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The Mathematical Work of the 2006 Fields Medalists, Volume 54, Number 3

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The Mathematical Work of the 2006 Fields Medalists, Volume 54, Number 3 The Mathematical Work of the 2006 Fields Medalists The Notices solicited the following articles about the works of the four individuals to whom Fields Medals were awarded at the International Congress of Mathematicians in Madrid, Spain, in August 2006. (Grigory Perelman was awarded the medal but declined to accept it.) The International Mathematical Union also issued news releasesabout the medalists’work, and these appearedin the October2006 Notices . —Allyn Jackson The Work of Andrei Okounkov Nicolai Reshetikhin∗ Perhaps two basic words that can characterize the styleofAndreiOkounkovareclarityandvision. His research is focused on problems that are at the junction of several areas of mathematics and mathematicalphysics.Ifonechoosesrandomlyone of his papers, it will almost certainly involve more than one subject and very likely will have a solu- tion to a problem from one area of mathematics by techniques from another area. Many of his papers openedupnew perspectivesonhow geometry,rep- resentation theory, combinatorics, and probability interactwitheachotherandwithotherfields. Typically, one also finds among the results of eachofhispapersabeautifulexplicitformula. The variety of tools Okounkov uses is very im- Figure 1. A random surface representing a pressive.Hehastheratheruniquequalityofmoving random dimer configuration on a region of freely from analysis and combinatorics to algebraic hexagonal lattice. geometry, numerical computations, and represen- tationtheory. Letmefocusonsomerepresentativeexamples. One of his remarkable results is the Gromov- where matched vertices are connected by edges. In Witten and Donaldson-Thomas correspondence, the 1960s, Kasteleyn and Fisher computed the par- which is an identification of two geometric enu- tition function of a dimer model on a planar graph merative theories. These results are intrinsically and the local correlation functions in terms of Pfaf- related to his works on Gromov-Witten invariants fians of the so-called Kasteleynmatrix [3]. Since that for curves, on random matrices, and on dimer time,dimermodelshaveplayedaprominentrolein models. statisticalmechanics. A pair of dimerconfigurations on a bipartitepla- Works on Dimers nar graph defines a stepped surface that projects Dimer configurations are well known in combina- bijectivelytothegraph.Thus,randomdimerconfig- torics as perfect matchings on vertices of a graph urationsonbipartiteplanargraphscanberegarded as random stepped surfaces. A snapshot of such ∗Nicolai Reshetikhin is professor of mathematics at the University of California, Berkeley. His email address is a random surface for dimer configurations on a do- [email protected]. maininahexagonallatticeisshowninFigure1. 388 Notices of the AMS Volume 54, Number 3 Looking at this picture, the following is clear: on variety, and let Mg,n(X, β) be the space of isomor- the scale comparable to the size of the system, such phism classes of triples C, p1; ...,pn; f where C { } random stepped surfaces are deterministic. This is a complex projective connected nodal curve of phenomenon is by its nature close to determinis- genus g with n marked smooth points p1,...,pn tic limits in statistical mechanics (also known as and f : C X is a stable mapping such that → hydrodynamical limits), to the semiclassical limit [f (C)] β. Here stability means that components = in quantum mechanics, and to the large deviation of C that are pre-images of points with respect to phenomenon in probability theory. It is also clear the mapping f have finite automorphism groups. from this snapshot that fluctuations take place on Let evj : (C, p1; ...,pn; f ) f (pj ) be the evalua- the smaller scale. → tion mapping. Denote by ev∗α H∗(Mg,n(X, β) The nature of fluctuations changes depending j ∈ the pull-back of a class α H∗(X). Let Lj be a onthepointinthelimitshape.Forexampleitisclear ∈ line bundle on M (X, β) whose fiber over the that the fluctuations in the bulk of the limit shape g,n point C, p1; ...,pn; f is T ∗ C. The Gromov-Witten are quite different from those near a generic point { } pj at the boundary of the limit shape, those near sin- invariantsofX areintersectionnumbers gular points at the boundary of the limit shape, or those near the points where the limit shape touches X (1) <τk (α1)...τk (αn) > the boundary. These empirical observations are 1 n β,g = confirmed now and quantified by precise mathe- n kj j 1c1(Lj ) evj∗(αj ). matical statements. Okounkov made an essential Z[Mg,n(X,β)] ∧ = contributiontotheseresults. In joint work with Kenyon and Sheffield [4] Ok- Here the (virtual) fundamental class [Mg,n(X, β)] ounkov proved that the limit shape for periodically was constructed in the works of Behrend-Fantechi weighteddimersisthegraphoftheRonkinfunction andLi-Tian. ofthespectralcurveofthemodel.Asimilardescrip- When X is a point, Witten [15] conjectured that tion was obtained for local correlation functions in the generating function the bulk. This work was based on Kasteleyn’s re- sults. ∞ 1 In subsequent papers with Kenyon, Okounkov (2) f (t0,t1,...) = nX0 n! gaveacompletedescriptionofrealalgebraiccurves = that describe the boundary of the limit shape in a n <τ ...