Proceedings of The International Congress of Mathematicians 2010 (ICM 2010): Vol. IV

Contents......Page 6
Section 7 Lie Theory and Generalizations......Page 10
1. Introduction, Conjectures, and Results......Page 12
2. Quasi-isometries are Height Respecting......Page 15
3. Geometry of Sol......Page 17
4.1. Behavior of quasi-geodesics......Page 19
4.2. Averaging......Page 21
4.3. The scheme of the proof of Theorem 2.1......Page 22
4.5. Remarks on coarse differentiation......Page 24
4.6. Deduction of rigidity results......Page 27
5.1. Geometry of Diestel–Leader graphs......Page 28
5.2. Low dimensional topology and geometry......Page 30
5.4. Distortion of embeddings and multi-commodity flow problems......Page 31
References......Page 32
1. Introduction......Page 36
2. Definitions......Page 37
3. Resolutions and Deformations......Page 38
4. Representations and Hecke Algebras......Page 42
5. Reduction and Localisation......Page 45
References......Page 48
1. Introduction......Page 53
2. Notation......Page 55
3. Some Basic Results......Page 57
4. Root Components in the Tensor Product......Page 60
5. Proof of Parthasarathy-Ranga Rao-Varadarajan-Kostant Conjecture......Page 61
6. Determination of the Saturated Tensor Cone......Page 67
7. Special Isogenies and Tensor Product Multiplicities......Page 77
8. Saturation Problem......Page 81
9. Generalization of Littlewood-Richardson Formula......Page 83
References......Page 85
Introduction......Page 89
1.2. Regularization......Page 90
1.3. Geometric side......Page 91
1.4. The spectral side......Page 94
1.5. The Weyl law......Page 96
1.6. Limit multiplicities......Page 98
2. Periods of Automorphic Forms......Page 99
2.1. Unitary periods......Page 101
References......Page 104
1. Introduction......Page 108
Notation and conventions......Page 110
2.1. Kostant\'s results: the case of a principal nilpotent element......Page 111
2.4. Classical counterpart: the Slodowy slice......Page 112
2.5. Ramifications......Page 113
2.6. Additional properties of U(g, e)......Page 114
3.1. Fedosov quantization......Page 115
3.2. Equivariant Slodowy slices and W-algebras......Page 117
3.3. Decomposition theorem......Page 118
4.1. Whittaker modules and Skryabin\'s equivalence......Page 119
4.2. Category O for W-algebras......Page 120
5.1. Map •† : Jd(U) → Jd(W)......Page 123
5.2. Map •† : Jd(W) → Jd(U)......Page 124
5.3. Classification of finite dimensional irreducible W-modules......Page 125
6.2. Classical algebras......Page 126
6.4. 1-dimensional representations via category O(θ)......Page 127
7.1. W-algebras vs Yangians......Page 129
7.2. Other results......Page 130
References......Page 131
1. Introduction......Page 135
2. Preliminaries......Page 136
3. Counting and Distribution of Circles in the Plane......Page 139
3.1. Apollonian circle packings in the plane......Page 141
3.2. More circle packings......Page 144
4. Circle Packings on the Sphere......Page 145
5. Integral Apollonian Packings: Primes and Twin Primes......Page 147
6. Equidistribution in Geometrically Finite Hyperbolic Manifolds......Page 149
7. Further Remarks and Questions......Page 153
References......Page 154
1. Introduction......Page 159
2. Counting Integral Points on Varieties and Translates of Closed Orbits of Subgroups......Page 160
2.1. Expanding translates of smooth measures on horospherical leaves......Page 161
3. Limits of measures on stretching translates of submanifolds......Page 162
3.1. Translates of a finite arc under geodesic flow......Page 163
4. Applications to Diophantine approximation......Page 164
4.1. Multiplicative Dirichlet-Minkowski approximation......Page 165
5. Unipotent flows, Linearization and Linear dynamics......Page 167
References......Page 168
Introduction......Page 171
1. Classical and Quantum Schur-Weyl Duality......Page 174
