Contents

Clay Mathematics Institute 2007 James A. Carlson Letter from the President 2

Annual Meeting Clay Research Conference 3

Recognizing Achievement Clay Research Awards 6

Researchers, Workshops Summary of 2007 Research Activities 8 & Conferences

Profile Interview with Research Fellow Mircea Mustata 11

Events Julia Robinson and Hilbert’s Tenth Problem: Conference and Film 15

Summer School Homogeneous Flows, Moduli Spaces and Arithmetic, Pisa, Italy 22

Program Overview Clay Lectures on Mathematics at the Tata Institute of Fundamental Research 25

Institute News George Mackey Library 28 2007 Olympiad Scholar Andrew Geng

Publications Selected Articles by Research Fellows 29

Books & Videos 30

Activities 2008 Institute Calendar 32

2007 1 Ben Green and Terry Tao that there exist arbitrarily long arithmetic progressions in the primes. Our era is indeed a golden one for mathematics!

In 2006, CMI inaugurated the Clay Lectures in Mathematics, a series of talks by former Clay Research Fellows on topics of current interest. Ben Green and Akshay Venkatesh delivered the first series at Cambridge University in November of 2006. Elon Lindenstrauss and Mircea Mustata delivered the Letter from the president second series in December 2007 at the Tata Institute for Fundamental Research in Mumbai.

James Carlson I am pleased to announce the formation of an editorial board for the Clay Mathematics Institute Dear Friends of Mathematics, Monograph series, published with the American Mathematical Society. The Editors in Chief In 2007 the Clay Mathematics Institute made major for the series are Simon Donaldson and Andrew changes in the format and content of its Annual Wiles. I will serve as managing editor, and there is Meeting. The date was changed from November to a distinguished board of associate editors (see www. May, and the program was expanded to a two-day claymath.org/monographs). The third volume in the series of lectures on recent research developments. series, Ricci Flow and the Poincaré Conjecture, by Presentation of the Clay Research Awards, which John Morgan and Gang Tian, appeared in August, had always been the focus of the Annual Meeting, 2007. The series publishes selected expositions of took place on the afternoon of the first day. The 2007 recent developments, both in emerging areas and Research Conference was held at Harvard on May in older subjects transformed by new insights or 14 and 15. The 2008 Conference will be held at MIT unifying ideas. CMI takes great care in the editing on May 12 and 13. As in the past, the conference and presentation of the final manuscript and in will alternate between MIT and Harvard. Videos of supporting a well-produced book. Authors with a the 2007 lectures are posted on the CMI website, and project in mind are encouraged to contact an editor. future talks will appear there as well. In closing, I would like to draw attention to The inaugural Clay Research Conference marked CMI’s program of workshops held in Cambridge, an exceptional year for mathematics. Christopher Massachusetts. Ten have been held to date, and Hacon and James McKernan received the Clay we generally schedule four to five per year. Our Research Award for their inductive proof of the workshops are intended to be small, informal, and existence of flips for algebraic varieties in dimension structured in whatever way the organizers deem greater than three. This is a crucial step towards best. CMI initiates workshops on its own and also realization of the Mori minimal model program in seeks proposals. Proposals should be short, and the all dimensions, which is one of the great outstanding same is true of the lead-time between proposal and problems in algebraic geometry. Michael Harris and workshop. Richard Taylor received the award for their proof of the Sato–Tate conjecture for elliptic curves with Sincerely, non-integral j-invariants. And Alex Eskin received the award for his proof of the quasi-isometric rigidity of sol, a longstanding problem in geometric group theory. These results follow closely other recent exceptional events in mathematics, e.g., Perelman’s solution of the Poincaré conjecture and the proof by James A. Carlson President

2 CMI ANNUAL REPORT Annual meeting

Clay Research Conference

Barron, culminating in the solution of the Sato-Tate conjecture for elliptic curves with non-integral j-invariants.

The inaugural Clay Research Conference Conference speakers were Peter Ozsváth, was held at Harvard University on May 14 and Holomorphic disks and knot invariants; William 15 at the Science Center. The lectures covered a Thurston, What is the future for 3-dimensional wide range of fields: topology, algebraic geometry, geometry and topology?; Shigefumi Mori, Recent number theory, real and complex dynamics, and progress in higher dimensional algebraic geometry geometric group theory. The Clay Research Awards I; Alessio Corti, Recent progress in higher (see sidebar and following article) were presented dimensional algebraic geometry II; Mark Kisin, on the afternoon of May 14. Awardees were: Modularity of 2-dimensional Galois representations; Richard Taylor, The Sato-Tate conjecture; Curtis Alex Eskin (University of Chicago) for his work on McMullen, Algebraic dynamics on surfaces; Alex rational billiards and geometric group theory, in Eskin, Dynamics of rational billiards; and David particular, his crucial contribution to joint work with Fisher, Coarse differentiation and quasi-isometries David Fisher and Kevin Whyte in establishing the of solvable groups. quasi-isometric rigidity of sol. Christopher Hacon (University of Utah) and James McKernan (UC Santa Barbara) for their work in advancing our understanding of the birational Clay Research Awards geometry of algebraic varieties in dimension greater Previous recipients of the award, in reverse chronological order are: than three, in particular, for their inductive proof of 2005 Manjul Bhargava (Princeton University) the existence of flips. Nils Dencker (Lund University, Sweden) Michael Harris (Université de Paris VII) and Richard 2004 Ben Green (Cambridge University) Gérard Laumon (Université de Paris-Sud, Orsay) Taylor (Harvard University) for their work on local and Bao-Châu Ngô (Université de Paris-Sud, Orsay) global Galois representations, partly in collaboration 2003 Richard Hamilton (Columbia University) with Laurent Clozel and Nicholas Shepherd- Terence Tao (University of California, Los Angeles) 2002 Oded Schramm (Theory Group, Microsoft Research) Manindra Agrawal (Indian Institute of Technology, Kanpur) 2001 Edward Witten (Institute for Advanced Study) Stanislav Smirnov (Royal Institute of Technology, Stockholm) 2000 Alain Connes (College de France, IHES, Vanderbilt University) Laurent Lafforgue (Institut des Hautes Études Scientifiques) 1999 Andrew Wiles (Princeton University)

The Clay Mathematics Institute presents the Clay Research Award annually to recognize major breakthroughs in mathematical research. Awardees receive the bronze sculpture “Figureight Knot Complement vii/CMI” by Helaman Ferguson and are named Clay Research Scholars for a period of one year. As such they receive substantial, flexible research support. Awardees have used their research support to organize a conference or workshop, to bring in Landon Clay, joined by James Carlson, congratulating Christopher one or more collaborators, to travel to work with a collaborator, and Hacon (right front) and James McKernan (right rear) after they received for other endeavors. the Clay Research Award.

2007 3 Clay Research Conference Annual meeting

The recipients of the 2007 Clay Research Awards with Landon and Lavinia Clay. From left to right: Richard Taylor, Michael Harris, Lavinia Clay, Landon Clay, Alex Eskin, Christopher Hacon, and James McKernan.

Alessio Corti was unable to attend the conference CLAY RESEARCH because of security difficulties in London. Shigefumi Mori, on very short notice, graciously agreed to give Conference Corti’s lecture. David Fisher presented the work of Eskin, Fisher, and Whyte on the quasi-isometric May 14—15, 2007 Speakers rigidity of sol, a major problem in geometric group Alessio Corti, Imperial College, London HARVARD SCIENCE CENTER Alex Eskin, University of Chicago theory. Alex Eskin and Curtis McMullen spoke David Fisher, Indiana University LECTURE HALL C about problems in dynamics: rational billiards Mark Kisin, University of Chicago ONE OXFORD STREET Curt McMullen, Harvard University Shigefumi Mori, RIMS, Kyoto CAMBRIDGE, MA 021 3 8 and the dynamics on complex algebraic surfaces, Peter Ozsváth, Columbia University Richard Taylor, Harvard University respectively. These are just two of the many active inaugurates an expanded format William Thurston, Cornell University The Clay Research Conference for the Clay Mathematics Institute’s annual meeting. The conference, hosted this year areas in the field of dynamics, which has important by the Harvard University Mathematics Department, consists of a two-day series of Schedule lectures on recent research developments and presentation of the Clay Research Awards. points of contact with number theory, representation Monday, May 1 4 theory, and other areas. Peter Ozsváth and William 9:15 Coffee 9:45 Peter Ozsváth, Holomorphic disks and knot invariants Thurston spoke about the past, present, and future of 11:00 William Thurston, TBA 2:00 Clay Research Awards topology: Ozsváth on new knot invariants that have 2:30 Shigefumi Mori, Recent progress in higher dimensional algebraic geometry I 3:45 Alessio Corti, Recent progress in higher dimensional algebraic geometry II resolved many long-standing problems in the field; 5:15 Reception at CMI Tuesday, May 15 Thurston on the geometrization conjecture and what 9:15 Coffee remains to be understood after the groundbreaking 9:45 Mark Kisin, Modularity of 2-dimensional Galois representations 11:00 Richard Taylor, The Sato-Tate conjecture work of Perelman. 1:30 Curt McMullen, Algebraic dynamics on surfaces 2:45 Alex Eskin, Dynamics of rational billiards 4:00 David Fisher, Coarse differentiation and quasi-isometries of solvable groups

www.claymath.org Videos of the lectures are available at www.claymath.org/publications/videos

CLAY MATHEMATICS INSTITUTE • ONE BOW STREET, CAMBRIDGE, MA 02138 • T. 617 995 2600 • [email protected] • WWW.CLAYMATH.ORG

4 CMI ANNUAL REPORT Annual meeting

Abstracts of Talks volume, organized; what are the deeper connections with number theory via the field of traces associated Peter Ozsváth (Columbia University) with the length of geodesic loops? Holomorphic disks and knot invariants Shigefumi Mori (RIMS, Kyoto) Heegaard Floer homology is an invariant for three- Recent progress in higher dimensional algebraic and four-manifolds defined using techniques from geometry I symplectic geometry. More specifically, a Heegaard diagram is used to set up an associated symplectic Higher dimensional algebraic geometry has manifold equipped with a pair of Lagrangian recently undergone major developments related submanifolds. The Heegaard Floer homology to the minimal model program. Mori reviewed the groups of the three-manifold are then defined as basic definitions of the minimal model program the homology groups of a chain complex whose and surveyed some of the recent achievements differential counts pseudo-holomorphic disks in the and their applications. symplectic manifold. These methods also lead to invariants for knots, links, and four-manifolds. Alessio Corti (Imperial College, London) Recent progress in higher dimensional algebraic Ozsváth discussed applications of this theory, geometry II along with some recent calculational advances Corti explained some of the key ideas in the recent that have rendered the knot Floer homology work of Hacon and McKernan on the higher groups purely combinatorial. Heegaard Floer dimensional minimal model program for algebraic homology was defined in joint work with varieties. Zoltan Szabo, in addition to further work with other collaborators, including Ciprian Manolescu, Mark Kisin (University of Chicago) Sucharit Sarkar, and Dylan Thurston. Modularity of 2-dimensional Galois representations William Thurston (Cornell University) Kisin discussed the recently proved conjecture of What is the future for 3-dimensional geometry and Serre on 2-dimensional mod p Galois representations, topology? and its implications for modularity of 2-dimensional motives and p-adic Galois representations. Though the geometrization conjecture is now proved, there is still much to be understood in achieving a Richard Taylor (Harvard University) simple big picture of three-manifold topology, e.g., The Sato-Tate conjecture how is the set of hyperbolic manifolds, ordered by A fixed elliptic curve over the rational numbers is known to have approximately p points modulo p for any prime number p. In about 1960, Sato and Tate gave a conjectural distribution for the error term. Laurent Clozel, Michael Harris, Nicholas Shepherd- Barron and Taylor recently proved this conjecture in the case that the elliptic curve has somewhere multiplicative reduction.

Curtis McMullen (Harvard University) Algebraic dynamics on surfaces McMullen discussed the role of Hodge theory, Salem numbers, and Coxeter groups in the construction of new dynamical systems on compact complex surfaces.

William Thurston delivering his talk at the conference. 2007 5 Clay Research Conference Clay Research Awards

Alex Eskin, (University of Chicago) The Clay Research Awards surface X and a point p on it. One may replace Dynamics of rational billiards the point by the set of tangent lines through p to Below is a brief account of the mathematics of the obtain a new surface Y . The set of tangent lines is Eskin called his talk “a short and extremely biased work for which each of the three Clay Research an algebraic curve E isomophic to one-dimensional Awards were given. – jc projective space CP1. Since X p and Y E survey of recent developments in the study of rational − { } − billiards and Teichmüller dynamics.” are isomorphic dense open sets in X and Y , re- spectively, the latter two varieties have isomorphic David Fisher (Indiana University) 1. Minimal Models in Algebraic Geometry function fields. In algebraic geometry we say that Coarse differentiation and quasi-isometries of Y is obtained from X by blowing up p. In more Let X be projective algebraic variety over the com- solvable groups topological language, we say that Y is obtained plex numbers, that is, the set of common zeroes from X by surgery: cut out the point p, and glue Annual meeting In the early 1980s Gromov initiated a program to of a system of homogeneous polynomial equations. in the projective line E. What is important here is study finitely generated groups up to quasi-isometry. The meromorphic functions on X form a field, the that the surgery is an operation on algebraic vari- function field of X. For the Riemann sphere (the eties. This program was motivated by rigidity properties 1 of lattices in Lie groups. A lattice T in a group G projective line CP ) this field is C(t), the field of rational functions in one variable. For an elliptic More generally, we can (and will) consider surgeries is a discrete subgroup where the quotient G/T has curve y2 = x3 + ax + b, it is the field obtained by of the form “cut out a subvariety A and paste in a finite volume. Gromov’s own major theorem in this adjoining the algebraic function y = √x3 + ax + b variety B.” More formally, we have varieties X and direction is a rigidity result for lattices in nilpotent Y such that X A is isomorphic to Y B, where to C(x). Two varieties are birationally equivalent − − Lie groups. if they have isomorphic function fields. we say that Y is obtained from X by surgery. Since X A and Y B are dense open sets, the function In the 1990s, a series of dramatic results led to the The birational equivalence problem is a fundamen- fields− of X and− Y are isomorphic. The partially completion of the Gromov program for lattices in tal one in algebraic geometry. Given two varieties defined map X  Y induces the isomorphism of semisimple Lie groups. The next natural class of X and Y , how do we recognize whether they are bi- function fields. examples to consider are lattices in solvable Lie rationally equivalent? In the case of elliptic curves, groups, and even results for the simplest examples there is an easy answer: the fields are isomorphic if The curve E obtained by blowing up p is a pro- 2 3 jective line with self-intersection number 1. Such were elusive for a considerable time. Fisher’s joint and only if the quantity b /a is the same in both − work with Eskin and Whyte in which they proved cases. What can we say about other varieties? On curves are known in the trade as “( 1) curves.” Any time one finds a ( 1) curve on− a surface Y , the first results on quasi-isometric classification what data does the birational equivalence class of − of lattices in solvable Lie groups was discussed. a variety depend? one can construct a smooth surface X and a map f : Y X that maps E to a point. This opera- The results were proven by a method of coarse Consider first the case of complex dimension one. tion is−→ called “blowing down,” or “contracting E.” differentiation, which was outlined. Every algebraic curve is birational to a smooth one, By successively contracting all the ( 1) curves in − Some interesting results concerning groups quasi- its normalization. Thus two curves are birational sight, one can construct from any algebraic surface if and only if their smooth models are isomorphic. isometric to homogeneous graphs that follow from S a smooth variety Smin devoid of such curves. Consequently, the birational equivalence problem the same methods will also be described. Let us call Smin a classical minimal model for S. is the same as the moduli problem. Take, for exam- Existence of classical minimal models was proved Satellite Workshop at the Clay Mathematics Insitute ple, the algebraic curves defined by the affine equa- by Castelnuovo and Enriques in 1901. They also tions x+y = 1, x2+y2 = 1 and x2+y2+x3+y3 = 0. May 16–17 showed that as long as S and S are not uniruled, The first two curves are smooth and isomorphic to they are birational if and only if Smin and Smin A satellite workshop held at the Clay Mathematics the Riemann sphere, as one sees by stereographic are isomorphic. In the non-uniruled case a classical minimal model X is topologically the simplest: Institute in the days following the Conference projection. The last curve has one singular point, min its second Betti number is smaller than that of any consisted of more detailed talks on recent progress but its smooth model is the Riemann sphere, as we smooth surface birationally equivalent to it. in higher dimensional algebraic geometry. On this see by the parametrization 1 occasion, Christopher Hacon and James McKernan x = (1 + t2)/(1 + t3),y= t(1 + t2)/(1 + t3). A variety X is uniruled if there is a map CP − − Y X whose image contains an open dense set.× spoke on the existence of flips and MMP scaling to −→ an audience of advanced graduate students in the Thus all three varieties are birationally equivalent, Thus, there is a curve birational to a projective line with function field C(t). field. passing through almost every point of X. Varieties of higher dimension are birationally equiv- A ruled surface, that is, a CP1 bundle over a curve, alent to a smooth one by Hironaka’s resolution of is uniruled. So is CP2. For uniruled surfaces, the singularities theorem. Nonetheless, this powerful minimal model is not unique. For example, blow up result does not answer the birational equivalence two points a and b on CP2. The proper transform problem. To see why, consider a smooth algebraic

6 CMI ANNUAL REPORT Recognizing achievement

continued on page 17

The Clay Research Awards surface X and a point p on it. One may replace of the line joining them is a ( 1) curve. Blow it The technical issues thus raised can all be success- the point by the set of tangent lines through p to down to obtain a new surface.− It is isomorphic fully dealt with; indeed, working in the larger cat- Below is a brief account of the mathematics of the obtain a new surface Y . The set of tangent lines is to CP1 CP1. Both CP2 and CP1 CP1 are egory of varieties with terminal singularities is cru- work for which each of the three Clay Research an algebraic curve E isomophic to one-dimensional classically× minimal, and both represent× the purely cial to the success of the minimal model program. Awards were given. – jc projective space CP1. Since X p and Y E transcendental function field C(x, y). Another cost associated with flips is the difficulty are isomorphic dense open sets− in{X} and Y ,− re- What can one say in dimension greater than two? of proving that they exist in sufficient generality. spectively, the latter two varieties have isomorphic The conjecture of Mori-Reid (see [4]) states the fol- One can construct motivating examples (see be- 1. Minimal Models in Algebraic Geometry function fields. In algebraic geometry we say that lowing: low), but even these are somewhat complicated. Y is obtained from X by blowing up p. In more Important special cases were proved by Tsunoda, Let X be projective algebraic variety over the com- topological language, we say that Y is obtained Shokurov, Mori, and Kawamata. Finally, Mori plex numbers, that is, the set of common zeroes from X by surgery: cut out the point p, and glue proved the general existence theorem for flips in of a system of homogeneous polynomial equations. in the projective line E. What is important here is ( ) Let X be an algebraic variety of ∗ dimension three [8], [7, p. 268]. The Clay Re- The meromorphic functions on X form a field, the that the surgery is an operation on algebraic vari- dimension n which is not uniruled. search Award was given to Hacon and McKernan function field of X. For the Riemann sphere (the eties. Then (a) it has a minimal model 1 for their proof of the existence of flips in dimen- projective line CP ) this field is C(t), the field of X and (b) it has a K¨ahlermet- More generally, we can (and will) consider surgeries min sion n assuming termination of flips in dimension rational functions in one variable. For an elliptic ric whose Ricci curvature is 0. 2 3 of the form “cut out a subvariety A and paste in a ≤ n 1. See [3]. This result suggests that one can curve y = x + ax + b, it is the field obtained by − adjoining the algebraic function y = √x3 + ax + b variety B.” More formally, we have varieties X and prove the existence of minimal models inductively, to C(x). Two varieties are birationally equivalent Y such that X A is isomorphic to Y B, where a program carried out in [1]. we say that Y is− obtained from X by surgery.− Since In the Mori-Reid conjecture, minimality is defined if they have isomorphic function fields. Let us now discuss minimality and flips in a more X A and Y B are dense open sets, the function in a different way, as a kind of algebro-geometric substantitve way. To say that X is minimal is to The birational equivalence problem is a fundamen- fields− of X and− Y are isomorphic. The partially positivity condition. We will discuss this notion in say that its canonical bundle is numerically effec- tal one in algebraic geometry. Given two varieties defined map X Y induces the isomorphism of greater detail below. For surfaces, it coincides with  tive, or nef. The canonical bundle is the line bundle X and Y , how do we recognize whether they are bi- function fields. the classical one: there are no ( 1) curves. For rationally equivalent? In the case of elliptic curves, higher dimensional varieties, minimality− as posi- K whose local sections are holomorphic n-forms A there is an easy answer: the fields are isomorphic if The curve E obtained by blowing up p is a pro- tivity signaled a major change in the way mathe- line bundle L is nef if the integral 2 3 jective line with self-intersection number 1. Such and only if the quantity b /a is the same in both − maticians viewed the birational equivalance prob- cases. What can we say about other varieties? On curves are known in the trade as “( 1) curves.” L C = ω − lem. The new line of investigation, initiated by · C what data does the birational equivalence class of Any time one finds a ( 1) curve on a surface Y , Shigefumi Mori, developed further by Kawamata, − is positive for all algebraic curves C on X, where a variety depend? one can construct a smooth surface X and a map Koll´ar,Mori, Reid, and Shokurov, culminated in ω is a differential form representing the first Chern f : Y X that maps E to a point. This opera- 1988 with Mori’s proof of (a) for varieties of di- Consider first the case of complex dimension one. −→ class of L. This integral represents the intersec- tion is called “blowing down,” or “contracting E.” mension three [8]. For this result, the goal of the Every algebraic curve is birational to a smooth one, tion number, which, as noted, may be a rational By successively contracting all the ( 1) curves in “minimal model program,” Mori received a Fields its normalization. Thus two curves are birational sight, one can construct from any algebraic− surface number. if and only if their smooth models are isomorphic. Medal in 1990. Although refereeing is still in pro- S a smooth variety Smin devoid of such curves. What is the relation of the new definition of mini- Consequently, the birational equivalence problem cess, it now appears that (a) is also a theorem for Let us call Smin a classical minimal model for S. mality to that of the Castelnuovo-Enriques theory? is the same as the moduli problem. Take, for exam- all dimensions. An algebraic approach has been Existence of classical minimal models was proved c By the adjunction formula, the value of the above ple, the algebraic curves defined by the affine equa- given by Birkar, Cascini, Hacon, and M Kernan by Castelnuovo and Enriques in 1901. They also [1] and an analytic approach has been give by Siu integral, i.e., the intersection number K C, is non- tions x+y = 1, x2+y2 = 1 and x2+y2+x3+y3 = 0. · showed that as long as S and S are not uniruled, [9]. negative. Thus a variety of dimension two whose The first two curves are smooth and isomorphic to they are birational if and only if Smin and Smin canonical class is nef has no ( 1) curves. Conse- In the remainder of this article we explain the mod- the Riemann sphere, as one sees by stereographic are isomorphic. In the non-uniruled case a classical quently, surfaces that are minimal− are classically ern notion of minimality and how it relates to the projection. The last curve has one singular point, minimal model Xmin is topologically the simplest: minimal. but its smooth model is the Riemann sphere, as we its second Betti number is smaller than that of any classical one. We then touch on just one of the cru- Following [7, Example 12.1], we give an example, see by the parametrization smooth surface birationally equivalent to it. cial parts of the proof. This is the existence of flips 1 and flops. These are surgeries that alter a variety in first observed by Atiyah, of a flop. We then modify x = (1 + t2)/(1 + t3),y= t(1 + t2)/(1 + t3). A variety X is uniruled if there is a map CP codimension two. Flops leave the positivity of the the example to give a flip. Consider the affine vari- − − Y X whose image contains an open dense set.× −→ canonical bundle unchanged, whereas flips make it ety C0 given by the quadratic equation xy uv = Thus all three varieties are birationally equivalent, 4 − Thus, there is a curve birational to a projective line it more positive, transforming the variety to one 0. Let C be its closure in CP ; it is the cone with function field C(t). passing through almost every point of X. that is closer to minimal. One cost of introduc- over a quadric CP1 CP1. Blow up the vertex 1 × Varieties of higher dimension are birationally equiv- A ruled surface, that is, a CP bundle over a curve, ing a flip is that certain mild singularities must be of the cone to obtain a variety C12. The result- 2 alent to a smooth one by Hironaka’s resolution of is uniruled. So is CP . For uniruled surfaces, the admitted. These are the so-called “terminal singu- ing exceptional set F = F1 F2 is isomorphic to 1 1 × singularities theorem. Nonetheless, this powerful minimal model is not unique. For example, blow up larities.” A consequence of working with singular CP CP . It is possible to blow down all the 2 × result does not answer the birational equivalence two points a and b on CP . The proper transform varieties is that the natural intersection numbers, fibers x F2 of F to obtain a variety C1 with { } × 1 problem. To see why, consider a smooth algebraic while well-defined, can be rational numbers. a subvariety E1 ∼= F1 ∼= CP . Likewise, we can

2007 7 Summary of 2007 Research Activities

The activities of CMI researchers and research programs are sketched below. Researchers and programs are selected by the Scientific Advisory Board (see inside back cover).

