Proceedings of the International Congress of Mathematicians, Madrid 2006 Volume 2

1. Introduction......Page 1
2. Preliminaries......Page 2
3. Three approaches to randomness......Page 3
4. Calibrating randomness......Page 14
5. Lowness and triviality......Page 17
References......Page 21
1. Determinacy......Page 27
2. Large cardinals......Page 31
3. Larger cardinals, longer games......Page 37
References......Page 42
1. Introduction......Page 44
2. Ordinal analyses of systems of second order arithmetic and set theory......Page 55
3. Beyond admissible proof theory......Page 61
4. A large cardinal notion......Page 63
References......Page 66
1. Introduction......Page 69
2. Foundations of analytic and difference structure......Page 71
3. AKE theorems for analytically difference henselian rings......Page 78
-differential geometry......Page 85
References......Page 89
1. Introduction......Page 91
2. Superrigidity......Page 99
3. The classificatio problem for the torsion-free abelian groups of finit rank......Page 105
References......Page 112
Introduction......Page 115
1. Preprojective algebras......Page 117
2. Weighted projective lines......Page 119
3. Hall algebras......Page 121
4. The Deligne–Simpson problem......Page 123
References......Page 124
1. Introduction......Page 128
-adic formalism......Page 134
-adic and adelic cone integrals......Page 135
4. The local factors: variation with......Page 138
5. Functional equations of the local factors......Page 139
6. Examples......Page 141
7. Variation......Page 143
References......Page 144
1. Introduction......Page 147
2. Definition......Page 149
3. The derived category of a dg category......Page 152
4. The homotopy category of small dg categories......Page 163
5. Invariants......Page 172
References......Page 178
1. Introduction......Page 187
2. Broué’s abelian defect group conjecture......Page 189
3. Invariants......Page 202
4. Categorification......Page 207
References......Page 212
1. Introduction......Page 218
-machines......Page 219
3. Dehn functions and the word problem......Page 223
4. Higman embeddings......Page 229
5. Non-amenable finitel presented groups......Page 231
References......Page 237
1. Introduction......Page 240
2. Permutation groups......Page 241
3. Matrix groups......Page 244
4. A new data structure......Page 249
References......Page 251
2. Nil rings......Page 254
3. Algebraic algebras......Page 258
4. Algebras with finit Gelfand–Kirillov dimension......Page 259
5. Simple rings......Page 261
References......Page 262
1. Introduction......Page 265
2. The parametrization of algebraic structures......Page 267
3. The story of the cube......Page 270
4. Cubic analogues of Gauss composition......Page 274
5. The parametrization of quartic and quintic rings......Page 278
6. Counting number field of low degree......Page 279
7. Related and future work......Page 285
References......Page 286
1. Introduction......Page 289
2. Hecke symmetry on modular varieties......Page 291
3. Leaves and the Hecke orbit conjecture......Page 293
4. Canonical coordinates on leaves......Page 294
5. Hypersymmetric points......Page 298
6. Action of stabilizer subgroups and rigidity......Page 299
7. Open questions and outlook......Page 301
References......Page 304
1. Introduction......Page 307
2. Elliptic curves over......Page 309
3. Elliptic curves over totally real elds......Page 319
4. Stark–Heegner points......Page 328
References......Page 336
1. Introduction......Page 340
2. Non-abelian class fiel theory......Page 341
3. Galois deformations and nearly ordinary Hecke algebras for GL......Page 343
4. Taylor–Wiles systems: the formalism......Page 346
5. Taylor–Wiles systems: a strategy for the construction......Page 350
6. Geometric Jacquet–Langlands correspondence......Page 353
7. Concluding remarks......Page 361
References......Page 362
1. Introduction......Page 365
2. The Hardy–Littlewood heuristic......Page 367
3. The Hardy–Littlewood method for primes......Page 369
4. Exponential sums with Möbius......Page 371
5. Proving the Möbius randomness law......Page 373
6. The insuf ciency of harmonic analysis......Page 376
