The Mathematical Work of the 1998 Fields Medalists James Lepowsky, Joram Lindenstrauss, Yuri I

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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 “discovered” 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-

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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,

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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. On the other hand, progress [B1] R. E. BORCHERDS, Vertex algebras, Kac-Moody algebras, on the structure theory of infinite- and the Monster, Proc. Natl. Acad. Sci. U.S.A. 83 dimensional Banach spaces was (1986), 3068–3071. rather slow till recently, and many William Timothy Gowers [B2] ———, Generalized Kac-Moody algebras, J. Algebra 115 natural problems (most of them (1988), 501–512. going back to Banach and his school [B3] ———, Monstrous moonshine and monstrous Lie su- in the 1930s) remained open. peralgebras, Invent. Math. 109 (1992), 405–444. Let me recall some background material. A se- [B4] , What is moonshine?, Documenta Mathemat- ——— quence of vectors xi i∞=1 is said to be a (Schauder) ica, extra volume ICM 1998 I. basis of a Banach {space} X if every x X has a [CN] J. H. CONWAY and S. P. NORTON, Monstrous moonshine, ∈ unique representation of the form x = ∞ aixi Bull. London Math. Soc. 11 (1979), 308–339. i=1 with ai scalars. The basis is called unconditional [FLM] I. B. FRENKEL, J. LEPOWSKY, and A. MEURMAN, Vertex Op- P if for every choice of signs = i ∞ the series erator Algebras and the Monster, Academic Press, { }i=1 1988; see also the announcement by I. B. Frenkel, i∞=1 θiaixi converges whenever i∞=1 aixi does (or J. Lepowsky, and A. Meurman, A natural represen- equivalently,P the operator P tation of the Fischer-Griess Monster with the modular function J as character, Proc. Natl. Acad. Sci. ∞ ∞ U.S.A. 81 (1984), 3256–3260. T aixi = θiaixi [Ge] R. W. GEBERT, Introduction to vertex algebras, iX=1 iX=1 Borcherds algebras and the monster Lie algebra, In- ternat. J. Modern Phys. A8 (1993), 5441–5503. is bounded). In the common separable Banach [Go] P. GODDARD, The work of R. E. Borcherds, laudation spaces it is quite easy to find bases, and it is also delivered at the International Congress of Math- easy to prove that every infinite-dimensional Ba- ematicians in Berlin following the award of the Fields nach space has an infinite-dimensional subspace Medal to Richard Borcherds, Documenta Math- with a basis. A famous result of P. Enflo is that not ematica, extra volume ICM 1998 I. every separable Banach space has a basis (and in [Gr] R. L. GRIESS JR., The Friendly Giant, Invent. Math. 69 fact does not even have the so-called approxima- (1982), 1–102. tion property which is implied by the existence of [JLW] E. JURISICH, J. LEPOWSKY, and R. L. WILSON, Realizations a basis). As for unconditional bases, it is not hard of the Monster Lie algebra, Selecta Math., New Se- to prove that several common spaces (like L (0, 1) ries 1 (1995), 129–161. 1 or C(0, 1)) fail to have unconditional bases. It was [K] M. KONTSEVICH, Product formulas for modular forms an open problem whether every infinite-dimen- on O(2,n) [after R. Borcherds], Sém. Bourbaki, Vol. 1996–97, Exp. No. 821, November 1996. sional Banach space has an infinite-dimensional subspace with an unconditional basis. For a long time it was hoped that every infinite-dimensional The Work of William Banach space might even have the stronger property of containing a subspace isomorphic to c0 or to `p for some 1 p< (in some sense this Timothy Gowers could have been viewed as∞ an infinite-dimensional Joram Lindenstrauss version of Dvoretzky’s theorem). This hope was put to rest by B. S. Tsirelson, who constructed a re- William Timothy Gowers has worked in two flexive space not containing a subspace isomorphic areas: Banach space theory and combinatorics. to `p for any 1

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ample and produced a space that is “arbitrarily dis- Y are Banach spaces, and assume that each of tortable” (I do not define this notion, since it will them is isomorphic to a complemented subspace not be needed in the sequel). Using Schlumprecht’s (i.e., a subspace on which there is a bounded lin- result, Gowers and B. Maurey [7] constructed a ear projection) of the other. Must X be isomorphic separable Banach space Y that does not have any to Y? With mild additional assumptions it was infinite-dimensional subspace with an uncondi- known that the answer is positive (and this has tional basis. In fact, this space Y turned out to have many applications). Gowers [3] showed, however, a stronger property: If Z is any subspace of Y, then that in general the answer