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). The conjecture of Maulik, Nekrasov, Okounkov, gram, J. Amer. Math. Soc. 16 (2003), no. 3, 581–603, arXiv math.CO/0107056. and Pandharipande [8] relates suitably normalized [13] A. Okounkov and N. Reshetikhin, Random skew generating functions of Gromov-Witten invariants plane partitions and the Pearcey process, Comm. forMg,0(X, β)andDonaldson-Thomasinvariantsof Math. Phys., 269 no. 3, February 2007, arXiv math. Hilb(X; β,χ). This conjecture was proved for local CO/0503508. curvesandwhenX isthetotalspaceofthecanonical [14] A. Okounkov, N. Reshetikhin, and C. Vafa, Quan- bundleofatoricsurface. tum Calabi-Yau and Classical Crystals, in The Unity of Mathematics (P. Etingof, V. Retakh, and I. M. One can argue that these results were inspired Singer, eds.), Progress in Mathematics, Vol. 244, by observations from [14] and from the computa- Birkhäuser (2006) pp. 597–618, hep-th/0309208. tion of the Seiberg-Witten potential in [9], and that [15] E. Witten, Two-dimensional gravity and inter- they can be regarded as a generalization of the for- section theory on moduli space, Surveys in Diff. mula (3) for Gromov-Witten invariants of a curve. It Geometry 1 (1991), 243–310. is also remarkable that combinatorics of partitions came up as a computational tool in this topological subject. This brief and incomplete description of the achievements of Andrei Okounkov might underes- timate one important feature of his work, namely, its unifying quality: Seemingly unrelated subjects have become parts of one field. In this sense his contributions also have an important organizing value.

390 Notices of the AMS Volume 54, Number 3 n n 1 n The Work of Andrei Okounkov and sn(α, β) k∞ 1[αk ( 1) − βk ], n > 1, are su- pernewtonian= sums.P = The+ characters− are multiplica- A. M. Vershik∗ tive with respect to decomposition of the permuta- tion on the cycles, and the value on a cycle of length In spite of his youth, Andrei Okounkov has gone n ofthepreviouscharacterissn(α, β). through several periods in his fruitful creative Thoma’s proof was based on hard analysis and work: he started in asymptotic representation the- the theory of positive-definite functions. It had ory, and then extremely quickly went one by one nothing in common with proper representation through several areas, subsequently enlarging his theory. In the 1970s the author with S. Kerov gave knowledge of a number of subjects in algebraic a “representation-theoretic” proof of the theorem geometry, symmetric functions, combinatorics, in the framework of the general approach to the topology, integrable systems, statistical physics, approximations of those characters with the ir- quantum field theory, etc.—modern topics of great reducible characters of finite symmetric groups, interest to all mathematicians. Each change in topic some ergodic ideas, and an important interpreta- had a prehistory in the previous one and brought tion of those parameters α,β above as frequencies useful erudition. The influence of the first period of the rows and columns in growing random Young of Okounkov’s activity—symmetric functions, as- tableaux. Then, in 1995, Okounkov suggested in ymptotic representation theory, Young diagrams, his thesis a completely new idea and gave a third partitions, and so on—one can clearly feel. I want to proof, based on a nice analysis of the elementary say a few words only about his first achievements operators in the space of the representation. This (and not even about all of them), which are perhaps proof immediately gives a new interpretation of not so well known and which are right now a little the parameters as eigenvalues of some operators. bitinthe shadowofhisgreatsubsequentsuccesses Moreover, his goal was the solution of a more of the last several years. These subjects also are general problem that had been posed by his ad- closer to me because in some sense I started with visor G. Olshansky, namely, the description of all them. so-called admissible representations of the bisym- I will mention several of the first of Okounkov’s metric group (which includes the problem about results: a new proof of Thoma’s theorem giving the the characters above). This is a problem about the solutionoftheproblemaboutadmissiblerepresen- complete list of the irreducible representations tations of S ; some distinguished formulas; con- of the group S S , whose restriction to the ∞ tributions to the asymptotic Plancherel measure on diagonal subgroup∞ × is a∞ direct sum of tensor rep- Young tableaux, in particular, the analysis of fluctu- resentations of S ; or perhaps it is better to say ationsofrandomYoungdiagramsandthelinkwith that those restrictions∞ could be extended to the random matricesand matrix problems;and several group of all (infinite) permutations S∞. A typical resultsonsymmetricfunctionsandcharacters. example of such a representation is the irreducible right-left (regular) representation of S S in Asymptotic Representation Theory: Finite l2(S ), whose restrictions to each multiplier∞ × ∞ are ∞ and Infinite Symmetric Groups type II1 factor representations. Olshansky gave a The modern theory of the representations of infi- method of description, but the whole list of ad- nite symmetric groups started with the celebrated missible representations and a model for it was and fundamentaltheorem of E. Thoma (1964) about obtained by Okounkov. The answer is very natural normalized characters of S (the group of all fi- and related to actions of the bisymmetric group in nite permutations of a countable∞ set). It asserts a groupoid. This was one of the first examples of a that an indecomposable normalized character (= a non-locally compact (“big”) group of type 1 with a positive-definite central function χ(.), χ(1) 1) very rich set (of infinite dimension) of irreducible has the form = representations. By the way, the identification of the left and right parts of some of those represen- ∞ rn(g) χα,β(g) sn(α, β) , tations with von Neumann factors is still an open = nY2 question. = The description of the characters in Thoma’s where rn(g), n > 1, is the (finite) number of cycles of length n in the permutation g S ; the param- theorem mentioned above is equivalent to the eters of the character (“Thoma’s∈ simplex”)∞ are α description of so-called central measures of the = space of Young tableaux (= space of paths of (α1,...) and β (β1,...), with α1 α2 0 = ≥ ≥···≥ Young graphs)—or to the description of homomor- and β1 β2 0. The sums k∞ 1 αk βk 1 ≥ ≥···≥ P = + ≤ phisms of the ring of symmetric functions Λ to the scalars that are positive on the Schur functions ∗A. M. Vershik is affiliated with the St. Petersburg depart- sµ (.) Λ. Each deformation of a Young graph ment of the Steklov Institute of Mathematics and also ∈ St. Petersburg University, Russia. His email address is (or one can substitute other symmetric functions [email protected]. for the Schur polynomials) gives a new problem