8. The Gowers norms and inverse theorems......Page 378
9. Nilsequences......Page 381
10. Working with the primes......Page 384
11. Möbius and nilsequences......Page 386
12. Future directions......Page 388
References......Page 389
1. Introduction......Page 392
2. Groupes réductifs......Page 393
3. Intégrales orbitales......Page 394
-Intégrales orbitales......Page 396
5. Dualité de Langlands......Page 398
6. Groupes endoscopiques......Page 400
7. Lemme Fondamental......Page 401
8. Résultats......Page 402
9. Fibres de Springer af nes......Page 403
10. L’approche de Goresky, Kottwitz et MacPherson......Page 404
11. Notre approche avec Ngô......Page 406
Références......Page 409
1. Linnik’s problems......Page 411
2. Linnik’s problems via harmonic analysis......Page 414
3. The subconvexity problem......Page 420
4. Subconvexity via periods of automorphic forms......Page 429
5. Applications......Page 434
6. Linnik’s ergodic method: a modern perspective......Page 437
7. Ergodic theory vs. harmonic analysis......Page 442
References......Page 444
1. Introduction......Page 448
2. K-theory......Page 449
3. Motivic cohomology......Page 452
-adic Hodge theory......Page 455
References......Page 460
Introduction......Page 462
-functions......Page 465
-adic deformations of automorphic representations......Page 471
3. Deformations of Eisenstein series......Page 475
4. Galois representations and applications to Selmer groups......Page 477
5. Higher order vanishing and higher rank Selmer groups......Page 485
References......Page 488
1. Introduction......Page 490
-adic families......Page 492
References......Page 501
Introduction......Page 504
1. Definitio of stable pairs and maps......Page 505
2. Minimal Model Program construction......Page 507
3. Surfaces......Page 508
4. Toric and spherical varieties......Page 509
5. Abelian varieties......Page 514
6. Grassmannians......Page 520
7. Higher Gromov–Witten theory......Page 522
References......Page 523
1. Introduction......Page 526
2. Algebraic formal germs and auxiliary polynomials......Page 529
3. An algebraicity criterion for smooth formal germs in varieties over function field......Page 532
-adic and global field......Page 535
5. Condition L and canonical semi-norms......Page 539
6. An algebraicity criterion for smooth formal germs in varieties over number field......Page 544
7. An algebraicity criterion for smooth formal curves in varieties over number field......Page 546
References......Page 549
1. Introduction......Page 552
2. Some basic problems......Page 553
3. Threefold flop......Page 557
4. Stability conditions......Page 560
5. Stability conditions and threefold flop......Page 562
6. Stability conditions on K3 surfaces......Page 565
7. Derived categories and the minimal model programme......Page 567
References......Page 569
1. Introduction......Page 572
2. Multiplier ideals......Page 573
3. Applications of multiplier ideals......Page 576
4. Bounds on log canonical thresholds and birational rigidity......Page 577
5. Bernstein–Sato polynomials......Page 578
6. Spaces of arcs and contact loci......Page 581
7. Invariants in positive characteristic......Page 586
References......Page 589
2. Classical results......Page 592
3. Rationally connected varieties......Page 593
4. Rational points on rationally connected varieties......Page 595
5. Higher rational connectivity......Page 597
References......Page 599
1. Introduction......Page 601
2. Geometric structures arising from minimal rational curves......Page 603
3. Deformation rigidity of rational homogeneous spaces......Page 607
4. The Campana–Peternell conjecture......Page 610
References......Page 612
1. Introduction......Page 615
1......Page 616
4. Mixed Tate motives and Grothendieck–Teichmüller group......Page 618
5. Harmonic shuffl relation......Page 620
6. Fake Hodge realization and harmonic shuffl relation......Page 621
References......Page 622
1. Introduction......Page 624
2. Classificatio schemes......Page 625
3. Potential density......Page 626
4. Points of bounded height......Page 628
5. Integral points......Page 631