Researchers Clay Research Fellows Mohammed Abouzaid began his five-year Clay Research Fellow Mohammed Abouzaid appointment in July 2007. He received his Ph.D. from the University of Chicago where he worked Andras Stipsicz (Alfréd Rényi Institute of under the direction of Paul Seidel. Abouzaid is Mathematics, Budapest). September 1–December currently a postdoctoral fellow at MIT. 31, 2007 at Columbia University. Soren Galatius, a native of Denmark, graduated from Senior Scholars the University of Aarhus in 2004, where he worked under Ib Madsen. He is currently Assistant Professor Alex Eskin (University of Chicago). August 20– of mathematics at Stanford University. He has a three- December 14, 2007. MSRI Program on Teichmüller year appointment that began in September 2007. Theory and Kleinian Groups. Davesh Maulik recently completed his doctorate in Gerhard Huisken (Max Planck Institute, Potsdam, mathematics at Princeton University. He began his Germany). March 12­–22, 2007. MSRI Program on five-year appointment in July 2007 at Columbia Geometric Evolution Equations. University. Peng Lu (University of Oregon). January 8–March 30, 2007. MSRI Program on Geometric Evolution Teruyoshi Yoshida is a graduate of Harvard Equations. University, where he is currently conducting his work in algebraic number theory as a member of the Andrei Okounkov (Princeton University). July 1–21, Society of Fellows. Yoshida began his three-year 2007. PCMI Program on Statistical Mechanics. appointment in December 2007. Abouzaid, Galatius, Maulik and Yoshida joined CMI’s current group of research fellows Artur Avila (IMPA Brazil), Daniel Biss (University of Chicago), Maria Chudnovsky (Columbia University), Bo’az Klartag (Princeton University), Ciprian Manolescu (Columbia University), Maryam Mirzakhani (Princeton University), Sophie Morel (Institute for Advanced Study), Samuel Payne (Stanford University), and David Speyer (MIT). Research Scholars Mihalis Dafermos (DPMMS, University of Cambridge). December 31, 2006–December 30, 2007 at University of Cambridge. Dipendra Prasad (TIFR, Mumbai). September 24, 2007–June 13, 2008 at University of California, San Diego. Clay Research Fellow Soren Galatius 8 CMI ANNUAL REPORT Workshops & conferences

Gang Tian (Princeton University). January 1– March March 19–23. Motives and Algebraic Cycles: A 31, 2007. MSRI Program on Geometric Evolution Conference dedicated to the Mathematical Heritage Equations. of Spencer J. Bloch at the Fields Institute. Srinivasa Varadhan (Courant Institute at NYU). April 2–7. Noncommutative Geometry at IHES, July 1–21, 2007. PCMI Program on Statistical Bur-sur-Yvette. Mechanics. April 16–20. Minimal and Canonical Models at Liftoff Fellows MSRI in Berkeley, California. April 20–22. Workshop on Symplectic Topology at CMI appointed seventeen Liftoff Fellows for the CMI. summer of 2007. Clay Liftoff Fellows are recent Ph.D. recipients who receive one month of summer April 29–May 5. Advances in Algebra and Geometry salary and travel funds before their first academic Conference at MSRI in Berkeley, California. position. See www.claymath.org/liftoff May 14–15. Clay Research Conference, Harvard Dmitriy Boyarchenko University Science Center, Cambridge. Hans Christianson Pavlo Pylyavskyy June 2. Developments in Algebraic Geometry Jason DeBlois Shahab Shahibi Conference in Honor of David Mumford at Brown Adrian Ioana Sug Woo Shin University. Anthony Licata Jeffrey Streets June 7–11. Geometry and Imagination Conference Grace Lyo Junecue Suh at Princeton University. Joshua Sussan Elizabeth Meckes June 7–20. Summer School on Serre’s Modularity Yi Ni Ben Webster Conjecture at CIRM in Marseille, France. Jeehoon Park Josephine Yu June 11–July 6. CMI 2007 Summer School on Research Programs Organized and Supported Homogeneous Flows, Moduli Spaces and Arithmetic by CMI in Pisa, Italy.

January, Spring Semester. Semester-Long Program in Symplectic Topology at MIT. January 8–27. School and Workshop in the Geometry and Topology of Singularities in Cuernavaca, Mexico. January 29–Feb 2. Diophantine and Analytic Problems in Number Theory Conference at Moscow Lomonosov University. January 1–April 30. Homological Mirror Symmetry and Applications Conference at IAS. March 5–9. Workshop on Hopf Algebras and Props at CMI. March 18–21. Dynamics Workshop in Honor of Hillel Furstenberg at the University of Maryland. March 15–16. Conference on Hilbert’s 10th Problem at CMI, including a preview screening of George Csicsery’s film on Julia Robinson at the Museum of Bjorn Poonen presenting his talk “Speculations about Rational Curves Science, Boston. on Varieties over Countable Fields” at CMI’s Rational Curves Workshop.

2007 9 Summary of 2007 Research Activities

July 23–27. Infinite Dimensional Algebras and Quantum Integrable Systems II Conference at the University of the Algarve in Faro, Portugal. July 30–August 3. Conference On Certain L-Functions at Purdue University. August 12–17. IV IberoAmerican Conference on Complex Geometry at Centro Metalurgico, Ouro Preto in Minas Gerais, Brazil.

Workshops & conferences August 27–31. Colloque JPB at IHES and L’Ecole Polytechnique in Paris. September 18–21. Solvability and Spectral Instability at CMI. September 30. SAGE Days 5: Computational Kirsten Eisentraeger (Penn State) discussing “Hilbert’s Tenth Problem for Function Fields of Varieties over C” at CMI’s Rational Curves Workshop. Arithmetic Geometry at CMI. November 2–4. Workshop on Geometry of Moduli Program Allocation Spaces of Rational Curves with Applications to Estimated number of persons supported by CMI in Deophantine Problems over Function Fields at selected scientific programs for calendar year 2007: CMI. December 11–14. Clay Lecture Series at the Tata Research Fellows, Research Awardees, Institute of Fundamental Research (TIFR) in Senior Scholars, Research Scholars, Mumbai, India. Book Fellows, Liftoff Fellows 45 Summer School participants and faculty 110 Student Programs, participants and faculty 140 CMI Workshops 95 Participants attending joint programs and the Independent University of Moscow > 1000

IRS Qualifying Charitable Expenses of CMI Since Inception IRSIRS Qualifying Qualifying Charitable Charitable Expenses Expenses of CMIof CMI Since Since Inception Inception

4.0 4.0 4.0 3.5 3.5 3.5 3.0 3.0 3.0 2.5 2.5 2.5 2.0 2.0 2.0 1.5 1.5 1.5 1.0 1.0 1.0 0.5 Izzet Coskun (MIT) presenting his talk on“Vanishing of Quantum 0.5 0.5 Cohomology” at CMI’s Rational Curves Workshop. 0.0 0.0 0.01990 20001990 20012000 20022001 20032002 20042003 20052004 20062005 20072006 2007 1990 2000 2001 2002 2003 2004 2005 2006 2007

Total through September 30, 2007: Over $28 million TotalTotal through through September September 30, 30,2007: 2007: Over Over $28 $28 million million

10 CMI ANNUAL REPORT Profile Interview with Research Fellow Mircea Mustata

James Carlson and Mircea Mustata at the Tata Institute of Fundamental Research in Mumbai where CMI’s 2007“Clay Lectures on Mathematics” took place.

What first drew you to mathematics? What are some math, and that this was not something completely of your earliest memories of mathematics? out of ordinary. On the downside, I was never really good at these competitions, and at some point this got I am afraid that I don’t have any math related a bit frustrating. With hindsight, I think I shouldn’t memories from my early childhood. I began to show have spent this much time just with the olympiads, an interest in mathematics in elementary school, though this is what kept me being interested in math around the sixth grade. At that point more challenging all through high school. IRS Qualifying Charitable Expenses of CMI Since Inception problems started to come up, and one started doing rigorous proofs in plane Euclidean geometry. I was Did you have a mentor? Who helped you develop 4.0 reasonably good at it, and there were interesting your interest in mathematics, and how? 3.5 problems around, so I enjoyed doing it. I don’t think that I had a real mentor while growing- 3.0 Could you talk about your mathematical education? up, though there have always been people around 2.5 What experiences and people were especially who influenced me. A key role was a tutor I had in 2.0 influential? the last grade in high school. That year I failed in 1.5 one of the early stages of the math olympiad, and 1.0 Maybe the turning point was when I started going to to cheer me up he gave me some books to read: 0.5 math olympiads, and to the circles organized around some general topology, some real and complex 0.0 1990 2000 2001 2002 2003 2004 2005 2006 2007 these contests. The main upshot was being around analysis. This was my first encounter with serious other kids who were enthusiastic about math. I then mathematics, and it was an exciting experience. It realized that I enjoyed spending my spare time doing certainly made the transition from high school to Total through September 30, 2007: Over $28 million college much smoother.

2007 11 Interview with Research Fellow Mircea Mustata

It would have been good to have a mentor during the working on a particular problem. Most of the time, first college years, but unfortunately I didn’t have one. it is because it relates to other things I have thought It would have helped about before. Of course, it helps One thing that I think is good in Romania is that in getting a better if I believe that it is a problem kids encounter mathematical reasoning at an earlier view of mathematics other people would care about. stage (through axiomatic Euclidean geometry, for as a whole, and it In general, I find that problems example). Another feature of the Romanian education might have motivated that seem to have connections is that study during college is very focussed: with a me to look at certain with other fields are particularly couple of minor exceptions, all courses I have taken directions that I didn’t appealing. At least, they motivate were in math... know they existed me to try to learn some new Profile until much later. things (though I have to say that in many cases it turned out that the new things were too hard and I got stuck). You were educated in Romania. Could you comment on the differences between mathematical education Can you describe your research in accessible terms? in Romania and in the US? Does it have applications to other areas?

One thing that I think is good in Romania is that Most of my research deals with singularities of kids encounter mathematical reasoning at an earlier algebraic varieties. These varieties are geometric stage (through axiomatic Euclidean geometry, for objects defined by polynomial equations. It turns example). Another feature of the Romanian education out that in many cases of interest, these objects are is that study during college is very focussed: with a not “smooth-shaped”—they have singularities. In couple of minor exceptions, all courses I have taken fact, what makes the local structure interesting are were in math (with the embarrassing outcome that precisely these singularities. In my research, I deal I did not take any college level course in physics). with invariants that measure the singularities. Part of With so many math courses, one could get a strong the motivation for what I do comes from questions background. The downside is that one insists maybe that come up in the classification theory of algebraic a bit too much on learning a lot of material, and on varieties. Besides this, I don’t know whether this reading many books, and not really on using this topic has applications to other areas, but I believe that information. What I like in the US is that it has strong connections students can get a taste of research at a I always have a hard time saying which with various fields. In very early stage. Myself, I started working problems I will work on in six months fact, the invariants that on a problem only after I graduated from (which makes writing NSF proposals I have mentioned have University. I was at the time a master’s a bit tricky). On the other hand, I have all different origins student in Bucharest. some long-term projects that I think (in valuation theory, about on and off over the years. commutative algebra or What attracted you to the particular the theory of differential problems you have studied? operators), but they turn out to be all related in ways that are still to be In my case, chance played a bit role. For example, understood. I learned about the problem that influenced most of my work from my advisor, David Eisenbud, and from What research problems and areas are you likely to Edward Frenkel. At the time, by looking at several explore in the future? examples, they came up with a bold and unexpected conjecture. They very generously shared it with me, I always have a hard time saying which problems and this got me started in a completely new direction I will work on in six months (which makes writing from what I had been doing. NSF proposals a bit tricky). On the other hand, I have some long-term projects that I think about on In general, there are various reasons why I end up and off over the years. One problem that’s been on

12 CMI ANNUAL REPORT Profile

my mind for a while has to do with the connections Probably the most important thing was that for three between some invariants of singularities that are years I had the freedom to choose where I want to be. defined in characteristic zero using valuations (or I think that in general it made my postdoc years less equivalently, using resolutions of singularities), and stressful than they might have been: for example, other invariants in positive characteristic that came I had no teaching duties (I ended up teaching two out of an area in commutative algebra called tight courses during this time, but this was my own choice, closure theory. There is evidence that the connections and actually got me much more enthusiastic about between these invariants are quite subtle, and they teaching). And of course, the stipend being pretty have to do with the arithmetic properties of the generous, I could appreciate the improvement in my varieties involved. At this point it is not clear whether life after being a graduate student. there is a good framework for understanding these connections, but at least, this motivates me to learn a What advice would you give to young people bit of number theory. starting out in math (i.e., high school students and young researchers)? Could you comment on collaboration versus solo work as a research style? Are I think that Maybe the most rewarding experience is when you certain kinds of problems better suited to it is good to get an intuition how certain disparate pieces might fit collaboration? get involved together. I am not talking about figuring out how to in research prove a precise result, usually by the time the details I believe that collaboration has more to do early on, are cleared the enthusiasm cools down. But rather with personality than with the particular finding an about the moment when you realize that certain things problems. Most of my work was done in easy, but might be connected in a way you hadn’t expected. collaboration, and I think that in general, interesting this social aspect is a very rewarding one p r o b l e m in our work (this is probably one of the things I got to work on. On the other hand, once you figure out from my advisor). In my case, however, it is not so a field you want to work in, it might be good to much a matter of choice: I realized that I get most of allow time to get also a broader view of that field, the ideas by talking to people --even when I don’t in addition to the technical mastery required for fully understand what they are saying. On the other working on a specific problem. It is true that this hand, I enjoy also the part of the work that’s done in might be hard to put in practice nowadays because private, and I always need to take the time to think of the way the programs are structured. about a problem on my own. What advice would you give lay persons who would What do you find most rewarding or productive? like to know more about mathematics — what it is, what its role in our society has been and so on? Maybe the most rewarding experience is when you What should they read? How should they proceed? get an intuition how certain disparate pieces might A direct way of figuring things out would be by fit together. I am not talking about figuring out talking to mathematicians (though convincing them how to prove a precise result, usually by the time to discuss math with a non-mathematician might the details are cleared the enthusiasm cools down. be considered a personal achievement). The good But rather about the moment when you realize that news is that there are by now several popularization certain things might be connected in a way you books, either about famous mathematicians or about hadn’t expected. Of course, this does not happen famous mathematical problems. I believe they can very often, and sometimes this intuition is wrong, convey what “doing mathematics” means, why but it is always exhilarating, and it pays off for all certain problems are important, and sometimes the moments when I hit a dead end. they can even put forward and explain interesting mathematical concepts. How has the Clay Fellowship made a difference for you? How do you think mathematics benefits culture and society? 2007 13 Interview with Research Fellow Mircea Mustata

Like other sciences, mathematics fulfils a need for figuring out the world around us. The special place of mathematics is due to the fact that it deals with an abstract realm. It is a common misconception that because of this fact, mathematics is out of touch with reality. It is indeed true that mathematical constructions do not need to be validated by “reality” (and I personally find this aspect very appealing). However, one should keep in mind that as physics taught us in the last hundred years, even very abstract Profile models can help us to understand our own world.

Please tell us about things you enjoy when not doing mathematics.?

Whenever I have time, I enjoy hiking, reading fiction or watching movies, though my favorite pastime lately has been playing with my daughter. Another thing I enjoy a lot, which is one of the perks Mircea Mustata at TIFR, Mumbai in December of 2007. of a mathematician’s life, is traveling. I always dreamt about traveling when I was a kid, but until my final years in grad school, I didn’t realize that Recent Research Articles this was the way to go. “Invariants of singularities of pairs,” with Lawrence Ein, International Congress of Mathematicians, Vol. Mircea Mustata, a native of Rumania, finished II, 583–602, Eur. Math. Soc., Zurich, 2006. the Ph.D program at University of California, Berkeley under the direction of David Eisenbud. Immediately “Asymptotic invariants of base loci,” with Lawrence afterwards he began his position as a Clay Research Ein, Robert Lazarsfeld, Michael Nakamaye, and Fellow. He held this position from July 2001, to August Mihnea Popa, Ann. Inst. Fourier (Grenoble) 56 2004. During his time he visited Université de Nice, (2006), 1701–1734. the Isaac Newton Institute (Cambridge), and Harvard University. In Septermber 2004, Mustata became “Bernstein-Sato polynomials of arbitrary varieties,” Associate Professor of Mathematics at the University with Nero Budur and Morihiko Saito, Compos. of Michigan. His reasearch is supported by the NSF Math. 142 (2006), 779–797. and a Packard Fellowship.

His main research interest is in algebraic geometry, “Ehrhart polynomials and stringy Betti numbers,” in particular in various invariants of singularities of with Sam Payne, Math. Ann. 333 (2005), 787–795. algebraic varieties, such as minimal log discrepancies, log canonical thresholds, multiplier ideals, Bernstein- “F-thresholds and Bernstein-Sato polynomials,”with Sato polynomials or F-thresholds. Various points Shunsuke Takagi and Kei-ichi Watanabe, European of view and techniques come in the picture when Congress of Mathematics, 341–364, Eur. Math. studying these invariants: resolutions of singularities, Soc., Zurich, 2005. jet schemes, D-modules or positive characteristic methods. Other interests include birational geometry, “Inversion of adjunction for local complete inter- asymptotic base loci and invariants of divisors, and section varieties,” with Lawrence Ein, Amer. J. toric varieties. Math. 126 (2004), 1355–1365.

14 CMI ANNUAL REPORT Events Julia Robinson and Hilbert’s Tenth Problem: Conference and Film

in modern cryptography: elliptic curves help keep your credit card data safe. In 1957, Selmer showed that the second important equation has no integer solutions.