March 2007 Notices of the AMS 391 of this type. With S. Kerov and G. Olshansky, Ok- and A. Borodin’s beautiful answer to the question ounkov gave the solution of such a problem for posed by Deift-Its about the existence of a formula Jack polynomials Pµ(x, θ)θ 0; in this case the for the determinant of a Toeplitz matrix as a Fred- ≥ formula above is changed as follows. For the su- holm determinant. Borodin and Okounkov gave pernewtonian sums sn(α, β) one must substitute such a formula with an elegant proof. It happened n n 1 n the sums sn,θ k∞ 1[αk ( θ) − βk ] where α that it had been proved before (with a different = = + − and β have the sameP meaning and where the pa- proof) by Jeronimo-Case, but this does not reduce rameter θ [0, ). Similar problems about central theimportanceoftheirresult. measures∈ on the∞ space of paths of the graded graphs (Bratelli diagram)—in another words, the Asymptotic Statistics of Young Diagrams question of how to describe the traces on locally with Respect to Plancherel Measure semisimple algebras—are difficult. Such problems This is one of the most impressive developments were extensively studied by S. Kerov, G. Olshansky, during the last few years in analysis and represen- A. Okounkov, the author, and others, but are not tationtheory. solvedformanynaturalexamples.Theyareveryim- The study of the statistics of Young diagrams portant for the asymptotic representation theory withPlancherelmeasure and the limitshape of typi- of classical groups and for the theory of symmetric functions. It is worth mentioning that the asymp- caldiagramswasstartedinthepapersoftheauthor totic point of view on the infinite symmetric group and S. Kerov in the 1970s. At the end of the 1990s, (also on the other groups of that type), produced a the remarkable link between those questions and new approach to the theory of representations of thebehavioroftheeigenvaluesofGaussianrandom the finite symmetric groups—in some sense this is matrices was discovered. Great progress has been alookfrominfinitytofiniteness.Thisapproachwas made in the last ten years, and the area has simul- started in the paper of Okounkov and the author taneously been connected to random matrices, (1996) and gave later many natural explanations of matrix problems, orthogonal polynomials, Young the classicalresults. diagrams, integrable systems, determinant point processes,etc. Several Formulas Many mathematicians took partinthis progress: Okounkov is the author (or coauthor) of papers con- J. Baik, P. Deift, K. Johansson, C. Tracy, H. Widom, taining remarkable concrete formulas that concern and others. Using serious analytical tools, those veryclassicalsubjects. Among other results by him authors obtained striking and deep results about about symmetric functions and their application distributions of the fluctuations of the maximal to various problems of analysis and representa- eigenvalue of Gaussian matrices (Tracy-Widom) tions, I want to mention two examplesof important and the first and second row of a Young diagram formulas. In 1996-7 with G. Olshansky (following with respect to Plancherel measure (Baik-Deift- Olshansky-Kerov and other authors), Okounkov Johansson). The contribution of Okounkov to sub- systematically studied the theory of shifted Schur sequent progress was very important: he suggested functions and its role in representation theory a strategy for considering all such problems. and obtained an elegant new formula for the num- This strategy, together with previous ideas of his ber of standard tableaux of a skew Young diagram coauthors G. Olshansky and A. Borodin, led to com- λ n,µ k, λ µ: pletely new proofs and simplifications of previous ⊢ ⊢ ⊃ results on the one hand, and on the other hand sµ∗(λ) dim λ/µ . provided a jumping-off point for him to move from = n(n 1) . . . (n k 1) − − + those problems to random surfaces, to algebraic Thisformula ismainly basedon the surprisingvan- curves, to the Gromov-Witten/Hurwitz correspon- ishingtheoremofOkounkov: dence, etc. He was able to join together problems that had previously been very far from each oth- sµ∗(λ) 0 unless µ λ sµ∗(µ) h(α). = ⊂ = αYµ er. One can say that this was a continuation of ∈ Here h(α) is the hook length of the cell α of the dia- the “representation-theoretic” and the “partition- gram µ. The reformulation of that theory of shifted combinatorial” approachto the problems,together Schur functions in terms of Frobenius coordinates withideasfrommodernmathematicalphysics. was given later (by Olshansky-Regev-Vershik) and I will mention only the main result. The concrete led to the so-called inhomogeneous Frobenius- problem was to refine old results about the limit Schur symmetric functions. These are nothing but distribution of the spectrum of random Gaussian reformulations of the shifted Schur functions in matrices (Wigner’s semicircle law) and the limit Frobeniuscoordinates. shape of random Young diagrams with Plancherel Another example is a formula that attracted the measure (Vershik-Kerov, Logan-Shepp), and to find attention of many mathematicians because of its the distribution of the fluctuations of the maxi- importance. This formula came out of Okounkov’s mal eigenvalue of random matrices (or the first

392 Notices of the AMS Volume 54, Number 3 several rows of Young diagrams). In a few prelim- The Work of Grigory Perelman inary papers Okounkov prepared the exclusively natural approach to the precise calculation of the John W. Morgan∗ correlationfunctions. The correlation functions for random point Introduction process had been studied earlier for so-called z- I will report on the work of Grigory Perelman for measures on Young diagrams, in several papers which he was awarded the Fields Medal at the of Borodin-Olshansky. The idea was to extend International Congress in the summer of 2006. their calculation to Plancherel measure. Consider Perelman posted three preprints on the arXiv be- “poissonization”(=passagetoagrandcanonicalen- tween November 2002 and July 2003, [14, 16, 15]. semble) ofthe Plancherelmeasure onthe diagrams. In these preprints he gave a complete, albeit high- The so-called “Russian way” to look at a Young ly condensed, proof of the Poincaré Conjecture. diagram is to turn the diagram 135◦, and then to Furthermore, at the end of the second preprint project it onto the lattice Z, or, more exactly, onto he stated a theorem about three-manifolds with the lattice Z (1/2). When diagrams are equipped + curvature bounded below which are sufficient- with poissonization of the Plancherel measure, the ly collapsed. He showed how, from what he had “Russian way” gives a remarkable random point established in the first two of his preprints, this col- process on the lattice, which is a determinant pro- lapsing result would imply the vast generalization cess (even before the limit procedure!). This is the of the Poincaré Conjecture, known as Thurston’s main point, and it gives the possibility of calculat- Geometrization Conjecture. He stated that he ing correlation functions of the processes and of would provide another manuscript proving the obtaining a remarkable solution of the problem collapsing result, and in private conversations, about the fluctuation of the Plancherel Young dia- he indicated that the proof used ideas contained grams in the middle of the diagrams and near the in an earlier unpublished manuscript of his from edges. It gives correspondingly the Airy and Bessel 1992 and a then recently circulated manuscript of ensemble as a determinant process on the lattice Shioya-Yamaguchi (which has now appeared [20]). Z (1/2) and produces asapartialresult the proof + To date, Perelman has not posted the follow-up of Deift’s conjecture about joint distributions of paper establishing the collapsing result he stated at the fluctuations of finitely many of the rows of a theendofhissecondpreprint. Young diagram with respect to Plancherel mea- In this article, I will explain a little of the histo- sure. This technique avoids the complicated tools ry and significance of the Poincaré Conjecture and of the Riemann-Hilbert problem as well as other Thurston’s Geometrization Conjecture. Then I will tools that were used before, and brings to light the describe briefly the methods Perelman used to es- essence of the effects. Okounkov’s idea about a tablish these results, and I will discuss my view of direct link with random matrices was realized via the current state of the Geometrization Conjecture. matrix problems and ramified covering of surfaces. Lastly, I will speculate on future directions that may We can say now that the asymptotic theory of the arise out of Perelman’s work. For a survey on the Plancherel measure on Young diagrams is more or PoincaréConjecturesee[12].Formoredetailsofthe lesscomplete. ideasandresultswesketchhere,thereadercancon- In order to give a flavor of the ideas in those sult[10],[2],and[13]and[23]. papers of Okounkov I mention some links and preliminary and hidden ideas: the action of SL(2) The Poincaré Conjecture and Thurston’s on the partitions (started by S. Kerov), the infinite Geometrization Conjecture wedge model, boson-fermion correspondence; In 1904 Poincaré asked whether every closed (i.e., the very important new idea of Schur measures compact without boundary) simply connected on the diagrams and, later, its generalization to three-manifold is homeomorphic to the three- Schur processes, which allowed one to consider sphere, see [18]. (It has long been known that this three-dimensional Young diagrams and their limit is equivalent to asking whether a simply connected shapes,asymptoticsofenumerationoftheramified smooth three-manifold is diffeomorphic to the coverings of surfaces, etc. Of course this gigantic three-sphere.) What has come to be known as the body of work still is not in complete order, and Poincaré Conjecture is the conjecture that the an- many ideas must be made clearer. But the number swer to this question is “yes”. Since its posing, the ofresultsandfutureprospectsareimpressive. Poincaré Conjecture has been a central problem in topology, and most of the advances in study of the topology of manifolds, both in dimensions three andinhigherdimensions,overthelastonehundred

∗John Morgan is professor of mathematics at Columbia University. His email address is [email protected].