is negative. He pro- any bounded linear projection P on Z is trivial (i.e., duced a Banach space X so that X is isomorphic either dim PZ < or dim Z/PZ < ). If a space to X X X but not to X X. ∞ ∞ Z has an unconditional basis zi ∞ , then there In [8] Gowers and Maurey develop a general { }i=1 are many nontrivial bounded projections on Z method for constructing Banach spaces for which (for any subset M of the integers a certain given class of linear maps (say, a shift in a sequence space) are bounded but so that any ∞ bounded linear operator on the space essentially PM aizi = aizi belongs to the algebra generated by those given op- iX=1 iXM ∈ erators. This method has far-reaching implications is a bounded linear projection). A space Y with such and will certainly be of much use in the future. It a property was called in [7] hereditarily indecom- is shown in [8] how to derive the results of [1] and posable (H.I.). Before [7] it was a well-known prob- [3] from this general method. lem whether there exists at all an infinite-dimen- 2. The dichotomy theorem. sional Banach space X that cannot be represented In [4] Gowers proved a general dichotomy theorem as a direct sum X = X1 X2 with dim X1 = for Banach spaces which, in particular, says the fol- dim X2 = (i.e., on which there is no nontrivial ∞ lowing: Every infinite-dimensional Banach space bounded linear projection). has a subspace that is either an H.I. space (i.e., a H.I. spaces have many additional unexpected space with very few operators) or a space with an properties. As shown in [7], if Y is an H.I. space and unconditional basis (i.e., a space with a very rich T is a bounded linear operator from Y into itself, structure). The proof is combinatorial, using infi- then T = IY + S for some scalar and a strictly singular S (an operator is called strictly singular nite Ramsey theory. The theory of finite-dimen- if its restriction to any infinite-dimensional sub- sional Banach spaces mainly uses arguments based space is not an isomorphism). Thus every such T on measure (volume) and probability. These tools is either a Fredholm operator with index 0 (if =0) are not naturally available in the infinite-dimen- 6 or is strictly singular and thus not Fredholm (if sional setting. Ramsey type arguments turn out to =0). Consequently, an H.I. space is not isomor- be an important tool in the infinite-dimensional set- phic to any of its proper subspaces. This solves in ting, where they can sometimes replace arguments particular the classical “hyperplane problem”, using measure. which asks whether any infinite-dimensional Ba- The dichotomy result makes it clear that H.I. nach space is isomorphic to its hyperplanes (it is spaces are of importance in the structure theory trivial that all hyperplanes are mutually isomorphic, of general Banach spaces (and not just pathologi- and it is easy to see that the common infinite-di- cal counterexamples). The dichotomy result led im- mensional spaces are isomorphic to their hyper- mediately to the solution (this time in the positive planes). The hyperplane problem was actually first direction) of the classical “homogeneous space” solved by Gowers [1] before it became clear that problem. A Banach space X is called homogeneous the example in [7] does the same. The example in if it is isomorphic to all its infinite-dimensional sub- [1] does have an unconditional basis and other in- spaces. A short time before the dichotomy result teresting properties. was proved, R. Komorowski and N. Tomczak- It is a relatively easy result due to R. C. James Jaegermann proved the following result: A Banach that a space with an unconditional basis is either space X of “finite cotype” that does not contain a reflexive or it contains a subspace isomorphic to subspace isomorphic to `2 must have a subspace c0 or `1. A considerable amount of work was done without an unconditional basis. By combining this on the question of whether every infinite-dimen- result with the dichotomy theorem, one deduces sional Banach space has a subspace that is either that the only homogeneous Banach space is `2. In- reflexive or c0 or `1. Several promising positive re- deed, if X is homogeneous it must have “finite co- sults were found on this question. However, Gow- type” by known arguments related to the approx- ers [2] showed that in general the answer is nega- imation property. Hence, if it is not isomorphic to tive and thus gave a stronger version of the result `2 (and thus does not contain `2), it must contain in [7] on unconditional bases. (and thus be) an H.I. space. But H.I. spaces are not Another well-known open problem in Banach isomorphic to any proper subspace of themselves space theory was the following: Assume that X and and thus are certainly not homogeneous.

