fields.qxp 11/13/98 11:22 AM Page 17 The Mathematical Work of the 1998 Fields Medalists James Lepowsky, Joram Lindenstrauss, Yuri I. Manin, and John Milnor vide the order of the “Monster” sporadic finite The Work of simple group M, a group of order about 1054 “dis- covered” by B. Fischer and R. Griess but not yet proved to exist at the time. Ogg offered a bottle of Richard E. Borcherds Jack Daniels whiskey for an explanation. James Lepowsky In 1978–79, J. McKay, J. Thompson, J. H. Con- way, and S. Norton explosively enriched this nu- Much of Richard Borcherds’s brilliant work is merology [CN] and in particular conjectured the ex- related to the remarkable subject of Monstrous istence of a natural infinite-dimensional Z-graded Moonshine. This started quietly in the 1970s when \ \ representation (let us call it V = n 1 Vn ) of A. Ogg noticed a curious coincidence spanning ≥− the conjectured group M that wouldL have the two apparently unrelated areas of mathematics— following property: For each of the 194 conjugacy the theory of modular functions and the theory of classes in M, choose a representative g M, ∈ finite simple groups. and consider the “graded trace” Jg(q)= There are fifteen prime numbers p for which the n 2i n 1(tr g V \ )q , where q = e , in the upper normalizer of the congruence subgroup (p) in ≥− | n 0 half-plane.P Then the McKay-Thompson series Jg SL(2, R) has the “genus-zero property”; that is, the should be a (specified) Hauptmodul for a suitable Γ compactification of the upper half-plane modulo discrete subgroup of SL(2, R) with the genus-zero this normalizer is a Riemann surface of genus property, and, in particular, J1 (corresponding to zero, so that the field of modular functions in- 1 M ) should be the modular function ∈ variant under this discrete group is generated by J(q)=q 1 + 196884q + . This existence was only one function (a Hauptmodul). The surprise was soon essentially (and nonconstructively) proved by that these coincide with the fifteen primes that di- Thompson, A. O. L. Atkin, P. Fong, and S. Smith, and the problem was to uncover the deeper story. James Lepowsky is professor of mathematics at Rutgers Griess [Gr] then proved the existence of the University, New Brunswick. His e-mail address is Monster by constructing it as an automorphism [email protected]. group of a remarkable new algebra of dimension Photographs used in this article are courtesy of ICM98. 196884. Later I. Frenkel, J. Lepowsky, and A. Meur- man gave a construction, incorporating a vertex op- Editor’s Note: The 1998 Fields Medalists are erator realization of the Griess algebra, of a “moon- Richard E. Borcherds, William Timothy Gowers, shine module” V \ for M whose McKay-Thompson Maxim Kontsevich, and Curtis T. McMullen. An ar- series for 1 M was indeed J(q). Only some, and ticle giving biographical information about the ∈ far from all, of the McKay-Thompson series for this medalists and a brief summary of the work of each structure V \ could be computed directly. This appears in the November 1998 Notices, pp. construction was reinterpreted by physicists, dur- 1158–1160. ing the resurgence of string theory in the mid- JANUARY 1999 NOTICES OF THE AMS 17 fields.qxp 11/13/98 11:22 AM Page 18 1980s, as a “toy model” physical the root multiplicities are exactly the coefficients theory of a 26-dimensional bosonic of certain automorphic forms. string compactified on a 24-dimen- Borcherds’s remarkable achievement concern- sional toral “orbifold” associated ing moonshine followed, in his strikingly original with the Leech lattice. Thus the Mon- proof [B3] that all of the McKay-Thompson series ster turned out to be the symmetry for V \ do in fact agree with the 194 series written group of an idealized physical the- down by Conway and Norton and in particular sat- ory. The term “vertex operator” isfy the genus-zero property; that is, the Conway- comes from the early days of string Norton conjecture holds for V \. His strategy was theory, when operators of this type to tensor V \ with a rank-two vertex algebra to were used to describe interactions form a rank-26 vertex algebra on which M acts at a “vertex”. Affine Lie algebras canonically, and he drew on a rich variety of ideas, were constructed via what turned among them ideas from vertex algebra theory, the out to be certain variants of physi- Richard E. Borcherds theory of Borcherds algebras (particularly his sin- cal vertex operators. gularly interesting “Monster Lie algebra”), string Then came a penetrating insight of Borcherds: theory (especially, critical 26-dimensional string He introduced his axiomatic notion of “vertex al- theory and the “no-ghost theorem” of R. C. Brower, gebra” [B1] and perceived among many other things P. Goddard, and C. Thorn), and modular function that the moonshine module could be endowed