March 2007 Notices of the AMS 393 years have been related to the Poincaré Conjecture that the manifold is the union of two three-balls, or its various generalizations. Prior to Perelman’s which then implies that it is homeomorphic to work, analogues of the conjecture had been formu- the three-sphere. While these types of topological lated inalldimensions and the topologicalversions arguments have proved many beautiful results of these analogues had been established (see [21] about three-dimensional manifolds, including for and [6]) in all dimensions except dimension three. example that a knot in the three-sphere is trivial Counterexamples to the analogue of the Poincaré (i.e., unknotted) if and only if the fundamental Conjecture for smooth manifolds of higher dimen- group of its complement is isomorphic to Z, they sion were first given by Milnor, in [11], where he have said nothing about the Poincaré Conjecture. constructed exotic smooth structures on the 7- For a description of some of these approaches and dimensional sphere. The smooth four-dimensional why they failed see [22]. More recently, there have Poincaré Conjecture remains open. Perelman’s been approaches to the Geometrization Conjecture workresolvesoneremainingcaseofthetopological along the following lines. Givenany three-manifold, version of the question. The Poincaré Conjecture is the complement of a sufficiently general knot in alsothefirstofthesevenClayMillenniumproblems the manifold will have a complete hyperbolic struc- tobesolved. ture of finite volume. Work from here back to the Poincaré was motivated to ask his question original manifold by navigating in the space of by an attempt to characterize the simplest of all hyperbolic structures with cone-like singularities. three-manifolds, the three-sphere. But there is no This approach, pioneered by Thurston, proved a reason to restrict only to this three-manifold. In beautiful result, called the orbifold theorem [5], 1982 Thurston formulated a general conjecture, that proves a special case of the Geometrization known as Thurston’s Geometrization Conjec- Conjecture, but has not led to a proof of the full ture, which, if true, would (essentially) classify all conjecture. closed three-manifolds, see [24] and [19]. Unlike the Poincaré Conjecture, whichhas a purely topological Perelman’s Method: Ricci Flow conclusion, Thurston’s Conjecture has a geomet- How did Perelman do it? He used, and general- ric conclusion: In brief, it says that every closed ized, the work of Richard Hamilton on Ricci flow. three-manifold has an essentially unique two-step For an introduction to Ricci flow see [3] and for a decomposition into simpler pieces. The first step collection of Hamilton’s papers on Ricci flow see is to cut the manifold open along a certain family [1]. Hamilton showed, at least in good cases, that of embeddedtwo-spheres and cap off the resulting the Ricci flow gives an evolution of a Riemannian boundaries with three-balls. The second step is to metric on a 3-manifold that converges to a locally cut the manifold open along a certain family of homogeneous metric (e.g., a metric of constant two-tori. The conjecture posits that at the end of positive curvature), see [7] and [9]. But Hamilton thisprocesseachoftheresultingpieceswilladmita also showed (see [8]) that there are other possibil- complete, finite-volume, Riemannian metric locally ities for the effect of Ricci flow. It can happen, and modeled on one of the eight homogeneous three- does happen, that the Ricci flow develops finite- dimensional geometries. All manifolds modeled on time singularities. These finite-time singularities sevenoftheeightgeometriesarecompletelyunder- impede the search for a good metric; for, in order stood and easily listed. The eighth homogeneous to find the good metric one needs to continue the three-dimensional geometry is hyperbolic geom- flow for all positive time—the Riemannian metrics etry. It is the richest class of examples and is the one is looking for in general only appear as limits most interesting case. Manifolds modeled on this as time goes to infinity. In fact, the finite-time sin- are given as quotients of hyperbolic three-space by gularities are needed in order to do the first layer torsion-free lattices in SL(2, C) of finite co-volume. of cutting (along two-spheres and filling in balls) as The classification of these manifolds is equivalent required in Thurston’s Conjecture. These cuttings to the classification of such lattices, a problem that along two-spheres and filling in balls have been has not yet been completely solved. Thurston’s long known to be necessary in order to produce a Conjecture includes the Poincaré Conjecture as manifoldthatadmitsaRiemannianmetricmodeled a special case, since the only geometry that can ononeofthehomogeneousgeometries.Asweshall model a simply connected manifold is the spher- seepresently,thesecondstepinthedecomposition ical geometry (constant positive curvature), and proposed by Thurston, along two-tori, happens as t simply connected manifolds of constant positive limits to infinity. curvature are easily shown to be isometric to the The Ricci flow equation, as introduced and first three-sphere. studiedbyHamiltonin[7],isaweaklyparabolicpar- Over the last one hundred years there have been tial differential equation for the evolution of a Rie- many attempts to prove the Poincaré Conjecture. mannian metric g on a smooth manifold. It is given Most have been direct topological attacks which by: involve simplifying surfaces and/or loops in a ∂g(t) 2Ric(g(t)), simply connected three-manifold in order to show ∂t = −

394 Notices of the AMS Volume 54, Number 3 where Ric(g(t)) is the Ricci curvature of g(t). One (2) Using(1)heprovedthatforeveryfinitetime should view this equation as a (weakly) nonlinear interval I, a Ricci flow of compact mani- version of the heat equation for symmetric two- folds parameterizedby I are non-collapsed tensors on a manifold. Hamilton laid down the where both the measure of non-collapsing basics of the theory of this equation—short-time and the scale depend only on the interval existence and uniqueness of solutions. Further- andthe geometryof the initial manifold. more, using a version of the maximum principle (3) Using (1) and results of Hamilton’s on sin- he established various important differential in- gularity development, he classified, at least equalities, including a Harnack-type inequality, qualitatively, all models for singularity that are crucial in Perelman’s work. Hamilton al- development for Ricci flows of compact so understood that, in dimension three, surgery three-manifolds at finite times. These would be a crucial ingredient for two reasons: (i) it models are the three-dimensional, ancient is the way to deal with the finite-time singularities solutions (i.e., defined for < t 0) of −∞ ≤ and (ii) because of the necessity in general to do non-negative, bounded curvature that are cutting on a given three-manifold in order to find non-collapsed on all scales (r0 ). = ∞ the good metric. With this motivation, Hamilton (4) He showed that any sequence of points introduced the geometric surgeries that Perelman (xn,tn) in the space-time of a three-dimen- employs. sional Ricci flow whose times tn are uni- Fromthepointofviewofthepreviousparagraph, formly bounded above and such that the an n-dimensionalRicciflowisaone-parameterfam- norms of the Riemannian curvature ten- ilyofmetricsonasmoothmanifold.Itisalsofruitful sors Rm(xn,tn) go to infinity (a so-called to take the perspective of space-time and define an “finite-time blow-up sequence”) has a sub- n-dimensional Ricci flow as a horizontal metric on sequence converging geometrically to one M [a, b] where the metric on M t is the metric of the models from (3). This result gives × ×{ } g(t) on M. This point of view allows for a natural neighborhoods, called “canonical neigh- generalization: a generalized n-dimensional Ricci borhoods”, for points of sufficiently high flowisan (n 1)-dimensionalspace-time equipped scalar curvature. These neighborhoods + with a time function which is a submersion onto an are geometrically close to corresponding interval, a vector field which will allow us to differ- neighborhoods in non-collapsed ancient entiate in the “time direction”, and a metric on the solutions as in (3). Their existence is a cru- “horizontal distribution” (which is the kernel of the cial ingredient necessary to establish the differential of the time function). We require that analytic and geometric results required to these data belocally isomorphicto a Ricci flowon a carryout andto repeatsurgery. product M (a, b), with the vector field becoming (5) From the classification in (3) and the blow- × ∂/∂t inthelocalproductstructure. up result in (4), he showed that surgery, as envisioned by Hamilton, was always possi- Perelman’s First Two Preprints ble for Ricci flows starting with a compact Here we give more details on Perelman’s first two three-manifold. preprints, [14] and [16]. As we indicated above, the (6) He extended all the previous results from workinthesepreprintsisbuiltonthefoundationof the category of Ricci flows to a category Ricci flow as developed by Hamilton. A central con- of certain well-controlled Ricci flows with cept is the notion of an n-dimensional Ricci flow, surgery. or generalizedRicci flow, being κ-non-collapsed on (7) He showed that, starting with any compact scales r0 for some κ > 0. Let p beapointofspace- three-manifold, repeatedly doing surgery ≤ time and let t be the time of p. Denote by B(p,t,r) and restarting the Ricci flow leads to a well- themetricballofradiusr inthet time-sliceofspace- controlled Ricci flow with surgery defined time, and denote by P(x,t,r, r 2) the backwards for all positive time. − parabolic neighborhood consisting of all flow lines (8) He studied the geometric properties of the backwardsintimefromtime t to time t r 2 starting limits as t goes to infinity of the Ricci flows − at points of B(x, t, r). Thismeans that for any point with surgery. p and any r r0 the following holds. If the norm of Let us examine these in more detail. Perel- ≤ 2 the Riemannian curvature tensor on P(p,t,r, r ) man’s length function is a striking new idea, one 2 − isatmost r − then the volume of B(x, t, r) isatleast that he has shown to be extremely powerful, for n κr .Herearethemainnewcontributionsofhisfirst example, allowing him to prove non-collapsing twopreprints: results. Let (M, g(t)), a t T , be a Ricci flow. ≤ ≤ (1) He introduced an integral functional, called The reduced -length functional is defined on L the reduced -length for paths in the space- paths γ : [0, τ]¯ M [a, T ] that are param- L → × timeofaRicciflow. eterized by backwards time, namely satisfying