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3. Szemerédi’s theorem. proof but do not describe the actual situation. The theorem states the following: If >0 and an This is the case, e.g., in Van der Waerden’s theo- integer k are given, then there is an N(k, ) so that rem mentioned above. Very surprisingly, Gowers every subset of 1, 2, ,n containing more than obtained in [5] a lower bound for K(ε, ε, ε) which { } n elements must contain an arithmetic progres- is also of the form of a tower of 2’s (but of a sion of length k whenever n N(k, ). The theo- smaller height, proportional to log ε ). The exis- | | rem was proved for k =3 by K. Roth using tools tence of such large lower bounds is of great sig- from analytic number theory. For k>3 the theo- nificance in combinatorics and in the theory of rem was proved first by E. Szemerédi using an ex- complexity of computations. tremely intricate and ingenious combinatorial argument. Some years later another proof was found References by H. Fürstenberg using the structure theory of er- [1] W. T. GOWERS, A solution to Banach’s Hyperplane godic measure preserving transformations. Problem, Bull. London Math. Soc. 26 (1994), 523–530. The proof of Roth gave a reasonable estimate [2] ———, A space not containing co,`1, or a reflexive for N(3,). The proof of Szemerédi gave an enor- subspace, Trans. Amer. Math. Soc. 344 (1994), mous bound for N(k, ). One reason for this is that 407–420. Szemerédi used in his proof the (much weaker) Van [3] ———, A solution to the Schröder Bernstein Problem for Banach spaces, Bull. London Math. Soc. 28 (1996), der Waerden theorem. This theorem (in its quali- 297–304. tative form) states that in any partition of the in- [4] ———, A new dichotomy for Banach spaces, Geom. tegers into two sets one of those sets must con- Funct. Anal. 6 (1996), 1083–1093; an extended ver- tain arbitrarily long arithmetic progressions. The sion of this paper is available as an IHES preprint original proof of this result gave (in its quantita- from July 1994 entitled “Analytic sets and games in tive form) huge upper bounds—what the logicians Banach spaces”. call an Ackerman function. A more recent proof of [5] ———, Lower bounds of tower type for Szemerédi’s Van der Waerden’s theorem, due to S. Shelah, gave uniformity lemma, Geom. Funct. Anal. 7 (1997), a much improved upper bound which was, how- 322–337. ever, still enormous. Fürstenberg’s proof of Sze- [6] ———, A new proof of Szemerédi’s theorem for arith- merédi’s theorem gave no estimate on N(k, ). metic progressions of length four, Geom. Funct. Anal. 8 (1998), 529–551; a paper on a general length By using the basic approach of Roth, Gowers is in preparation. gave in [6] a proof of Szemerédi’s theorem for [7] W. T. Gowers and B. MAUREY, The unconditional every k. His main tool is a deep theorem of G. A. basic sequence problem, J. Amer. Math. Soc. 6 (1993), Frieman on the structure of sets of integers A so 851–874. that the cardinality A + A of the sum set A + A [8] , Banach spaces with small spaces of operators, | | ——— is at most C A for some constant C. The proof Math. Ann. 307 (1997), 543–568. | | of Gowers gives an estimate for N(k, ) of the form 2k+C log 2 22 | | The Work of for some constant C. This gives in particular the first “reasonable” estimate for the constant in Van Maxim Kontsevich der Waerden’s theorem. Yuri I. Manin The proof of Fürstenberg led to the creation of The mathematical achievements of Maxim Kont- an entire theory and to many generalizations of sevich have received worldwide recognition. He Szemerédi’s theorem. It is expected that Gowers’s has influenced a considerable body of research in proof will have a similar impact but in a somewhat different direction. mathematical physics, topology, and algebraic geometry. What follows is a brief report on some 4. Szemerédi’s uniformity lemma. of his work. One of the tools used by Szemerédi in his proof Kontsevich’s most famous paper is probably of the result mentioned above is a result on par- “Intersection theory on the moduli spaces of curves titioning general graphs into “uniform subsets”, and the matrix Airy function” (Comm. Math. Phys., and it is called Szemerédi’s uniformity lemma. 147 (1992), 1–23). It contains a complete proof of This lemma found many other important applica- Witten’s conjecture on the generating function of tions in graph theory. The statement of the lemma a family of characteristic numbers defined on the involves three (small) parameters ε, , and an es- moduli spaces of curves with marked points. Such timate K(ε, , ) for the number of sets in the partition. The upper bound found by Szemerédi in the Yuri I. Manin is a director of the Max-Planck-Institut case ε = = is a tower of 2’s of height propor- für Mathematik in Bonn. His e-mail address is 6 tional to ε . [email protected]. Versions of this segment are In most cases in combinatorics such large esti- being published simultaneously by the Notices and the mates are just a consequence of the method of Gazette des Mathématiciens.