Hilbert, in posing his Tenth Problem, asked whether it was possible “to devise a process according to which it can be determined in a finite number of operations whether the equation is solvable in rational integers.” What is sought is a general method applicable to all Diophantine equations, not just to specific equations like those above, or even specific classes of equations such as ax2 + by2 + cz2 + dz2= 0. (*) Today we would ask whether “the solubility of Diophantine equations is decidable.” That is, we ask whether there is an algorithm or computer program which, given the equation, runs for a finite amount of time and then prints out the answer “yes, it is soluble” or “no, it is not soluble.” From top to bottom: Julia Robinson, courtesy of Constance Reid; Yuri Matiyasevich, photo by George Csicsery; David Hilbert, courtesy The Solution AK Peters, Ltd. The story of the solution of Hilbert’s Tenth Problem On March 15 and 16, 2007, CMI held a small is one of great intellectual challenge, adventure, conference at its Cambridge office on Hilbert’s and accomplishment. Hilbert himself worked on it, Tenth Problem. Participants included Martin Davis, and probably thought that it could be solved in the Hilary Putnam, Yuri Matiyasevich, and Constance affirmative. He knew, of course that the solvability Reid, sister of Julia Robinson. The conference was of equations of the form ( ) could be determined coupled with a screening at the Museum of Science in * by an algorithm of his colleague Minkowski. Boston of a pre-release version of George Csicsery’s Nevertheless, the first real progress came in the film Julia Robinson and Hilbert’s Tenth Problem. 1930s with the work of Gödel on the undecidability The Problem of arithmetic. This work, which gave a negative solution of Hilbert’s First Problem, presaged the At the 1900 International Congress of Mathema- solution of the Tenth Problem. Later in that decade ticians in Paris, David Hilbert presented a list of came the work of Alan Turing and a group of twenty- three problems that he felt were important logicians: Church, Kleene, Post, and Rosser. A key for the progress of mathematics. Tenth on the list discovery was the existence of sets of numbers that was a question about Diophantine equations. These could be listed by a Turing machine but for which no are polynomial equations like Turing machine could answer the problem “is x an element of the set” for all x. x2 + y2 = z2 or 3x3 + 4y3 + 5z3 = 0 that have integer coefficients and for which we The discovery of listable but non-computable sets seek integer solutions. The first equation, which set the stage for the contributions of Martin Davis comes from the Pythagorean theorem, was known and Hilary Putnam, and later Julia Robinson. By to the Babylonians and the Greeks. It has infinitely the late 1940s Davis had made substantial progress, many solutions, of which the smallest is 3, 4, 5. The and he formulated a bold conjecture relating listable second, which defines an “elliptic curve,” is the kind sets with those defined by Diophantine equations. of object that played a crucial role in Wiles’ proof of Julia Robinson, working independently, had been Fermat’s last theorem and which is also important studying a seemingly simple question of the great

2007 15 Julia Robinson and Hilbert’s Tenth Problem: Conference and Film

Polish logician Alfred Tarski: can the set of powers of two be defined via Diophantine equations? Robinson was not able to solve the problem, but in a 1952 paper, she reduced it to the question of whether there was a set of pairs of numbers (a,b) that (i) grows exponentially (but not too fast) and (ii) is definable by Diophantine equations. Dubbed “JR” by Martin Davis, this hypothesis was to play a decisive role. Indeed, in a 1961 paper, Davis, Hilary Putnam, and Robinson reduced the solution of Hilbert’s Tenth Events Problem to the problem of proving JR.

The last, crucial step was taken by Yuri Matiyasevich shortly after New Year’s Day, 1970. As a sophomore Panelists George Csicsery and Constance Reid at Leningrad State University, Matiyasevich had taken up Hilbert’s Tenth Problem, but after several Csicsery, Kirsten Eisentrager, Martin Davis, Yuri years of frustration, set it aside, vowing never to Matiyasevich, Hilary Putnam, and Constance Reid. look at it again. Nevertheless, when asked to review a new paper by Julia Robinson, he saw almost Julia Robinson and Hilbert’s Tenth Problem has immediately a way of proving JR, and within a few now been released and is available on DVD from days had done so. AK Peters (www.akpeters.com). It was shown to an enthusiastic audience at the winter meeting of the With Matiyasevich’s work, Hilbert’s Tenth Problem American Mathematical Society in San Diego on was at last solved. Despite the difficulties of January 6, 2008. communication during the Cold War, the good news quickly traveled from the USSR to the USA. The Conference Robinson and Matiyasevich exchanged letters, and thus began a long, fruitful, and generous Held at CMI’s offices at One Bow Street in collaboration among Davis, Matiyasevich, Putnam, Cambridge, the conference brought together all the and Robinson. For more about both the history living participants in the solution of the problem: and the mathematics, see “Decidability in Number Yuri Matiyasevich, Martin Davis, and Hilary Putnam. Theory,” by Bjorn Poonen, The Notices of the AMS, The talks given were as follows: November 2008. Constance Reid, Genesis of the Hilbert Problems The Film George Csicsery, Film clip on life and work of Julia Robinson The threads of this story form the warp and weave Bjorn Poonen, Why number theory is hard? of the film Julia Robinson and Hilbert’s Tenth Yuri Matiyasevich, My collaboration with Julia Robinson Problem, produced and directed by George Csicsery Martin Davis, My collaboration with Hilary Putnam with major support from CMI and Will Hearst III. Yuri Matiyasevich, Hilbert’s Tenth Problem: What was All the main players – Davis, Matiyasevich, Putnam, done and what is to be done and Robinson — appear, as does Julia Robinson’s Bjorn Poonen, Thoughts about the analogue for rational numbers sister, Constance Reid, author of the well-known Alexandra Shlapentokh, Diophantine generation, horizontal biography of David Hilbert. and vertical problems, and the weak vertical method CMI organized a screening of a preliminary version Yuri Matiyasevich, Computation paradigms in the light of of the film at the Museum of Science in Boston on Hilbert’s Tenth Problem March 15, in conjunction with a two-day conference, Gunther Cornelisson, Hard number-theoretical problems held March 15 and 16, on Hilbert’s Tenth Problem. and elliptic curves Following the film was a panel discussion Kirsten Eisentrager, Hilbert’s Tenth Problem for algebraic moderated by Jim Carlson; panelists were George function fields

16 CMI ANNUAL REPORT Recognizing achievement

Clay Research Awards continued from page 7

of the line joining them is a ( 1) curve. Blow it The technical issues thus raised can all be success- blow down the fibers F1 y to obtain a vari- For part (b) of the Mori-Reid conjecture, the dif- − × { } 1 down to obtain a new surface. It is isomorphic fully dealt with; indeed, working in the larger cat- ety C2 with a subvariety E2 ∼= F2 ∼= CP . Thus ferential geometric form of positivity for minimal to CP1 CP1. Both CP2 and CP1 CP1 are egory of varieties with terminal singularities is cru- C E = C F = C E . The birational models, consider first the case of algebraic curves. × × 1 − 1 ∼ 12 − ∼ 2 − 2 classically minimal, and both represent the purely cial to the success of the minimal model program. map C2  C1 is the flop obtained by the surgery Via the uniformization theorem, every Riemann X transcendental function field C(x, y). “cut out E and glue in E .” If q : C C is has as universal cover which is a space of constant Another cost associated with flips is the difficulty 2 1 i 12 i the canonical projection, then the birational→ map curvature +1, 0, or 1: the sphere in the case of What can one say in dimension greater than two? of proving that they exist in sufficient generality. 1 − C C is just the composition q q− . In this genus zero, the plane in the case of genus one, and The conjecture of Mori-Reid (see [4]) states the fol- One can construct motivating examples (see be- 2  1 1 2 case, the piece cut out and the piece glued in are the upper half plane in the case of higher genus. lowing: low), but even these are somewhat complicated. both projective lines. Flops do not affect the inter- For genus g 0, the non-uniruled case, the cur- Important special cases were proved by Tsunoda, ≥ section number with the canonical divisor. vature 0 or 1 condition is a strong differential- Shokurov, Mori, and Kawamata. Finally, Mori geometric form− of nefness. In higher dimension, proved the general existence theorem for flips in To understand this flop better, consider the family ( ) Let X be an algebraic variety of suppose that X is a smooth minimal variety of ∗ dimension three [8], [7, p. 268]. The Clay Re- of planes P on C given by x = λu, v = λy. There dimension n which is not uniruled. λ general type. By the base-point-free theorem, its search Award was given to Hacon and McKernan is a corresponding family of projective planes in C Then (a) it has a minimal model canonical bundle is semi-ample, and so the first for their proof of the existence of flips in dimen- and in C12, and a corresponding family of hyper- Xmin and (b) it has a K¨ahlermet- Chern class of the canonical bundle is represented sion n assuming termination of flips in dimension surfaces P˜ in C . Each P˜ is a projective plane ric whose Ricci curvature is 0. λ 12 λ by a semi-positive (1, 1) form. It follows from the ≤ n 1. See [3]. This result suggests that one can with a point blown up. The blowup of the point is − solution of the Calabi Conjecture, independently prove the existence of minimal models inductively, one of the fibers x F . 2 given by Aubin and Yau in their work on the Monge- a program carried out in [1]. { } × Consider now the surfaces q (P˜ ) C . Obtained Amp`ereequation, that X has a metric whose Ricci In the Mori-Reid conjecture, minimality is defined 1 λ ⊂ 1 in a different way, as a kind of algebro-geometric Let us now discuss minimality and flips in a more by blowing down x F2, these surfaces are curvature is negative ( 0). Alternatively, the { } × 2 ≤ positivity condition. We will discuss this notion in substantitve way. To say that X is minimal is to disjoint for distinct λ, isomorphic to CP , and each Ricci form of the metric is positive-semidefinite. ˜ greater detail below. For surfaces, it coincides with say that its canonical bundle is numerically effec- one meets E1 in a single point. Thus q1(Pλ) E1 = As noted, one generally has to admit terminal sin- tive, or nef. The canonical bundle is the line bundle ˜ · gularities in the minimal model. The best results the classical one: there are no ( 1) curves. For 1. A more subtle computation yields q2(Pλ) E2 = · higher dimensional varieties, minimality− as posi- K whose local sections are holomorphic n-forms A 1. Thus the flop changes the intersection number for (b) in the general case are due to Eyssidieux, line bundle L is nef if the integral − Guedj, and Zeriahi, generalizing the work of Aubin tivity signaled a major change in the way mathe- of qi(Pλ) with the surgery loci Ei. In the example, maticians viewed the birational equivalance prob- the canonical bundle of C is defined and trivial and Yau on Monge-Ampre equations. L C = ω lem. The new line of investigation, initiated by · at the vertex of the cone and so also trivial on C Shigefumi Mori, developed further by Kawamata, the exceptional set F . Therefore K Ei = 0: the is positive for all algebraic curves C on X, where · Acknowledgments. The author relied heavily on Koll´ar,Mori, Reid, and Shokurov, culminated in flop does not affect the positivity of the canonical ω is a differential form representing the first Chern [2], [4], and [7] for this article, and would like to 1988 with Mori’s proof of (a) for varieties of di- bundle. class of L. This integral represents the intersec- thank Herb Clemens and J´anosKoll´arfor their mension three [8]. For this result, the goal of the tion number, which, as noted, may be a rational For the second example, we follow [7, Example help in its preparation. “minimal model program,” Mori received a Fields number. 12.5]. Factor C0 by the Z2 action given by the map Medal in 1990. Although refereeing is still in pro- (x,y,u,v) (x, y, u, v). The Z2 actions make cess, it now appears that (a) is also a theorem for What is the relation of the new definition of mini- −→ − − References sense on C1, C2, and C12. However, there is a sym- all dimensions. An algebraic approach has been mality to that of the Castelnuovo-Enriques theory? metry that is broken. In addition to the family of [1] Caucher Birkar, Paolo Cascini, Christopher D. Ha- given by Birkar, Cascini, Hacon, and McKernan By the adjunction formula, the value of the above con, and James McKernan, Existence of Mini- planes Pλ there is a family of planes Qλ defined by [1] and an analytic approach has been give by Siu integral, i.e., the intersection number K C, is non- mal Models for Varieties of Log General Type, x = λv, u = λy. They are interchanged by the map arXiv:math/AG/0610203. negative. Thus a variety of dimension· two whose [9]. permuting u and v. This global symmetry is the [2] Alessio Corti, What is ... a flip?, Notices of the AMS canonical class is nef has no ( 1) curves. Conse- 51, number 11 (2004), pp. 1350-1351. In the remainder of this article we explain the mod- − reason why C1 = C2. However, the permutation quently, surfaces that are minimal are classically ∼ [3] Christopher Hacon and James McKernan, Extension ern notion of minimality and how it relates to the of coordinates does not commute with the Z2 ac- minimal. theorems and the existence of flips. Flips for 3-folds and classical one. We then touch on just one of the cru- tion. As a result, one finds that C1/Z2 and C2/Z2 4-folds, 76–110, Oxford Lecture Ser. Math. Appl., 35, cial parts of the proof. This is the existence of flips Following [7, Example 12.1], we give an example, are not isomorphic. Indeed, the intersection of the Oxford Univ. Press, Oxford, 2007. and flops. These are surgeries that alter a variety in first observed by Atiyah, of a flop. We then modify canonical class with E2/Z2 is negative whereas its [4] J´anosKoll´ar,What is ... a minimal model? Notices of the AMS 57, number 3 (2007), pp. 370-371. codimension two. Flops leave the positivity of the the example to give a flip. Consider the affine vari- intersection with E1/Z2 is positive. The natural [5] J´anosKoll´ar, Flips, Flops, Minimal Models, etc., Sur- canonical bundle unchanged, whereas flips make it ety C0 given by the quadratic equation xy uv = birational map C2/Z2  C1/Z2 can be described 4 − veys in Differential Geometry (Cambridge, MA, 1990), it more positive, transforming the variety to one 0. Let C be its closure in CP ; it is the cone as “cut out E2/Z2 and replace it by E1/Z1.” The 113-199, Lehigh Univ. Bethlehem PA, 1991. 1 1 that is closer to minimal. One cost of introduc- over a quadric CP CP . Blow up the vertex quotients Ei/Z2 are projective lines. Thus, as in [6] J´anosKoll´arand Shigefumi Mori, Birational geometry × ing a flip is that certain mild singularities must be of the cone to obtain a variety C12. The result- the first example, the surgery is obtained by re- of algebraic varieties, Cambridge University Press, Cam- bridge, UK, 1998. admitted. These are the so-called “terminal singu- ing exceptional set F = F1 F2 is isomorphic to moving a projective line and glueing it back in a 1 1 × [7] J´anosKoll´ar,The Structure of Algebraic Threefolds: an larities.” A consequence of working with singular CP CP . It is possible to blow down all the different way. But in this case, the geometry of the × Introduction to Mori’s Program. Bulletin (New Series) of varieties is that the natural intersection numbers, fibers x F2 of F to obtain a variety C1 with variety subjected to surgery changes and positivity the American Mathematical Society, Volume 17, Num- { } × 1 of the canonical bundle increases. while well-defined, can be rational numbers. a subvariety E1 ∼= F1 ∼= CP . Likewise, we can ber 2, October 1987.

2007 17 Clay Research Awards

blow down the fibers F1 y to obtain a vari- For part (b) of the Mori-Reid conjecture, the dif- [8] Shigefumi Mori, Flip theorem and the existence of min- According to Hasse’s theorem, this normalized mea- × { } 1 ety C2 with a subvariety E2 = F2 = CP . Thus ferential geometric form of positivity for minimal imal models for 3-folds, J. Amer. Math. Soc. 1 (1988), sure of the deviation of the number of solutions ∼ ∼ Number 1, 117–253. C1 E1 ∼= C12 F ∼= C2 E2. The birational models, consider first the case of algebraic curves. modulo p from its expected value is a number in − − − Via the uniformization theorem, every Riemann X [9] Yum-Tong Siu, A General Non-Vanishing Theorem and map C2  C1 is the flop obtained by the surgery an Analytic Proof of the Finite Generation of the Canon- the range [ 1, 1]. A deeper question, then, is the “cut out E and glue in E .” If q : C C is has as universal cover which is a space of constant nature of the− probability law governing the distri- 2 1 i 12 → i ical Ring, 68 pp., arXiv:math.AG/0610740. the canonical projection, then the birational map curvature +1, 0, or 1: the sphere in the case of [10] S.-T. Yau, Calabi’s Conjecture and Some New Re- bution of the numbers δ. 1 − C C is just the composition q q− . In this genus zero, the plane in the case of genus one, and sults in Algebraic Geometry, Proceedings of the National 2  1 1 2 Around 1960, Mikio Sato and John Tate indepen- case, the piece cut out and the piece glued in are the upper half plane in the case of higher genus. Academy of Science 74 (May 1977) pp. 1798-1799. An announcement of the proof of the Calabi conjecture and dently conjectured that the probability law for el- both projective lines. Flops do not affect the inter- For genus g 0, the non-uniruled case, the cur- ≥ some applications of it. liptic curves without complex multiplication (“ex- section number with the canonical divisor. vature 0 or 1 condition is a strong differential- tra symmetry”) is given by the function f(δ)= geometric form− of nefness. In higher dimension, 2 To understand this flop better, consider the familyRecognizing achievement √ suppose that X is a smooth minimal variety of 2. The Sato-Tate Conjecture (2/π) 1 δ . Sato was led to the conjecture by of planes P on C given by x = λu, v = λy. There − λ general type. By the base-point-free theorem, its experimental evidence. Although the documentary is a corresponding family of projective planes in C canonical bundle is semi-ample, and so the first Since the time of the Greeks, the study of Dio- record is sparse, there is still extant a letter from and in C , and a corresponding family of hyper- 12 Chern class of the canonical bundle is represented phantine equations has driven some of the most John Tate to Jean-Pierre Serre dated August 5, surfaces P˜ in C . Each P˜ is a projective plane λ 12 λ by a semi-positive (1, 1) form. It follows from the important developments of mathematics. Of par- 1963, about Tate’s thoughts on the conjecture. In with a point blown up. The blowup of the point is ticular significance are the cubic equations in two this letter Tate adds, “Mumford tells me that Sato solution of the Calabi Conjecture, independently 2 one of the fibers x F2. variables, which we may put in the form ( ) y2 = has found f(θ)=(2/π)sin (θ) experimentally on { } × given by Aubin and Yau in their work on the Monge- 3 ∗ one curve with thousands of p.” The angle θ is Consider now the surfaces q (P˜ ) C . Obtained Amp`ereequation, that X has a metric whose Ricci x +ax+b. The set of complex solutions of ( ) plus 1 λ 1 ∗ cos 1 δ; the formulations in terms of δ and θ are by blowing down x F , these⊂ surfaces are curvature is negative ( 0). Alternatively, the the point at infinity is a torus; in general the solu- − { } × 2 ≤ tion set, including the point at infinity, is called an equivalent. See [11] for computations now accessi- disjoint for distinct λ, isomorphic to CP2, and each Ricci form of the metric is positive-semidefinite. “elliptic curve,” written E. The name is something ble to anyone. one meets E in a single point. Thus q (P˜ ) E = As noted, one generally has to admit terminal sin- 1 1 λ · 1 of a historical accident having do with the prob- 1. A more subtle computation yields q (P˜ ) E = gularities in the minimal model. The best results Tate was led to the conjecture on theoretical grounds 2 λ 2 lem of computing the lengths of arcs on ellipses. 1. Thus the flop changes the intersection number· for (b) in the general case are due to Eyssidieux, having to do with the connection between alebraic Its solution leads to integrals of the form dx/y, of− q (P ) with the surgery loci E . In the example, Guedj, and Zeriahi, generalizing the work of Aubin cycles and the zeroes and poles of L-functions. Start- i λ i where y = √x3 + ax + b. Of course the differen- the canonical bundle of C is defined and trivial and Yau on Monge-Ampre equations. ing from the Hasse-Weil function L(s, E) of the at the vertex of the cone and so also trivial on tial form dx/y plays a leading role in the modern elliptic curve, J.-P. Serre defined [6] a natural se- theory of elliptic curves, e.g., by determining its n the exceptional set F . Therefore K Ei = 0: the Acknowledgments. The author relied heavily on quence of functions L(s, E, sym ) associated to the flop does not affect the positivity of· the canonical Hodge structure. irreducible representations of SU(2). When n = 1, [2], [4], and [7] for this article, and would like to n bundle. thank Herb Clemens and J´anosKoll´arfor their It was Gauss, in his Disquisitiones Arithmetica, the L(s, E, sym )=L(s, E). These functions were help in its preparation. who formally introduced the idea of studying Dio- variants of those considered by Tate [7]. Serre For the second example, we follow [7, Example showed, as Tate had predicted, that the Sato-Tate 12.5]. Factor C by the Z action given by the map phantine equations by examining their reduction 0 2 modulo a prime p. The notion certainly predates conjecture would follow from the assertion that (x,y,u,v) (x, y, u, v). The Z actions make n −→ − − 2 References Gauss, however, and goes back at least as far as L(s, E, sym ) has an extension to an analytic func- sense on C1, C2, and C12. However, there is a sym- tion in the half-plane (s) n +1/2 and is non- metry that is broken. In addition to the family of [1] Caucher Birkar, Paolo Cascini, Christopher D. Ha- Fermat. The central problem is to count the num-  ≥ con, and James McKernan, Existence of Mini- vanishing there. The work of Wiles [10] and of planes P there is a family of planes Q defined by ber of points N on E modulo p. By N we mean λ λ mal Models for Varieties of Log General Type, Taylor and Wiles [9] on Fermat’s last theorem, and x = λv, u = λy. They are interchanged by the map the number of solutions of the equation ( ) mod- arXiv:math/AG/0610203. ulo p, plus one for the point at infinity. A∗ classical finally the work of Breuil, Conrad, Diamond, and permuting u and v. This global symmetry is the [2] Alessio Corti, What is ... a flip?, Notices of the AMS theorem of Hasse tells us how many solutions to Taylor, [1] established the crucial fact that the func- reason why C1 = C2. However, the permutation 51, number 11 (2004), pp. 1350-1351. ∼ [3] Christopher Hacon and James McKernan, Extension expect: tion L(s, E) extends to an analytic function in the of coordinates does not commute with the Z2 ac- theorems and the existence of flips. Flips for 3-folds and N (p + 1) 2√p. half-plane (s) 3/2 and is non-vanishing there. tion. As a result, one finds that C1/Z2 and C2/Z2 4-folds, 76–110, Oxford Lecture Ser. Math. Appl., 35, | − | ≤  ≥ are not isomorphic. Indeed, the intersection of the Oxford Univ. Press, Oxford, 2007. Consider, for example, the elliptic curve defined by One important ingredient in the proof is an exten- 2 3 canonical class with E /Z is negative whereas its [4] J´anosKoll´ar,What is ... a minimal model? Notices of y = x x 1. Modulo 3 there is just one solution, sion of Wiles’ technique for identifying L-functions 2 2 − − intersection with E1/Z2 is positive. The natural the AMS 57, number 3 (2007), pp. 370-371. namely the point at infinity. Modulo 5 there are of elliptic curves with L-functions of modular forms. [5] J´anosKoll´ar, Flips, Flops, Minimal Models, etc., Sur- birational map C2/Z2  C1/Z2 can be described eight. Further experiment reveals the behavior of This was begun in the paper [2] by Clozel, Har- veys in Differential Geometry (Cambridge, MA, 1990), N as a function of p to be quite random, suggesting ris, and Taylor and completed in later paper by as “cut out E2/Z2 and replace it by E1/Z1.” The 113-199, Lehigh Univ. Bethlehem PA, 1991. quotients Ei/Z2 are projective lines. Thus, as in [6] J´anosKoll´arand Shigefumi Mori, Birational geometry a statistical interpretation: the expected value of N Taylor [8]. Another is an extension of an idea of the first example, the surgery is obtained by re- of algebraic varieties, Cambridge University Press, Cam- is p + 1 and the standard deviation is proportional Taylor for proving meromorphic continuation of L- moving a projective line and glueing it back in a bridge, UK, 1998. to √p. To make more precise statement, consider functions associated to two-dimensional Galois rep- different way. But in this case, the geometry of the [7] J´anosKoll´ar,The Structure of Algebraic Threefolds: an the quantity resentations by applying to moduli spaces an ap- Introduction to Mori’s Program. Bulletin (New Series) of variety subjected to surgery changes and positivity N (p + 1) proximation theorem of Moret-Baily. Execution of the American Mathematical Society, Volume 17, Num- δ = − . of the canonical bundle increases. ber 2, October 1987. 2√p