March 2007 Notices of the AMS 395 γ(τ) M T τ for all τ [0, τ]¯ . One (when they are rescaled to have scalar curvature considers∈ × { − } ∈ 1 at some point), and (v) a non-compact manifold τ¯ of positive curvature which is a union of a cap 1 2 l(γ) √τ R(γ(τ)) Xγ (τ) dτ. of bounded geometry (modulo rescaling) and a = 2√τ¯ Z0 +| |
cylindrical neck (approximately S2 times a line) at Here, R(γ(τ)) is the scalar curvature of the metric infinity. g(T τ) at the point in M that is the image under Using the non-collapsing result and delicate geo- the projection− into M of γ(τ), and X (τ) is the γ metric limit arguments, Perelman showed that in a projection into TM of the tangent vector to the Ricci flow on compact three-manifolds any finite- path γ at γ(τ). The norm of X (τ) is measured γ time blow-up sequence has a subsequence which, using g(T τ). It is fruitful to view this functional after rescaling to make the scalar curvature equal as the analogue− of the energy functional for paths to one at the base point, converges geometrically in a Riemannian manifold. Indeed, there are - to a non-collapsed, ancient solution of bounded, geodesics, -Jacobi fields, and a function l (q,L t) (x,T ) non-negative curvature. (Geometric convergence thatmeasuresL the minimalreduced -length of any means the following: Given a finite-time blow-up -geodesic from (x, T ) to (q, t). FixingL (x, T ), the sequence (x ,t ), after replacing the sequence by Lfunction l (q, t) is a Lipschitz function of (q, t) n n (x,T ) a subsequence, there is a model solution with base andissmoothalmosteverywhere. point and scalar curvature at the base point equal Perelman proves an extremely important mono- to one such that for any R < the balls of radius tonicity result along -geodesics for a function R centered at the x in the nth∞ rescaled flow con- related to this reducedL length functional. Namely, n verge smoothly to a ball of radius R centered at suppose that W is an open subset in the time-slice the base point in the given model.) For example, we T τ¯, and each point of W is the endpoint of a couldhaveacomponentofpositiveRiccicurvature. unique− minimizing -geodesic.Thenhedefinedthe According to Hamilton’s result [7] this manifold reducedvolumeofWLtobe contracts to a point at the singular time and as it n/2 l (w,T τ)¯ (τ)¯ − e− (x,T ) − dvol. doessoitapproachesaround(i.e.,constantpositive ZW curvature) metric. Thus, the model in this case is a For each τ (0, τ]¯ let W (τ) bethe resultofflowing round manifold with the same fundamental group. ∈ W alongtheuniqueminimalgeodesicstotimeT τ. Perelman’s result implies that there is a threshold − Then the reduced volume of W (τ) is a monotone so that all points of scalar curvature above the non-increasing function of τ. This is the basis of threshold have canonical neighborhoods modeled his proof of the non-collapsing result: Fix a point on corresponding neighborhoods in non-collapsed (x, T ) satisfying the hypothesis of non-collapsing ancient solutions. This leads to geometric and for some r r0. The monotonicity allows him to analytic control in these neighborhoods, which in transfer lower≤ bounds on reduced volume (from turn is crucial for the later arguments showing that (x, T )) at times near the initial time (which is auto- surgeryisalwayspossible. matic from compactness) to lower bounds on the Now let us turn to surgery. An n-dimensional reducedvolumetimesnear T .Fromthis,oneproves Ricci flow with surgery is a more general object. It thenon-collapsingresultat(x, T ). consists of a space-time which has a time function. Perelman then turned to the classification of The level sets of the time function are called the ancient three-dimensional solutions (ancient in the time-slices and they are compact n-manifolds. This sense of being defined for all time < t 0) space-time contains an open dense subset that is a of bounded, non-negative curvature−∞ that are≤ non- smoothmanifoldofdimension n 1 equipped with collapsed on all scales. By geometric arguments a vector field and a horizontal metric+ that make he established that the space of based solutions this open dense set a generalized n-dimensional of this type is compact, up to rescaling. Here the Ricci flow. At the singular times, the time-slices non-collapsed condition is crucial: After we rescale have points not included in the open subset which to make the scalar curvature one at the base point, is a generalized Ricci flow. This allows the topology this condition implies that the injectivity radius at of the time-slices to change as the time evolves the base point is bounded away from zero. There past a singular time. For example, in the case just are severaltypes of these ancient solutions. A fixed described above, when a component is shrinking time section is of one of the following types—(i) a to a point, the effect of surgery is to remove entire- compact round three-sphere or a Riemannian man- ly that component. A more delicate case is when ifold finitely covered by a round three-sphere, (ii) a the singularity is modeled on a manifold with a cylinder which is a product of a round two-sphere long, almost cylindrical tube in it. In this case, at with the line or a Riemannian manifold finitely the singular time, the singularities are developing covered by this product, (iii) a compact manifold inside the tube. Following Hamilton, at the singular diffeomorphic to S3 or RP 3 that contains a long time Perelman cuts off the tube near its ends and neck which is approximately cylindrical, (iv) a com- sews in a predetermined metric on the three-ball, pact manifold of bounded diameter and volume using a partition of unity. The effect of surgery is

396 Notices of the AMS Volume 54, Number 3 to produce a new, usually topologically distinct a complete constant negatively curved metric, or manifold at the singular, or surgery, time. After the metric is arbitrarily collapsed on the scale of having produced this new closed, smooth mani- its curvature. Components of the first type clearly fold, one restarts the Ricci flow using that manifold support hyperbolic metrics of finite volume. It is to as the initial conditions. This then is the surgery deal with the components of the second type that process: remove some components of positive cur- Perelman states the proposed result on collapsed vature and those fibered over circles by manifolds manifoldswithcurvatureboundedbelow. of positive curvature (all of which are topologically standard)andsurgerothersalongtwo-spheres,and Completion of the Proof of the Poincaré thenrestartRicciflow. Conjecture Perelman then showed the entire Ricci flow As we have already remarked, the Poincaré Con- analysis described above extends to Ricci flows jecture follows as a special case of Thurston’s with surgery, provided that the surgery is done in a sufficiently controlled manner. Thus, one is able Geometrization Conjecture. Thus, the results of to repeat the argument ad infinitum and construct the first two of Perelman’s preprints together with a Ricci flow with surgery defined for all time. Fur- the collapsing result stated at the end of the second thermore, one has fairly good control on both the preprint, give a proof of the Poincaré Conjecture. change in the topology and the change in the ge- In a third preprint [15] Perelman gave a different ometry as one passes the surgery times. Here then argument, avoiding the collapsing result, proving is the main result of the first two preprints taken the Poincaré Conjecture but not the entire Ge- together: ometrizationConjecture.Foradetailedproofalong these lines, see Chapter 18 of [13]. One shows that Theorem. Let (M, g0) be a compact, orientable Rie- if one begins with a homotopy three-sphere, or mannian three-manifold. Then there is a Ricci flow indeed any manifold whose fundamental group is with surgery ( , G) defined for all positive time a free product of finite groups and infinite cyclic whose zero-timeM slice is (M, g ). This Ricci flow 0 groups, then the Ricci flow with surgery, which with surgery has only finitely many surgery times by the results of Perelman’s first two preprints is in any compact interval. As one passes a surgery defined for all positive time, becomes extinct after time, the topology of the time-slices changes in the a finite time. That is to say, for all t sufficiently following manner. One does a finite number of large, the manifold at time t is empty. We conclude surgeries along disjointly embedded two-spheres (removing an open collar neighborhood of the that the original manifold is a connected sum of 3 two-spheres and gluing three-balls along each of spherical space-forms (quotients of the round S 2 the resulting boundary two-spheres) and removes by finite groups of isometries acting freely) and S - 1 a finite number of topologically standard compo- spherebundlesover S . Itfollows thatif the original nents (i.e., components admitting round metrics manifold is simply connected, then it is diffeo- and components finitely covered by S2 S1). morphic to a connected sum of three-spheres, and × hence is itself diffeomorphic to the three-sphere. Acoupleofremarksareinorder: This shows that the Poincaré Conjecture follows (i) Itis clear from the constructionthatif Thurston’s from this finite-time extinction result together Geometrization Conjecture holds for the manifold with the existence for all time of a Ricci flow with at time t then it holds for the manifolds at all previ- surgery. oustimes. Let us sketch how the finite-time extinction re- (ii) In this theorem one does not need orientability, sult is established in [15]. (There is a parallel ap- only the weaker condition that every projective proach in [4] using areas of harmonic two-spheres plane has non-trivial normal bundle. To extend of non-minimal type instead of area minimizing 2- the results to cover manifolds admitting pro- disks.) We consider the case of a homotopy three- jective planes with trivial normal bundle one takes a double covering and works equivariant- sphere M (though the same ideas easily generalize ly. This fits perfectly with the formulation of the to cover all cases stated above). The fact that the GeometrizationConjectureforsuchmanifolds. manifold is a homotopy three-sphere implies that In addition, Perelman established strong geo- π3(M) isnon-trivial.Perelman’sargumentistocon- sider a non-trivialelementin ξ π3(M). Represent metric control over the nature of the metrics on the ∈ time-slices as time tends to infinity. In particular, this element by a two-sphere family of homotopi- at the end of the second preprint he showed that cally trivial loops. Then for each loop take the in- for t sufficiently large, the t time-slice contains a fimum of the areas of spanning disks for the loop, finite number of incompressible tori (incompress- maximize over the loops in family and then mini- ible means π1 injective) that divide the manifold mize over all representative families for ξ. The re- into pieces. Each component that results from the sultis aninvariant W (ξ). One asks what happens to cutting process has a metric of one of two types. this invariant under Ricci flow. The result (follow- Either the metric is, after rescaling, converging to ing similar arguments that go back to Hamilton [9])