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a generating function appeared in bels these cells, and a, b become the labels of { } the context of topological quantum the cells adjoining e. field theory, and Witten’s identities To prove this, Kontsevich analyzes the re- reflected a highly speculative con- markable cell complex representation of the mod- jecture that different approaches to uli space and reinterprets complex analytic inte- the quantization lead to identical grals via combinatorial data. A paradoxical property results. of his identity is the cancellation of poles in the To state a part of Kontsevich’s right-hand side, which is not obvious even in the results, we need some notation. Let simplest cases. Mg,n be the moduli space of stable The ideas contained in this paper were devel- n-pointed curves of genus g. The in- oped in many works of physicists; the matrix Airy tersection theory of these spaces is function Kontsevich introduced in order to sum the understood in the sense of orb- generating function is an important ingredient of ifolds, or stacks. The algebro-geo- Maxim Kontsevich what are now called Kontsevich’s models. metric study of the Chow ring of On the other hand, the experience he acquired Mg,0 was initiated by D. Mumford. in dealing with the geometry of moduli spaces al- Put lowed him to introduce several decisive ideas in 2 the very active area of quantum cohomology and n,i := (c1(ωC/M)) H (Mg,n,Q), i ∈ the Mirror Conjecture. The Mirror Conjecture is by now a series of where Ψ : M C are the structure sections of i g,n n stunning, interrelated insights in the geometry of the universal curve.→ complex manifolds with vanishing canonical class, Following the notation of Witten, the integrals which are motivated by a conjectural duality in of top degree monomials in n,i are denoted quantum string theory. Only some of these in- a1 an sights are formulated as precise mathematical con- a1 ...an = Ψn,1 ... n,n. h i ZMg,n jectures. The first of them was what we call the Mir- Ψ Ψ ror Identity: an equality of two formal series, one Kontsevich’s Main Lemma gives an infinite family of which is a generating function for the number of identities that allows one to calculate these num- of rational curves of various degrees on a three- bers algorithmically and to sum an appropriate gen- dimensional quintic, and another is produced from erating function (I omit this part, but the result is periods of the mirror dual variety. extremely beautiful). The identities have the fol- Kontsevich’s paper “Enumeration of rational lowing structure. Fix (g,n), put d =3g 3+n, − curves via torus actions” (The Moduli Space of and choose n independent variables l1,...,ln. Then Curves, R. Dijkgraaf, C. Faber, and G. Van der Geer, n eds., Progress in Mathematics, vol. 129, Birkhäuser, (2di 1)!! Boston, 1995, pp. 120–139) consists of two parts. τd ...τd − h 1 n i 2di +1 d=d + +d i=1 li The first part contains the definition and study of 1X··· n Y V what are now called Kontsevich stable maps. These 2−| τ | 2 = . are systems (C; x1,...,xn; f ) where C is a projec- Aut τ l0(e)+l00(e) tive curve with only cusps as singularities, x are XGg,n eYEτ i ∈ ∈ pairwise distinct smooth points on it, and Γ f : C V is a map without infinitesimal auto- Here Gg,n is the set of the isomorphism classes of → triples =(,c,f) where: morphisms to a smooth projective manifold V. (i) is a connected graph with all vertices v V Such maps with a fixed image = f (C) H2(V) ∈ ∈ of valencyΓ 3 and with no tails; form a Deligne-Mumford stack Mg,n(V,). Kont- (ii) c is a family of cyclic orders on all F (v) sevich’s insight was that such a stack carried a where F (v) is the set of flags adjoining v; Chow class that he called a virtual fundamental (iii) f is a bijection between 1,...,n and the class. Using the image of this class as a corre- { } n set of all cycles of . A cycle is a cyclically ordered spondence between V and Mg,n, he defined the sequence of edges (without repetitions) very strong motivic version of Gromov-Witten in- (e1,e2,...,ek) such that for every i, ei and ei+1 variants. Some of these invariants are essentially have a common vertex vi and the flag (ei,vi) fol- numbers, among which the numbers of rational lows the flag (ei+1,vi) as specified by c; curves of various degrees on a quintic threefold are (iv) for any edge e E, l0(e),l00(e) = la,lb , contained. The conjecture that an appropriate gen- where a, b 1,...,n∈ are{ the f-labels} { of the} erating function for them expresses a variation of { }{ } two cycles to which e belongs. Hodge structure for the dual quintic was the first If is embedded into a closed Riemann surface case of the Mirror Conjecture. X that is oriented compatibly with c, the cycles of The second part of Kontsevich’s paper gives become the boundaries of the oriented con- explicit formulas for Gromov-Witten numbers for nected components of X (2-cells). Then f la- complete intersections in (products of) projective \| |

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spaces, in particular for quintics. The structure of In a recent preprint, “Formal quantization of the formulas is similar to the one above (summa- Poisson manifolds”, Kontsevich solved a long- tion over trees). However, their origin is very dif- standing problem showing that any Poisson man- ferent. ifold admits a formal quantization. In the flat case One can get recursive formulas for Gromov- he produced an explicit quantization introducing Witten numbers of, say, a projective space by using a beautiful new class of integrals. This work has degenerations of stable maps of rational curves to great potential. it. However, on for example a quintic, rational curves tend to be rigid, so that there is nothing to degenerate. Kontsevich’s idea is to degenerate the The Work of quintic itself, replacing it by a union of five hyperplanes. The problem of how to calculate the weights of individual stable maps to such a sim- Curtis T. McMullen plex is solved by the creative application of Bott’s John Milnor residue formula to the stack of stable maps, cor- Curt McMullen has made important contribu- responding to the standard torus action on the am- tions to the study of Kleinian groups, hyperbolic bient projective space. It so happens that only 3-manifolds, and holomorphic dynamics. Indeed, very degenerate curves mapping onto the 1-skele- following the lead of Dennis Sulli- ton of the simplex contribute. Their combinator- van, he clearly regards these three ial types are marked trees. areas as different facets of one uni- Kontsevich thus exhibited a precise formula for fied branch of mathematics. Fol- the left-hand side of the conjectural Mirror Iden- lowing are descriptions of a few se- tity (counting curves). It is worth stressing that he lected topics. I hope that these will has supplied the first ever algebro-geometric def- illustrate the variety and depth of inition of this function: all previous work on the his work. However, by all means the Mirror Conjecture dealt with a vague “physical” no- reader should look at the original tion of it. Kontsevich stopped short of proving papers, since he is a master expos- this case of the Mirror Conjecture, which he re- itor. See especially his two books duced to an explicit identity. A. Givental completed and his Berlin lecture [Mc10]. the proof, introducing new ideas: in particular, Solving the Quintic the technique of equivariant cohomology. His first work was on Smale’s the- In the paper “Vassiliev’s knot invariants” (Adv. ory of purely iterative algorithms. in Soviet Math. 16 (1993), 137–150), Kontsevich in- By definition, these are numerical al- vented a generalization of Gauss’s integral for- gorithms which can be carried out Curtis T. McMullen mula for linking numbers, thereby supplying si- by iterating a single rational func- multaneously all of Vassiliev’s invariants of knots. tion, without allowing any “if , then ” A parametrized knot is a map K : S1 R3. The ··· ··· 3 → branching. In [Mc1] he showed that the roots of a space R will be represented as Cz Rt. Put polynomial of degree n can be computed by a gen- ∞ erally convergent, purely iterative algorithm if and Z(K)=1+ (2πi) m − only if n 3. With Peter Doyle [DMc] he showed mX=1 that these≤ roots can be computed by a tower of fi- m dzi dzi0 nitely many such algorithms if and only if n 5. DP − . ≤ × t <...