1.2. Skein relations and crossings......Page 175
2. Categorification and Functorial Knot Invariants......Page 176
2.2. Braid group action and Serre functor......Page 178
2.3. Khovanov homology......Page 179
2.4. Knot invariants for other types......Page 180
3.1. Step 1: categorifications of certain gl∞-modules......Page 181
3.2. Step 2: Higher structure: cyclotomic Hecke algebras......Page 183
4.1. Khovanov-Lauda-Rouquier-Varagnolo-Vasserot algebras......Page 185
5.1. The category of finite dimensional GL(m|n)-modules......Page 186
5.3. The equivalence......Page 187
References......Page 188
1. Introduction......Page 193
2. Notation and Statements......Page 194
2.1. Applications to cup-products......Page 196
2.3. Mumford-Tate Groups......Page 197
3. The Action of the Cohomology of the Compact Dual......Page 198
4. Non-Hermitian Case......Page 199
References......Page 200
Section 8 Analysis......Page 204
1. Differentiability of Lipschitz Functions......Page 206
2. Structure of Null Sets and Other Problems......Page 214
2.2. Covering by Lipschitz slabs and intersecting by curves......Page 215
2.3. Mappings onto balls and weak derivatives......Page 216
2.4. Tangents of measures......Page 217
3. Combinatorial Connections......Page 219
References......Page 221
1. Introduction......Page 222
2. Szegö and Fisher-Hartwig Asymptotics......Page 223
3. The Riemann-Hilbert Method......Page 227
4. Fisher-Hartwig Asymptotics for Hankel Determinants......Page 229
5. Toeplitz + Hankel Determinants. The L - functions......Page 230
6. Transition Asymptotics and Painlevé Functions......Page 232
References......Page 235
1. Planar Sobolev Mappings......Page 238
2. Planar BV-mappings......Page 241
References......Page 242
1.1. Random matrix theory......Page 244
1.2. Unitary ensembles and orthogonal polynomials......Page 246
1.3. This paper......Page 247
2.2. Correlation kernel and RH problem......Page 248
3.1. Non-intersecting Brownian motion......Page 250
3.2. Non-intersecting squared Bessel paths......Page 251
4.1. Random matrices with external source......Page 252
5.1. Non-intersecting Brownian motion......Page 253
5.2. Random matrices with external source......Page 255
5.3. Two matrix model......Page 256
References......Page 257
1. Introduction......Page 260
2.2. Rigidity n ≥ 3......Page 262
2.5. Beltrami systems: Regularity......Page 263
3. Uniformly Quasiregular Mappings......Page 264
4.1. Smooth uqr mappings......Page 265
5. Dynamics of UQR mappings......Page 266
6.1. The Lichnerowicz problem......Page 267
6.3. Proper open surjections and π1......Page 269
7. Quasiregular Mappings Between Hyperbolic Manifolds......Page 270
8. Endomorphisms of Hyperbolic Groups......Page 271
8.1. Topological rigidity results......Page 272
9. Quasiregular Maps and Rigidity of Hyperbolic Manifolds......Page 273
References......Page 274
Random Complex Zeroes and Random Nodal Lines......Page 277
Part I. Random Complex Zeroes......Page 278
1. Linear Statistics......Page 279
2. Uniformity of Spreading of Random Complex Zeroes Over the Plane......Page 286
3. Almost Independence and Correlations......Page 291
4. Gaussian Spherical Harmonic and Gaussian Plane Wave......Page 295
5. Nodal Portrait......Page 298
6. The Sketch of the Proof of the Theorem on the Number of Nodal Domains......Page 302
Acknowledgements......Page 307
References......Page 308
1. Introduction......Page 312
2. Regularity of Elliptic Measure on Rough Domains......Page 314
3. Boundary Structure and Size Are Determined by Harmonic Measure......Page 319
References......Page 322
Section 9 Functional Analysis and Applications......Page 326
1. Introduction......Page 328
2. Setting and Examples......Page 329
3. The Full Group......Page 332
4. Associated von Neumann Algebra......Page 333
5. Strong Ergodicity......Page 334
6. Graphings......Page 336
7. Dimensions......Page 338