March 2007 Notices of the AMS 397 is that future may hold. First of all, an affirmative resolu- dW (ξ) 1 tion of the one-hundred-year-old Poincaré Conjec- 2π Rmin(t)W (ξ). ture is an accomplishment rarely equaled or sur- dt ≤ − − 2 passed in mathematics. An affirmative resolution Here Rmin(t) is the minimum of the scalar curvature of the Geometrization Conjecture will surely lead at time t. Since one of Hamilton’s results using the to a complete and reasonably effective classifica- maximum principle is that Rmin(t) 6/(4t α) ≥ − + tion of all closed three-dimensional manifolds. It for some positive constant α, it is easy to see that is hardtooverestimate the progress thatthis repre- anyfunctionsatisfyingEquation(6)goesnegativein sents.TheGeometrizationConjectureisagoalinits finite time. Perelman shows that the same equation own right, leading as it (essentially) does to a clas- holdsinRicciflowswithsurgeryaslongasthemani- sification of three-manifolds. Paradoxically, the ef- fold continues to existinthe Ricci flowwithsurgery. fectofPerelman’sworkonthree-manifoldtopology On the other hand, W (ξ) is always non-negative. It willbeminimal.Almostallworkersinthefieldwere follows that the homotopy three-sphere must dis- already assuming that the Geometrization Conjec- appearinfinitetime. ture is true and were working modulo that assump- This then completes Perelman’s proof of the tion, or else they were working directly on hyper- Poincaré Conjecture. Start with a homotopy three- bolic three-manifolds, which obviously satisfy the sphere. Run the Ricci flow with surgery until the GeometrizationConjecture. manifold disappears. Hence, it is a connected sum The largest effects of Perelman’s work will lie of manifolds admitting constant positive curvature in other applications of his results and methods. metrics. That is to say it is a connected sum of I think there are possibilities for applying Ricci three-spheres, and hence itself diffeomorphic to flow to four-manifolds. Four-manifolds are terra the three-sphere. incognita compared to three-manifolds. In dimen- sion four there is not even a guess as to what the Status of the Geometrization Conjecture possibilities are. Much less is known, and what is What about the general geometrization theorem? known suggests a far more complicated landscape Perelman’s preprints show that proving this result indimensionfourthanindimensionthree.Inorder has been reduced to proving a statement about to apply Ricci flow and Perelman’s techniques to manifolds with curvature locally bounded below four-manifolds there are many hurdles to over- that are collapsed in the sense that they have come. Nevertheless,this is an area that, in my view, short loops through every point. There are results showspromise. along these lines by Shioya-Yamaguchi [20]. The There are also applications of Ricci flow to Käh- full result that Perelman needs can be found in an ler manifolds, where already Perelman’s results are appendix to [20], except for the issue of allowing a having an effect—see for example [17] and the ref- boundary. The theorem that Perelman states in his erences therein. second preprint allows for boundary tori, whereas As the above description should make clear, the statement in [20] is for closed manifolds. Perel- Perelman’s great advance has been in finding a man has indicated privately that the arguments way to control and qualitatively classify the sin- easily extend to cover this more general case. But gularities that develop in the Ricci flow evolution in any event, invoking deep results from three- equation. There are many evolution equations in manifold topology, it suffices to consider only the mathematics and in the study of physical phenom- closed case in order to derive the full Geometriza- ena that are of the same general nature as the Ricci tion Conjecture. The paper [20] relies on an earlier, flow equation. Some, such as the mean curvature unpublishedworkbyPerelman. flow, are related quite closely to the Ricci flow. All in all, Perelman’s collapsing result, in the Singularity development is an important aspect of full generality that he stated it, seems eminently the study of almost all of these equations both in plausible. Still, to my mind the entire collapsing mathematics and in physical applications. It is not space theory has not yet received the same careful yet understood if the type of analysis that Perelman scrutiny that Perelman’s preprints have received. carriedout for the Ricci flowhas analoguesin some So,whileIseenoseriousissueslooming,Ipersonal- of these other contexts, but if there are analogues, lyamnotreadytosaythatgeometrizationhasbeen the effect of these on the study of those equations completely checked in detail. I am confident that couldbequiteremarkable. it is only a matter of time before these issues are satisfactorilyexplored. References [1] H. D. Cao, B. Chow, S. C. Chu, and S. T. Yau, eds., The Effect of Perelman’s Work Collected papers on Ricci flow, volume 37 of Se- Letmesayafewwordsaboutthelargersignificance ries in Geometry and Topology, International Press, of what Perelman has accomplished and what the Somerville, MA, 2003.

398 Notices of the AMS Volume 54, Number 3 [2] Huai-Dong Cao and Xi-Ping Zhu, A com- [22] John Stallings, How not to prove the Poincaré plete proof of the Poincaré and Geometriza- Conjecture, In Topology Seminar of Wisconsin, tion conjectures—Application of the Hamilton- volume 60 of Annals of Math Studies,1965. Perleman theory of the Ricci flow, Asian J. of Math [23] Terry Tao, Perelman’s proof of the Poincaré (2006), 169–492. Conjecture—a nonlinear PDE perspective, arX- [3] Bennett Chow and Dan Knopf, The Ricci flow: an iv:math.DG/0610903, 2006. introduction, volume 110 of Mathematical Surveys [24] William P. Thurston, Hyperbolic structures and Monographs, American Mathematical Society, on 3-manifolds. I, Deformation of acylindrical Providence, RI, 2004. manifolds, Ann. of Math. 124(2) (1986), 203–246. [4] Tobias H. Colding and William P. Minicozzi II, Estimates for the extinction time for the Ricci flow on certain 3-manifolds and a question of Perel- man, J. Amer. Math. Soc. 18(3) (2005), 561–569 The Work of Terence Tao (electronic). [5] Daryl Cooper, Craig D. Hodgson, and Steven P. Jean Bourgain∗ Kerckhoff, Three-dimensional orbifolds and cone- manifolds, volume 5 of MSJ Memoirs, Mathematical The work of Terence Tao includes major contribu- Society of Japan, Tokyo, 2000, with a postface by tions to analysis, number theory, and aspects of Sadayoshi Kojima. representation theory. Perhaps his most spectacu- [6] Michael Hartley Freedman, The topology of four-dimensional manifolds, J. Differential Geom. lar achievement to date is the proof that the set of 17(3) (1982), 357–453. prime numbers contains arbitrarily long arithmetic [7] Richard S. Hamilton, Three-manifolds with pos- progressions (jointly with Ben Green). A reasonably itive Ricci curvature, J. Differential Geom. 17(2) detailed account on this result would already easily (1982), 255–306. make up the full article. But it certainly would not [8] , The formation of singularities in the Ric- give a picture of the unique scope and diversity ci flow, in Surveys in differential geometry, Vol. II in Tao’s opus. The number of areas that he has (Cambridge, MA, 1993), Internat. Press, Cambridge, marked either by solving the main questions or MA, 1995, pp. 1–136. making them progress in a decisive way is utterly [9] , Non-singular solutions of the Ricci flow on three-manifolds, Comm. Anal. Geom. 7(4) (1999), astonishing. This unique problem-solving ability 695–729. relies not only on supreme technical strength but [10] Bruce Kleiner and John Lott, Notes on also on a deep global understanding of large parts Perelman’s papers, arXiv:math.DG/0605667, 2006. of mathematics. To illustrate this, I plan to report [11] John Milnor, On manifolds homeomorphic to the below also on some of his contributions to har- 7-sphere, Ann. of Math. (2) (1956), 399–405. monic analysis, partial differential equations, and [12] , Towards the Poincaré conjecture and the representation theory. Because of lack of space, classification of 3-manifolds, Notices Amer. Math. the results will be formulated with little or no Soc. 50(10) (2003), 1226–1233. background discussion or how they fit in the larger [13] John Morgan and Gang Tian, Ricci flow and the Poincaré Conjecture, arXiv:math.DG/0607607, pictureofTao’sresearch. 2006. 1)Taohasthestrongestresultstodateonseveral [14] Grisha Perelman, The entropy formula for central conjectures in higher-dimensional Fourier the Ricci flow and its geometric applications, analysis, first formulated by E. Stein. These conjec- arXiv:math.DG/0211159, 2002. tures express mapping properties of the Fourier [15] , Finite extinction time for the solutions transform when restricted to hypersurfaces (in to the Ricci flow on certain three-manifolds, particular, the Fourier restriction conjecture to arXiv:math.DG/0307245, 2003. spheres and the Bochner-Riesz conjectures belong [16] , Ricci flow with surgery on three-manifolds, arXiv:math.DG/0303109, 2003. to this class of problems). Those issues were ba- [17] Duong H. Phong and Jacob Sturm, On stability sically understood in dimension 2 already in the and the convergence of the Kähler-Ricci flow, J. 1970s, but they turn out to be much more resistant Differential Geom. 72(1) (2006), 149–168. starting from dimension 3 (see [1] for the current [18] Henri Poincaré, Cinquième complément à state of affairs). They are intricately connected to l’analysis situs, In Œuvres. Tome VI, Les Grands combinatorial questions such as the dimension Classiques Gauthier-Villars, [Gauthier-Villars Great of higher-dimensional “Kakeya sets”. Those are Classics], pages v+541, Éditions Jacques Gabay, compact subsets in Rn containing a line segment in Sceaux, 1996, reprint of the 1953 edition. every direction. Such sets may have zero Lebesgue [19] Peter Scott, The geometries of 3-manifolds, Bull. measure (the well-known Besicovitch construction London Math. Soc. 15(5) (1983), 401–487. [20] Takashi Shioya and Takao Yamaguchi, Volume in the plane) but it is believed that the Hausdorff collapsed three-manifolds with a lower curvature dimension is always maximal (i.e., equals n). This bound, Math. Ann. 333(1) (2005), 131–155. [21] Stephen Smale, Generalized Poincaré’s conjecture ∗Jean Bourgain is professor of mathematics at the in dimensions greater than four, Ann. of Math. Institute for Advanced Study. His email address is 74(2) (1961), 391–406. [email protected].