18 CMI ANNUAL REPORT Recognizing achievement

[8] Shigefumi Mori, Flip theorem and the existence of min- According to Hasse’s theorem, this normalized mea- this plan relies on the existence of a suitable mod- translation. The Cayley graph has a natural metric imal models for 3-folds, J. Amer. Math. Soc. 1 (1988), sure of the deviation of the number of solutions uli space. Harris, Shepherd-Barron, and Taylor [4] where each edge is isometric to a unit interval, and Number 1, 117–253. modulo p from its expected value is a number in found one suitable for studying n-dimensional rep- the distance between points is given by the length [9] Yum-Tong Siu, A General Non-Vanishing Theorem and the range [ 1, 1]. A deeper question, then, is the resentations for any even n. It is a twisted form of a shortest path joining them. an Analytic Proof of the Finite Generation of the Canon- − nature of the probability law governing the distri- of the moduli space for Calabi-Yau manifolds orig- ical Ring, 68 pp., arXiv:math.AG/0610740. The Cayley graph depends on the choice of gen- [10] S.-T. Yau, Calabi’s Conjecture and Some New Re- bution of the numbers δ. inally studied by Dwork in certain cases and now erating set and so its geometry is not intrinsic to sults in Algebraic Geometry, Proceedings of the National an important part of string theory. Academy of Science 74 (May 1977) pp. 1798-1799. An Around 1960, Mikio Sato and John Tate indepen- the group Γ. However, it turns out that there is a announcement of the proof of the Calabi conjecture and dently conjectured that the probability law for el- The kind of probability distribution proved for el- natural equivalence relation, quasi-isometry, relat- some applications of it. liptic curves without complex multiplication (“ex- liptic curves is conjecturally far more general. See ing graphs defined by different sets of generators. tra symmetry”) is given by the function f(δ)= Barry Mazur’s article in Nature [5]. We say that two metric spaces X and Y are quasi- (2/π)√1 δ2. Sato was led to the conjecture by isometric if there is a map f : X Y that does 2. The Sato-Tate Conjecture Acknowledgments. The author relied heavily on experimental− evidence. Although the documentary not distort distances too much. To→ make a precise the account in [3] in preparing this article. Since the time of the Greeks, the study of Dio- record is sparse, there is still extant a letter from statement, let dX and dY denote the metrics. Sup- phantine equations has driven some of the most John Tate to Jean-Pierre Serre dated August 5, pose that there are constants K 1 and C 0 ≥ ≥ important developments of mathematics. Of par- 1963, about Tate’s thoughts on the conjecture. In References such that for every x1,x2 X this letter Tate adds, “Mumford tells me that Sato ∈ ticular significance are the cubic equations in two [1] Christophe Breuil, Brian Conrad, Fred Diamond, and 1 variables, which we may put in the form ( ) y2 = has found f(θ)=(2/π)sin2(θ) experimentally on dX (x1,x2) C dY (f(x1,f(x2) ∗ Richard Taylor, On the modularity of elliptic curves over K − ≤ x3 +ax+b. The set of complex solutions of ( ) plus one curve with thousands of p.” The angle θ is Q: Wild 3-adic exercises, Journal of the American Math- ∗ 1 and the point at infinity is a torus; in general the solu- cos− δ; the formulations in terms of δ and θ are ematical Society 14 (2001), pp. 843-939. equivalent. See [11] for computations now accessi- [2] Laurent Clozel, Michael Harris, and Richard Taylor, Au- dY (f(x1,f(x2) KdX (x1,x2)+C tion set, including the point at infinity, is called an ≤ ble to anyone. tomorphy for some l-adic lifts of automorphic mod l rep- “elliptic curve,” written E. The name is something resentations. and such that the C-neighborhood of f(X) is all of a historical accident having do with the prob- Tate was led to the conjecture on theoretical grounds [3] Michael Harris, The Sato-Tate Conjecture: Introduction of Y . Such a map is a (K, C) quasi-isometry. Two lem of computing the lengths of arcs on ellipses. having to do with the connection between alebraic to the Proof, preprint (Societ Math´ematiquede France). spaces are said to be quasi-isometric if there is a [4] Michael Harris, Nicholas Shepherd-Barron, and Richard Its solution leads to integrals of the form dx/y, cycles and the zeroes and poles of L-functions. Start- (K, C) quasi-isometry between them for some K where y = √x3 + ax + b. Of course the differen- Taylor, Ihara’s Lemma and Potential Automorphy ing from the Hasse-Weil function L(s, E) of the [5] Barry Mazur, Controlling our errors, Nature, vol. 443, and C. tial form dx/y plays a leading role in the modern elliptic curve, J.-P. Serre defined [6] a natural se- September 2006, pp 38-40. The central question that Gromov raised was the theory of elliptic curves, e.g., by determining its n [6] John Tate, appendix to Chapter I Abelian l-adic Rep- quence of functions L(s, E, sym ) associated to the classification of groups up to quasi-isometry, i.e., Hodge structure. irreducible representations of SU(2). When n = 1, resentations and Elliptic Curves, Jean-Pierre Serre, n W.A.Benjamin,1968. the enumeration and characterization of the quasi- It was Gauss, in his Disquisitiones Arithmetica, the L(s, E, sym )=L(s, E). These functions were [7] John Tate, Algebraic Cycles and Poles of Zeta Func- isometry classes of finitely generated groups. who formally introduced the idea of studying Dio- variants of those considered by Tate [7]. Serre tions, in Arithmetic Algebraic Geometry, Proceedings phantine equations by examining their reduction showed, as Tate had predicted, that the Sato-Tate of conference held at Purdue University, December 5-7, A modest beginning is to note that all finite groups modulo a prime p. The notion certainly predates conjecture would follow from the assertion that 1963. Harper & Row, 1965. are quasi-isometric, that is, quasi-isometric to a L(s, E, symn) has an extension to an analytic func- [8] Richard Taylor, Automorphy for some l-adic lifts of au- point. Thus Gromov’s theory is a theory of infinite Gauss, however, and goes back at least as far as tomorphic mod l representations II. tion in the half-plane (s) n +1/2 and is non- groups. For nontrivial examples, consider a group Fermat. The central problem is to count the num-  ≥ [9] Richard Taylor and Andrew Wiles, ”Ring-theoretic ber of points N on E modulo p. By N we mean vanishing there. The work of Wiles [10] and of properties of certain Hecke algebras”. Annals of Mathe- Γ that acts properly discontinuously by isometries the number of solutions of the equation ( ) mod- Taylor and Wiles [9] on Fermat’s last theorem, and matics 141 (3), 553-572 (1995). on a nice space X, such as a connected Riemann- ulo p, plus one for the point at infinity. A∗ classical finally the work of Breuil, Conrad, Diamond, and [10] Andrew Wiles, ”Modular elliptic curves and Fermat’s ian manifold. Suppose further that the quotient Last Theorem”. Annals of Mathematics 141 (3), 443-551 theorem of Hasse tells us how many solutions to Taylor, [1] established the crucial fact that the func- X/Γ is compact. (We say that Γ is “co-compact.”) (1995). Then Γ and X are quasi-isometric. For example, expect: tion L(s, E) extends to an analytic function in the [11] http://sage.math.washington.edu/sato-tate/ the lattice Zd in Rd is quasi-isometric to Rd — N (p + 1) 2√p. half-plane (s) 3/2 and is non-vanishing there. | − | ≤  ≥ which is not quasi-isometric to a point. More gen- Consider, for example, the elliptic curve defined by One important ingredient in the proof is an exten- 3. Virtual Quasi-Isometric Rigidity of Sol erally, consider a lattice Γ in a Lie group G:a y2 = x3 x 1. Modulo 3 there is just one solution, sion of Wiles’ technique for identifying L-functions − − discrete subgroup such that the quotient G/Γ has namely the point at infinity. Modulo 5 there are of elliptic curves with L-functions of modular forms. In his 1983 ICM address, Mikhael Gromov pro- finite volume relative to a left invariant Haar mea- eight. Further experiment reveals the behavior of This was begun in the paper [2] by Clozel, Har- posed a program for studying finitely generated sure on G. If Γ is co-compact, then it is quasi- N as a function of p to be quite random, suggesting ris, and Taylor and completed in later paper by groups as geometric objects [6]. The story begins isometric to G, when G is equipped with any left a statistical interpretation: the expected value of N Taylor [8]. Another is an extension of an idea of with the Cayley graph, a metric space CS(Γ) asso- invariant distance function. Moreover, if K is a is p + 1 and the standard deviation is proportional Taylor for proving meromorphic continuation of L- ciated to a group Γ and a set of generators S. The compact subgroup of G and X = G/K is the quo- to √p. To make more precise statement, consider functions associated to two-dimensional Galois rep- set of vertices is Γ itself. Two vertices are con- tient, then Γ is quasi-isometric to X. Any cocom- the quantity resentations by applying to moduli spaces an ap- nected by an edge if right multiplication by some pact lattice in SO(1,n) or PSO(1,n), for example, N (p + 1) proximation theorem of Moret-Baily. Execution of generator maps one to the other. There is a natu- is quasi-isometric to real hyperbolic n-space. Like- δ = − . 2√p ral action of the group on this graph given by left wise, any two such lattices are quasi-isometric to

2007 19 Clay Research Awards this plan relies on the existence of a suitable mod- translation. The Cayley graph has a natural metric each other. In particular the fundamental groups The tractability of semisimple Lie groups comes uli space. Harris, Shepherd-Barron, and Taylor [4] where each edge is isometric to a unit interval, and Γ and Γ of compact Riemann surfaces S and S from their curvature properties: they act on non- found one suitable for studying n-dimensional rep- the distance between points is given by the length are quasi-isometric, so long as both surfaces have positively curved spaces, and for such manifolds resentations for any even n. It is a twisted form of a shortest path joining them. genus at least two. powerful tools have been forged. Nilpotent groups of the moduli space for Calabi-Yau manifolds orig- are tractable for a different reason. Their asymp- The Cayley graph depends on the choice of gen- Since any finitely generated group isomorphic to inally studied by Dwork in certain cases and now totic structure is well approximated by simple scale erating set and so its geometry is not intrinsic to a cocompact lattice in a Lie group G is quasi- an important part of string theory. invariant models which are also nilpotent Lie groups. the group Γ. However, it turns out that there is a isometric to G, it is natural to ask whether the con- On the other hand, solvable Lie groups can fail The kind of probability distribution proved for el- natural equivalence relation, quasi-isometry, relat- verse is true: whether any group quasi-isometric to have either of these simplifying characteristics. liptic curves is conjecturally far more general. See ing graphs defined by different sets of generators. to G is a cocompact lattice. However, this fails The easiest such example is the group Sol, given Barry Mazur’s article in Nature [5]. We say that two metric spaces X and Y are quasi- for trivial reasons because passing to finite index by 3 3 matrices Recognizing achievement isometric if there is a map f : X Y that does subgroups or finite extensions does not change the × Acknowledgments. The author relied heavily on not distort distances too much. To→ make a precise quasi-isometry class of a group. If we say that ez/2 x 0 the account in [3] in preparing this article. statement, let d and d denote the metrics. Sup- two groups are weakly commensurable if they are 01 0 . X Y   pose that there are constants K 1 and C 0 the same, modulo applying a finite sequence of 0 yez/2 ≥ ≥ − References such that for every x1,x2 X these two operations, then the most one can hope   ∈ to show is that a finitely generated group quasi- This group, with the invariant metric [1] Christophe Breuil, Brian Conrad, Fred Diamond, and 1 dX (x1,x2) C dY (f(x1,f(x2) isometric to G is weakly commensurable to a co- 2 z 2 z 2 2 Richard Taylor, On the modularity of elliptic curves over K − ≤ ds = e− dx + e dy + dz compact lattice. Proving this statement is the qua- Q: Wild 3-adic exercises, Journal of the American Math- and ematical Society 14 (2001), pp. 843-939. si-isometric rigidity problem for G. For G a semi- gives one of the seven geometries in Thurston’s ge- [2] Laurent Clozel, Michael Harris, and Richard Taylor, Au- d (f(x ,f(x ) Kd (x ,x )+C simple Lie group, quasi-isometric rigidity holds: a ometrization program. It also represented a key Y 1 2 ≤ X 1 2 tomorphy for some l-adic lifts of automorphic mod l rep- deep theorem that is the result of the work of many obstacle in Gromov’s program. Indeed, the ques- resentations. and such that the C-neighborhood of f(X) is all people, including Sullivan, Tukia, Gromov, Pansu tion of whether a group quasi-isometric to a latice [3] Michael Harris, The Sato-Tate Conjecture: Introduction of Y . Such a map is a (K, C) quasi-isometry. Two [9], Casson-Jungreis, Gabai, Schwartz [10], Kleiner- in Sol is virtually a lattice in Sol became known to the Proof, preprint (Societ Math´ematiquede France). spaces are said to be quasi-isometric if there is a [4] Michael Harris, Nicholas Shepherd-Barron, and Richard Leeb [7], Eskin [2], and Eskin-Farb [4]. as the Farb-Mosher conjecture. (K, C) quasi-isometry between them for some K Taylor, Ihara’s Lemma and Potential Automorphy The Farb-Mosher conjecture was proved in the af- [5] Barry Mazur, Controlling our errors, Nature, vol. 443, and C. A landmark in the development of Gromov’s pro- September 2006, pp 38-40. gram was his polynomial growth theorem [5]. To firmative by recent work of Alex Eskin, David Fi- The central question that Gromov raised was the [6] John Tate, appendix to Chapter I Abelian l-adic Rep- state it, let N(r) be the number of group elements sher, and Kevin Whyte [3]. One of the main tools classification of groups up to quasi-isometry, i.e., resentations and Elliptic Curves, Jean-Pierre Serre, within distance r of the identity element relative was the notion of coarse differentiation, which has W.A.Benjamin,1968. the enumeration and characterization of the quasi- d found application to other areas, e.g., the geome- [7] John Tate, Algebraic Cycles and Poles of Zeta Func- to the word metric. For the group Z , the function isometry classes of finitely generated groups. d try of Banach spaces [1] and combinatorics [8], was tions, in Arithmetic Algebraic Geometry, Proceedings N(r) is bounded by a constant times r . A group of conference held at Purdue University, December 5-7, A modest beginning is to note that all finite groups for which N(r) Crd is said to have polynomial introduced by Eskin to the problem around 2005. 1963. Harper & Row, 1965. are quasi-isometric, that is, quasi-isometric to a growth. The least≤ integer d for which the preced- Coarse differentiation may be viewed as a coarse [8] Richard Taylor, Automorphy for some l-adic lifts of au- point. Thus Gromov’s theory is a theory of infinite ing estimate holds is independent of the generating variant of the theorem of Rademacher, which states tomorphic mod l representations II. n groups. For nontrivial examples, consider a group set, so this notion depends on the group alone. It is that a Lipschitz function R R is differentiable [9] Richard Taylor and Andrew Wiles, ”Ring-theoretic almost everywhere; instead−→ of stating that at al- properties of certain Hecke algebras”. Annals of Mathe- Γ that acts properly discontinuously by isometries not hard to prove that a nilpotent group, as an it- matics 141 (3), 553-572 (1995). on a nice space X, such as a connected Riemann- erated extension of abelian groups, has polynomial most every point a Lipschitz function has linear [10] Andrew Wiles, ”Modular elliptic curves and Fermat’s ian manifold. Suppose further that the quotient growth. For example, the group of matrices behavior on small scales, coarse differentation says Last Theorem”. Annals of Mathematics 141 (3), 443-551 X/Γ is compact. (We say that Γ is “co-compact.”) that in a quantitative sense, Lipschitz functions 1 ac (1995). Then Γ and X are quasi-isometric. For example, Sol R have linear behavior at large scales, at [11] http://sage.math.washington.edu/sato-tate/ 01b → the lattice Zd in Rd is quasi-isometric to Rd — many points. Proving such a result requires one to 001 which is not quasi-isometric to a point. More gen- develop a quantitative version of the Rademacher   3. Virtual Quasi-Isometric Rigidity of Sol erally, consider a lattice Γ in a Lie group G:a with integer coefficients, has polynomial growth of theorem. discrete subgroup such that the quotient G/Γ has degree 4, one greater than the the dimension of the Acknowledgments. The author thanks Bruce Klei- In his 1983 ICM address, Mikhael Gromov pro- finite volume relative to a left invariant Haar mea- corresponding nilpotent Lie group (the Heisenberg ner and Domingo Toledo for their help in the prepa- posed a program for studying finitely generated sure on G. If Γ is co-compact, then it is quasi- group) that contains it. Gromov’s theorem gave a ration of this article. groups as geometric objects [6]. The story begins isometric to G, when G is equipped with any left converse: that any group of polynomial growth is with the Cayley graph, a metric space CS(Γ) asso- invariant distance function. Moreover, if K is a virtually nilpotent, that is, is weakly commensu- ciated to a group Γ and a set of generators S. The compact subgroup of G and X = G/K is the quo- rable with a cocompact lattice in a nilpotent Lie References set of vertices is Γ itself. Two vertices are con- tient, then Γ is quasi-isometric to X. Any cocom- group. This theorem can then be applied to prove [1] Yoav Benyamini and Joram Lindenstrauss, Geometric nected by an edge if right multiplication by some pact lattice in SO(1,n) or PSO(1,n), for example, quasi-isometric rigidity for many nilpotent groups. nonlinear functional analysis, Vol. 1, American Mathe- generator maps one to the other. There is a natu- is quasi-isometric to real hyperbolic n-space. Like- The classification of general nilpotent groups up to matical Society Colloquium Publications, 48, American ral action of the group on this graph given by left wise, any two such lattices are quasi-isometric to quasi-isometry is still wide a open problem. Mathematical Society, Providence, RI.

