g h workout in the very first days. (One visitor expressed great surprise on seeing the blackboards, but in due course she was convinced that they are indispensable.) Every afternoon tea is set up between the blackboards. To the right is the administrative complex. To the left are the other things vital for the functioning of the

Institute. The adjective I think of for the library is "inviting". The uncluttered look, the numerous browsing nooks a n d the instantly u n d e r s t o o d shelving plan combine to make it my favorite place in the Institute (that is, after my office). Of course, if gifts and acquisitions continue at their fine pace, the uncluttered look will come to an end, say in ten years. No one is worrying about that yet. A fully equipped kitchen adjoins a room set up for brown-bag lunching and snacking (there are vending machines). I have noticed the tables doubling as work desks while several members hammer out a recalcitrant theorem over a cup of coffee. The unitable seminar room and lecture hall are on the opposite side, safely out of the way of traffic. Patios on every floor have been heavily used since the weather has made them pleasant. From them there is a spectacular view of the university, the city of Berkeley, the skylines of Oakland and San Francisco, and (if the day is quite clear) four bridges. The fireplace will probably not be used till the winter of 1985-86 sets in, but then should be cozy on a blustery day (there are some in the Bay area). The top two floors are devoted to the 53 offices for members. On the west they have the Bay view which I just described. On the east there is a forested landscape that goes well with the building exterior. Of course there have been problems but I think it is fair to describe them as minor. Members and visitors have been lavish in their praise and the outlook seems bright.

Epilogue After three years of MSRI there remains the question of whether a research institute can thrive in the long run if the only quasi-permanent faculty consists of two administrators, with everyone else being a visitor. Cal Moore [7] suggested that this actually carries with it the advantage of " . . . a constant process of renewal and invigoration built in with the changing programs and the changing staff each year." I agree. Apparently a large part of the mathematical c o m m u n i t y also agrees, for a serious problem MSRI currently faces is reluctantly saying no to many worthwhile prospective members, for fear of overwhelming the available resources.

References 1. An interview with Michael Atiyah, Math. Intell. 6 (1984), 9-19. 2. N.S.F. considers the establishment of a mathematical sciences research institute, Notices AMS, vol. 25, Nov. 1978, 481-488. 3. Letters in the Notices of the Amer. Math. Soc.: 25, 489494; 26, 61-63; 27, 117 and 633. 4. James A. Krumhansl, Support of Mathematical Sciences, same Notices, 26 (1979), 118.

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THE MATHEMATICAL INTELLIGENCER VOL. 7, NO. 4, 1985

5. NSF funds two research institutes, same Notices, 28 (1981), 427-428. 6. William G. Rosen, Mathematical Sciences Research Institutes and other modes of support for the mathematical sciences, same Notices, 29 (1982), 2-14. 7. Calvin C. Moore, Mathematical Sciences Research Institute, Berkeley, California, Math. Intell. 6 (1984), 59-64. 8. M. Boyle, Shift equivalence and the Jordan form away from zero, Ergodic Theory and Dynamical Systems 4 (1984), 367-379. 9. S. K. Donaldson, An application of gauge theory to four dimensional topology, J. of Diff. Geom. 18 (1983), 279315. 10. Michael Freedman, The topology of four-dimensional manifolds, J. of Diff. Geom. 17 (1982), 357-453. 11. Robert E. Gompf, Three exotic R4's and other anomalies, J. of Diff. Geom. 18 (1983), 317-328. 12. V. F. R. Jones, Index for subfactors, Inventiones Math. 72 (1983), 1-25. 13. , A polynomial invariant for knots via von Neumann algebras, Bull. Amer. Math. Soc. 12 (1985), 103-111. 14. Gina Kolata, Topologists startled by new results, Science 217 (1982), 432-433. 15. Ronald J. Stern, Instantons and the topology of 4-manifolds, Math. Intell. 5 (1983), 39-44.