8. L2-Betti Numbers......Page 339
9. Measure Equivalence......Page 340
10. Non-orbit Equivalent Actions for a Given Group......Page 342
11. Relative Property (T)......Page 344
12. Some Rigidity Results......Page 345
Acknowledgements......Page 346
References......Page 347
1. Introduction......Page 355
2.1. Injective factors......Page 358
2.2. The classification of group actions......Page 359
2.3. Galois correspondence......Page 360
3.1. K-theory......Page 361
3.2. Classifiable C*-algebras......Page 363
3.3. The Rohlin property......Page 365
3.4. Finite group actions......Page 367
3.5. ℤN-actions......Page 369
3.6. Conjectures......Page 370
References......Page 372
1. Introduction......Page 376
2. Embeddings......Page 378
3. L1 as a Metric Space......Page 379
4. The Sparsest Cut Problem......Page 381
4.1. Reformulation as an optimization problem over L1......Page 382
4.2. The linear program......Page 383
4.3. The semidefinite program......Page 384
5. Embeddings of the Heisenberg Group......Page 393
5.2. Pansu differentiability......Page 394
5.3. Cheeger-Kleiner differentiability......Page 395
5.4. Compression bounds for L1 embeddings of the Heisenberg group......Page 396
References......Page 398
1. Asymptotic and Non-asymptotic Problems on Random Matrices......Page 403
Asymptotic behavior of extreme singular values......Page 406
Non-asymptotic behavior of extreme singular values......Page 408
Tracy-Widom fluctuations......Page 409
Quantitative invertibility problem......Page 411
Smallest singular values of general random matrices......Page 412
Universality of the smallest singular values......Page 413
Sparsity and invertibility: a geometric proof of Theorem 3.2......Page 414
Small ball probabilities and additive structure......Page 415
New results on Lévy concentration function......Page 417
5. Applications......Page 419
Circular law......Page 420
Compressed Sensing......Page 421
Short Khinchin\'s inequality and Kashin\'s subspaces......Page 423
References......Page 425
1. Introduction......Page 430
2.1. Non-commutative probability spaces......Page 432
2.3. Non-commutative laws......Page 433
3.1. Non-commutative laws with quantum symmetry......Page 434
3.2. The standard invariant of a subfactor: spaces of intertwiners......Page 435
3.4. Planar algebras......Page 436
3.6. Examples of planar algebras......Page 438
3.7. Algebras and non-commutative probability spaces arising from planar algebras......Page 439
3.9. The Voiculescu trace on (P, Λ0)......Page 440
3.13. Application: constructing a subfactor realizing a given planar algebra......Page 441
4.1. GUE and the Voiculescu trace τTL......Page 442
4.3. The case of a general planar algebra......Page 443
4.6. Random matrix ensembles......Page 444
4.7. Combinatorial properties of the laws τV(N)......Page 445
4.12. Example: O(n) models......Page 446
4.13. Properties of the limit laws τV......Page 447
References......Page 448
1. Classifying II1 Factors, a Panoramic Overview......Page 451
1.2. (Non)-isomorphism of II1 factors......Page 452
1.3. Popa\'s deformation/rigidity theory......Page 453
1.5. Outer automorphisms and generalized symmetries......Page 454
1.6. W*-superrigidity and uniqueness of Cartan subalgebras......Page 455
Organization of the paper......Page 457
Group measure space construction......Page 458
Completely positive maps and bimodules......Page 459
Cartan subalgebras and equivalence relations......Page 460
3. Popa\'s Deformation/Rigidity Theory......Page 461
Combining deformation and rigidity......Page 462
4. Fundamental Groups of II1 Factors......Page 464
II1 factors with uncountable fundamental group [PV08a, PV08c]......Page 465
How to get equality in (1)......Page 466
How wild can mod CentrAut Y (Λ) be......Page 467
Non-uniqueness of Cartan subalgebras......Page 468