20 CMI ANNUAL REPORT Recognizing achievement

each other. In particular the fundamental groups The tractability of semisimple Lie groups comes [2] Alex Eskin, Quasi-isometric rigidity of nonuniform lat- Γ and Γ of compact Riemann surfaces S and S from their curvature properties: they act on non- tices in higher rank symmetric spaces, J. Amer. Math. are quasi-isometric, so long as both surfaces have positively curved spaces, and for such manifolds Soc.. 11 (1998), no. 2, 321-361. [3] Alex Eskin, David Fisher, and Kevin Whyte, genus at least two. powerful tools have been forged. Nilpotent groups Quasi-isometries and rigidity of the solvable groups, are tractable for a different reason. Their asymp- Since any finitely generated group isomorphic to arXiv:math.GR/05111647v3 totic structure is well approximated by simple scale [4] Alex Eskin, Benson Farb, Quasiflats and rigidity in a cocompact lattice in a Lie group G is quasi- invariant models which are also nilpotent Lie groups. higher rank symmetric spaces, J. Amer. Math. Soc., 10 isometric to G, it is natural to ask whether the con- On the other hand, solvable Lie groups can fail (1997), no. 3, 653-692. verse is true: whether any group quasi-isometric [5] Mikhael Gromov, Groups of polynomial growth and ex- to have either of these simplifying characteristics. to G is a cocompact lattice. However, this fails panding maps, Inst. des Hautes Etudes´ Scientifiques The easiest such example is the group Sol, given for trivial reasons because passing to finite index Publ. Math., No. 53 (1981), 53-73. by 3 3 matrices subgroups or finite extensions does not change the [6] Mikhael Gromov, Infinite groups as geometric objects, × Proceedings of the International Congress of Mathemati- z/2 quasi-isometry class of a group. If we say that e x 0 cians, Vol 1, 2 (Warsaw, 1983), 385-392, PWN, Warsaw two groups are weakly commensurable if they are 01 0 . 1984.  z/2 the same, modulo applying a finite sequence of 0 ye− [7] Bruce Kleiner and Bernhard Leeb, Rigidity of quasi- these two operations, then the most one can hope isometries for symmetric spaces and Euclidean buildings,   ´ to show is that a finitely generated group quasi- This group, with the invariant metric Inst. des Hautes Etudes Scientifiques Publ. Math., No. 86 (1998), 115-197. 2 z 2 z 2 2 isometric to G is weakly commensurable to a co- ds = e− dx + e dy + dz [8] Matouˇsek,JiˇrOn embedding trees into uniformly convex compact lattice. Proving this statement is the qua- Banach spaces, Israel J. Math. 114 (1999), 221–237. si-isometric rigidity problem for G. For G a semi- gives one of the seven geometries in Thurston’s ge- [9] Pierre Pansu, Metriques de Carnot-Carath´eodory et simple Lie group, quasi-isometric rigidity holds: a ometrization program. It also represented a key quasisom´etries des esapces sym´etriques de rang un. deep theorem that is the result of the work of many obstacle in Gromov’s program. Indeed, the ques- (French) (Carnot-Caratheodory metrics and quasi- tion of whether a group quasi-isometric to a latice isometries of rank-one symmetric spaces), Ann. of people, including Sullivan, Tukia, Gromov, Pansu Math., (2) 129 (1989), no 1. 1-60. [9], Casson-Jungreis, Gabai, Schwartz [10], Kleiner- in Sol is virtually a lattice in Sol became known [10] Richard Schwartz, The quasi-isometry classification of Leeb [7], Eskin [2], and Eskin-Farb [4]. as the Farb-Mosher conjecture. rank one lattices, Inst. des Hautes Etudes´ Scientifiques Publ. Math., No. 82 (1995), 133-168. A landmark in the development of Gromov’s pro- The Farb-Mosher conjecture was proved in the af- gram was his polynomial growth theorem [5]. To firmative by recent work of Alex Eskin, David Fi- state it, let N(r) be the number of group elements sher, and Kevin Whyte [3]. One of the main tools within distance r of the identity element relative was the notion of coarse differentiation, which has to the word metric. For the group Zd, the function found application to other areas, e.g., the geome- N(r) is bounded by a constant times rd. A group try of Banach spaces [1] and combinatorics [8], was for which N(r) Crd is said to have polynomial introduced by Eskin to the problem around 2005. growth. The least≤ integer d for which the preced- Coarse differentiation may be viewed as a coarse ing estimate holds is independent of the generating variant of the theorem of Rademacher, which states that a Lipschitz function Rn R is differentiable set, so this notion depends on the group alone. It is −→ not hard to prove that a nilpotent group, as an it- almost everywhere; instead of stating that at al- erated extension of abelian groups, has polynomial most every point a Lipschitz function has linear growth. For example, the group of matrices behavior on small scales, coarse differentation says that in a quantitative sense, Lipschitz functions 1 ac Sol R have linear behavior at large scales, at 01b →   many points. Proving such a result requires one to 001 develop a quantitative version of the Rademacher with integer coefficients, has polynomial growth of theorem. degree 4, one greater than the the dimension of the Acknowledgments. The author thanks Bruce Klei- corresponding nilpotent Lie group (the Heisenberg ner and Domingo Toledo for their help in the prepa- group) that contains it. Gromov’s theorem gave a ration of this article. converse: that any group of polynomial growth is The group [6,4] is the triangle group generated by reflections in a right virtually nilpotent, that is, is weakly commensu- triangle with angles π/6 and π/4. The figure above shows a Hamiltonian path in the Cayley graph of this group relative to the standard set of rable with a cocompact lattice in a nilpotent Lie References generators. The heavy line segments, both solid and dashed, represent group. This theorem can then be applied to prove the Cayley graph; the light lines show the triangle tessellation. The [1] Yoav Benyamini and Joram Lindenstrauss, Geometric quasi-isometric rigidity for many nilpotent groups. Hamiltonian path consists of the solid line segments. Figure and text nonlinear functional analysis, Vol. 1, American Mathe- credit: Douglas Dunham, “Creating Repeating Hyperbolic Patterns— The classification of general nilpotent groups up to matical Society Colloquium Publications, 48, American Old and New,” Notices of the AMS, Volume 50, Number 4, April 2003, quasi-isometry is still wide a open problem. Mathematical Society, Providence, RI. p. 453.

2007 21 Homogeneous Flows, Moduli Spaces and Arithmetic, Pisa, Italy

Clay Mathematics Institute 2007 Summer Eigenfunctions of the laplacian: a semi- Summer school School on Homogeneous Flows, Moduli • classical study by Nalini Anantharaman; Spaces and Arithmetic Equidistribution on homogeneous spaces The Centro di Ricerca Matematica Ennio De • and the analytic theory of L-functions by Giorgi in Pisa, Italy provided a wonderful setting Akshay Venkatesh; for the 2007 Clay Mathematics Institute Summer School. The school was designed to serve as a Counting and equidistribution on homoge- comprehensive introduction to the theory of flows • neous spaces, via mixing and unipotent on homogeneous spaces, moduli spaces and their flows by Hee Oh; many applications. These flows give concrete ex- amples of dynamical systems with highly interest- Informal introduction to unipotent flows by ing behavior and a rich and powerful theory. They • Gregory Margulis; are also a source of many interesting problems and conjectures. Furthermore, understanding the dy- Modular shadows by Yuri Manin; namics of such concrete system lends to numer- • ous applications in number theory and geometry On the regularity of solutions of the coho- regarding equidistributions, diophantine approxi- • mological equation for IET’s and transla- mations, rational billiards and automorphic forms. tion flows Modular Shadows by Giovanni Aerial view of Pisa. © Peter Adams. The program was built around three foundation Forni; courses: Clay Mathematics Institute 2007 Summer Eigenfunctions of the laplacian: a semi- The Distribution of free path lengths • • School on Homogeneous Flows, Moduli (1) Unipotentclassical study flowsby and Nalini applications Anantharaman;by Alex in the Periodic Lorentz Gas by J. Marklof; Spaces and Arithmetic Eskin and Dmitry Kleinbock; Equidistribution on homogeneous spaces Uniform spectral gap bounds and arithmetic • • The Centro di Ricerca Matematica Ennio De (2) Diagonalizableand the analytic actions theory and of L-functions arithmetic ap-by applications by A. Gamburd; Giorgi in Pisa, Italy provided a wonderful setting plicationsAkshay Venkatesh;by Manfred Einsiedler and Elon Multi-valued Hamiltonians and Birkhoff for the 2007 Clay Mathematics Institute Summer Lindenstrauss; • School. The school was designed to serve as a Counting and equidistribution on homoge- sums over rotations and IET by • comprehensive introduction to the theory of flows (3) Intervalneous spaces, exchange via mapsmixing and and translation unipotent sur- C. Ulcigrai; on homogeneous spaces, moduli spaces and their facesflows byby Jean-Christophe Hee Oh; Yoccoz. Random hyperbolic surfaces and measured many applications. These flows give concrete ex- • amples of dynamical systems with highly interest- Informal introduction to unipotent flows by laminations by M. Mirzakhani. These• were supplemented by various Short Courses ing behavior and a rich and powerful theory. They and AdvancedGregory Mini Margulis; Courses and Lectures: are also a source of many interesting problems and conjectures. Furthermore, understanding the dy- Modular shadows by Yuri Manin; namics of such concrete system lends to numer- • Equidistribution and L-Functions by Gerge- • ly Harcos; ous applications in number theory and geometry On the regularity of solutions of the coho- One way to orientate oneself within the formida- • regarding equidistributions, diophantine approxi- mological equation for IET’s and transla- ble mathematical landscape explored in the school Reveiw of Vatsal’s work on equidistribution mations, rational billiards and automorphic forms. tion flows Modular Shadows by Giovanni is to consider the familiar space SL(2, )/SL(2, ): • and non-vanishing L-functions by Nicolas R Z The program was built around three foundation Forni; to a hyperbolic geometer, it is the unit tangent Templier; courses: bundle of a hyperbolic surface; to a number the- The Distribution of free path lengths orist, it is the space of elliptic curves; to a low- • Homogeneous flows, buildings and tilings 22 CMI ANNUAL REPORT in the Periodic Lorentz Gas by J. Marklof; dimensional topologist it is the moduli space of flat (1) Unipotent flows and applications by Alex • by Shahar Mozes; Eskin and Dmitry Kleinbock; metrics (with an associated vector field) on genus 1 Uniform spectral gap bounds and arithmetic surfaces; to those who study Diophantine approx- • Fuchsian groups, geodesic flows on surfaces applications by A. Gamburd; imation, it is the space of unimodular lattices in (2) Diagonalizable actions and arithmetic ap- • of constant negative curvature and symbolic 2; and to Lie theorists it is a motivating example plications by Manfred Einsiedler and Elon coding of geodesics by Svetlana Katok R Multi-valued Hamiltonians and Birkhoff of a finite volume homogeneous space G/Γ, that is, Lindenstrauss; • sums over rotations and IET by of a lattice Γ inside a Lie group G. ChaoticityC. Ulcigrai; of the Teichmuller flow by Ar- (3) Interval exchange maps and translation sur- • tur Avila; Two principal generalizations provided the set- faces by Jean-Christophe Yoccoz. Random hyperbolic surfaces and measured ting for much of the material presented at the school: • laminations by M. Mirzakhani. These were supplemented by various Short Courses and Advanced Mini Courses and Lectures:

