Proceedings, vol. 2

The International Congress of Mathematicians (ICM) is held every four years. It is a major scientific event, bringing together mathematicians from all over the world and demonstrating the vital role that mathematics play in our society. In particular, the Fields Medals are awarded to recognize outstanding mathematical achievement. At the same time, the International Mathematical Union awards the Nevanlinna Prize for work in the field of theoretical computer science. The proceedings of ICM 2006, published as a three-volume set, present an overview of current research in all areas of mathematics and provide a permanent record the congress. The first volume features the works of Fields Medallists and the Nevanlinna Prize winner, the plenary lectures, and the speeches and pictures of the opening and closing ceremonies and award sessions. The other two volumes present the invited lectures, arranged according to their mathematical subject. A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society. 1. Introduction......Page 8 2. Preliminaries......Page 9 3. Three approaches to randomness......Page 10 4. Calibrating randomness......Page 21 5. Lowness and triviality......Page 24 References......Page 28 1. Determinacy......Page 34 2. Large cardinals......Page 38 3. Larger cardinals, longer games......Page 44 References......Page 49 1. Introduction......Page 51 2. Ordinal analyses of systems of second order arithmetic and set theory......Page 62 3. Beyond admissible proof theory......Page 68 4. A large cardinal notion......Page 70 References......Page 73 1. Introduction......Page 76 2. Foundations of analytic and difference structure......Page 78 3. AKE theorems for analytically difference henselian rings......Page 85 -differential geometry......Page 92 References......Page 96 1. Introduction......Page 98 2. Superrigidity......Page 106 3. The classificatio problem for the torsion-free abelian groups of finit rank......Page 112 References......Page 119 Introduction......Page 122 1. Preprojective algebras......Page 124 2. Weighted projective lines......Page 126 3. Hall algebras......Page 128 4. The Deligne–Simpson problem......Page 130 References......Page 131 1. Introduction......Page 135 -adic formalism......Page 141 -adic and adelic cone integrals......Page 142 4. The local factors: variation with......Page 145 5. Functional equations of the local factors......Page 146 6. Examples......Page 148 7. Variation......Page 150 References......Page 151 1. Introduction......Page 154 2. Definition......Page 156 3. The derived category of a dg category......Page 159 4. The homotopy category of small dg categories......Page 170 5. Invariants......Page 179 References......Page 185 1. Introduction......Page 194 2. Broué’s abelian defect group conjecture......Page 196 3. Invariants......Page 209 4. Categorification......Page 214 References......Page 219 1. Introduction......Page 225 -machines......Page 226 3. Dehn functions and the word problem......Page 230 4. Higman embeddings......Page 236 5. Non-amenable finitel presented groups......Page 238 References......Page 244 1. Introduction......Page 247 2. Permutation groups......Page 248 3. Matrix groups......Page 251 4. A new data structure......Page 256 References......Page 258 2. Nil rings......Page 261 3. Algebraic algebras......Page 265 4. Algebras with finit Gelfand–Kirillov dimension......Page 266 5. Simple rings......Page 268 References......Page 269 1. Introduction......Page 272 2. The parametrization of algebraic structures......Page 274 3. The story of the cube......Page 277 4. Cubic analogues of Gauss composition......Page 281 5. The parametrization of quartic and quintic rings......Page 285 6. Counting number field of low degree......Page 286 7. Related and future work......Page 292 References......Page 293 1. Introduction......Page 296 2. Hecke symmetry on modular varieties......Page 298 3. Leaves and the Hecke orbit conjecture......Page 300 4. Canonical coordinates on leaves......Page 301 5. Hypersymmetric points......Page 305 6. Action of stabilizer subgroups and rigidity......Page 306 7. Open questions and outlook......Page 308 References......Page 311 1. Introduction......Page 314 2. Elliptic curves over......Page 316 3. Elliptic curves over totally real elds......Page 326 4. Stark–Heegner points......Page 335 References......Page 343 1. Introduction......Page 347 2. Non-abelian class fiel theory......Page 348 3. Galois deformations and nearly ordinary Hecke algebras for GL......Page 350 4. Taylor–Wiles systems: the formalism......Page 353 5. Taylor–Wiles systems: a strategy for the construction......Page 357 6. Geometric Jacquet–Langlands correspondence......Page 360 7. Concluding remarks......Page 368 References......Page 369 