Uniqueness of Cartan subalgebras......Page 469
6. Superrigidity for Group Measure Space Factors......Page 472
References......Page 473
Section 10 Dynamical Systems and Ordinary Differential Equations......Page 478
1. Introduction......Page 480
2.1. Well-known facts for twist maps......Page 483
2.2. Well-known facts for Hamiltonians......Page 484
2.3. Construction of the Green Bundles, first properties......Page 485
2.4. A dynamical criterion and some consequences......Page 486
Remark......Page 487
3.1. The link between the Green bundles and the number of zero Lyapunov exponents......Page 489
3.2. Lower and upper bounds for the positive Lyapunov exponents in the Hamiltonian case......Page 491
3.3. The non negative Lyapunov exponent for twist maps......Page 493
4.1. Two notions of C1 regularity......Page 494
4.2. Tangent cones and Green bundles......Page 495
4.3. Regularity of invariant C0-Lagrangian graphs......Page 496
5. The Link Between the Shape of the Aubry-Mather Sets and Their Lyapunov Exponents......Page 498
6. Weak KAM Theory......Page 499
6.1. The Lax-Oleinik semigroup and its interpretation on pseudographs......Page 500
6.3. Mather, Aubry and Mañé sets......Page 501
7. Weak KAM Solutions and Green Bundles......Page 503
References......Page 504
1. Introduction......Page 507
2. The Example of Arnold and Some Extensions......Page 510
2.1. Homoclinic orbits......Page 511
2.2. Heteroclinic orbits......Page 512
2.3. Poincaré-Melnikov approximation......Page 513
2.4. Bessi’s variational mechanism......Page 515
2.5. Remarks on estimates......Page 516
2.6. Higher dimensions......Page 517
3.1. The Large Gap Problem......Page 521
3.2. Normally hyperbolic invariant cylinder......Page 522
4. Back to the a priori Stable Case......Page 524
References......Page 525
1. Introduction......Page 528
2. Quadratic Polynomials with an Indifferent Fixed Point......Page 530
3. Strategy of the Proof......Page 532
4. The Set S......Page 533
5. McMullen’s Results on Siegel Disks of Bounded Type......Page 534
7. The Control of the Cycle......Page 536
8. The Density of Perturbed Siegel Disks......Page 537
10. Further Questions......Page 538
References......Page 539
1. Introduction......Page 541
2. Local Connecting Orbits......Page 542
3. Global Connecting Orbits......Page 546
4. A Priori Unstable Systems......Page 547
5. Barrier Functions and Elementary Weak-KAM......Page 548
6. A Priori Stable Systems......Page 550
References......Page 552
Generic Dynamics of Geodesic Flows......Page 556
2. Twist Maps......Page 557
3. The Kupka-Smale Theorem......Page 558
4. Many Closed Geodesics......Page 559
6. The Perturbation Lemma......Page 560
7. Elliptic Geodesics in the Sphere......Page 563
References......Page 564
1. Introduction......Page 567
2. Arithmetic Quantum Unique Ergodicity......Page 570
3. Diophantine Approximation for Points in Fractals......Page 572
4. Non-uniformity of Bad Approximations......Page 575
5. Littlewood’s Conjecture......Page 576
6. Compact Orbits and Ideal Classes......Page 577
7. Counting Rational Points......Page 579
8. Divisibility Properties of Hamiltonian Quaternions......Page 580
9. Open Problems......Page 581
References......Page 583
1. Introduction......Page 587
2. Rigidity of Actions of Higher Rank Groups......Page 589
3. Partially Hyperbolic Systems......Page 592
4. Product Measures and Entropy Formula......Page 595
References......Page 599
1.1. Motivation......Page 604
1.2. Basic Definitions......Page 605
1.3. Measure Rigidity......Page 606
2.1. Definition......Page 607
2.2. Horocycle flows on hyperbolic surfaces with finite genus......Page 609
2.3. Invariant measures in infinite genus......Page 611
2.4. Generic points......Page 615
2.5. Conditional unique ergodicity......Page 616
3.1. Group extensions......Page 618
3.2. Cylinder Transformations......Page 620