6. Arithmetic over function field of curves......Page 633
7. Geometry over finit field......Page 634
References......Page 635
1. Introduction......Page 639
2. Birational cobordisms......Page 641
3. Toric varieties......Page 649
4. Polyhedral cobordisms of Morelli......Page 650
-desingularization of birational cobordisms......Page 655
References......Page 667
1. Introduction......Page 669
2. Preliminaries......Page 671
3. On the proof of Theorem 2......Page 672
4. On the proof of Theorem 1......Page 673
References......Page 675
1. The uniformization theorem and the Ricci flw in dimension 2......Page 677
2. The Yamabe problem......Page 679
3. The Yamabe flw......Page 680
4. Convergence of the Yamabe flw in dimension greater or equal to 6......Page 684
5. Compactness of the set of constant scalar curvature metrics in a given conformal class......Page 686
References......Page 688
1. Tight vs. overtwisted......Page 691
2. Open book decompositions......Page 693
3. Right-veering......Page 695
4. Contact homology......Page 697
References......Page 700
1. Introduction......Page 704
2. Metric spaces modelled on Coxeter complexes......Page 705
3. Generalized triangle inequalities......Page 707
4. Algebraic problems......Page 709
5. Geometry behind the proofs......Page 715
6. Other developments......Page 721
References......Page 724
1. Introduction......Page 727
2. Rigidity and geometrization in geometric group theory......Page 728
3. Gromov hyperbolic spaces and their boundaries......Page 731
4. Quasiconformal homeomorphisms......Page 735
5. Applications to rigidity......Page 739
6. Uniformization......Page 741
7. Geometrization......Page 745
8. Open problems......Page 748
References......Page 749
1. Introduction......Page 753
2. Exact Lagrangian submanifolds......Page 754
3. The cluster complex......Page 758
4. Fine Floer homology......Page 763
5. Applications of cluster homology......Page 768
6. The emerging fiel of real symplectic topology......Page 771
References......Page 772
1. Introduction......Page 774
2. Tautological relations and universal equations......Page 775
3. The Virasoro conjecture......Page 784
4. Universal equations and spin curves......Page 789
References......Page 793
1. Introduction......Page 796
2. Stability for manifolds in algebraic geometry......Page 797
3. The Hitchin–Kobayashi correspondence and its manifold analogue......Page 798
4. The asymptotic Bergman kernel......Page 799
5. Balanced metrics......Page 801
6. A simple heuristic proof of Donaldson’s theorem......Page 803
admits symmetries......Page 804
8. Concluding remarks......Page 805
References......Page 807
1. Introduction......Page 810
2. Tropical algebra......Page 811
3. Geometry: tropical varieties......Page 812
4. Tropical intersection theory......Page 818
5. Tropical curves......Page 822
, their phases and amoebas......Page 828
7. Applications......Page 830
References......Page 833
1. Introduction......Page 836
2. Minimal surfaces......Page 838
3. Embedded minimal surfaces with fixe genus......Page 840
4. [8]: Compactness of embedded minimal surfaces with fixe genus......Page 841
5. The structure of embedded minimal annuli......Page 847
6. Properness and removable singularities for minimal laminations......Page 849
7. The uniqueness of the helicoid......Page 852
8. Quasiperiodicity of properly embedded minimal planar domains......Page 854
B. The lamination theorem and one-sided curvature estimate......Page 856
References......Page 858
1. Prologue......Page 861
2. Floer theory of Hamiltonian fixe points......Page 862
3. Towards topological Hamiltonian dynamics......Page 867
4. Floer theory of Lagrangian intersections......Page 869
5. Displaceable Lagrangian submanifolds......Page 880
6. Applications to mirror symmetry......Page 881
References......Page 884
1. Introduction......Page 888
2. Geometry of the ends......Page 889
3. Minimal surfaces with finit topology in......Page 890
4. The periodic case......Page 893
5. Vertical flu......Page 894
6. Compactness and limit configuration......Page 896
7. Smoothness of moduli spaces......Page 899
8. Classificatio results......Page 901