March 2007 Notices of the AMS 399 problem, which underlies Stein’s conjectures, has extension of Szemerédi’s theorem, claiming that been the object of intensive research over the the same conclusion holds if now we consider last few decades. Although still unsolved even for subsets of positive density in a “pseudo-random” n 3, also here Tao’s work has led to some of the set. Here one should be more precise about what = most significant developments, often of interest “pseudo-random”means,butletusjustsaythatthe in their own right. Among these is his joint paper prime numbers fit the model well, and the required with N. Katz and I. Laba [2] and the discovery of pseudo-random “container” is simply provided by the “sum-product phenomenon” in finite fields looking at an appropriate set of pseudo-primes [3]. (these are integers without small divisors). Follow- 2) Tao’s work on nonlinear dispersive equations ing Furstenberg’s structural approach (in a finite is too extensive to describe here in any detail. But setting), Green and Tao construct in this pseudo- the two papers [4], [5] particularly stand out. Both random setting a “characteristic factor” (in the prove a final result, contain new ideas, and are sense of Furstenberg’s theory) that is of positive technically speaking a real tour de force. The paper density in the integers and hence allows applica- [4] (jointly with J. Colliander, M. Keel, G. Staffilani tion of Szemerédi’s theorem. Their analysis uses and H. Takaoka) solves the longstanding conjec- strongly the so-called “Gowers norms” introduced ture of global wellposedness and scattering for the in [8] to manipulate higher-order correlations. The defocusing energy-critical Schrödinger equation in importance of [8] is even more prominent in Green R3, thus the equation iu ∆u u u 4 0. This t and Tao’s subsequent papers (see [9] in particular) is the counterpart for the+ nonlinear− | | Schrödinger= where [7] is combined with Gowers’ analysisinvolv- equation of M. Grillakis’ theorem [6] on nonlinear wave equations. In both cases, the local theory was ing generalized spectra and classical techniques understood for at least a decade, and the main from analytic number theory. The authors establish difficulty was to rule out “energy concentrations”. inparticular the validity of the expected asymptotic Tao and his collaborators resolve this issue by formula for the number of length 4 progressions proving a new type of “Morawetz-inequality” where in the primes. Such a formula had been obtained the solution u is studied simultaneously in physical by Van der Corput in 1939 for prime triplets, using and frequency space. In the paper [5] the wave the circle method and Vinogradov’s work. Besides map equation with spherical target is studied in solving questions of historical magnitude, the work dimension 2, in which case the energy norm is the of Green and Tao had a strong unifying effect on critical Sobolev norm for the local wellposedness variousfields. of the Cauchy problem. This result is the main 4) I conclude this report with one more line of theorem of the paper. Its proof (which builds upon research in a completely different direction: Tao’s several earlier works) involves a careful study of solution [10] (jointly with A. Knutsen) of the “satu- the “null-structures” in the nonlinearity and a novel ration conjecture” for the Littlewood-Richardson renormalization argument. One should mention coefficients in representation theory, formulated that large data may create blowup behavior al- by A. A. Klyachko in 1998, and Horn’s conjecture though this is not expected to happen if the sphere for the eigenvalues of hermitian matrices under is replaced by a hyperbolic target (S. Klainerman’s addition. This was one of Tao’s earlier achieve- conjecture). ments (with a later follow-up). Again it has greatly 3) Tao’s proof (in joint work with B. Green) impressedexpertsinthefield. that the prime numbers contain arbitrarily long Somuchinsuchshorttime.ToparaphraseC.Fef- arithmetic progressions(a problem considered out ferman, “What’s next?” of reach even for length-four progressions) has come as a real surprise to the experts, because the References initial breakthrough here is one in ergodic theo- [1] T. Tao, Recent progress on the Restriction ry rather than in our analytical understanding of conjecture, Park City Proceedings, submitted. prime numbers. Of great relevance is Szemerédi’s [2] N. Katz, I. Laba, and T. Tao, An improved bound theorem, which roughly states that arbitrary sets on the Minkowski dimension of Besicovitch sets in of integers of positive density contain arithmetic R3, Ann. Math. 152 (2000), 383–446. progressionsofanylength(asconjecturedbyErdos˝ [3] J. Bourgain, N. Katz, and T. Tao, A sum-product and Turán). Szemerédi’s original proof was com- estimate in finite fields and applications, Geom. binatorial. Subsequent developments came with Funct. Anal. 13 (2003), 334–365. H. Furstenberg’s new proof through ergodic theory [4] J. Colliander, M. Keel, G. Staffilani, H. Takao- ka, and T. Tao, Global well-posedness and and the concept of multiple recurrence and later scattering in the energy space for the critical non- T. Gowers’ approach using harmonic analysis, clos- linear Schrödinger equation in R3, to appear in Ann. er in spirit to K. Roth’s work. All of them are crucial Math. for the discussion of the Green-Tao achievement. [5] T. Tao, Global well-posedness of wave maps, 2. In their first paper [7], the existence of progres- Small energy in two dimensions, Commun. Math. sions in the primes is derived from a remarkable Phys. 224 (2001), 443–544.