3.3. Hajian-Ito-Kakutani Maps......Page 623
References......Page 627
Richness of Chaos in the Absolute Newhouse Domain......Page 631
References......Page 641
Introduction......Page 643
1. Partial Hyperbolicity......Page 647
2. Stable Ergodicity and the Pugh-Shub Conjectures......Page 648
3. Accessibility......Page 650
4. Ergodicity......Page 651
5. Exponents......Page 652
6. Pathology......Page 654
7. Rigidity......Page 656
8. Summary, Questions......Page 657
References......Page 659
Section 11 Partial Differential Equations......Page 664
1. Introduction......Page 666
2. The Hyperbolic Dispersion Estimate......Page 667
2.1. Propagation of a single plane wave......Page 668
2.2. Estimating the norm of Pn o . . . o P2 o P1......Page 670
3.1. Statement of the conjecture......Page 671
3.2. Entropy of semiclassical measures on hyperbolic manifolds......Page 674
3.3. Generalization to higher rank symmetric spaces of nonpositive curvature......Page 677
4. Resonances, Local Smoothing and Strichartz Estimates......Page 682
References......Page 686
1. Introduction......Page 689
2.2. Proof of Theorem 2.1.......Page 692
2.3. Random series on manifolds and on the line......Page 695
3.1. Local theory......Page 697
3.2. A global existence result......Page 699
4. Non Linear Harmonic Oscillators......Page 700
4.1. Bilinear estimates......Page 702
5.1. Construction of the measure......Page 705
5.2. Improved Sobolev embeddings......Page 706
References......Page 707
1. Introduction......Page 711
2. Characteristic Boundary Value Problems......Page 712
3. 1-D Like Problems with M-D Perturbation – Fan-shaped Wave Structure......Page 713
4. Essentially M-D Problems – Flower-shaped Wave Structure......Page 716
5. Global Theory and Mixed Type Equations......Page 721
References......Page 724
Finite Morse Index and Linearized Stable Solutions on Bounded and Unbounded Domains......Page 728
1. Linearized Stable Solutions on RN......Page 729
2. Finite Morse Index Solutions on RN......Page 731
3. Application to Bounded Domain Problems......Page 733
References......Page 735
1.1. The regularity theory for area-minimizing currents......Page 737
1.2. Branching......Page 739
2.1. De Giorgi’s excess decay......Page 740
2.2. Again branching......Page 742
3.1. The metric space of unordered Q-tuples......Page 743
3.2. Almgren’s extrinsic maps......Page 744
3.3. The generalized Dirichlet energy......Page 745
3.4. The cornerstones of the theory of Dir-minimizers......Page 746
4.1. An intrinsic approach......Page 747
4.2. Intrinsic definition of the Dirichlet energy......Page 748
5.2. Higher integrability......Page 749
6.1. Almgren’s main approximation theorem......Page 750
6.2. Higher integrability for area-minimizing currents......Page 751
6.3. Some new techniques coming from metric analysis......Page 752
7. Center Manifold: A Case Study......Page 754
7.1. Higher regularity “without PDEs”......Page 755
8. Open Problems......Page 756
References......Page 757
1. Introduction......Page 761
2. The Allen-Cahn Equation......Page 762
2.1. Formal asymptotic behavior of vε......Page 763
3. From Bernstein’s to De Giorgi’s Conjecture......Page 766
3.1. Outline of the proof......Page 768
4.1. Embedded minimal surfaces of finite total curvature......Page 770
4.2. A general statement......Page 771
4.3. Further comments......Page 772
5.1. Solutions with multiply connected nodal set......Page 773
6.1. The standing wave problem for NLS......Page 775
6.2. The Yamabe equation in RN......Page 778
References......Page 779
1. Introduction......Page 785
2. History......Page 788
2.1. Principally normal operators......Page 789
2.2. The Nirenberg–Treves conjecture......Page 791
2.3. The Beals–Fefferman localization......Page 793
2.4. Lerner’s counterexample......Page 795
3. The Resolution of the Nirenberg–Treves Conjecture......Page 797