9. Least area surfaces......Page 903
References......Page 904
1. Introduction......Page 908
2. Soliton equations associated to simple Lie algebras......Page 910
3. Soliton equations in submanifold geometry......Page 912
4. The space-time monopole equation......Page 914
-hierarchy......Page 917
6. Direct scattering for the space-time monopole equation......Page 918
-hierarchy via loop group factorizations......Page 919
8. The inverse scattering for monopole equations......Page 922
9. Birkhoff factorization and local solutions......Page 923
-hierarchy......Page 925
11. Bäcklund transformations for the space-time monopole equation......Page 927
References......Page 929
1. Introduction......Page 932
2. Conformal volume of orbifolds......Page 933
3. Finite subgroups of O(3)......Page 934
4. Eigenvalue bounds......Page 935
5. Congruence arithmetic hyperbolic 3-orbifolds......Page 936
6. Finiteness of arithmetic Kleinian maximal reflectio groups......Page 937
7. Conclusion......Page 938
References......Page 940
Introduction......Page 942
1. The universe of finitel presented groups......Page 943
spaces and their isometries......Page 948
3. Non-positively curved groups......Page 951
4. Word problems and fillin invariants......Page 955
5. Subdirect products of hyperbolic groups......Page 960
6. Two questions of Grothendieck......Page 962
References......Page 964
1. Introduction......Page 969
2. A categorificatio of the Jones polynomial......Page 971
3. Extensions to tangles......Page 972
link homology and matrix factorizations......Page 974
5. Triply-graded link homology and beyond......Page 976
References......Page 977
1. Disjoint curves in surfaces......Page 980
2. Curve complexes......Page 984
3. Nested structure......Page 988
4. Coarse geometry of......Page 992
5. Hyperbolic geometry and ending laminations......Page 997
6. Heegaard splittings......Page 1002
References......Page 1006
1. The Brouwer degree......Page 1013
2. A quick recollection on A1-homotopy......Page 1016
3. A1-homotopy and A1-homology: the basic theorems......Page 1018
4. A1-homotopy and A1-homology: computations involving Milnor– Witt K-theory......Page 1022
5. Some results on classifying spaces in A1-homotopy theory......Page 1029
6. Miscellaneous......Page 1033
References......Page 1035
1. Introduction......Page 1038
2. Floer theory for symplectomorphisms......Page 1039
3. Floer theory for Lagrangian submanifolds......Page 1045
References......Page 1056
1. Heegaard–Floer homology of three-manifolds......Page 1060
2. Heegaard–Floer homology of knots......Page 1062
3. Heegaard–Floer homology for links......Page 1063
4. Basic properties......Page 1069
5. Three applications......Page 1070
References......Page 1074
1. Introduction......Page 1077
2. Outer space and homological finitenes results......Page 1078
3. The bordificatio and duality......Page 1081
4. The Degree Theorem and rational homology stability......Page 1082
5. Sphere complexes and integral homology stability......Page 1084
6. Graph complexes and unstable homology......Page 1087
7. IA automorphisms and the IA quotient of Outer space......Page 1090
8. Further reading......Page 1091
References......Page 1092
1. Introduction......Page 1094
2. Noncommutative resolutions and braid group actions......Page 1098
-modules in positive characteristic and localization theorem......Page 1107
4. Perverse sheaves on affin flag of the dual group (local geometric Langlands)......Page 1113
References......Page 1117
1. Introduction......Page 1120
2. Definitio of quasi-maps......Page 1122
3. Quasi-maps into fla varieties and semi-infinit Schubert varieties......Page 1126
and geometric Eisenstein series......Page 1132
5. Quasi-maps into affin fla varieties and Uhlenbeck compactifica tions......Page 1135
6. Applications to gauge theory and quantum cohomology of (affine fla manifolds......Page 1139
7. Some open problems......Page 1141
References......Page 1143
1. Smooth representations......Page 1146
3. The Jacquet–Langlands correspondence......Page 1147
4. Extending the Jacquet–Langlands correspondence......Page 1149