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Figure 1. (A different and more complex structures on a hyperbolic Riemann sur- complicated example was given face of finite area by adding an ideal boundary con- in [EL].) sisting of algebraic limits of associated Kleinian The Kra Conjecture groups. He conjectured that the “cusps”, corre- To any Riemann surface X one sponding to ideal limits in which some simple can associate the Banach space closed curve has been pinched to a point, are every- Q(X) consisting of all holo- where dense in this boundary. (Compare Figure 2.) morphic quadratic differentials This was proved by McMullen [Mc4] in 1991, using = φ(z) dz2 for which the a careful estimate for the change in the associated norm = φ dz dz is finite. group representation π1(S) PSL2(C) as some k k | | → AnyΦ covering map f : X Y in- simple closed curve in the surface S shrinks to a R → duces aΦ push-forward operation point. f : Q(X) Q(Y ), where the Thurston Geometrization ∗ → image differential at a point y The still unproved Thurston Geometrization Con- Figure 1. The Julia set for is obtained by summing over jecture asserts that every compact 3-manifold can z sin(z), shown in black, the points of f 1(y). This op- 7→ − be cut up along spheres and tori into pieces, each has positive area but no eration can never increase of which admits a simple geometric structure. interior points. norms, f ( ) . In the Here eight possible geometries must be allowed. k ∗ k≤k k special case of the universal cov- For six of these eight geometries, the problem is ering f : Y Y of a hyperbolicΦ surfaceΦ of finite now well understood, but difficulties remain in the → area, Irwin Kra conjectured in 1972 that there is hyperbolic case, while the spherical case, includ- always somee definite amount of cancellation being the classical Poincaré conjecture, is still in- tween the different preimages of a point of Y, so tractable. For references, compare [Mi2]. that f is strictly less than 1. This was proved Thurston outlined proofs that a 3-manifold ad- k ∗k by McMullen [Mc3, Mc5]. In fact, McMullen con- mits a hyperbolic structure in two important spe- sidered a completely arbitrary covering map cial cases. First suppose that M is a Haken mani- f : X Y, showing that f < 1 if and only if fold, that is, suppose that M is S2-irreducible and → k ∗k this covering is nonamenable. can be built up inductively from 3-balls by gluing Cusps in the Boundary of Teichmüller Space together submanifolds of the boundary, taking Next, a problem in Kleinian groups. In 1970 Lip- care that no essential simple closed curve in this man Bers compactified the Teichmüller space of submanifold bounds a disk in the manifold. Thurston showed that M can be given a hyperbolic structure if and only if every Z Z in its ⊕ fundamental group comes from a boundary torus. McMullen [Mc6] used his work on the Kra conjecture to give a new and explicitly worked out proof of this theorem. (The details are quite complicated.) The second case handled by Thurston con- cerned 3-manifolds which fiber over the circle. Again, McMullen gave a new proof, which will be discussed below. Renormalization Let f be a smooth even map from the closed interval I =[ 1, 1] into itself with a nondegenerate − Figure 2 (courtesy of David Wright). Dense cusps in the critical point at the origin and with no other crit- boundary of Teichmüller space for a punctured torus, using the ical points. We will say that f is renormalizable if Maskit embedding. Teichmüller space is the region underneath there is an integer n 2 so that the n-fold iterate ≥ this boundary curve. g = f n maps the subinterval x ; x g(0) into ◦ { | |≤| |} itself with only one nondegenerate critical point. If we rescale by setting f (x)=g(αx)/α where α = g(0), then f will be a new map from the interval I into itself satisfying the originalb hypothesis. This f is called theb renormalization n(f ). In 1978 R Mitchell Feigenbaum, and independently Pierre Coulletb and Charles Tresser, considered the special case n =2and studied maps f which are infinitely renormalizable, so that we can form a se- 3 Figure 3. A quadratic map f, its iterate f ◦ , and its quence of iterated renormalizations 2 renormalization 3(f ). The right-hand box is obtained from the f, 2f, ◦ f , ... , each mapping I to itself with R R R2 small box in the middle by magnifying and rotating 180◦. one critical point. They observed empirically that