τ > t dimer model. It turns out that natural equivalence k1 kn g kj k1 kXn 3g 3 n jY1 classes of such curves form the moduli space of +···+ = − + = Harnack curves [5]. It also turned out that such is a tau-function for the KdV integrable hierarchy curves describe a special class of solutions to the satisfying the additional“string equation”. The first complexBurgersequation[6]. proof of this conjecture was given by Kontsevitch Fluctuations near the generic and special points in [7]. An alternative derivation of the key formula oftheboundaryaredescribedin[12]and[13].Inthis from[7]waslatergivenbyOkounkovandPandhari- workthedimermodelwasreformulatedintermsof pande. theSchurprocessandthenintermsofvertexopera- When X is a curve, the Gromov-Witteninvariants tors for gl .Somekeyideasusedinthesepapersgo were described completely by Okounkov and Pand- backtopreviousworksofOkounkovonrepresenta-∞ haripande [10][11]. They showed that when X P1 tiontheory. = the generating function for the Gromov-Witten in- All these results are characteristic of the style of variants is a tau-function for the Toda hierarchy, Okounkov: the problem in statistical mechanics is solvedusingtoolsfromalgebraicgeometryandrep- againwitha specialconstraintsimilartothe “string resentation theory. The moduli space of Harnack equations”. They also showed that the case when curves is described in terms of Gibbs measures on X is a point can be obtained by taking a limit of the X P1 case. dimer configurations. The same space is identified = as the space of certain algebraic solutions to the Another remarkable result by Okounkov and complexBurgersequation. Pandharipande is the following explicit formula for GW-invariants when X is a curve.Let β d[X], and = Gromov-Witten Theory suppose all αi are equalto ω (the Poincaré dualof a Gromov-Witten theory studies enumerative ge- point). Then ometry of moduli spaces of mappings of algebraic X curves(Riemannsurfaces)intosomefixedalgebraic (3) <τk (ω)...τkn (ω) > 1 d[X],g = variety. 2 2g dim(λ) − pki 1(λ) Let us recall the idea of the Gromov-Witten in- + . d! (k 1)! λXd jY1 i variant. Let X be a nonsingular complex projective | |= = + March 2007 Notices of the AMS 389 Herethesumistakenoverallpartitionsofd and References [1] S. Bloch and A. Okounkov, The character of k k the infinite wedge representation, Adv. Math. 149 (4) pk(λ) (λj j 1/2) ( j 1/2) (2000), 1–60, arXiv alg-geom/9712009. = jX1 − + − − + ≥ [2] A. Eskin and A. Okounkov, Asymptotics of num- k (1 2− )ζ( k). bers of branched coverings of a torus and volumes + − − of moduli spaces of holomorphic differentials, This formula for GW-invariants of curves is root- Invent. Math. 145 (2001), 59–103. ed in the relation between the GW-invariants and [3] P. W. Kasteleyn, Graph theory and crystal physics, Hurwitznumbers.Recallthatthelatterarethenum- in Graph theory and Theoretical Physics, Academic bers of branched coverings of X with given ramifi- Press, London, 1967, pp. 43–110. [4] R. Kenyon, A. Okounkov, and S. Sheffield, cationtype atgivenpoints. The branched coverings Dimers and amoebae, Ann. of Math.(2) 163 (2006), of X were studied in [1], [2], where the problem was 1019–1056. resolved essentially by using representation theory [5] R. Kenyon, and A. Okounkov, Planar Dimers and of S( ). Harnack curves, Duke Math. J. 131 (2006), 499–524. ∞ [6] R. Kenyon, and A. Okounkov, Limit shapes Donaldson-Thomas Invariants and the complex Burgers equation, arXiv math- ph/0507007. Let X be a three-dimensional algebraic variety. Al- [7] M. Kontsevich, Intersection theory on moduli gebraic curves C X of arithmetic genus g with the space of curves and the matrix Airy function, ⊂ fundamental class β H2(X) are parameterized Comm. Math. Phys. 164 (1992), 1–23. ∈ [8] D. Maulik, N. Nekrasov, A. Okounkov, and R. by the Hilbert scheme Hilb(X; β, 1 g). Let c2(γ) − Pandharipande, Gromov-Witten and Donaldson- be the coefficient of γ H∗(X) in the Künneth de- ∈ Thomas theory, I, arXiv math.AG/0312059; II arXiv composition of the second Chern class of the uni- math.AG/0406092. versal ideal sheaf : Hilb(X; β,χ) X X. The J × → [9] N. Nekrasov and A. Okounkov, Seiberg-Witten Donaldson-Thomasinvariantsare: theory and random partitions, in The Unity of Math- ematics (P. Etingof, V. Retakh, and I. M. Singer, eds.) Progress in Mathematics, Vol. 244, Birkhauser (5) <γ1,...,γn >β,χ = (2006), pp. 525–594, hep-th/0306238. n [10] A. Okounkov and R. Pandharipande, Gromov- c (γ ). Z 2 j Witten theory, Hurwitz theory, and completed [Hilb(X;β,χ)] jY1 = cycles, Ann. of Math.(2) 163 (2006), pp. 517–560; [11] A. Okounkov and R. Pandharipande, The equi- Here [Hilb(X; β, χ)] is the (virtual) fundamental variant Gromov-Witten theory of P1, Ann. of Math. class of Hilb(X; β,χ) constructed by R. Thomas. It (2) 163 (2006), pp. 561–605. is of dimension βKX where KX is the fundamen- [12] A. Okounkov and N. Reshetikhin, Correlation − tal class of X. This is the same dimension as the function of Schur process with application to local geometry of a random 3-dimensional Young dia- dimensionofMg,0(X, β).
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