3.1. The Proof......Page 798
3.2. The localization......Page 799
3.3. The weight......Page 800
3.4. The Wick Calculus......Page 802
4.1. Solvability of systems......Page 803
4.2. Spectral instability......Page 806
4.3. Non-linear equations......Page 807
4.4. Open Problems......Page 808
References......Page 809
1. Introduction......Page 812
2. Existence and Regularity of Equilibrium Configurations......Page 815
3.1. The second variation......Page 820
3.2. Local and global minimizers......Page 824
References......Page 826
1. Introduction......Page 828
2. Stochastic Processes and Viscosity Solutions of Linear Elliptic Equations......Page 831
3. Nonclassical Solutions to Fully Nonlinear Elliptic Equations......Page 832
4. Trialities, Quaternions, Octonions and Hessian Equations......Page 836
5. Special Lagrangian Equation......Page 841
References......Page 843
Section 12 Mathematical Physics......Page 846
1. Introduction......Page 848
2.1. Extended TFT in two dimensions......Page 850
2.2. Extended TFT in three dimensions......Page 853
2.3. Extended TFT in n dimensions......Page 856
3.1. Definition and basic properties......Page 858
3.2. Monoidal deformations of the derived category of coherent sheaves......Page 861
4.1. Electric-magnetic duality and Topological Gauge Theory......Page 864
4.2. From Topological Gauge Theory to Geometric Lang-lands Duality......Page 865
5. Open Questions......Page 867
References......Page 868
1. Diffusion from Conservative Dynamics......Page 871
2. Coupled Oscillators......Page 872
4. Coupled Chaotic Maps......Page 875
6. Quenched Diffusion......Page 877
7. Random Walk in Nonlinear Random Environment......Page 878
8. Renormalization Group for Random Coupled Maps......Page 880
9. Towards Hamiltonian Systems......Page 882
References......Page 883
1. Introduction......Page 884
2. Elliptic Curves and Noncommutative Tori......Page 885
3.1. Noncommutative tori with real multiplication......Page 888
3.2. Analytic versus algebraic model......Page 889
3.3. Stark numbers and L-functions......Page 890
3.4. Solvmanifolds and noncommutative spaces......Page 891
3.6. The Shimizu L-function and noncommutative tori......Page 892
3.8. Quantum field theory and noncommutative tori......Page 893
4. The Noncommutative Boundary of Modular Curves......Page 894
4.1. Modular shadow play......Page 895
4.2. Modular shadows and the Kronecker limit formula......Page 897
4.3. Quantum modular forms......Page 898
5. Quantum Statistical Mechanics and Number Fields......Page 899
5.3. Quantum statistical mechanical systems for number fields......Page 900
5.4. Noncommutative geometry and anabelian geometry......Page 901
References......Page 902
1. Phase Transitions and Critical Phenomena......Page 905
2. Universality......Page 908
3. Grassmann Integrals Representation......Page 912
4. Renormalization Group and Multiscale Decomposition......Page 915
5. The Extended Scaling Relations......Page 921
6. Quantum Spin Chains......Page 924
7. Conclusions and Open Problems......Page 927
References......Page 929
1. Introduction......Page 932
2. Blowup Techniques, Bounded Ancient Solutions......Page 938
3. Backward Uniqueness for Navier-Stokes Equations......Page 942
4. How Does L3-norm Approach Potential Blowup?......Page 946
References......Page 952
1. Introduction......Page 955
2. Finite vs. Infinite Volume, Continuum vs. Lattice, Equilibrium Measures......Page 957
3. Kinetic Limit......Page 960
4. Equilibrium Time Correlations......Page 964
5. Fluctuation Field......Page 967
References......Page 969
Introduction......Page 971
1. Some Background on Singularities......Page 973
2. Some Background on Conformal Field Theory......Page 979
3. Some Insights from Topological Field Theory......Page 985
4. Further Directions......Page 990
References......Page 993
Author Index......Page 998