5. The Langlands correspondence......Page 1150
6. Explicit Langlands correspondence in the tame case......Page 1152
8. Construction and explicit Jacquet–Langlands correspondence, 1......Page 1153
References......Page 1154
1. Introduction......Page 1158
2. Three ways to stumble upon H......Page 1159
3. The rôle of amenability......Page 1167
4. Rigidity......Page 1170
5. Randomorphisms......Page 1173
6. Additional questions......Page 1177
References......Page 1179
1. Commentaires historiques......Page 1187
2. Fibration de Hitchin......Page 1188
3. Stabilisation de la partie anisotrope......Page 1191
4. Symétries de la bration de Hitchin......Page 1194
5. Groupes endoscopiques......Page 1196
Références......Page 1198
1. Introduction......Page 1200
2. Affin Hecke algebras......Page 1203
-theory and abstract Plancherel theorem......Page 1205
4. The Plancherel measure......Page 1206
5. The structure of the Schwartz algebra......Page 1214
6. Smooth families of tempered representations......Page 1218
-theory of the Schwartz algebra......Page 1220
8. Index functions......Page 1223
References......Page 1229
1. Motivation......Page 1233
2. Banach space representations......Page 1236
3. Locally analytic representation......Page 1239
4. Analytic vectors......Page 1246
5. Unramified p-adic functoriality......Page 1247
References......Page 1253
1. Introduction......Page 1255
2. First algebraization results: bounded generation and general rings......Page 1258
3. The algebraization of property (T) for nite groups: expanders......Page 1263
4. Reduced cohomology and property (T) for elementary linear groups......Page 1271
5. Some concluding remarks, questions, and speculations......Page 1274
References......Page 1277
1. Introduction......Page 1283
2. The descent method and applications......Page 1286
-functions for orthogonal groups; non-generic representations......Page 1291
References......Page 1296
1. Introduction......Page 1298
and automorphic representations......Page 1299
3. Locally symmetric spaces in the adelic language......Page 1302
4. Modular symbols and automorphic representations......Page 1303
5. A conjecture......Page 1304
References......Page 1305
1. Introduction......Page 1307
-action on a compactificatio......Page 1309
-action......Page 1311
4. Character sheaves on......Page 1315
References......Page 1316
1. Introduction......Page 1319
2. Quasiconformal and quasisymmetric maps......Page 1320
3. The quasisymmetric uniformization problem......Page 1323
4. Gromov hyperbolic spaces and quasisymmetric maps......Page 1325
5. Cannon’s conjecture and fractal 2-spheres......Page 1327
6. Post-critically finit rational maps......Page 1329
7. Sierpinski´ carpets......Page 1333
8. Rigidity of square carpets......Page 1337
9. Conclusion......Page 1340
References......Page 1341
1. Introduction......Page 1344
2. Local Tb theorems for square functions and applications......Page 1348
3. Local Tb theorems for singular integrals and applications......Page 1353
References......Page 1359
1. Introduction......Page 1362
2. Convergence of the sequence of all partial sums......Page 1364
3. Convergence of subsequences of the sequence of partial sums......Page 1365
4. Ul’yanov’s problem......Page 1367
5. Strong summability......Page 1368
References......Page 1370
1. Introduction and notation......Page 1373
2. Iterated Segre mappings......Page 1375
3. Nondegeneracy conditions for generic submanifolds......Page 1377
4. Transversality of mappings......Page 1379
5. Finite jet determination......Page 1381
6. Stability groups......Page 1382
7. Algebraicity of mappings......Page 1384
References......Page 1385
1. Introduction......Page 1388
2. Lattice models......Page 1391
3. Schramm–Loewner evolution......Page 1398
4. SLE as a scaling limit......Page 1402
5. Ising model and beyond......Page 1407
6. Conclusion......Page 1415
References......Page 1416
1. Introduction......Page 1419
2. Compactness......Page 1423
3. Global regularity......Page 1432
References......Page 1439
1. Introduction. Historical remarks......Page 1445
2. Greedy algorithms with regard to bases......Page 1450