1. Introduction......Page 372 2. The Hardy–Littlewood heuristic......Page 374 3. The Hardy–Littlewood method for primes......Page 376 4. Exponential sums with Möbius......Page 378 5. Proving the Möbius randomness law......Page 380 6. The insuf ciency of harmonic analysis......Page 383 8. The Gowers norms and inverse theorems......Page 385 9. Nilsequences......Page 388 10. Working with the primes......Page 391 11. Möbius and nilsequences......Page 393 12. Future directions......Page 395 References......Page 396 1. Introduction......Page 399 2. Groupes réductifs......Page 400 3. Intégrales orbitales......Page 401 -Intégrales orbitales......Page 403 5. Dualité de Langlands......Page 405 6. Groupes endoscopiques......Page 407 7. Lemme Fondamental......Page 408 8. Résultats......Page 409 9. Fibres de Springer af nes......Page 410 10. L’approche de Goresky, Kottwitz et MacPherson......Page 411 11. Notre approche avec Ngô......Page 413 Références......Page 416 1. Linnik’s problems......Page 418 2. Linnik’s problems via harmonic analysis......Page 421 3. The subconvexity problem......Page 427 4. Subconvexity via periods of automorphic forms......Page 436 5. Applications......Page 441 6. Linnik’s ergodic method: a modern perspective......Page 444 7. Ergodic theory vs. harmonic analysis......Page 449 References......Page 451 1. Introduction......Page 455 2. K-theory......Page 456 3. Motivic cohomology......Page 459 -adic Hodge theory......Page 462 References......Page 467 Introduction......Page 469 -functions......Page 472 -adic deformations of automorphic representations......Page 478 3. Deformations of Eisenstein series......Page 482 4. Galois representations and applications to Selmer groups......Page 484 5. Higher order vanishing and higher rank Selmer groups......Page 492 References......Page 495 1. Introduction......Page 497 -adic families......Page 499 References......Page 508 Introduction......Page 511 1. Definitio of stable pairs and maps......Page 512 2. Minimal Model Program construction......Page 514 3. Surfaces......Page 515 4. Toric and spherical varieties......Page 516 5. Abelian varieties......Page 521 6. Grassmannians......Page 527 7. Higher Gromov–Witten theory......Page 529 References......Page 530 1. Introduction......Page 533 2. Algebraic formal germs and auxiliary polynomials......Page 536 3. An algebraicity criterion for smooth formal germs in varieties over function field......Page 539 -adic and global field......Page 542 5. Condition L and canonical semi-norms......Page 546 6. An algebraicity criterion for smooth formal germs in varieties over number field......Page 551 7. An algebraicity criterion for smooth formal curves in varieties over number field......Page 553 References......Page 556 1. Introduction......Page 559 2. Some basic problems......Page 560 3. Threefold flop......Page 564 4. Stability conditions......Page 567 5. Stability conditions and threefold flop......Page 569 6. Stability conditions on K3 surfaces......Page 572 7. Derived categories and the minimal model programme......Page 574 References......Page 576 1. Introduction......Page 579 2. Multiplier ideals......Page 580 3. Applications of multiplier ideals......Page 583 4. Bounds on log canonical thresholds and birational rigidity......Page 584 5. Bernstein–Sato polynomials......Page 585 6. Spaces of arcs and contact loci......Page 588 7. Invariants in positive characteristic......Page 593 References......Page 596 2. Classical results......Page 599 3. Rationally connected varieties......Page 600 4. Rational points on rationally connected varieties......Page 602 5. Higher rational connectivity......Page 604 References......Page 606 1. Introduction......Page 608 2. Geometric structures arising from minimal rational curves......Page 610 3. Deformation rigidity of rational homogeneous spaces......Page 614 4. The Campana–Peternell conjecture......Page 617 References......Page 619 1. Introduction......Page 622 1......Page 623 4. Mixed Tate motives and Grothendieck–Teichmüller group......Page 625 5. Harmonic shuffl relation......Page 627 6. Fake Hodge realization and harmonic shuffl relation......Page 628 References......Page 629 1. Introduction......Page 631 2. Classificatio schemes......Page 632 3. Potential density......Page 633 4. Points of bounded height......Page 635 5. Integral points......Page 638 6. Arithmetic over function field of curves......Page 640 7. Geometry over finit field......Page 641 References......Page 642 1. Introduction......Page 646 2. Birational cobordisms......Page 648 3. Toric varieties......Page 656 4. Polyhedral cobordisms of Morelli......Page 657 -desingularization of birational cobordisms......Page 662 References......Page 674 1. Introduction......Page 676 2. Preliminaries......Page 