Equidistribution and L-Functions by Gerge- • ly Harcos; One way to orientate oneself within the formida- ble mathematical landscape explored in the school Reveiw of Vatsal’s work on equidistribution is to consider the familiar space SL(2, )/SL(2, ): • and non-vanishing L-functions by Nicolas R Z to a hyperbolic geometer, it is the unit tangent Templier; bundle of a hyperbolic surface; to a number the- orist, it is the space of elliptic curves; to a low- Homogeneous flows, buildings and tilings dimensional topologist it is the moduli space of flat • by Shahar Mozes; metrics (with an associated vector field) on genus 1 surfaces; to those who study Diophantine approx- Fuchsian groups, geodesic flows on surfaces imation, it is the space of unimodular lattices in • of constant negative curvature and symbolic 2; and to Lie theorists it is a motivating example coding of geodesics by Svetlana Katok R of a finite volume homogeneous space G/Γ, that is, of a lattice Γ inside a Lie group G. Chaoticity of the Teichmuller flow by Ar- • tur Avila; Two principal generalizations provided the set- ting for much of the material presented at the school: Clay Mathematics Institute 2007 Summer Eigenfunctions of the laplacian: a semi- Clay Mathematics Institute 2007 Summer • Eigenfunctions of the laplacian: a semi- School on Homogeneous Flows, Moduli • classical study by Nalini Anantharaman; School onSpaces Homogeneous and Arithmetic Flows, Moduli classical study by Nalini Anantharaman; Spaces and Arithmetic Equidistribution on homogeneous spaces • Equidistribution on homogeneous spaces Summer school The Centro di Ricerca Matematica Ennio De • and the analytic theory of L-functions by GiorgiThe in Centro Pisa, Italydi Ricerca provided Matematica a wonderful Ennio setting De Akshayand the Venkatesh; analytic theory of L-functions by Organizers: David Ellwood (CMI), Manfred Einsiedler (Ohio State), Alex Eskin (Chicago), Dmitry Kleinbock (Brandeis), forGiorgi the in 2007 Pisa, Clay Italy Mathematics provided a Institute wonderful Summer setting Akshay Venkatesh; Elon Lindenstrauss, chair (Princeton), Gregory Margulis (Yale), Stefano Marmi (La Scuola Normale Superiore di Pisa), School.for the 2007 The Clay school Mathematics was designed Institute to serve Summer as a Counting and equidistribution on homoge- PeterSchool. Sarnak The (Princeton), school was Jean-Christophe designed to Yoccoz serve as(Collège a de France), Don• Counting Zagier (MIT) and equidistribution on homoge- comprehensive introduction to the theory of flows • neous spaces, via mixing and unipotent oncomprehensive homogeneous introduction spaces, moduli to the spaces theory and of flowstheir flowsneousby spaces, Hee Oh; via mixing and unipotent manyon homogeneous applications. spaces, These moduli flows give spaces concrete and their ex- flows by Hee Oh; amplesmany applications. of dynamical These systems flows with give highly concrete interest- ex- Informal introduction to unipotent flows by amples of dynamical systems with highly interest- • Informal introduction to unipotent flows by ing behavior and a rich and powerful theory. They • Gregory Margulis; areing alsobehavior a source and of a rich many and interesting powerful problems theory. They and Gregory Margulis; conjectures.are also a source Furthermore, of many interesting understanding problems the and dy- Modular shadows by Yuri Manin; conjectures. Furthermore, understanding the dy- • Modular shadows by Yuri Manin; namics of such concrete system lends to numer- • ousnamics applications of such concrete in number system theory lends and to geometry numer- On the regularity of solutions of the coho- ous applications in number theory and geometry • On the regularity of solutions of the coho- regarding equidistributions, diophantine approxi- • mological equation for IET’s and transla- mations,regarding rational equidistributions, billiards and diophantine automorphic approxi- forms. tionmological flows equation Modular for Shadows IET’sby and Giovanni transla- Themations, program rational was billiards built around and automorphic three foundation forms. Forni;tion flows Modular Shadows by Giovanni courses:The program was built around three foundation Forni; courses: The Distribution of free path lengths • The Distribution of free path lengths (1) Unipotent flows and applications by Alex in the Periodic Lorentz Gas by J. Marklof; • in the Periodic Lorentz Gas by J. Marklof; (1) EskinUnipotent and Dmitryflows and Kleinbock; applications by Alex Eskin and Dmitry Kleinbock; Uniform spectral gap bounds and arithmetic • Uniform spectral gap bounds and arithmetic (2) Diagonalizable actions and arithmetic ap- applications by A. Gamburd; • applications by A. Gamburd; (2) plicationsDiagonalizableby Manfred actions Einsiedler and arithmetic and Elon ap- plications by Manfred Einsiedler and Elon Multi-valued Hamiltonians and Birkhoff Lindenstrauss; • Lindenstrauss; sumsMulti-valued over rotations Hamiltonians and IET andby Birkhoff • sums over rotations and IET by (3) Interval exchange maps and translation sur- C. Ulcigrai; C. Ulcigrai; (3) facesIntervalby exchange Jean-Christophe maps and Yoccoz. translation sur- faces by Jean-Christophe Yoccoz. Random hyperbolic surfaces and measured • laminationsRandom hyperbolicby M. Mirzakhani. surfaces and measured These were supplemented by various Short Courses • laminations by M. Mirzakhani. andThese Advanced were supplemented Mini Courses by variousand Lectures: Short Courses and Advanced Mini Courses and Lectures: Equidistribution and L-Functions by Gerge- • lyEquidistribution Harcos; and L-Functions by Gerge- • ly Harcos; One way to orientate oneself within the formida- bleOne mathematical way to orientate landscape oneself explored within in the the formida- school Reveiw of Vatsal’s work on equidistribution ble mathematical landscape explored in the school • Reveiw of Vatsal’s work on equidistribution is to consider the familiar space SL(2, R)/SL(2, Z): • and non-vanishing L-functions by Nicolas is to consider the familiar space SL(2, )/SL(2, ): Photo courtesyand Centro non-vanishing di Ricerca Matematica L-functions Ennio De Giorgi.by Nicolas to a hyperbolic geometer, it is theR unit tangentZ Templier; to a hyperbolic geometer, it is the unit tangent Templier; bundle of a hyperbolic surface; to a number the- orist,bundle it of is a the hyperbolic space of surface; elliptic tocurves; a number to a low- the- Homogeneous flows, buildings and tilings orist, it is the space of elliptic curves; to a low- • Homogeneous flows, buildings and tilings dimensional topologist it is the moduli space of flat by Shahar Mozes; dimensional topologist it is the moduli space of flat • by Shahar Mozes; metrics (with an associated vector field) on genus 1 surfaces;metrics (with to those an associated who study vector Diophantine field) on approx- genus 1 Fuchsian groups, geodesic flows on surfaces surfaces; to those who study Diophantine approx- • Fuchsian groups, geodesic flows on surfaces imation, it is the space of unimodular lattices in of constant negative curvature and symbolic imation,2 it is the space of unimodular lattices in • of constant negative curvature and symbolic R ; and to Lie theorists it is a motivating example coding of geodesics by Svetlana Katok 2; and to Lie theorists it is a motivating example coding of geodesics by Svetlana Katok ofR a finite volume homogeneous space G/Γ, that is, of a finitelattice volumeΓ inside homogeneous a Lie group spaceG. G/Γ, that is, Chaoticity of the Teichmuller flow by Ar- of a lattice Γ inside a Lie group G. • Chaoticity of the Teichmuller flow by Ar- • tur Avila; Two principal generalizations provided the set- tur Avila; tingTwo for much principal of the generalizations material presented provided at the the school: set- ting for much of the material presented at the school: Clay Mathematics Institute 2007 Summer Eigenfunctions of the laplacian: a semi- Dynamics on the space of lattices: Let X = Dynamics on the moduli space of flat sur- School on Homogeneous Flows, Moduli • classical study by Nalini Anantharaman; n SL(n, )/SL(n, ). This is the space of unimodu- faces: Instead of considering the space X of lat- Spaces and Arithmetic R Z n lar lattices in n, as well as a homogeneous space tices in higher dimensions, we can consider the Equidistribution on homogeneous spaces R G/Γ. On our motivating example X there are two moduli spaces of flat metrics (with associated The Centro di Ricerca Matematica Ennio De • and the analytic theory of L-functions by 2 important dynamical systems arising from the left vector fields) onH surfaces of higher genus g 2. Giorgi in Pisa, Italy provided a wonderful setting Akshay Venkatesh; action of one-parameter subgroups: the geodesic There is a natural SL(2, ) action on this space,≥ for the 2007 Clay Mathematics Institute Summer R flow, given by action of the diagonal subgroup coming from the linear action on 2. The action School. The school was designed to serve as a Counting and equidistribution on homoge- R of the subgroup A is called Teichm¨uller geodesic comprehensive introduction to the theory of flows • neous spaces, via mixing and unipotent et 0 flow, and the orbit of a point x under this on homogeneous spaces, moduli spaces and their flows by Hee Oh; A = g = ; ∈ H t 0 e t flow yields information about the ergodic proper- many applications. These flows give concrete ex- − t   ∈R ties of the associated vector field. amples of dynamical systems with highly interest- Informal introduction to unipotent flows by • 2007 23 ing behavior and a rich and powerful theory. They Gregory Margulis; and the horocycle flow, given by action of the unipo- A seemingly unrelated family of dynamical sys- are also a source of many interesting problems and tent subgroup tems are interval exchange maps: given a partition conjectures. Furthermore, understanding the dy- Modular shadows by Yuri Manin; of the unit interval into n labeled subintervals, re- • namics of such concrete system lends to numer- 1 t arrange them according to a permutation π S . U = ht = . ∈ n ous applications in number theory and geometry On the regularity of solutions of the coho- 01 t However, if we take a first return map for the flow •   ∈R regarding equidistributions, diophantine approxi- mological equation for IET’s and transla- associated to a vector field x to a transverse mations, rational billiards and automorphic forms. tion flows Modular Shadows by Giovanni ∈ H Two sets of lectures focused primarily on gen- interval, we obtain exactly one of these exchange The program was built around three foundation Forni; eralizations of each of these actions: Alex Eskin maps. courses: and Dmitry Kleinbock (with a two lecture prequel In the last foundational course, Jean-Chrstophe The Distribution of free path lengths by Grigorii Margulis) delivered a lecture series on Yoccoz explored this connection from the perspec- • in the Periodic Lorentz Gas by J. Marklof; (1) Unipotent flows and applications by Alex the action of unipotent subgroups H on homoge- tive of combinatorics and dynamics of interval ex- Eskin and Dmitry Kleinbock; neous spaces G/Γ, exploring the results of Dani, changes. In particular, he showed how to use the Uniform spectral gap bounds and arithmetic Margulis, and others on non-divergence of orbits; ergodicity of Teichm¨ullerflow and associated renor- • applications by A. Gamburd; (2) Diagonalizable actions and arithmetic ap- Margulis’s use of unipotent dynamics to prove the malization procedures on the space of interval ex- plications by Manfred Einsiedler and Elon Oppenheim conjecture on the values of quadratic changes to show the resolution (by Masur & Veech) Multi-valued Hamiltonians and Birkhoff Lindenstrauss; forms; and Ratner’s classification of orbit closures of the Keane conjecture that almost every interval • sums over rotations and IET by and invariant measures. Here, even in the case exchange map is uniquely ergodic. C. Ulcigrai; (3) Interval exchange maps and translation sur- n = 2, the situation is quite rigid: for example, all faces by Jean-Christophe Yoccoz. orbits of the group U are either periodic or dense. Following Yoccoz’s lectures, Artur Avila, Gio- Random hyperbolic surfaces and measured vanni Forni, and Maryam Mirzakhani gave further • laminations by M. Mirzakhani. In contrast, the action of A for n = 2 is re- talks on Teichm¨ullerdynamics, exploring applica- These were supplemented by various Short Courses markably chaotic: given any 1 α 3, one can tions to the ergodic theory of polygonal billiards, and Advanced Mini Courses and Lectures: ≤ ≤ produce an orbit whose closure has Hausdorff di- and studying closely related spaces of foliations and mension α. However, for n 3, there are conjec- laminations on surfaces. Equidistribution and L-Functions by Gerge- tures of Margulis on the rigidity≥ of the action of • ly Harcos; the diagonal subgroup. Two of the main contrib- All the lecturers made a special effort to ensure One way to orientate oneself within the formida- utors to this field, Manfred Einsiedler and Elon their presentations would be accessible to all the ble mathematical landscape explored in the school Reveiw of Vatsal’s work on equidistribution Lindenstrauss, gave lectures on the progress made participants in the summer school, from beginning • is to consider the familiar space SL(2, R)/SL(2, Z): and non-vanishing L-functions by Nicolas toward these conjectures, focusing on the theory of graduate students on upwards. Ninety-seven young to a hyperbolic geometer, it is the unit tangent Templier; entropy; the applications to the Littlewood conjec- mathematicians participated, from Europe, Asia, bundle of a hyperbolic surface; to a number the- ture on simultaneous diophantine approximation; the Middle East, and the Americas. In addition orist, it is the space of elliptic curves; to a low- Homogeneous flows, buildings and tilings and the theory of quantum unique ergodicity. to bringing many of the participants in these fields • dimensional topologist it is the moduli space of flat by Shahar Mozes; together to exchange ideas, the summer school has metrics (with an associated vector field) on genus 1 Building on these lectures, Nalini Ananthara- hopefully helped spark the interest of a new gener- surfaces; to those who study Diophantine approx- man, Gergely Harcos, Hee Oh, and Akshay Venka- Fuchsian groups, geodesic flows on surfaces ation of mathematicians in these beautiful areas. imation, it is the space of unimodular lattices in tesh gave shorter series of more advanced lectures, • of constant negative curvature and symbolic 2; and to Lie theorists it is a motivating example indicating applications to quantum chaos, auto- coding of geodesics by Svetlana Katok R of a finite volume homogeneous space G/Γ, that is, morphic forms, and counting points on varieties. In of a lattice Γ inside a Lie group G. addition, a special session on Diophantine approxi- Chaoticity of the Teichmuller flow by Ar- mation was organized by Dmitry Kleinbock, allow- • tur Avila; Two principal generalizations provided the set- ing many of the younger participants to present ting for much of the material presented at the school: their recent results in the subject. Dynamics on the space of lattices: Let Xn = Dynamics on the moduli space of flat sur- SL(n, R)/SL(n, Z). This is the space of unimodu- faces: Instead of considering the space Xn of lat- lar lattices in Rn, as well as a homogeneous space tices in higher dimensions, we can consider the G/Γ. On our motivating example X2 there are two moduli spaces of flat metrics (with associated important dynamical systems arising from the left vector fields) onH surfaces of higher genus g 2. ≥ action of one-parameter subgroups: the geodesic There is a natural SL(2, R) action on this space, Homogeneousflow, given by action of the diagonal Flows, subgroup Modulicoming Spaces from the linear and action on R2. The action of the subgroup A is called Teichm¨uller geodesic et 0 flow, and the orbit of a point x under this Arithmetic,A = g = Pisa, Italy; t t flow yields information about the∈ ergodicH proper- 0 e− t   ∈R ties of the associated vector field. and the horocycle flow, given by action of the unipo- A seemingly unrelated family of dynamical sys- tent subgroup tems are interval exchange maps: given a partition of the unit interval into n labeled subintervals, re- 1 t arrange them according to a permutation π S . U = ht = . ∈ n 01 t However, if we take a first return map for the flow   ∈R associated to a vector field x to a transverse interval, we obtain exactly one∈ ofH these exchange DynamicsTwo sets on of lectures the space focused of lattices: primarilyLet onX gen-= Dynamics on the moduli space of flat sur- n maps. eralizationsSL(n, R)/SL of(n, eachZ). Thisof these is the actions: space of Alex unimodu- Eskin faces: Instead of considering the space Xn of lat- andlar lattices Dmitry in KleinbockRn, as well (with as a a homogeneous two lecture prequel space ticesIn thein higher last foundational dimensions, course, we can Jean-Chrstophe consider the byG/Γ Grigorii. On our Margulis) motivating delivered example aX lecture2 there series are two on Yoccozmoduli explored spaces thisof flatconnection metrics from (with the associated perspec- Summer school theimportant action dynamical of unipotent systems subgroups arisingH fromon homoge- the left tivevector of combinatoricsfields) onH surfaces and dynamics of higher ofgenus intervalg ex-2. ≥ neousaction spaces of one-parameterG/Γ, exploring subgroups: the results the ofgeodesic Dani, changes.There is a In natural particular,SL(2 he, R showed) action how on this to use space, the Margulis,flow, given and by action others of on the non-divergence diagonal subgroup of orbits; ergodicitycoming from of Teichm¨ullerflow the linear action and on associatedR2. The actionrenor- Margulis’s use of unipotent dynamics to prove the malizationof the subgroup proceduresA is calledon theTeichm¨uller space of interval geodesic ex- Oppenheim conjecture onet the0 values of quadratic changesflow, and to the show orbit the resolution of a point (byx Masurunder & Veech) this A = g = ; forms; and Ratner’st classificationt of orbit closures offlow the yields Keane information conjecture about that almost the∈ ergodic everyH interval proper- 0 e− t and invariant measures. Here, even∈R in the case exchangeElonties Lindenstrauss of the map associated conducting is uniquely a session vector in ergodic. Pisa. field. nand= the2, thehorocycle situation flow is, quitegiven rigid:by action for ofexample, the unipo- all orbits of the group U are either periodic or dense. FollowingA seemingly Yoccoz’s unrelated lectures, family Artur of dynamical Avila, Gio- sys- tent subgroup vannitems are Forni,interval andMaryam exchange Mirzakhani maps: given gave a partition further In contrast, the action of A for n = 2 is re- talksof the on unit Teichm¨ullerdynamics, interval into n labeled exploring subintervals, applica- re- markably chaotic: given any1 t 1 α 3, one can tionsarrange to them the ergodic according theory to a of permutation polygonal billiards,π S . U = ht = ≤ ≤. ∈ n produce an orbit whose closure01 hast Hausdorff di- andHowever, studying if we closely take arelated first return spaces map of foliations for the flow and   ∈R mension α. However, for n 3, there are conjec- laminationsassociated to on a surfaces. vector field x to a transverse tures of Margulis on the rigidity≥ of the action of ∈ H Two sets of lectures focused primarily on gen- interval, we obtain exactly one of these exchange the diagonal subgroup. Two of the main contrib- All the lecturers made a special effort to ensure eralizations of each of these actions: Alex Eskin maps. utors to this field, Manfred Einsiedler and Elon their presentations would be accessible to all the and Dmitry Kleinbock (with a two lecture prequel Lindenstrauss, gave lectures on the progress made participantsIn the last in foundational the summer course, school, Jean-Chrstophe from beginning by Grigorii Margulis) delivered a lecture series on toward these conjectures, focusing on the theory of graduateYoccoz explored students this on upwards.connection Ninety-seven from the perspec- young the action of unipotent subgroups H on homoge- entropy; the applications to the Littlewood conjec- mathematicianstive of combinatorics participated, and dynamics from Europe,of interval Asia, ex- neous spaces G/Γ, exploring the results of Dani, ture on simultaneous diophantine approximation; thechanges. Middle In East, particular, and the he Americas.showed how In to addition use the Margulis, and others on non-divergence of orbits; and the theory of quantum unique ergodicity. toergodicity bringing of many Teichm¨ullerflow of the participants and associated in these renor- fields Margulis’s use of unipotent dynamics to prove the togethermalization to procedures exchange ideas, on the the space summer of interval school has ex- OppenheimBuilding onconjecture these lectures, on the values Nalini of Ananthara- quadratic hopefullychanges to helped show the spark resolution the interest (by Masur of a new & Veech) gener- man,forms; Gergely and Ratner’s Harcos, classification Hee Oh, and of Akshay orbit closures Venka- ationof the of Keane mathematicians conjecture that in these almost beautiful every interval areas. teshand gaveinvariant shorter measures. series of more Here, advanced even in thelectures, case exchange map is uniquely ergodic. indicatingn = 2, thesituation applications is quite to quantum rigid: for chaos, example, auto- all morphicorbits of forms, the group andU countingare either points periodic on varieties. or dense. In Following Yoccoz’s lectures, Artur Avila, Gio- addition, a special session on Diophantine approxi- vanni Forni, and Maryam Mirzakhani gave further In contrast, the action of A for n = 2 is re- mation was organized by Dmitry Kleinbock, allow- talks on Teichm¨ullerdynamics, exploring applica- markably chaotic: given any 1 α 3, one can ing many of the younger participants to present tions to the ergodic theory of polygonal billiards, produce an orbit whose closure≤ has Hausdor≤ ff di- their recent results in the subject. and studying closely related spaces of foliations and mension α. However, for n 3, there are conjec- laminations on surfaces. ≥ Dynamics on the space of lattices: Let Xn = Dynamicstures of Margulis on the on moduli the rigidity space of the of flat action sur- of SL(n, R)/SL(n, Z). This is the space of unimodu- faces:the diagonalInstead subgroup. of considering Two of the the space mainXn contrib-of lat- All the lecturers made a special effort to ensure lar lattices in Rn, as well as a homogeneous space ticesutors in to higher this field, dimensions, Manfred we Einsiedler can consider and Elon the their presentations would be accessible to all the G/Γ. On our motivating example X2 there are two moduliLindenstrauss, spaces gaveof lectures flat metrics on the (with progress associated made participants in the summer school, from beginning important dynamical systems arising from the left vectortoward fields) these conjectures, onH surfaces focusing of higher ongenus the theoryg 2. of graduate students on upwards. Ninety-seven young ≥ action of one-parameter subgroups: the geodesic Thereentropy; is the a natural applicationsSL(2, toR) the action Littlewood on this conjec- space, mathematicians participated, from Europe, Asia, flow, given by action of the diagonal subgroup comingture on from simultaneous the linear diophantine action on R approximation;2. The action the Middle East, and the Americas. In addition ofand the the subgroup theory ofA quantumis called uniqueTeichm¨uller ergodicity. geodesic to bringing many of the participants in these fields et 0 flow, and the orbit of a point x under this together to exchange ideas, the summer school has A = g = ; Building on these lectures, Nalini∈ H Ananthara- t 0 e t flow yields information about the ergodic proper- hopefully helped spark the interest of a new gener- − t man, Gergely Harcos, Hee Oh, and Akshay Venka-   ∈R ties of the associated vector field. ation of mathematicians in these beautiful areas. tesh gave shorter series of more advanced lectures, and the horocycle flow, given by action of the unipo- 24 CMI indicatingANNUALA seemingly REPORT applications unrelated to family quantum of dynamical chaos, auto- sys- tent subgroup temsmorphic are forms,interval and exchange counting maps points: given on varieties. a partition In ofaddition, the unit a specialinterval session into n onlabeled Diophantine subintervals, approxi- re- 1 t arrangemation was them organized according by to Dmitry a permutation Kleinbock,π allow-S . U = ht = . ∈ n 01 t However,ing many if of we the take younger a firstreturn participants map for to the present flow   ∈R associatedtheir recent to results a vector in the field subject.x to a transverse ∈ H Two sets of lectures focused primarily on gen- interval, we obtain exactly one of these exchange eralizations of each of these actions: Alex Eskin maps. and Dmitry Kleinbock (with a two lecture prequel In the last foundational course, Jean-Chrstophe by Grigorii Margulis) delivered a lecture series on Yoccoz explored this connection from the perspec- the action of unipotent subgroups H on homoge- tive of combinatorics and dynamics of interval ex- neous spaces G/Γ, exploring the results of Dani, changes. In particular, he showed how to use the Margulis, and others on non-divergence of orbits; ergodicity of Teichm¨ullerflow and associated renor- Margulis’s use of unipotent dynamics to prove the malization procedures on the space of interval ex- Oppenheim conjecture on the values of quadratic changes to show the resolution (by Masur & Veech) forms; and Ratner’s classification of orbit closures of the Keane conjecture that almost every interval and invariant measures. Here, even in the case exchange map is uniquely ergodic. n = 2, the situation is quite rigid: for example, all orbits of the group U are either periodic or dense. Following Yoccoz’s lectures, Artur Avila, Gio- vanni Forni, and Maryam Mirzakhani gave further In contrast, the action of A for n = 2 is re- talks on Teichm¨ullerdynamics, exploring applica- markably chaotic: given any 1 α 3, one can tions to the ergodic theory of polygonal billiards, ≤ ≤ produce an orbit whose closure has Hausdorff di- and studying closely related spaces of foliations and mension α. However, for n 3, there are conjec- laminations on surfaces. tures of Margulis on the rigidity≥ of the action of the diagonal subgroup. Two of the main contrib- All the lecturers made a special effort to ensure utors to this field, Manfred Einsiedler and Elon their presentations would be accessible to all the Lindenstrauss, gave lectures on the progress made participants in the summer school, from beginning toward these conjectures, focusing on the theory of graduate students on upwards. Ninety-seven young entropy; the applications to the Littlewood conjec- mathematicians participated, from Europe, Asia, ture on simultaneous diophantine approximation; the Middle East, and the Americas. In addition and the theory of quantum unique ergodicity. to bringing many of the participants in these fields together to exchange ideas, the summer school has Building on these lectures, Nalini Ananthara- hopefully helped spark the interest of a new gener- man, Gergely Harcos, Hee Oh, and Akshay Venka- ation of mathematicians in these beautiful areas. tesh gave shorter series of more advanced lectures, indicating applications to quantum chaos, auto- morphic forms, and counting points on varieties. In addition, a special session on Diophantine approxi- mation was organized by Dmitry Kleinbock, allow- ing many of the younger participants to present their recent results in the subject. Clay Lectures on Mathematics, Mumbai Integrals connected to singularities (complex powers, p-adic zeta functions): Let K be a field The 2007 Clay Lectures on Mathematics were hosted with an absolute value . and a measure. Con- by the School of Mathematics, Tata Institute of sider the corresponding| product| measure on Kn. Fundamental Research (TIFR) in India from De- Roughly speaking, the goal is to relate the singu- cember 10 through 14, 2007. larities of a polynomial f K[x1, ,xn] with the asymptotic behavior of µ∈( x K···n f(x) < ), The lecturers were Mircea Mustat¸ˇaof the Univer- when goes to zero. One{ way∈ of encoding|| | this} a- sity of Michigan and Elon Lindenstrauss of Prince- symptotic behavior is by studying certain integrals. ton University. The key is to use a log resolution of singularities Mustat¸ˇadelivered a series of three lectures: for f and some version of the Change of Variable Formula. The cases K = C, K = Qp (or more Singularities in the Minimal Model • general p-adic fields) and K = C((t)) are the main Program examples. The first case that was discussed was Arc Spaces and Motivic Integration theProgram overview Archimedean case (that is, when K = R or C), • Singularities in Positive Characteristic • which goes back to a question of Gelfand, subse- Clay Lectures on Mathematics at the quently proved by Atiyah, Bernstein and Gelfand. and one public lecture Mustat¸ˇathen gave an overview of the p-adic side of Tata Institute of FundamentalIntegrals, Research the Change of Variable Formula, the theory. There was a discussion of p-adic inte- • and Singularities. gration, the Igusa zeta function and the main result of Igusa about the rationality of the zeta function. Lindenstrauss delivered a parallel series of three In this context, he emphasized the connection be- lectures: tween the largest pole of the zeta function and the log canonical threshold. Flows from Unipotent to Diagonalizable (and • what is in between) Values of some Integral and Non-integral Invariants of singularities (the log canoni- • Forms cal threshold) in birational geometry: In this Equidistribution and Stationary Measures lecture Mustat¸ˇadiscussed the role that singulari- • on the Torus ties play in higher dimensional birational geometry. The first part covered the role of vanishing theo- and one public lecture rems and the canonical divisor. This motivated the definition of various classes of singularities. After The Geometry and Dynamics of Numbers. a brief overview of the basic setup in the Minimal • Model Program, and of the recent progress in this The following is a summary of Mustat¸ˇa’sand Lin- field, Mustat¸ˇaexplained a conjecture of Shokurov denstrauss’s lectures. and its relevance in this setting.

Spaces of arcs and singularities: The talk cov- 1. Singularities in Algebraic Geometry: ered some basic facts about spaces of arcs, with an Mircea Mustat¸aˇ emphasis on geometric aspects and applications to singularities. One result that was discussed was a The role of singularities in the program of classify- theorem of Kontsevich relating the spaces of arcs ing higher-dimensional algebraic varieties is well- of two smooth varieties X and Y , with a proper known. The public lecture gave an introduction birational morphism between them. This result is to some results relating invariants of singularities the key ingredient in the theory of motivic integra- (like the log canonical threshold) with various inte- tion. In the rest of the talk, Mustat¸ˇaexplained how gration theories. The other lectures covered vari- this result can be used to relate the log canonical ous aspects of invariants of singularities, discussing threshold to the codimension of certain subsets in some of the recent results, as well as the main open spaces of arcs. This result can be considered an problems and conjectures in the field. The content analogue in the context of spaces of arcs of Igusa’s of the talks was roughly the following:

Clay Lectures on Mathematics, Mumbai Integrals connected to singularities (complex Clay Lectures on Mathematics, Mumbai powers,Integralsp-adic connected zeta functions) to singularities: Let K (complexbe a field The 2007 Clay Lectures on Mathematics were hosted withpowers, an absolutep-adic zeta value functions). and a: Letmeasure.K be a Con- field The 2007 Clay Lectures on Mathematics were hosted with an absolute value | . | and a measure. Con-n by the School of Mathematics, Tata Institute of sider the corresponding| product| measure on K n. Fundamentalby the School Research of Mathematics, (TIFR) in Tata India Institute from De- of Roughlysider the speaking, corresponding the goal product is to relatemeasure the on singu-K . Fundamental Research (TIFR) in India from De- Roughly speaking, the goal is to relate the singu- cember 10 through 14, 2007. larities of a polynomial f K[x1, ,xn] with the cember 10 through 14, 2007. asymptoticlarities of a polynomialbehavior of fµ∈( xK[x1K,···n ,xf(nx]) with< the), The lecturers were Mircea Mustat¸ˇaof the Univer- asymptotic behavior of µ∈({x ∈ K···n||f(x)| < }), The lecturers were Mircea Mustat¸ˇaof the Univer- when goes to zero. One way of encoding this a- sity of Michigan and Elon Lindenstrauss of Prince- when goes to zero. One{ way∈ of encoding|| | this} a- sity of Michigan and Elon Lindenstrauss of Prince- symptotic behavior is by studying certain integrals. ton University. symptotic behavior is by studying certain integrals. ton University. The key is to use a log resolution of singularities Mustat¸ˇadelivered a series of three lectures: forThef keyand is some to use version a log of resolution the Change of singularities of Variable Mustat¸ˇadelivered a series of three lectures: for f and some version of the Change of Variable Formula. The cases K = C, K = Qp (or more Singularities in the Minimal Model Formula. The cases K = C, K = Qp (or more • Singularities in the Minimal Model general p-adic fields) and K = C((t)) are the main • Program examples.general p-adic The fields) first caseand K that= C was((t)) discussed are the main was Program examples. The first case that was discussed was Arc Spaces and Motivic Integration the Archimedean case (that is, when K = R or C), • SingularitiesArc Spaces and in Positive Motivic Characteristic Integration the Archimedean case (that is, when K = R or C), • Singularities in Positive Characteristic which goes back to a question of Gelfand, subse- • quentlywhich goes proved back by to Atiyah, a question Bernstein of Gelfand, and Gelfand. subse- and one public lecture quently proved by Atiyah, Bernstein and Gelfand. and one public lecture Mustat¸ˇathen gave an overview of the p-adic side of Mustat¸ˇathen gave an overview of the p-adic side of Integrals, the Change of Variable Formula, the theory. There was a discussion of p-adic inte- • Integrals, the Change of Variable Formula, gration,the theory. the Igusa There zeta was function a discussion and the of p main-adic result inte- • and Singularities. and Singularities. ofgration, Igusa theabout Igusa the zeta rationality function of and the the zeta main function. result of Igusa about the rationality of the zeta function. Lindenstrauss delivered a parallel series of three In this context, he emphasized the connection be- In this context, he emphasized the connection be- lectures:Lindenstrauss delivered a parallel series of three tween the largest pole of the zeta function and the2007 25 lectures: logtween canonical the largest threshold. pole of the zeta function and the Flows from Unipotent to Diagonalizable (and log canonical threshold. • Flows from Unipotent to Diagonalizable (and • what is in between) Valueswhat is of in some between) Integral and Non-integral Invariants of singularities (the log canoni- • Values of some Integral and Non-integral calInvariants threshold) of singularitiesin birational geometry(the log: canoni- In this • Forms cal threshold) in birational geometry: In this EquidistributionForms and Stationary Measures lecture Mustat¸ˇadiscussed the role that singulari- • Equidistribution and Stationary Measures tieslecture play Mustat¸ˇadiscussed in higher dimensional the birational role that geometry. singulari- • on the Torus on the Torus Theties play first in part higher covered dimensional the role birational of vanishing geometry. theo- The first part covered the role of vanishing theo- and one public lecture rems and the canonical divisor. This motivated the and one public lecture definitionrems and the of various canonical classes divisor. of singularities. This motivated After the The Geometry and Dynamics of Numbers. adefinition brief overview of various of the classes basic of setup singularities. in the Minimal After • The Geometry and Dynamics of Numbers. a brief overview of the basic setup in the Minimal • Model Program, and of the recent progress in this The following is a summary of Mustat¸ˇa’sand Lin- Model Program, and of the recent progress in this The following is a summary of Mustat¸ˇa’sand Lin- field, Mustat¸ˇaexplained a conjecture of Shokurov denstrauss’s lectures. andfield, its Mustat¸ˇaexplained relevance in this setting. a conjecture of Shokurov denstrauss’s lectures. and its relevance in this setting.