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this sequence of maps always seems to converge nentially close to an to a fixed smooth limit map. Their ideas, motivated isometry as we pene- by renormalization ideas from statistical mechanics trate deeper into the and by attempts to understand the onset of tur- convex core. Closely bulence in fluid mechanics, now occupy a central related is the state- role in one-dimensional dynamics, since the infi- ment that actions of nitely renormalizable maps are the most difficult π1(M) and π1(M0) on ones to understand. the 2-sphere at infinity This construction was extended to the complex for hyperbolic 3-space case by Douady and Hubbard, using the idea of a are quasiconformally quadratic-like map, that is, a proper holomorphic conjugate and that map f : U V of degree two, where U and V are this quasiconformal → simply connected open sets in C and U is a com- conjugacy is actually pact subset of V. This has led to important work conformal at every by mathematicians such as Dennis Sullivan, Curt deep point of the limit Figure 4. The critical point (at the center McMullen, and Mikhail Lyubich. Most subsequent set for this action. of symmetry) is a deep point of the Julia progress in understanding the real case has been set for the Feigenbaum infinitely As an application, 2 based on complex methods. (One exception is McMullen gave a new renormalizable map z 1 az , where a =1.401155189 . This7→ Julia− set has Martens [Ma].) McMullen’s book [Mc7] provided proof of the second ··· the first careful presentation of the foundations Thurston geometriza- no interior points. of renormalization. As one example, he was the first tion theorem. To any to notice the possibility of an aberrant form of surface diffeomor- “crossed renormalization” in the complex case phism : S S we which does not fit into the usual pattern. He used can associate the→ map- his work on renormalization to obtain partial re- ping torus T, that is, sults on the generic hyperbolicity conjecture for the quotient of S R × real quadratic maps, that is, the conjecture that under the Z action every such map can be approximated by one with which is generated by an attracting periodic orbit. For example, he showed (x, t) ((x),t +1). If 7→ that every component of the interior of the Man- S has genus two or delbrot set which meets the real axis is hyperbolic. more and is The full conjecture was later proved by Lyubich [L1] pseudo-Anosov, then and by Graczyk and ´Swia˛tek [GS]. Thurston showed that McMullen’s second book [Mc8] developed renor- T is a hyperbolic 3- malization theory further and tied it up with manifold. McMullen Mostow rigidity and also with Thurston geometriza- proved this by using tion. (See also [Mc10].) He introduced the concept his inflexibility result of a “deep point” in a fractal subset X C. By de- to construct a hyper- ⊂ finition, p is deep if there are positive constants bolic structure on Figure 5. Filled Julia set associated and c so that the distance from an arbitrary S R which is invari- with the golden mean Siegel disk, with × point q to X is at most c p q 1+. Taking p =0 ant under the given Z rotation number ρ =1/ | − | for convenience, we can understand this concept action. (1 + 1/(1 + 1/(1 + ))). The Siegel disk is the large region to··· the lower left. by zooming in on the origin so as to magnify the Next he applied set X by some large constant λ. Replacing X by these ideas to renor- the magnified copy λX, we can replace the constant malization. One basic result is a rigidity theorem c by c/λ, which tends to zero as λ . In other for bi-infinite “towers” of renormalizations. We →∞ words, these magnified images will fill out the can think of such a tower as a bi-infinite sequence complex plane more and more densely, with gaps ( ,q 1,q0,q1,q2, ) of quadratic-like maps ··· − ··· which become smaller and smaller as λ becomes qj : Uj Vj, where each qj+1 is a renormalization → large. (Compare [Mi1].) n (qj ). He showed that if the renormalization pe- R j Now consider a hyperbolic 3-manifold which riods nj are bounded and if the annuli Vj r Uj may have infinite volume. The convex core of such have modulus bounded away from zero, then the a manifold M can be described as the smallest ge- entire tower is uniquely determined up to a suit- odesically convex subset which is a strong defor- able isomorphism relation by its quasiconformal mation retract of M. Assuming both upper and conjugacy class. lower bounds for the injectivity radius at points Consider an infinitely renormalizable real qua- of the convex core, McMullen’s inflexibility theo- dratic map f with periodic combinatorics. Using the rem asserts that two such manifolds M and M0 complex theory, McMullen showed that the suc- which are “pseudo-isometric” must actually be re- cessive renormalizations converge exponentially lated by a diffeomorphism which becomes expo- fast to a map which is periodic under renormal-