678 3. On the proof of Theorem 2......Page 679 4. On the proof of Theorem 1......Page 680 References......Page 682 1. The uniformization theorem and the Ricci flw in dimension 2......Page 684 2. The Yamabe problem......Page 686 3. The Yamabe flw......Page 687 4. Convergence of the Yamabe flw in dimension greater or equal to 6......Page 691 5. Compactness of the set of constant scalar curvature metrics in a given conformal class......Page 693 References......Page 695 1. Tight vs. overtwisted......Page 698 2. Open book decompositions......Page 700 3. Right-veering......Page 702 4. Contact homology......Page 704 References......Page 707 1. Introduction......Page 711 2. Metric spaces modelled on Coxeter complexes......Page 712 3. Generalized triangle inequalities......Page 714 4. Algebraic problems......Page 716 5. Geometry behind the proofs......Page 722 6. Other developments......Page 728 References......Page 731 1. Introduction......Page 734 2. Rigidity and geometrization in geometric group theory......Page 735 3. Gromov hyperbolic spaces and their boundaries......Page 738 4. Quasiconformal homeomorphisms......Page 742 5. Applications to rigidity......Page 746 6. Uniformization......Page 748 7. Geometrization......Page 752 8. Open problems......Page 755 References......Page 756 1. Introduction......Page 760 2. Exact Lagrangian submanifolds......Page 761 3. The cluster complex......Page 765 4. Fine Floer homology......Page 770 5. Applications of cluster homology......Page 775 6. The emerging fiel of real symplectic topology......Page 778 References......Page 779 1. Introduction......Page 781 2. Tautological relations and universal equations......Page 782 3. The Virasoro conjecture......Page 791 4. Universal equations and spin curves......Page 796 References......Page 800 1. Introduction......Page 803 2. Stability for manifolds in algebraic geometry......Page 804 3. The Hitchin–Kobayashi correspondence and its manifold analogue......Page 805 4. The asymptotic Bergman kernel......Page 806 5. Balanced metrics......Page 808 6. A simple heuristic proof of Donaldson’s theorem......Page 810 admits symmetries......Page 811 8. Concluding remarks......Page 812 References......Page 814 1. Introduction......Page 817 2. Tropical algebra......Page 818 3. Geometry: tropical varieties......Page 819 4. Tropical intersection theory......Page 825 5. Tropical curves......Page 829 , their phases and amoebas......Page 835 7. Applications......Page 837 References......Page 840 1. Introduction......Page 843 2. Minimal surfaces......Page 845 3. Embedded minimal surfaces with fixe genus......Page 847 4. [8]: Compactness of embedded minimal surfaces with fixe genus......Page 848 5. The structure of embedded minimal annuli......Page 854 6. Properness and removable singularities for minimal laminations......Page 856 7. The uniqueness of the helicoid......Page 859 8. Quasiperiodicity of properly embedded minimal planar domains......Page 861 B. The lamination theorem and one-sided curvature estimate......Page 863 References......Page 865 1. Prologue......Page 868 2. Floer theory of Hamiltonian fixe points......Page 869 3. Towards topological Hamiltonian dynamics......Page 874 4. Floer theory of Lagrangian intersections......Page 876 5. Displaceable Lagrangian submanifolds......Page 887 6. Applications to mirror symmetry......Page 888 References......Page 891 1. Introduction......Page 895 2. Geometry of the ends......Page 896 3. Minimal surfaces with finit topology in......Page 897 4. The periodic case......Page 900 5. Vertical flu......Page 901 6. Compactness and limit configuration......Page 903 7. Smoothness of moduli spaces......Page 906 8. Classificatio results......Page 908 9. Least area surfaces......Page 910 References......Page 911 1. Introduction......Page 915 2. Soliton equations associated to simple Lie algebras......Page 917 3. Soliton equations in submanifold geometry......Page 919 4. The space-time monopole equation......Page 921 -hierarchy......Page 924 6. Direct scattering for the space-time monopole equation......Page 925 -hierarchy via loop group factorizations......Page 926 8. The inverse scattering for monopole equations......Page 929 9. Birkhoff factorization and local solutions......Page 930 -hierarchy......Page 932 11. Bäcklund transformations for the space-time monopole equation......Page 934 References......Page 936 1. Introduction......Page 939 2. Conformal volume of orbifolds......Page 940 3. Finite subgroups of O(3)......Page 941 4. Eigenvalue bounds......Page 942 5. Congruence arithmetic hyperbolic 3-orbifolds......Page 943 6. Finiteness of arithmetic Kleinian maximal reflectio groups......Page 944 7. Conclusion......Page 945 References......Page 947 Introduction......Page 949 1. The universe of finitel