theory. He established a twisted denominator for- with an M-invariant vertex algebra structure. The mula for the Monster Lie algebra by exploiting the concept of vertex algebra is a mathematically pre- homology of a suitable subalgebra, and he con- cise algebraic counterpart of the concept of “chi- cluded that the series for V \ satisfy the “replica- ral algebra” in two-dimensional conformal quan- tion formulas” of Conway-Norton and thus, as a tum field theory as formalized by A. Belavin, result of a verification of initial data, agree with A. Polyakov, and A. Zamolodchikov (a physical the Conway-Norton series. theory foundational in string theory and in two- The fact that the root multiplicities of the Mon- dimensional statistical mechanics). This funda- ster Lie algebra are the coefficients of J(q) and the mental notion reflects deep features of the tradi- relation of this fact to the denominator formula are tional notions of commutative associative algebra just the tip of an iceberg: When Borcherds pursued and at the same time of Lie algebra. A vertex op- this idea for a wide range of Borcherds algebras, erator algebra structure (a variant of vertex alge- he discovered a powerful and unexpected corre- bra structure) on V \ was given in [FLM] and allowed spondence between certain classical modular func- the possibility of characterizing V \, by a still-un- tions and meromorphic modular forms associated proved conjecture, as the unique vertex operator algebra satisfying a short list of natural condi- with arithmetic subgroups of SO(n, 2). The re- tions. The Fischer-Griess Monster, then, is the au- sulting infinite product expansions led to striking tomorphism group of a (conjecturally) unique new new results on moduli spaces of certain varieties. kind of mathematical object. The nonclassical fla- J. Harvey, G. Moore, Borcherds, and others have de- vor of the theory of vertex algebras in mathemat- veloped potentially far-reaching connections with ics can be thought of as analogous to the non- mirror symmetry and string duality. classical flavor of string theory in physics. Borcherds’s insights have influenced a wide range of works. For example, the value of having Borcherds was meanwhile developing his the- a conceptual notion of vertex (operator) algebra has ory of generalized Kac-Moody algebras [B2], now been immense. It becomes possible to formulate also called “Borcherds algebras”. Kac-Moody alge- new questions and to address new problems. Here bras form a very important class of Lie algebras generalizing the class of finite-dimensional semi- are some notable examples: the initiation of a pro- simple Lie algebras. Outside the fundamental class gram to construct (geometric) conformal field the- of affine Lie algebras, the infinite-dimensional Kac- ory using vertex operator algebras (I. Frenkel), so- Moody algebras have been notoriously difficult to lution of the problem of constructing “tree-level” construct concretely. Borcherds had the insight conformal field theory in the sense of G. Segal and to study systematically the phenomenon of “imag- M. Kontsevich (Y.-Z. Huang), a vertex-operator-al- inary simple roots”, and the resulting algebras en- gebra-theoretic proof of modular transformation compass a wide variety of striking examples whose properties of “characters” of modules (Y. Zhu), root multiplicities Borcherds determined com- and a natural approach to the construction of ver- pletely and which he in fact constructed directly tex (operator) algebras and their modules (devel- from suitable vertex algebras. He established what oped systematically by H. Li and others). are now called the “Weyl-Kac-Borcherds charac- The deepest mysteries of moonshine are still not ter” and “denominator formulas” for these alge- fully resolved. Some notable works in this direc- bras. Most importantly, he made the fundamental tion are Borcherds’s and A. Ryba’s investigation of discovery that for suitable families of examples, moonshine over finite fields and work of C. Dong, 18 NOTICES OF THE AMS VOLUME 46, NUMBER 1 fields.qxp 11/13/98 11:22 AM Page 19 H. Li, and G. Mason on Norton’s generalized moon- 1. Counterexamples to the main shine conjectures. open problems on the structure of Only part of Borcherds’s important work has infinite-dimensional Banach been touched on here. Discussions and treatments spaces. of many facets of his accomplishments and his It has become clear in recent ideas can be found in, for example, [FLM], [Ge], decades that there is a rich structure [JLW], [K], and Goddard’s and Borcherds’s talks at theory for Banach spaces of a high the International Congress [Go], [B4]. These works finite dimension, as exemplified by include listings of many of Borcherds’s papers. the theorem of A. Dvoretzky on the existence of almost Euclidean sec- References tions.