Spaces of arcs and singularities: The talk cov- 1. Singularities in Algebraic Geometry: eredSpaces some of basic arcs facts and about singularities spaces of: Thearcs, talk with cov- an 1. Singularities in Algebraic Geometry: Mircea Mustat¸aˇ emphasisered some on basic geometric facts about aspects spaces and of applications arcs, with anto Mircea Mustat¸aˇ singularities.emphasis on geometric One result aspects that was and discussed applications was to a The role of singularities in the program of classify- theoremsingularities. of Kontsevich One result relating that was the discussed spaces of was arcs a The role of singularities in the program of classify- ing higher-dimensional algebraic varieties is well- oftheorem two smooth of Kontsevich varieties relatingX and theY , with spaces a properof arcs ing higher-dimensional algebraic varieties is well- known. The public lecture gave an introduction birationalof two smooth morphism varieties betweenX and them.Y , with This a result proper is known. The public lecture gave an introduction to some results relating invariants of singularities thebirational key ingredient morphism in the between theory them. of motivic This resultintegra- is to some results relating invariants of singularities (like the log canonical threshold) with various inte- tion.the key In ingredientthe rest of the in the talk, theory Mustat¸ˇaexplained of motivic integra- how (like the log canonical threshold) with various inte- gration theories. The other lectures covered vari- thistion. result In the can rest be of used the talk, to relate Mustat¸ˇaexplained the log canonical how gration theories. The other lectures covered vari- ous aspects of invariants of singularities, discussing thresholdthis result to can the be codimension used to relate of certain the log subsets canonical in ous aspects of invariants of singularities, discussing some of the recent results, as well as the main open spacesthreshold of arcs. to the This codimension result can of certainbe considered subsets an in some of the recent results, as well as the main open problems and conjectures in the field. The content analoguespaces of in arcs. the context This result of spaces can be of arcs considered of Igusa’s an ofproblems the talks and was conjectures roughly the in thefollowing: field. The content analogue in the context of spaces of arcs of Igusa’s of the talks was roughly the following: Clay Lectures on Mathematics, Mumbai Integrals connected to singularities (complex powers, p-adic zeta functions): Let K be a field The 2007 Clay Lectures on Mathematics were hosted with an absolute value . and a measure. Con- by the School of Mathematics, Tata Institute of sider the corresponding| product| measure on Kn. Fundamental Research (TIFR) in India from De- Roughly speaking, the goal is to relate the singu- cember 10 through 14, 2007. larities of a polynomial f K[x1, ,xn] with the asymptotic behavior of µ∈( x K···n f(x) < ), The lecturers were Mircea Mustat¸ˇaof the Univer- when goes to zero. One{ way∈ of encoding|| | this} a- sity of Michigan and Elon Lindenstrauss of Prince- symptotic behavior is by studying certain integrals. ton University. The key is to use a log resolution of singularities Mustat¸ˇadelivered a series of three lectures: for f and some version of the Change of Variable Formula. The cases K = C, K = Qp (or more Singularities in the Minimal Model • general p-adic fields) and K = C((t)) are the main Program examples. The first case that was discussed was Arc Spaces and Motivic Integration the Archimedean case (that is, when K = R or C), • Singularities in Positive Characteristic Clay Lectures on Mathematics at the • which goes back to a question of Gelfand, subse- quently proved by Atiyah, Bernstein and Gelfand. and one public lecture TataMustat¸ˇathen Institute gave an overview of the Fundamentalp-adic side of Research Integrals, the Change of Variable Formula, the theory. There was a discussion of p-adic inte- • and Singularities. gration, the Igusa zeta function and the main result of Igusa about the rationality of the zeta function. Lindenstrauss delivered a parallel series of three In this context, he emphasized the connection be- results in the p-adic setting. Rigidity of unipotent and diagonal flows: The lectures: tween the largest pole of the zeta function and the general setting is always that of a homogeneous space Γ G where G is a linear algebraic group log canonical threshold. Invariants of singularities in positive char- Flows from Unipotent to Diagonalizable (and and Γ one\ of its discrete subgroups. From Rat- • acteristic: The invariants of singularities that ap- what is in between) ner’s work, it is well known that unipotent actions pear in birational geometry (in characteristic zero) Values of some Integral and Non-integral Invariants of singularities (the log canoni- are rigid: if H is a subgroup of G generated by • are defined via valuations, or equivalently, via res- Forms cal threshold) in birational geometry: In this unipotent one-parameter subgroups, then any H- olution of singularities. In positive characteristic, Equidistribution and Stationary Measures lecture Mustat¸ˇadiscussed the role that singulari- invariant probability measure on Γ G is homoge- • one can define invariants in a very algebraic way, on the Torus ties play in higher dimensional birational geometry. neous and any orbit of a one-parameter\ unipotent exploiting the Frobenius morphism. In this talk. The first part covered the role of vanishing theo- subgroup is equidistributed in the closure of the H

Program overview Mustat¸ˇadiscussed analogies, results, and conjec- and one public lecture rems and the canonical divisor. This motivated the orbit. A result of Mozes and Shah claims that for a tures relating the invariants such as the log canon- definition of various classes of singularities. After sequence µ of natural measures on periodic H- The Geometry and Dynamics of Numbers. a brief overview of the basic setup in the Minimal ical threshold with algebraic invariants in positive { i} • orbits, there would be a subsequence converging to characteristic. Model Program, and of the recent progress in this a homogeneous measure unless all points in their The following is a summary of Mustat¸ˇa’sand Lin- field, Mustat¸ˇaexplained a conjecture of Shokurov denstrauss’s lectures. orbits escape to infinity. There remain several chal- and its relevance in this setting. lenges in the domain of unipotent dynamics: spaces Γ G of infinite volume, polynomial trajectories and \ Spaces of arcs and singularities: The talk cov- effective estimates. Lindenstruass discussed these 1. Singularities in Algebraic Geometry: ered some basic facts about spaces of arcs, with an 2. Aspects of Homogeneous Dynamics challenges, and some of the recent progress that Mircea Mustat¸aˇ emphasis on geometric aspects and applications to has been made. The Geometry and Dynamics of Numbers: singularities. One result that was discussed was a The second part of the lecture focused on the rigid- The role of singularities in the program of classify- Lindenstrauss began his public lecture by explain- theorem of Kontsevich relating the spaces of arcs ity of diagonal flows. Although the linearization ing higher-dimensional algebraic varieties is well- of two smooth varieties X and Y , with a proper ingElon Minkowski’sLindenstrauss delivering idea thatone of thehis lectures study at of TIFR. lattice points n technique does not work in this case, there is a known. The public lecture gave an introduction in R implies many deep results about number birational morphism between them. This result is weaker analogue called isolation, that essentially to some results relating invariants of singularities fields. Given the determinant, some lattices be- the key ingredient in the theory of motivic integra- says that any point close enough to (but not in- (like the log canonical threshold) with various inte- have more “unboundedly” than others in the sense tion. In the rest of the talk, Mustat¸ˇaexplained how side) a periodic orbit has an unbounded orbit it- gration theories. The other lectures covered vari- this result can be used to relate the log canonical that they contain very short vectors. The group n self. However, isolation techniques are proven only ous aspects of invariants of singularities, discussing resultsthreshold in the to thep-adic codimension setting. of certain subsets in SLRigidity(n, R) acts of unipotent on the space and of diagonal lattices in flowsR while: The for G = SL(n, R). Let A be the full diagonal sub- some of the recent results, as well as the main open spaces of arcs. This result can be considered an preservinggeneral setting the determinant. is always that The of key a homogeneous observation group; when G = SL(2, R) the orbits are known to problems and conjectures in the field. The content analogue in the context of spaces of arcs of Igusa’s herespace is thatΓ G orbitswhere ofG actionsis a linear of large algebraic enough diago- group Invariants of singularities in positive char- \ behave in a nasty way. But for G = SL(n, R),n of the talks was roughly the following: results in the p-adic setting. nalandRigidity subgroupsΓ one of of unipotent of itsSL discrete(n, R) and have subgroups. diagonal to either flowsFrom contain: Rat- The an ≥ acteristic: The invariants of singularities that ap- 3, it is conjectured that any ergodic A-invariant “unbounded”ner’sgeneral work, setting it lattice is wellis always or known be periodic. that that of unipotent a Two homogeneous theorems actions pear in birational geometry (in characteristic zero) measure is homogeneous. Lindenstrauss discussed fromarespace rigid: numberΓ G ifwhereH theory,is aG subgroup Diophantineis a linear of G algebraic approximationgenerated group by areInvariants defined via of valuations, singularities or equivalently, in positive via char- res- \ high entropy techniques, as well as the recent work andunipotentand theΓ one existence one-parameter of its discrete of solutions subgroups, subgroups. to Pell’s then From equation, any Rat-H- olutionacteristic of: singularities. The invariants In of positive singularities characteristic, that ap- of Maucourant and its relation to this conjecture. caninvariantner’s be work, demonstrated probability it is well known as measure elegant that on examples unipotentΓ G is homoge-of actions these onepear can in birational define invariants geometry in (in a very characteristic algebraic zero) way, twoneousare situations. rigid: and anyif H orbitAfteris a of subgroupmentioning a one-parameter of theG\ threegenerated unipotent dimen- by exploitingare defined the via Frobenius valuations, morphism. or equivalently, In this via talk. res- sionalsubgroupunipotent setting is one-parameter equidistributed (Dirichlet Theorem), subgroups, in the closure Lindenstrauss then of any theHH- Values of integral and non-integral forms: Mustat¸ˇadiscussedolution of singularities. analogies, In positive results, characteristic, and conjec- discussedorbit.invariant A result Cassels probability of and Mozes Swinnerton-Dyer measure and Shah on claimsΓ G conjecture,is that homoge- for a The main focus of this lecture was on homoge- turesone can relating define the invariants invariants in such a very as thealgebraic log canon- way, \ whichsequenceneous was and reformulatedµ anyi of orbit natural of a by one-parametermeasures Margulis on in periodic the unipotent follow-H- neous polynomials F that can be written in the icalexploiting threshold the with Frobenius algebraic morphism. invariants In in this positive talk. n n ingorbits,subgroup way: there in{ is the equidistributed would} moduli be a space subsequence in of then-dimensional closure converging of the lat-H to form F = ( h x ) where H =(h ) is a characteristic.Mustat¸ˇadiscussed analogies, results, and conjec- i=1 i=1 ij j ij tices,aorbit. homogeneousSL A( resultn, Z) ofSL measure Mozes(n, R), and unless any Shah orbit all claims points of the that in full their for di- a non-degenerate n n real matrix. Lindenstrauss tures relating the invariants such as the log canon- \ × agonalorbitssequence escape subgroupµi toof infinity. of naturalSL(n, There measuresR) is remain either on several periodicunbounded chal-H- discussed the example of such forms arising as the ical threshold with algebraic invariants in positive { } ororbits, periodic. there It would was orbserved be a subsequence by Cassels converging and Swin- to norm of an element in a number field of degree n characteristic. lenges in the domain of unipotent dynamics: spaces nerton-Dyer(1955)Γa homogeneousG of infinite volume, measure and polynomial Margulis unless all (1997) trajectories points that in their this and given an integral basis. The objective is to study \ n conjectureeorbitsffective escape estimates. would to infinity. imply Lindenstruass the There famous remain Littlewood discussed several these chal-con- the distribution of F (Z ), which is preserved under 2 2. Aspects of Homogeneous Dynamics jecture:challenges,lenges in the(α and, domainβ) someR of, of unipotent inf then recentn dynamics:nα progressnβ spaces= that 0, the action of GL(n, ) given by γ.F (x)=F (xγ). ∀ ∈ ∈N    Z wherehasΓ G beenofx infinite made.= min volume,z x polynomialz . By studying trajectories dynamics and One observes that the dynamics of GL(n, ) on the \   ∈Z | − | Z The Geometry and Dynamics of Numbers: oneffectiveSL(n, Z estimates.) SL(n, R Lindenstruass), Einsiedler, Katok, discussed and these Lin- space Y of equivalence classes of F up to scaling The second\ part of the lecture focused on the rigid- n Lindenstrauss2. Aspects began of Homogeneous his public lecture Dynamics by explain- denstrausschallenges, recently and some proved of the that recent the exceptional progress that set is closely related to that of A on ing Minkowski’s idea that the study of lattice points ofityhas Littlewood’s of been diagonal made. conjecture flows. Although is of Hausdor the linearizationff dimesion n technique does not work in this case, there is a P GL(n, Z) P GL(n, R)=SL(n, Z) SL(n, R). inTheR Geometryimplies many and deep Dynamics results about of Numbers number: 0. Lindenstrauss also talked about applications of \ \ weakerThe second analogue part of called the lecture isolation, focused that on essentially the rigid- n fields.Lindenstrauss Given beganthe determinant, his public lecture some lattices by explain- be- geometry of numbers to the study of number fields. Conditions on the distribution of F (Z ) can be ity of diagonal flows. Although the linearization haveing Minkowski’s more “unboundedly” idea that the than study others of lattice in the points sense says that any point close enough to (but not in- translated into conditions on the A-orbit of [H] n technique does not work in this case, there is a ∈ 26 CMI thatinANNUALR they impliesREPORT contain many very deep short results vectors. about The number group side) a periodic orbit has an unbounded orbit it- weaker analogue called isolation, that essentially SLfields.(n, Given) acts onthe the determinant, space of lattices some in latticesn while be- self. However, isolation techniques are proven only R R says that any point close enough to (but not in- preservinghave more “unboundedly” the determinant. than The others key inobservation the sense for G = SL(n, R). Let A be the full diagonal sub- side) a periodic orbit has an unbounded orbit it- herethat is they that contain orbits of very actions short of vectors. large enough The diago- group group; when G = SL(2, R) the orbits are known to n self. However, isolation techniques are proven only nalSL( subgroupsn, R) acts of onSL the(n, space) have of lattices to either in containR while an behave in a nasty way. But for G = SL(n, R),n R for G = SL(n, ). Let A be the full diagonal sub-≥ “unbounded”preserving the lattice determinant. or be periodic. The key Two observation theorems 3, it is conjecturedR that any ergodic A-invariant group; when G = SL(2, ) the orbits are known to fromhere is number that orbits theory, of actions Diophantine of large approximation enough diago- measure is homogeneous.R Lindenstrauss discussed behave in a nasty way. But for G = SL(n, ),n andnal subgroups the existence of SL of(n, solutionsR) have to to either Pell’s contain equation, an high entropy techniques, as well as the recentR work 3, it is conjectured that any ergodic A-invariant≥ can“unbounded” be demonstrated lattice or as be elegant periodic. examples Two theorems of these of Maucourant and its relation to this conjecture. measure is homogeneous. Lindenstrauss discussed twofrom situations. number theory, After mentioning Diophantine the approximation three dimen- high entropy techniques, as well as the recent work sionaland the setting existence (Dirichlet of solutions Theorem), to Pell’s Lindenstrauss equation, Valuesof Maucourant of integral and its and relation non-integral to this conjecture. forms: discussedcan be demonstrated Cassels and Swinnerton-Dyeras elegant examples conjecture, of these The main focus of this lecture was on homoge- whichtwo situations. was reformulated After mentioning by Margulis the in three the dimen-follow- neous polynomials F that can be written in the ingsional way: setting in the (Dirichlet moduli space Theorem), of n-dimensional Lindenstrauss lat- Values of integraln n and non-integral forms: form F = i=1( i=1 hijxj) where H =(hij) is a tices,discussedSL( Casselsn, Z) SL and(n, Swinnerton-DyerR), any orbit of theconjecture, full di- non-degenerateThe main focusn ofn thisreal lecture matrix. was Lindenstrauss on homoge- \ agonalwhich was subgroup reformulated of SL(n, byR Margulis) is either in unbounded the follow- discussedneous polynomials the example×F that of such can forms be written arising inas the ing way: in the moduli space of n-dimensional lat- n n or periodic. It was orbserved by Cassels and Swin- normform F of= an elementi=1( i=1 inh aijx numberj) where fieldH =( of degreehij) isn a nerton-Dyer(1955)tices, SL(n, Z) SL( andn, R), Margulis any orbit (1997) of the that full this di- givennon-degenerate an integraln basis.n real The matrix. objective Lindenstrauss is to study \ × conjectureagonal subgroup would imply of SL( then, R famous) is either Littlewood unbounded con- thediscussed distribution the example of F (Zn of), such which forms is preserved arisingas under the or periodic. It was orbserved2 by Cassels and Swin- jecture: (α, β) R , infn N n nα nβ = 0, thenorm action of an of elementGL(n, Z in) a given number by γ field.F (x of)= degreeF (xγ).n nerton-Dyer(1955)∀ ∈ and Margulis∈  (1997) that this where x = minz Z x z . By studying dynamics Onegiven observes an integral that basis. the dynamics The objective of GL(n, isZ to) on study the conjecture  would imply∈ | − the| famous Littlewood con- n on SL(n, Z) SL(n, R), Einsiedler, Katok, and Lin- spacethe distributionYn of equivalence of F (Z ), classes which of is preservedF up to scaling under \ 2 denstraussjecture: ( recentlyα, β) provedR , inf thatn n thenα exceptionalnβ = set 0, the action of GL(n, ) given by γ.F (x)=F (xγ). ∀ ∈ ∈N    is closely related to thatZ of A on ofwhere Littlewood’sx = min conjecturez Z x z is. By of Hausdor studyingff dynamicsdimesion One observes that the dynamics of GL(n, Z) on the   ∈ | − | P GL(n, ) P GL(n, )=SL(n, ) SL(n, ). 0.on LindenstraussSL(n, Z) SL(n, alsoR), talked Einsiedler, about Katok, applications and Lin- of space Y ofZ equivalenceR classes of FZ up to scalingR \ n \ \ geometrydenstrauss of recently numbers proved to the that study the of exceptional number fields. set Conditionsis closely related on the todistribution that of A on of F (Zn) can be of Littlewood’s conjecture is of Hausdorff dimesion translated into conditions on the A-orbit of [H] P GL(n, Z) P GL(n, R)=SL(n, Z) SL(n, R). ∈ 0. Lindenstrauss also talked about applications of \ \ geometry of numbers to the study of number fields. Conditions on the distribution of F (Zn) can be translated into conditions on the A-orbit of [H] ∈ Program overview

results in the p-adic setting. Rigidity of unipotent and diagonal flows: The P GL(n, Z) P GL(n, R). By such translation, the general setting is always that of a homogeneous reformulation\ of Margulis mentioned in his public space Γ G where G is a linear algebraic group Invariants of singularities in positive char- lecture is equivalent to the original Cassels-Swinn- and Γ one\ of its discrete subgroups. From Rat- acteristic: The invariants of singularities that ap- erton-Dyer conjecture. Lindenstrauss explained his ner’s work, it is well known that unipotent actions pear in birational geometry (in characteristic zero) work with Einsiedler and Katok on the Hausdorff are rigid: if H is a subgroup of G generated by are defined via valuations, or equivalently, via res- dimension of the exceptional set as well as more re- unipotent one-parameter subgroups, then any H- olution of singularities. In positive characteristic, cent work with Einsiedler, Michel, and Venkatesh invariant probability measure on Γ G is homoge- one can define invariants in a very algebraic way, \ that gives an upper bound # orbit GL(n, Z).F neous and any orbit of a one-parameter unipotent { n ⊂ exploiting the Frobenius morphism. In this talk. Yn of determinant D : F (Z 0 ) [ δ, δ]= subgroup is equidistributed in the closure of the H \{ } ∩ − Mustat¸ˇadiscussed analogies, results, and conjec- << ,δ D , , δ,D. orbit. A result of Mozes and Shah claims that for a ∅} ∀ tures relating the invariants such as the log canon- sequence µ of natural measures on periodic H- ical threshold with algebraic invariants in positive i orbits, there{ would} be a subsequence converging to The Torus: Equidistribution and Stationary characteristic. a homogeneous measure unless all points in their Measures: Suppose S is the multiplicative semi- orbits escape to infinity. There remain several chal- group generated by two multiplicatively indepen- lenges in the domain of unipotent dynamics: spaces dent postive integers a and b. Let S act natu- Γ G of infinite volume, polynomial trajectories and rally on T = R/Z. The Furstenberg theorem as- eff\ective estimates. Lindenstruass discussed these serts that any closed S-invariant subset of T is ei- 2. Aspects of Homogeneous Dynamics challenges, and some of the recent progress that ther full or finite. Furstenburg also conjectured has been made. that any ergodic S-invariant probability measure The Geometry and Dynamics of Numbers: µ is either Lebesgue or finitely supported. Though Lindenstrauss began his public lecture by explain- The second part of the lecture focused on the rigid- the conjecture is still open, Rudolph and Johnson ing Minkowski’s idea that the study of lattice points ity of diagonal flows. Although the linearization showed if h(µ, a) > 0 then µ has to be Lebesgue n technique does not work in this case, there is a in R implies many deep results about number measure. Lindenstrauss discussed how an effective fields. Given the determinant, some lattices be- weaker analogue called isolation, that essentially Rudolph-Johnson theorem can be used to prove an have more “unboundedly” than others in the sense says that any point close enough to (but not in- effective version of Furstenberg’s theorem. He then that they contain very short vectors. The group side) a periodic orbit has an unbounded orbit it- discussed how higher-rank analogues of these prob- n self. However, isolation techniques are proven only d SL(n, R) acts on the space of lattices in R while lems on (R/Z) ,d>1 diverge into two cases. The preserving the determinant. The key observation for G = SL(n, R). Let A be the full diagonal sub- first type of problems deal with two commuting here is that orbits of actions of large enough diago- group; when G = SL(2, R) the orbits are known to matrices A, B SL(d, Z) acting on (R/Z)d, there behave in a nasty way. But for G = SL(n, R),n ∈ nal subgroups of SL(n, R) have to either contain an ≥ are results by Berend, Kalinin-Katok-Spatzier and “unbounded” lattice or be periodic. Two theorems 3, it is conjectured that any ergodic A-invariant Einsiedler-Lindenstrauss. The other extreme is a from number theory, Diophantine approximation measure is homogeneous. Lindenstrauss discussed pair of non-commuting matrices A and B generat- and the existence of solutions to Pell’s equation, high entropy techniques, as well as the recent work ing a Zariski-dense subgroup S in SL(d, Z). can be demonstrated as elegant examples of these of Maucourant and its relation to this conjecture. two situations. After mentioning the three dimen- sional setting (Dirichlet Theorem), Lindenstrauss Values of integral and non-integral forms: discussed Cassels and Swinnerton-Dyer conjecture, The main focus of this lecture was on homoge- which was reformulated by Margulis in the follow- neous polynomials F that can be written in the ing way: in the moduli space of n-dimensional lat- n n form F = i=1( i=1 hijxj) where H =(hij) is a tices, SL(n, Z) SL(n, R), any orbit of the full di- non-degenerate n n real matrix. Lindenstrauss \ agonal subgroup of SL(n, R) is either unbounded discussed the example× of such forms arising as the or periodic. It was orbserved by Cassels and Swin- norm of an element in a number field of degree n nerton-Dyer(1955) and Margulis (1997) that this given an integral basis. The objective is to study conjecture would imply the famous Littlewood con- the distribution of F (Zn), which is preserved under 2 jecture: (α, β) R , infn n nα nβ = 0, the action of GL(n, ) given by γ.F (x)=F (xγ). ∀ ∈ ∈N    Z where x = minz x z . By studying dynamics One observes that the dynamics of GL(n, ) on the   ∈Z | − | Z on SL(n, Z) SL(n, R), Einsiedler, Katok, and Lin- space Y of equivalence classes of F up to scaling \ n denstrauss recently proved that the exceptional set is closely related to that of A on of Littlewood’s conjecture is of Hausdorff dimesion P GL(n, Z) P GL(n, R)=SL(n, Z) SL(n, R). 0. Lindenstrauss also talked about applications of \ \ geometry of numbers to the study of number fields. Conditions on the distribution of F (Zn) can be translated into conditions on the A-orbit of [H] Mustata delivering one of his lectures at TIFR. ∈

2007 27

George Mackey Library

The Clay Mathematics Institute is honored to be the recipient of Professor George Mackey’s extensive mathematical library — a gift to the Institute by his widow Alice Mackey. The library adds 1220 volumes to the 720 previously received from the family of Raoul Bott in 2005. The addition of these books is a great benefit to CMI, both in carrying out its specific functions and in creating a general atmosphere that is conducive to doing mathematics, including the presentation of workshops at One Bow Institute news Street. George Mackey was born February 1, 1916, and George Mackey in his office at Harvard, c. 1970s. Photo courtesy of the Mackey family. died on March 15, 2006. He is survived by his widow, Alice Mackey, and his daughter, Ann under the direction of Marshall Stone. Following a Mackey. His mathematical work and legacy are one-year postdoctoral position at the Illinois Institute beautifully described in a twenty-seven page article of Technology, Mackey returned to Harvard as an in the Notices of the American Mathematical Society instructor. He was appointed the Landon T. Clay (Volume 54, number 7, August 2007), to which Professor of Mathematics and Theoretical Physics, thirteen authors, among them many of his twenty- and he remained at Harvard until he retired in 1985. three students, contributed. Mackey’s main areas of interest were in representation George Mackey received a bachelors degree theory, group actions, ergodic theory, functional in physics from the Rice Institute (now Rice analysis, and mathematical physics. He had a long University) in 1938, where he studied chemistry, and abiding interest in the mathematical foundations physics, and mathematics. As one of the top five of quantum mechanics, an interest that led to his William Lowell Putnam Contest winners, he earned now classic work, Mathematical Foundations of a scholarship to Harvard for graduate work, where Quantum Mechanics (1963). He was an influential he received his doctorate in mathematics in 1942 and beloved figure, and a mentor to many.