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ization. Closely related is the statement that the [Mc1] C. MCMULLEN, Families of rational maps and itera- critical point is a deep point for the Julia set of f. tive root finding algorithms, Ann. of Math. 125 (Figure 4.) (1987), 467–493. Now consider a quadratic map f (z)=z2 + c [Mc2] ———, Area and Hausdorff dimension of Julia sets of entire functions, Trans. Amer. Math. Soc. 300 which has a Siegel disk of rotation number ρ. That (1987), 329–342. is, choose the constant c so that the derivative of [Mc3] , Amenability, Poincaré series and quasicon- 2πiρ ——— f at one of its two fixed points is equal to e , formal maps, Invent. Math. 97 (1989), 95–127. where ρ RrQsatisfies a suitable Diophantine [Mc4] , Cusps are dense, Ann. of Math. 133 (1991), ∈ ——— condition. McMullen used similar ideas in [Mc9] to 425–454. show that this Siegel disk is “self-similar” about the [Mc5] ———, Amenable coverings of complex manifolds critical point 0 (the central point in Figure 5) if the and holomorphic probability measures, Invent. Math. continued fraction expansion of ρ is periodic. In 110 (1992), 29–37. [Mc6] , Iteration on Teichmüller space, Invent. Math. fact, his argument can be used to show that the ——— 99 (1990), 425-454; see also: Riemann surfaces and entire Julia set J of f is asymptotically self-similar the geometrization of 3-manifolds, Bull. Amer. Math. in the following sense: There is a scale factor λ with Soc. (N.S.) 27 (1992), 207–216. n λ > 1 so that the magnified images λ J con- [Mc7] , Complex Dynamics and Renormalization, | | ——— verge to a well-defined limit set J = λJ C as Ann. of Math. Stud., vol. 135, Princeton Univ. Press, n , using the Hausdorff topology for compact⊂ Princeton, NJ, 1994. →∞ subsets of the Riemann sphere. (In thisb particularb b [Mc8] ———, Renormalization and 3-Manifolds Which Fiber example, λ =1.8166 is real.) The correspond- over the Circle, Ann. of Math. Stud., vol. 142, Prince- ··· ton Univ. Press, Princeton, NJ, 1996. ing limit for the boundary of the Siegel disk is a [Mc9] , Self-similarity of Siegel disks and the Haus- quasicircle contained in J, while the correspond- ——— dorff dimension of Julia sets, Acta Math. 180 (1998), ing limit for the filled Julia set K(f ) (the union of 247–292. bounded orbits for f) is theb entire sphere . C [Mc10] ———, Rigidity and inflexibility in conformal dy- There has been very significant subsequent namics, Proc. Internat. Congr. Math. (Berlin) vol. 2 work in renormalization, based in part onb Mc- 1998, pp. 841–855. Mullen’s ideas. Compare the discussion in [Me]. [Me] W. DE MELO, Rigidity and renormalization in one di- Note in particular [L2], which implies that the mensional dynamical systems, Proc. Internat. Congr. boundary of the Mandelbrot set is asymptotically Math. (Berlin) vol. 2 1998, pp. 765–778. [Mi1] J. MILNOR, Self-similarity and hairiness in the Man- self-similar about the Feigenbaum point, and [L3], delbrot set, Computers in Geometry and Topology which proves existence of a full horseshoe struc- (M. Tangora, ed.), Lecture Notes Pure Appl. Math., ture for the real renormalization operator and Dekker, 1989, pp. 211–259. uses it, together with work of Martens and Now- [Mi2] ———, The fundamental group, Collected papers, Pub- icki, to prove that every real quadratic map out- lish or Perish, Vol. II, 1995. See pp. 93-95; but note side a set of measure zero has either a periodic at- that unpublished claims in the spherical case have tractor or an absolutely continuous asymptotic not been substantiated. measure.

References [DMc] P. DOYLE and C. MCMULLEN, Solving the quintic by iteration, Acta Math. 163 (1989), 151–180. [EL] A. EREMENKO and M. LYUBICH, Examples of entire func- About the Cover tions with pathological dynamics, J. London Math. Curt McMullen has shown that a quadratic Siegel disk Soc. 36 (1987), 458–468. is asymptotically self-similar about its critical point [GS] J. GRACZYK and G. ´SWIA˛TEK, Generic hyperbolicity in whenever the continued fraction expansion for its ro- the logistic family, Ann. of Math. 146 (1997), tation number is periodic. (Compare the discussion on 399–482. this page.) The cover figure (a color version of Figure 5) [L1] M. LYUBICH, Dynamics of complex polynomials, parts illustrates this result by showing part of the filled Julia I, II, Acta Math. 178 (1997), 185–297; see also: Geom- set (the union of bounded orbits) for the quadratic poly- etry of quadratic polynomials: Moduli, rigidity, and nomial map f (z)=z2 (0.3905 + i 0.5867 ), which local connectivity, Stony Brook I.M.S. preprint, 1993, − ··· · ··· has a Siegel disk with rotation number equal to the golden #9. mean ρ =(√5 1)/2. The critical point lies at the center [L2] , Feigenbaum-Coullet-Tresser universality and − ——— of this picture, while the Siegel disk is the large region Milnor’s hairiness conjecture, Ann. of Math. (to ap- to the lower left with emphasized boundary. Under f, pear). this disk maps homeomorphically onto itself with ro- [L3] , Regular and stochastic dynamics in the real ——— tation number ρ, and the symmetric region to the upper quadratic family, Proc. Nat. Acad. Sci. U.S.A. (to ap- right folds onto it. If we expand this figure repeatedly by pear); see also: Almost every real quadratic map is a fixed scale factor of 1.8166 , keeping the center point ··· either regular or stochastic, Stony Brook I.M.S. fixed, then the expanded images will converge to a well preprint, 1997, #8. defined limiting shape. [Ma] M. MARTENS, The periodic points of renormalization, —John Milnor Ann. of Math. 147 (1998), 543-584.

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