Mathematical Sciences Research Inst. 1000 Centennial Drive Berkeley, California 94720

First Case of Fermat's Last Theorem continued from page 47 all of the above w o r k o n the Brun-Titchmarsh Theorem, with its application to Y'2, was carried out before the possible relevance to FLT came to light. W h e n this n e w incentive a p p e a r e d , the problem was r e o p e n e d b y F o u v r y [5], w h o f o u n d further ways of applying " K l o o s t e r m a n i a " to the B r u n - T i t c h m a r s h T h e o r e m , a n d a f t e r m u c h e f f o r t h e p r o d u c e d the a d m i s s i b l e range 0 < 0.6687. T h u s the problem is solved: the set S is infinite. Over the last 15 years successive i m p r o v e m e n t s of the Brun-Titchmarsh T h e o r e m have given rise to m a n y different versions of the estimate (16), only some of which we have m e n t i o n e d . In particular, the values 0.58, 0.611, 0.619, 0.625, 0.638, 0.6563, 0.6578, 0.6587, a n d 0.6687 have a p p e a r e d in the literature or in private c o r r e s p o n d e n c e . T h e final v a l u e , t h e o n e d u e to Fouvry, incorporates five n e w estimates for C(r), for different ranges of r. It is often the case in certain areas of a n a l y t i c n u m b e r t h e o r y t h a t m u c h e f f o r t is exp e n d e d in i m p r o v i n g s o m e e x p o n e n t or other, in the w a y that the above figures show. To outsiders this can a p p e a r to be a refuge for w e a k e r researchers, as if the i m p r o v e m e n t s could be obtained merely b y m o r e care, m o r e pages of working, a n d more iterations of some e s t a b l i s h e d t e c h n i q u e . N o n e t h e l e s s , it is a fact that e a c h s u c h i m p r o v e m e n t , h o w e v e r small, r e q u i r e s a n e w idea. Fouvry's i m p r o v e m e n t from 0.6587 to 0.6687

We recently had translated a French dictionary of mathematics (by Bouvier and George). This dictionary, about 800 pages in length, is of an elementary nature, is mostly concerned with what one might call classical mathematics. It has a decidedly geometric flavor. We now need someone (or perhaps a small group of people) with a broad mathematical background to read the translation and make small corrections, correcting mistranslations, and generally smoothing it out. A reasonable fee for this seems to be around $3000. If you are interested in this project, contact Robert Torop Mathematics Editor Springer-Verlag New York, Inc. 175 Fifth Avenue New York, NY 10010

n e e d e d five n e w ideas to achieve a 1.5% increase. It must be said, h o w e v e r , that the p r e s e n t example is the only instance to date w h e r e this process of improvem e n t b y small s t e p s has t a k e n us across a critical threshold (namely 0 = 2/3 in this case) to produce a qualitatively n e w result.

References 1. L. M. Adleman and D. R. Heath-Brown, The first case of Fermat's Last Theorem, Invent. Math., 79 (1985), 409416. 2. P. D6nes, An extension of Legendre's criterion in connection with the first case of Fermat's Last Theorem, Publ. Math. Debrecen, 2 (1951), 115-120. 3. H. M. Edwards, Fermat's Last Theorem, (Springer, New York, 1977). 4. G. Faltings, Endlichkeitss~itze ffir abelsche Variet/iten fiber Zahlk6rpern, Invent. Math., 73 (1983), 349-366. 5. E. Fouvry, Th6or~me de Brun-Titchmarsh. Application au Th6or~me de Fermat, Invent. Math., 79 (1985), 383407.. 6. H. Halberstam and H.-E. Richert, Sieve Methods, (Academic Press, London, 1974). 7. H. Iwaniec, On the error term in the linear sieve, Acta Arith., 19 (1971), 1-30. 8. D. H. Lehmer, On Fermat's quotient, base two, Math. Comp., 36 (1981), 289-290. 9. P. Ribenboim, 13 Lectures on Fermat's Last Theorem, (Springer, New York, 1979). 10. S. S. Wagstaff, The irregular primes to 125000, Math. Comp., 32 (1978), 583-591.

Magdalen College Oxford, OX1 4AU England THE MATHEMATICAL 1NTELLIGENCER VOL. 7, NO, 4, 1985 5 5