3. Optimal methods in nonlinear approximation......Page 1456
4. The TGA with regard to the trigonometric system......Page 1458
5. Convergence of the TGA with regard to the trigonometric system......Page 1459
6. General greedy algorithms......Page 1462
References......Page 1468
1. Introduction......Page 1471
2. Analytic capacity and the Cauchy transform......Page 1473
3. Principal values for the Cauchy integral and related results......Page 1482
4. Lipschitz harmonic capacity and Riesz transforms......Page 1485
5. Some open problems......Page 1486
References......Page 1490
1. Introduction......Page 1494
2. Functional extensions, functional tools......Page 1496
3. Multilinear inequalities......Page 1499
4. Geometry in Gauss space......Page 1502
5. Shannon entropy......Page 1504
References......Page 1508
1. Introduction......Page 1512
2. Symmetrization of convex bodies......Page 1514
3. Volume distribution in convex bodies......Page 1518
4. Beyond Brunn–Minkowski and Santaló inequalities......Page 1522
References......Page 1524
1. Introduction......Page 1528
2. Amenable actions and exactness......Page 1529
3. Amenable compactification which are small......Page 1537
4. Application to von Neumann algebra theory......Page 1538
References......Page 1542
1. Introduction......Page 1546
-algebras......Page 1548
3. Elliott’s classificatio conjecture......Page 1553
4. Almost commuting self-adjoint matrices: an application of real rank zero and stable rank one......Page 1555
-algebras......Page 1557
References......Page 1562
1. Introduction......Page 1564
2. Metric entropy and its duality......Page 1566
3. Geometric complexity of convex bodies and their diversity......Page 1570
4. Algorithmic complexity and derandomization, pseudorandom matrices......Page 1574
References......Page 1580
1. Introduction......Page 1587
2. Higher index theory of elliptic operators......Page 1589
3. Geometry of groups and metric spaces......Page 1594
4. Main results......Page 1598
References......Page 1599
1. Introduction......Page 1604
2. General classical problems of the spectral ergodic theory......Page 1606
3. Weak operator convergence......Page 1607
4. The homogeneous spectrum problem of arbitrary multiplicity......Page 1608
5. Spectral rigidity of group actions......Page 1610
6. Spectral invariants in natural subclasses of dynamical systems......Page 1611
7. Some more problems......Page 1614
References......Page 1615
Introduction......Page 1617
1. A brief survey......Page 1618
2. Ramsey theory and multiple recurrence along polynomials......Page 1626
3. Ergodic Ramsey theory in a noncommutative setting......Page 1630
4. Generalized polynomials and dynamical systems on nilmanifolds......Page 1632
5. Amenable groups and ergodic Ramsey theory......Page 1635
References......Page 1638
1. Introduction......Page 1641
2. Dispersing billiards......Page 1643
3. Slow mixing and non-standard limit theorems......Page 1645
4. Transport coefficient......Page 1648
5. Interacting particles......Page 1650
6. Infinit measure systems......Page 1658
References......Page 1662
1. Introduction......Page 1667
2. A mathematical formulation of the instability problem......Page 1668
3. The example of [Arn64] and the large gap problem......Page 1672
4. The role of normally hyperbolic manifolds......Page 1673
5. Overcoming the large gap problem by the method of [DdlLS03b]......Page 1676
6. Perturbations of geodesic flws and of superlinear oscillators......Page 1678
7. The method of correctly aligned windows......Page 1680
8. The method of normally hyperbolic laminations......Page 1682
9. The scattering map and the obstruction mechanism......Page 1684
References......Page 1685
1. Introduction......Page 1692
2. Entropy and classificatio of invariant measures......Page 1694
3. Brief review of some elements of entropy theory......Page 1701
4. Entropy and the set of values obtained by products of linear forms......Page 1705
5. Entropy and arithmetic quantum unique ergodicity......Page 1708
6. Entropy and distribution of periodic orbits......Page 1712
References......Page 1716