400 Notices of the AMS Volume 54, Number 3 2 [6] M. Grillakis, Regularity and asymptotic behaviour Let S0 a. If S0, S1,...,Sk δZ K have already = ∈ \ of the wave equation with a critical nonlinearity, been chosen, pick Sk 1 uniformly from the four Ann. Math. 132 (1990), 485–509. 2+ neighbors of Sk in δZ . Stop this process at the ran- [7] B. Green and T. Tao, The primes contain arbitrar- dom time τ when S. first reaches a point in K. Then ily long arithmetic progressions, to appear in Ann. S ,...,S is a realization of simple random walk Math. 0 τ from a to K. A realization of loop-erased random [8] T. Gowers, A new proof of Szemerédi’s theorem, Geom. Funct. Anal. 11 (2001), 465–588. walk is obtainedbyerasingthe loops in S0,...,Sτ in [9] B. Green and T. Tao, Linear equations in primes, the order in which they arise. A formal description to appear in Ann. Math. ofthisprocesscanbefoundin[11]. [10] A. Knutsen and T. Tao, The honeycomb model of The resulting path will be a self-avoiding path n GL (C) tensor products 1, Proof of the saturation LE0, LE1,...,LEσ from a to K with LEj St(j) for conjecture, J. AMS 12 (1999), 1055–1090. some t(0) 0 < t(1) < < t(σ )= τ. The process of loop-erased= walks· · obtained · in= this way is completely described by its distribution function The Work of Wendelin Werner whichissomeprobabilitymeasureonself-avoiding paths from a to K, which we denote by LERW (a,K). Harry Kesten∗ Usually one wants to study the path as long as it is inside some domain D C, so one often takes K to be C D or an approximation⊂ to ∂D. How does a Wendelin Werner received the Fields Medal for \ work which is on the borderline of probability and “typical” loop-erased path look? Does there exist a limit in a suitable topology of LERW (a, K) as δ 0 statistical physics, and which confirmed and made ↓ rigorous quite a number of conjectures made by and is the limit simple to describe? Such a limit is usually called a “scaling limit”. Note that δ 0 au- mathematical physicists studying critical phenom- ↓ ena. In particular he proved so-called power laws tomatically means that one studies walks of more and gave the precise value ofcorresponding critical and more steps. It seems reasonable to expect that Z2 exponents. the lattice structure of δ is no longer visible in Many of Werner’s papers are joint papers with the limit and that the limit is rotation invariant. In one or more of G. Lawler, O. Schramm, and S. fact, it was believed, in the first place by physicists, Smirnov. Together they have developed entirely thatthescalinglimitexistsandiseven“conformally new tools for determining limits of lattice models invariant”. As Schramm puts it “conformal invari- as the lattice spacing goes to 0, and for establishing anceconjectures[were]‘floatingintheair’.”[8]gives conformal invariance of such limits. Even though the following precise formulation to the conjecture there are still many open problems, the work of (weskipanydiscussionofthetopologyinwhichthe Werner and his co-workers opened up an area in scalinglimitshouldbetaken): whichnorigorousargumentswereknown. Let D ⊊ C be a simply connected WernerreceivedaFieldsMedalforatheoryanda domain in C, and let a D. Then ∈ general method, rather than for a specific theorem. the scaling limit of LERW(a,∂D) We can only discusssome ofthis work; basicallywe exists. Moreover, suppose that restricted ourselves to Werner’s work from the year f : D D′ is a conformal homeo- → 1999 on. More details, and especially references to morphism onto a domain D′ C. ⊂ other contributors than Werner and his co-workers Then f µa,D µf(a),D , where µa,D ∗ = ′ canbefoundin[11],[12],and[3]. is the scaling limit measure of LERW(a,∂D) and µ is the f(a),D′ Background: Stochastic Loewner Evolu- scaling limit measure of LERW tions or Schramm-Loewner Evolutions (f (a), ∂D′). Schramm [8] invented a new kind of stochastic Schramm in [8] could not prove this conjecture, process to model certain two-dimensional random but assuming the conjecture to be true he did de- processes, in which the state at time t is the initial termine what µa,D had to be. (The conjecture has piece of a curve γ : [0, t] C, or more generally, been proven now in [7].) Let D ⊊ C be a simply con- a set S(t) C which increases→ with t. A principal nected domain containing 0 and let U be the open motivating⊂ example was the so-called scaling limit unit disc. By the Riemann mapping theorem there of loop-erased random walk on δZ2 as δ 0. Loop- exists a unique conformal homeomorphism ψ ↓ = erased random walk on δZ2 starting from a δZ2 ψD which takes D onto U and is such that ψ(0) 0 ∈ = and stopped at a set K is obtained as follows: First and ψ′(0) is realand strictly positive(the prime de- we choose a simple random walk path from a to K. notes differentiation with respect to the complex variable z). Now let η : [0, ] U and set Ut U ∞ → = ∗Harry Kesten is Emeritus Professor of Mathe- η[0, t] and gt ψUt . Then one can parametrize \ = t matics at Cornell University. His email address is the path η in such a way that g′(0) e . One can t = [email protected]. show that W (t) : limz η(t) gt (z) exists and lies in = →

March 2007 Notices of the AMS 401 ∂U,whereinthislimitz approachesη(t)frominside for q “packets” of Brownian motions. It was known Ut . Loewner’s slit mapping theorem implies that in that there is some exponent ξ(p1,...,pq ) > 0 such ξ(p1,...,pq )/2 the above parametrization of η, gt (z) satisfies the that f (p1,...,pq) t− .(a b means that 1 ≈ ≈ differentialequation [log b]− log a tends to 1 as an appropriate limit ∂ W (t) g (z) is taken; here it is the limit as t .) Lawler and g (z) g (z) t , → ∞ t t + Werner also considered f (p1,...,pq) which is de- ∂t = − W (t) gt (z) − (ℓ) fined by adding the conditione that all Bi [0, t] H ℓ ⊂ g0(z) z for all z U. = ∈ in (2). This has similar exponents ξ such that Here W : [0, ) ∂U is some continuous function ξ(p1,...,pq )/2 f (p1,...,pq ) t− . Duplantiere and Kwon which is called∞ → the driving function of the curve ≈ e predictede (on the basis of non-rigorous arguments η. Properties of η are tied to properties of W . If η from quantum field theory and quantum gravity) is a random curve, then W is also random. To get explicit values for many of the exponents. This back to loop-erased random walk, Schramm first model has conformal invariance built in, because if showed that µ0,U has to be concentrated on simple B( ) isa planarBrownianmotioninadomain D and · curves. Using a kind of Markov property of loop- Φ is a conformal homeomorphism from D onto D′, erased random walks and assuming the conjecture then Φ B( ) is atime changeofa Brownianmotion ◦ · in italics above, Schramm then shows that if η (or in D′. In [4], [5] the authors manage to compute the rather its reversal) is chosen according to µ0,U, then exponents ξ and ξ by relating Brownian excursions W (t) must be exp[iB(κt)] for a Brownian motion to SLE6. Ane important ingredient of their proof is B whose initial point is uniformly distributed on the singling out of SLE6 as the only SLE process ∂U and κ 0 some constant. In a way one can get with the “locality property”. Roughly speaking, ≥ back from the driving force to a curve η. An SLEκ this property says that SLE6 does not “feel the process is then the process of the η[0, t] (or of the boundary” of its domain as long as it does not hit sets U η[0, t]) when the driving function W (t) it. \ equals B(κt). What we described here is so-called Earlier Lawler had shown that the Hausdorff radial SLE. There is also a variant called chordal dimension of various sets defined in terms of SLE whichweshallnotdefinehere. planar Brownian motions could be expressed by What value of κ should one take for the scaling means of the exponents ξ, ξ or similar exponents. limit of loop-erased random walk? Schramm com- For instance, the Hausdorff dimension of the outer e paresthewindingnumberofSLE withthatofLERW boundaryofaplanarBrownianmotionequals2 η2. toconcludethatoneshouldtakeκ 2. Here η is the “disconnection” exponent defined− = 2 by The Work of Werner on Brownian Intersec- P B(1)[0,σ (1)] B(2)[0,σ (2)] tion Exponents { R ∪ R η2 does not disconnect 0 from R− , It has turned out that the representation of certain ∞} ≈ models by SLEκ is very helpful. Various questions with B(1)(0), B(2)(0) two independent two-dimen- reduce to questions about hitting probabilities of (ł) sional Brownian motions starting at 1, and σR is sets by SLE or behavior of functionals of SLE, and the first hitting time by B(ł) of the circle of radius R in many instances these can be reformulated into and center atthe origin. questions about stochastic differential equations In [4], [5] it is shown that η2 equals 2/3, thereby and stochastic integrals. One now clearly needs proving a conjecture by Mandelbrot of more than techniques to show in concrete cases that a model twenty years’ standing. can be analyzed by relating it to SLE. A case where Werner and his co-workers have been quite suc- Background: Conformal Invariance in Per- cessful is the calculation of Brownian intersection colation exponents. Percolation is one of the simplest models which Without using SLE Lawler and Werner had al- exhibits a phase transition and because of its sim- ready made good progress on determining the plicity is a favorite model among probabilists and asymptotic behavior of the “non-intersection prob- statistical physicists for studying critical phenom- ability” ena. One considers an infinite connected graph (2) and takes each edge of open or closed with p1 p2 G G (1) (2) probability p and 1 p, respectively. All edges are f (p1, p2) : P Bi [0, t] Bj [0, t] , − = n i[1 ∩ j[1 =∅o independent of each other. In the simple case this = = modelhas just the one parameter p. We write P for (ℓ) p where Bi , 1 i pℓ, ℓ 1, 2, are independent the corresponding probability measure on config- ≤ ≤ = (ℓ) planar Brownian motions with B (0) aℓ H : urations of open and closed edges. Let v be some i = ∈ = 0 R (0, ) with a1 a2. They also studied the anal- fixed vertex of G and consider the so-called perco- × ∞ 6= ogous non-intersection probabilities f (p1,...,pq) lation probability θ(p) θ(p,v0) : Pp there is = = {