2007 Olympiad Scholar Andrew Geng

On May 21, 2007, Andrew American Mathematics Competitions and sponsored Geng of Westford, Mass- by the Mathematical Association of America and achusetts received the several other organizations. ninth Clay Mathematics The Clay Mathematics Olympiad Scholar Award is Olympiad Scholar Award given for the solution to an Olympiad problem judged at the 36th USA most creative. It consists of a plaque and cash award Mathematical Olympiad to the recipient, and a cash award to the recipient’s Awards Ceremony in school. It is presented each year at the official awards Washington, DC, the dinner for the US American Mathematics Olympiad nation’s premier high (USAMO) held in June in Washington, DC at the school level mathematics State Department Ballroom. problem-solving compe- tition. The USAMO is a Andrew is currently enjoying classes at MIT where Andrew Geng two-day, nine-hour, six- he has chosen to major in mathematics and devotes question, essay-proof examination that is one of much of his free time to teaching in the Educational a series of national contests administered by the Studies Program.

28 CMI ANNUAL REPORT Publications

Selected Articles by Research Fellows

MODHAMMED ABOUZAID DAVESH MAULIK “On the Fukaya categories of higher genus surfaces,” “Quantum Cohomology of Hilbert schemes of points on Advances in Mathematics (2007). A_n resolutions,” with A. Oblomkov. To appear.

“Homogeneous coordinate rings and mirror symmetry for “Gromov-Witten theory and Noether-Lefschetz theory,” toric varieties,” Geometry & Topology 10 (2006), 1097- with R. Pandharipande. math.AG/07051653. 1157. “A Topological View of Gromov-Witten Theory,” with R. DANIEL BISS Pandharipande. Topology, Vol. 45, no. 5, 2006. “Large annihilators in Cayley-Dickson algebras II,” with J. Daniel Christensen, Daniel Dugger, and Daniel C. Isaksen. MARYAM MIRZAKHANI Submitted Feb 4, 2007 (v1), last revised Feb 22, 2007 “Ergodic Theory of the Earthquake Flow.” Int Math Res (this version, v2). arXiv:math/0702075v2 [math.RA]. Notices (2008) Vol. 2008.

ARTUR AVILA “Ergodic Theory of the Space of Measured Laminations,” with Elon Lindenstrauss. Int Math Res Notices (2008) “Simplicity of Lyapunov spectra: proof of the Zorich- Vol. 2008. Kontsevich conjecture,” with M. Viana. Acta Mathematica 198 (2007), 1-56. SOPHIE MOREL “Weak mixing for interval exchange transformations “Complexes pondérés sur les compactifications de Baily- and translation flows,” with Giovanni Forni. Annals of Borel: Le cas des variétés de Siegel.” J. Amer. Math. Soc. Mathematics 165 (2007), 637-664. 21 (2008), 23-61 ext.

MARIA CHUDNOVSKY SAMUEL PAYNE “The Strong Perfect Graph Theorem,” with N. Robertson, “Toric vector bundles, branched covers of fans, and the P. Seymour, and R. Thomas. Annals of Math Vol 164 resolution property.” To appear in J. Algebraic Geom. (2006), 51-229. arXiv:math/0605537.

“Coloring quasi-line graphs,” with Alexandra Ovetsky. “Moduli of toric vector bundles.” To appear in Compositio Journal of Graph Theory Vol. 54 (2007), 41-50. Math. arXiv:0705.0410.

SOREN GALATIUS DAVID SPEYER “Divisibility of the Stable Miller-Morita-Mumford Classes,” ”Cambrian Fans,” with Nathan Reading. To appear in with I. Madsen, and U. Tillmann. J. Amer. Math. Soc. 19 Journal of the European Mathematical Society. (2006), no. 4, 759–779. “A Kleiman-Bertini Theorem for sheaf tensor products” with “The Homotopy Type of the Cobordism Category,” with I. Ezra Miller. J. Algebraic Geom. 17 (2008), 335-340. Madsen, U. Tillmann, and M. Weiss. To appear, Acta Math. TERUYOSHI YOSHIDA BO’AZ KLARTAG “Compatibility of local and global Langlands correspon- “A Berry-Esseen type inequality for convex bodies with an dences,” with Richard Taylor. J. Amer. Math. Soc., 20-2 unconditional basis,” Manuscript. (2007), 467-493.

“Power-law estimates for the central limit theorem for “Local class field theory via Lubin-Tate theory.” To appear convex sets.” J. Funct. Anal., Vol. 245 (2007), 284–310. in Annales de la Faculté des Sciences de Toulouse.

CIPRIAN MANOLESCU “A combinatorial description of knot Floer homology,” with P. Ozsavth and S. Sarkar. To appear in the Annals of Mathematics (2008).

“On combinatorial link Floer homology,” with P. Ozsvath, D. Thurston, and Z. Szabo. Geometry and Topology, Vol. 12 (2007), 2339-2412. 2007 29 Books & Videos

Analytic Number Theory; A Tribute to Gauss and Dirichlet; Editors: William Duke, Yuri Tschinkel. CMI/ AMS, 2007, 265 pp. www.claymath.org/publications/Gauss_Dirichlet. This volume contains the proceedings of the Gauss–Dirichlet Conference held in Göttingen from June 20–24 in 2005, commemorating the 150th anniversary of the death of Gauss and the 200th anniversary of Dirichlet’s birth. It begins with a definitive summary of the life and work of Dirichlet by J. Elstrodt and continues with thirteen papers by leading experts on research topics of current interest within number theory that were directly influenced by Gauss and Dirichlet.

Clay Mathematics Monographs Ricci Flow and the Poincaré Conjecture; Authors: John Morgan, Gang Tian. CMI/AMS, 3 Volume 3 Ricci Flow and the Poincaré Conjecture Poincaré the and Flow Ricci 2007, 521 pp., www.claymath.org/publications/ricciflow. This book presents a complete Ricci Flow and the and detailed proof of the Poincaré Conjecture. This conjecture was formulated by Henri

Publications Poincaré Conjecture Poincaré in 1904 and has remained open until the recent work of Grigory Perelman. The arguments given in the book are a detailed version of those that appear in Perelman’s three

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˜`À?ÃÊ°Ê-̈«ÃˆVâÊ Lecture Notes on Motivic Cohomology; Authors: Carlo Mazza, Vladimir Voevodsky, ASMOREADVANCEDRESEARCH LEVELTALKSGIVENDURINGTHISPROGRAM 4HERESULTINGVOLUMEPROVIDESASTATE OF THE ARTINTRODUCTIONTO <œÌ?˜Ê-â>LÊ CURRENTRESEARCH COVERINGMATERIALFROM(EEGAARD&LOERHOMOLOGY `ˆÌœÀà CONTACTGEOMETRY SMOOTHFOUR MANIFOLDTOPOLOGY ANDSYMPLECTIC Charles Weibel. CMI/AMS, 2006, 210 pp., www.claymath.org/publications/Motivic_ FOUR MANIFOLDS Cohomology. This book provides an account of the triangulated theory of motives. Its purpose is to introduce the reader to Motivic Cohomology, to develop its main properties,

ÜÜÜ°>“ðœÀ} !MERICAN-ATHEMATICAL3OCIETY #-)0 !-3 ÜÜÜ°V>ޓ>Ì °œÀ} #-) #LAY-ATHEMATICS)NSTITUTE and finally to relate it to other known invariants of algebraic varieties and rings such as Milnor K-theory, étale cohomology and Chow groups.

Clay Mathematics Proceedings The modern theory of automorphic forms, embodied in 4 Volume 4 what has come to be known as the Langlands program, is an extraordinary unifying force in mathematics. It proposes fundamental relations that tie arithmetic Surveys in Noncommutative Geometry; Editors: Nigel Higson, Johninformation Roe. from number theory and algebraic CMI/AMS, geometry with analytic information from harmonic analysis and group representations. These “reciprocity laws”, and Shimura VarietiesShimura and

conjectured by Langlands, are still largely unproved. Analysis, Harmonic 2006, 189 pp., www.claymath.org/publications/Noncommutative_Geometry.However, their capacity to unite large Inareas of June Formula, ofTrace the mathematics insures that they will be a central area of study for years to come.

The goal of this volume is to provide an entry point into 2000, a summer school on Noncom- mutative Geometry, organized jointlythis exciting and challengingby field.the It is directed American on the one hand at graduate students and professional mathematicians who would like to work in the area. The longer articles in particular represent an attempt to Mathematical Society and the Clay Mathematics Institute, was held at Mountenable a reader toHolyoke master some of the more difficult College techniques. On the other hand, the book will also be useful to mathematicians who would like simply to Proceedings of the Clay Mathematics Institute understand something of the subject. They will be able 2003 Summer School, The Fields Institute in Massachusetts. The meeting centered around several series of expositoryto consult the expositorylectures portions of the various that articles. were Toronto, Canada, June 2–27, 2003 The volume is centered around the trace formula and Shimura varieties. These areas are at the heart of the intended to introduce key topics in noncommutative geometry to mathematicianssubject, but they have been especially difficult unfamiliar to learn because of a lack of expository material. The volume aims to rectify the problem. It is based on the courses

given at the 2003 Clay Mathematics Institute Summer Kottwitz, Arthur,and Ellwood with the subject. Those expository lectures have been edited and are reproducedSchool. However, many ofin the articles this have been volume. expanded into comprehensive introductions, either to the trace formula or the theory of Shimura varieties, or to some aspect of the interplay and application of the two areas.

James Arthur Harmonic Analysis, the Trace Formula and Shimura Varieties; Proceedings of the 2003 CMI David Ellwood Robert Kottwitz Summer School at Fields Institute, Toronto. Editors: James Arthur, David Ellwood, Robert Editors Kottwitz. CMI/AMS, 2005, 689 pp., www.claymath.org/publications/Harmonic_Analysis.CMIP/4 Editors

www.ams.org American Mathematical Society The subject of this volume is the trace formula and Shimura varieties. These areas haveAMS www.claymath.org CMI Clay Mathematics Institute been especially difficult to learn because of a lack of expository material. This volume aims 4-color process 704 pages • 31 5/16” spine 30 CMI ANNUAL REPORT The AMS will provide a discount of 20% to students purchasing Clay publications. Publications To receive the discount, students should provide the reference code CLAY MATH. Online Orders: enter “CLAY MATH” in the comments field for each Clay publication ordered. Phone Orders: give Customer Service the reference code and they will extend a 20% discount on each Clay publiction ordered. to rectify that problem. It is based on the courses given at the 2003 Clay Mathematics Institute Summer School. Many of the articles have been expanded into comprehensive introductions, either to the trace formula or the theory of Shimura varieties, or to some aspect of the interplay and application of the two areas.

Global Theory of Minimal Surfaces; Proceedings of the 2001 CMI Summer School Clay Mathematics Proceedings at MSRI. Editor: David Hoffman. CMI/AMS, 2005, 800 pp., www.claymath.org/2 Volume 2 During the Summer of 2001, MSRI

hosted the Clay Mathematics Institute Surfaces Minimal of Theory Global publications/Minimal_Surfaces. This book is the product of the 2001Summer School onCMI the Global Theory of Summer Minimal Surfaces, during which 150 mathematicians—undergraduates, post- doctoral students, young researchers, School held at MSRI. The subjects covered include minimal and constant-mean-curvatureand the world's experts—participated in GLOBAL the most extensive meeting ever held on the subject in its 250-year history. The unusual nature of the meeting has made THEORY OF submanifolds, geometric measure theory and the double-bubble conjecture,it possible to assemble a volume Lagrangian of expository lectures, together with some specialized reports that give a MINIMAL geometry, numerical simulation of geometric phenomena, applicationspanoramic of picture mean of a vital subject, curvature presented with care by the best people in the field. SURFACES

The subjects covered include minimal Proceedings of the to general relativity and Riemannian geometry, the isoperimetric problem,and constant-mean-curvature the geometry submanifolds, geometric measure theory Clay Mathematics Institute and the double-bubble conjecture, 2001 Summer School of fully nonlinear elliptic equations, and applications to the topology of Lagrangianthree-manifolds. geometry, numerical Mathematical Sciences Research Institute simulation of geometric phenomena, applications of mean curvature to Berkeley, California general relativity and Riemannian June 25 – July 27, 2001 geometry, the isoperimetric problem, the geometry of fully nonlinear elliptic equations, and applications to the David Hoffman topology of three manifolds. Editor , Editor Hoffman , Strings and Geometry; Proceedings of the 2002 CMI Summer

School held at the Isaac Newton Institute forCMIP/2 Mathematical

www.ams.org American Mathematical Society Sciences, UK. Editors: Michael Douglas, Jerome Gauntlett,AMS www.claymath.org CMI Clay Mathematics Institute Mark Gross. CMI/AMS publication, 376 pp., Paperback, ISBN 0-8218-3715-X. List: $69. AMS 4-colorMembers: process $55. Order 816 pages • 1 9/16" spine code: CMIP/3. To order, visit www.ams.org/bookstore.

Mirror Symmetry; Authors: Kentaro Hori, Sheldon Katz, Clay Mathematics Monographs MIRROR SYMMETRY 1 Volume 1 Kentaro Hori, Sheldon Katz, Albrecht Klemm, Albrecht Klemm, Rahul Pandharipande,Rahul Pandharipande, Richard Richard Thomas, Thomas, Cumrun Vafa, Ravi Vakil, Eric Zaslow Mirror Symmetry Mirror

Mirror symmetry is a phenomenon arising in string theory in which two very Ravi Vakil. Editors: Cumrun Vafa,different manifolds Eric give rise to equivalent Zaslow. physics. Such a correspondence CMI/AMS has significant mathematical consequences, the most familiar of which involves the enumeration of holomorphic curves inside complex manifolds by solving differential equations obtained from a “mirror” geometry. The publication, 929 pp., Hardcover. inclusionISBN of D-brane states in 0-8218-2955-6.the equivalence has led to further conjectures List: involving calibrated submanifolds of the mirror pairs and new (conjectural) invariants of complex manifolds: the Gopakumar Vafa invariants. This book aims to give a single, cohesive treatment of mirror symmetry $124. AMS Members: $99. Orderfrom both code:the mathematical and physicalCMIM/1. viewpoint. Parts I and II develop To order, the necessary mathematical and physical background “from scratch,” and are intended for readers trying to learn across disciplines. The treatment

is focussed, developing only the material most necessary for the task. In

Parts III and IV the physical and mathematical proofs of mirror symmetry visit www.ams.org/bookstore. MIRROR SYMMETRY are given. From the physics side, this means demonstrating that two different physical theories give isomorphic physics. Each physical theory MIRROR SYMMETRY can be described geometrically,

and thus mirror symmetry gives rise to a “pairing” of geometries. The VafaCumrun R 1/R proof involves applying ↔ circle duality to the phases of the fields in the gauged linear sigma model. The mathematics proof develops Kentaro Hori Gromov-Witten theory in the algebraic setting, beginning with the moduli Sheldon Katz spaces of curves and maps, and uses localization techniques to show that certain hypergeometric functions encode the Gromov-Witten invari- Albrecht Klemm Strings 2001; Authors: Atish Dabholkar, Sunil Mukhi, Spenta R.ants in genus Wadia. zero, as is predicted by mirrorTata symmetry. PartInstitute V is devoted of Rahul Pandharipande to advanced topics in mirror symmetry, including the role of D-branes in Richard Thomas the context of mirror symmetry, and some of their applications in physics and mathematics. and mathematics; topological strings and large N and Cumrun Vafa Fundamental Research. Editor: American Mathematical SocietyChern-Simons (AMS), theory; geometric engineering; 2002, mirror symmetry at489 higher pp., Ravi Vakil

genus; Gopakumar-Vafa invariants; and Kontsevich's formulation of the Editors Zaslov , Eric mirror phenomenon as an equivalence of categories. Eric Zaslow This book grew out of an intense, month-long course on mirror symmetry Paperback, ISBN 0-8218-2981-5, List $74. AMS Members: $59.at Pine Manor Order College, sponsored bycode: the Clay Mathematics Institute.CMIP/1. The To several lecturers have tried to summarize this course in a coherent, order, visit www.ams.org/bookstore unified text.

Box for ISBN, Barcode and Itemcode www.ams.org American Mathematical Society AMS www.claymath.org CMI Clay Mathematics Institute Video Cassettes

The CMI Millennium Meeting Collection; Authors: Michael Atiyah, Timothy Gowers, John Tate, François Tisseyre. Editors: Tom Apostol, Jean-Pierre Bourguignon, Michele Emmer, Hans-Christian Hege, Konrad Polthier. Springer VideoMATH, © Clay Mathematics Institute, 2002. Box set consists of four video cassettes: The CMI Millennium Meeting, a film by François Tisseyre; The Importance of Mathematics, a lecture by Timothy Gowers; The Mil- lennium Prize Problems, a lecture by Michael Atiyah; and The Millennium Prize Problems, a lecture by John Tate. VHS/NTSC or PAL. ISBN 3-540-92657-7. List: $119, EUR 104.95. To order, visit www.springer-ny.com (in the United States) or www.springer.de (in Europe). These videos document the Paris meeting at the Collège de France where CMI announced the Millennium Prize Problems. The videos are for anyone who wants to learn more about these seven grand challenges in mathematics.

Videos of the 2000 Millennium event are available online and in VHS format from Springer-Verlag. To order the box set or individual tapes visit www.springer.com.

2007 31 2008 Institute Calendar

Cycles, Motives and Shimura Varieties at TIFR, Mumbai, India. January 3–12. JANUARY Recent Progress on the Moduli Space of Curves at Banff International Research Station, Canada. FEBRAURY March 16–21. CMI Workshop on K3s: modular forms, moduli, and string theory. March 20–23. MARCH Conference on Algebraic Cycles II at Ohio State University, Columbus, Ohio. March 24–29.

Additive Combinatorics, Number Theory, and Harmonic Analysis at the Fields Institute, Toronto, Canada. April 5–13.

Activities APRIL “The Music of the Primes,” Clay Public Lecture by Marcus du Sautoy at MIT. May 8. Workshop on Global Riemannian Geometry National Autonomous University of Mexico, Cuernavaca (IMATE-UNAM Cuernavaca), Mexico. May 11–18. Clay Research Conference at MIT. May 12–13. MAY HIRZ80 at Emmy Noether Research Institute for Mathematics at Bar Ilan University, Ramat Gan, Israel. May 18–23.

Senior Scholars John Lott and Gang Tian at Institut Henri Poincaré. May 1–June 30.

Conference on Motives, Quantum Field Theory and Pseudodifferential Operators, Boston University. June 4–14. A Celebration of Raoul Bott’s Legacy in Mathematics at CRM, Montreal. June 9–20. JUNE Conference on Aspects of Moduli Theory at the De Giorgi Center, Scuola Normale Superior di Pisa June 16–28. Senior Scholar Rob Lazarsfeld at PCMI/IAS. July 6–26. JULY CMI Summer School: Evolution Equations at the ETH, Eidgenossische Technische Hochschule, Swiss Federal Institute of Technology, Zurich, Switzerland. June 23–July 18. Algebraic Geometry, D-modules, Foliations and their Interactions, Buenos Aires, Argentina. AUGUST July 14–24. 60 Miles: A conference in Honor of Miles Reid’s 60th Birthday, LMS, London. July 18–19.

Senior Scholar Henri Gillet at the Fields Institute Program on Arithmetic Geometry, Hyperbolic SEPTEMBER Geometry and Related Topics. September 1–November 30. Senior Scholar Fedor Bogomolov at the Centro di Ricerca Matematica Program on Groups in Algebraic Geometry. September 1–November 30. OCTOBER Senior Scholar Richard Schoen at the Mittag-Leffler Institute Program on Geometry, Analysis and General Relativity. September 1–December 15. NOVEMBER Senior Scholar Werner Mueller at the MSRI Program on Analysis of Singular Spaces. September 10–October 21. DECEMBER Senior Scholar Gunther Uhlmann at the MSRI Program on Analysis of Singular Spaces. September 16–December 15.

32 CMI ANNUAL REPORT