402 Notices of the AMS Volume 54, Number 3 an infinite self avoiding open path starting at v0}, A connection between critical percolation and where a path is called open if all its edges are open. SLE is suggested by Schramm in [8]. Schramm If Zd or is the triangular lattice in the plane, writes that assuming a conformal invariance hy- thenG = it was shownG by Broadbent and Hammersley pothesis, the exploration process should be dis- that there is some pc (0, 1) such that θ(p) 0 tributed as SLE6. The simplest version of the ∈ = for p < pc and θ(p) > 0 for p > pc . One can exploration process for critical percolation is as also attach the randomness to the vertices instead follows: Consider percolation on the hexagonal lat- of the edges of . Then the vertices are open or tice (the dual of the triangular lattice) and let each closed with probabilityG p or 1 p, respectively. hexagonal face be independently open or closed The critical probability is then defined− as before. with probability 1/2. Assume that the origin is the The different critical probabilities are sometimes center of some hexagon and all hexagons which denoted as pc ( ,edge) and pc ( ,site). The study intersect the open negative real axis (the closed of “critical phenomena”G usuallyG refers to the study positive real axis) are open (respectively, closed). of singularities of various functions near or at pc . Then the exploration curve is a curve γ on the For instance, let be the cluster of v0, that is the boundaries of the hexagonal faces and starting on set of all points whichG can be reached by an open the boundary of the hexagon containing the origin, path starting at v0. Let denote the number of such that γ always has an open hexagon on its left |C| vertices in , and Ep its expectation. Then it is andaclosedhexagononitsright. C |C| γ believed that Ep (pc p)− as p pc . Also, |C| ≈ − ↑ at criticality, Ppc n is expected to behave Werner’s Work: Power Laws and Critical ρ {|C| ≥ } like n− for some ρ. Constants like γ and ρ are Exponents for Percolation called critical exponents. Physicists not only be- Many power laws had been conjectured in the lieve that these exponents exist, but also that they physics literature for the asymptotic behavior of are universal, that is, that they depend only on functions at or near the critical point for percola- some simple characteristic of the graph , such tion. In fact, for a two-dimensional lattice even G as the dimension. For instance the exponent γ the exact value ofG the corresponding exponents is believed to be the same for the edge and site had been predicted. In the two articles [6], [10] problems on Z2 and the edge and site problem Werner and co-authors established most of these on the triangular lattice. This is not the case for power laws and rigorously confirmed the predicted the critical probabilities which usually vary with values for the critical exponents in the case of site . In order to explain this universality physicists percolation on the triangular lattice. Before this, G proposedthe“renormalizationgroup”,butthishas no approach which mathematicians deemed rig- not been made rigorous in the case of percolation. orous existed for these problems. Even now, the However, this led people to consider whether there results are restricted to percolation on the triangu- existed a scaling limit for critical percolation. More lar lattice because this is the only lattice for which specifically, if one considers percolation at pc ( ) conformal invariance of the scaling limit has been G on δ for a periodically imbedded graph in Z2, proven. G G can one find a limit of the pattern of open paths as Previous work, partly by the present author, δ 0? A concrete question is the following: Let D reduces the problem to the following: Consider ↓ be a “nice” simply connected domain in C and let critical site percolation on δ times the triangular A and B be two arcs in ∂D. Let Pδ(D, A, B) be the lattice; this is percolation at p 1/2 for this lattice. probability that in critical percolation on δ there Let b (δ, r, R) be the probability= that there exist j G j is an open connection in D between A and B. Does disjoint crossings of the annulus r z < R . { ≤ | | } P0(D, A, B) : limδ 0 Pδ(D, A, B) exist and what is Here a crossing may be either an open crossing = ↓ its value? Here too, it was conjectured that there or a closed crossing, but for j 2 we require ≥ would be conformal invariance in the limit. For the in bj that among the j crossings there are some question at hand that would mean that if f : D D′ open ones and some closed ones. Then show → is a conformal homeomorphism from D onto a do- that bj (r, R) limδ 0 bj (δ, r, R) exists and that =1 ↓ 2 main D′, then P0(D, A, B) P0(f (D), f (A), f (B)). limR [log R]− log bj (r, R) (j 1)/12 if = →∞ Using conformal invariance and some arguments j 2, and equals 5/48 if j 1.= An − analogous− half- ≥ = whichhavenotbeenmaderigorous,JohnCardy([2]) plane problem is to set aj (δ, r, R) the probability = even found explicit values for P0(D, A, B). Cardy that there exist j disjoint open crossings of the typically took the upper half plane for D and his semi-annulus z H : r z

March 2007 Notices of the AMS 403 that one cannot single out such a first crossing of [7] , Conformal invariance of loop-erased ran-

the full annulus is also the reason why in bj with dom walks and uniform spanning trees, Ann. j 2 one has to require the existence of both open Probab. 32 (2004), 939–995. ≥ O. Schramm and closed crossings, but in a they can be all of [8] , Scaling limits of loop-erased random j walks and uniform spanning trees, Israel J. Math. the same type (or, in fact, any mixture of types). 118 (2000) 221–288. The proofs of these limit relations in [6], [10] are [9] S. Smirnov, Critical percolation in the plane: based on the result of [9] that the scaling limit for conformal invariance, Cardy’s formula, scaling lim- the exploration process in critical percolation on its, C. R. Acad. Sci. Paris Sér. I 333 (2001),

the triangular lattice is an SLE6 curve. Actually, one 239–244; a longer version is available from needs convergence in a stronger sense than given http://www.math.kth.se/ stas/papers/. S. Smirnov W. Werner in [9], but this convergence has now been provided [10] and , Critical exponents for two-dimensional percolation, Math. Res. Lett. 8 in [1]. The limit relation for percolation is then (2002), 729–744 reduced to finding certain crossing probabilities [11] W. Werner, Random Planar Curves and Schramm- for SLE6 which already had been tackled in [4], Loewner Evolutions, pp. 107–195 in Lectures on [5]. Probability Theory and Statistics, Lecture Notes in Over the years the proofs have been simplified Math., vol. 1840, (J. Picard, ed.) Springer, 2004. quite a bit. [7] gives a general outline how to prove [12] , Conformal restriction and related ques- tions, Probab. Surveys 2 (2005), 145–190. that agivenprocesshas SLEκ as a scaling limit. One modelfor which this is still an open problem is self avoiding walk. It is believed that two-dimensional

self avoiding walk has SLE8/3 as scaling limit. An indication for this is the fact that SLE8/3 is the on- ly SLEκ process which has the so-called restriction property. This property relates SLE conditioned to stay in D with the same SLE process conditioned to

stay in D′, for a domain D and a subdomain D′ (see [12]). Now that the power of SLE to rigorously ana- lyze some models motivated by physics has been demonstrated, the properties of SLE are also being developedfortheirownsake(see[3]). ThereisalistofopenproblemsinSection11.2of Werner’s St. Flour Lecture Notes [11]. We may add to this that many of the problems mentioned here can also be formulated in dimension > 2 and the two- dimensional tools of conformal invariance and SLE areprobablyoflittlehelpthere. Acknowledgement: I thank Geoffrey Grimmett for helpfulcomments.

References [1] F. Camia and C. M. Newman, Critical percolation exploration path and SLE6: a proof of convergence, arXiv:math.PR/0604487 2006. [2] J. L. Cardy, Critical percolation in finite geome- tries, J. Physics A 25 (1992), L201-L206; see also lecture notes at arXiv:math-ph/0103018. [3] G. F. Lawler, Conformally Invariant Processes in the Plane, Math. Surveys and Monographs, vol. 114, Amer. Math. Soc., 2005. [4] G. Lawler, O. Schramm, and W. Werner, Values of Brownian intersection exponents, I: Half-plane exponents and II: Plane exponents, Acta Math. 187 (2001), 237–273 and 275–308. [5] , Analiticity of intersection exponents for planar Brownian motion, Acta Math. 189 (2002), 179–201. [6] , One-arm exponent for critical 2D per- colation, Electronic J. Probab. 7 (2002), paper #2.

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