presented groups......Page 950 spaces and their isometries......Page 955 3. Non-positively curved groups......Page 958 4. Word problems and fillin invariants......Page 962 5. Subdirect products of hyperbolic groups......Page 967 6. Two questions of Grothendieck......Page 969 References......Page 971 1. Introduction......Page 976 2. A categorificatio of the Jones polynomial......Page 978 3. Extensions to tangles......Page 979 link homology and matrix factorizations......Page 981 5. Triply-graded link homology and beyond......Page 983 References......Page 984 1. Disjoint curves in surfaces......Page 987 2. Curve complexes......Page 991 3. Nested structure......Page 995 4. Coarse geometry of......Page 999 5. Hyperbolic geometry and ending laminations......Page 1004 6. Heegaard splittings......Page 1009 References......Page 1013 1. The Brouwer degree......Page 1020 2. A quick recollection on A1-homotopy......Page 1023 3. A1-homotopy and A1-homology: the basic theorems......Page 1025 4. A1-homotopy and A1-homology: computations involving Milnor– Witt K-theory......Page 1029 5. Some results on classifying spaces in A1-homotopy theory......Page 1036 6. Miscellaneous......Page 1040 References......Page 1042 1. Introduction......Page 1045 2. Floer theory for symplectomorphisms......Page 1046 3. Floer theory for Lagrangian submanifolds......Page 1052 References......Page 1063 1. Heegaard–Floer homology of three-manifolds......Page 1067 2. Heegaard–Floer homology of knots......Page 1069 3. Heegaard–Floer homology for links......Page 1070 4. Basic properties......Page 1076 5. Three applications......Page 1077 References......Page 1081 1. Introduction......Page 1084 2. Outer space and homological finitenes results......Page 1085 3. The bordificatio and duality......Page 1088 4. The Degree Theorem and rational homology stability......Page 1089 5. Sphere complexes and integral homology stability......Page 1091 6. Graph complexes and unstable homology......Page 1094 7. IA automorphisms and the IA quotient of Outer space......Page 1097 8. Further reading......Page 1098 References......Page 1099 1. Introduction......Page 1101 2. Noncommutative resolutions and braid group actions......Page 1105 -modules in positive characteristic and localization theorem......Page 1114 4. Perverse sheaves on affin flag of the dual group (local geometric Langlands)......Page 1120 References......Page 1124 1. Introduction......Page 1127 2. Definitio of quasi-maps......Page 1129 3. Quasi-maps into fla varieties and semi-infinit Schubert varieties......Page 1133 and geometric Eisenstein series......Page 1139 5. Quasi-maps into affin fla varieties and Uhlenbeck compactifica tions......Page 1142 6. Applications to gauge theory and quantum cohomology of (affine fla manifolds......Page 1146 7. Some open problems......Page 1148 References......Page 1150 1. Smooth representations......Page 1153 3. The Jacquet–Langlands correspondence......Page 1154 4. Extending the Jacquet–Langlands correspondence......Page 1156 5. The Langlands correspondence......Page 1157 6. Explicit Langlands correspondence in the tame case......Page 1159 8. Construction and explicit Jacquet–Langlands correspondence, 1......Page 1160 References......Page 1161 1. Introduction......Page 1165 2. Three ways to stumble upon H......Page 1166 3. The rôle of amenability......Page 1174 4. Rigidity......Page 1177 5. Randomorphisms......Page 1180 6. Additional questions......Page 1184 References......Page 1186 1. Commentaires historiques......Page 1194 2. Fibration de Hitchin......Page 1195 3. Stabilisation de la partie anisotrope......Page 1198 4. Symétries de la bration de Hitchin......Page 1201 5. Groupes endoscopiques......Page 1203 Références......Page 1205 1. Introduction......Page 1207 2. Affin Hecke algebras......Page 1210 -theory and abstract Plancherel theorem......Page 1212 4. The Plancherel measure......Page 1213 5. The structure of the Schwartz algebra......Page 1221 6. Smooth families of tempered representations......Page 1225 -theory of the Schwartz algebra......Page 1227 8. Index functions......Page 1230 References......Page 1236 1. Motivation......Page 1240 2. Banach space representations......Page 1243 3. Locally analytic representation......Page 1246 4. Analytic vectors......Page 1253 5. Unramified p-adic functoriality......Page 1254 References......Page 1260 1. Introduction......Page 1262 2. First algebraization results: bounded generation and general rings......Page 1265 3. The algebraization of property (T) for nite groups: expanders......Page 1270 4. Reduced cohomology and property (T) for elementary linear groups......Page 1278 5. Some concluding remarks, questions, and speculations......Page 1281 References......Page 1284 1. Introduction......Page 1290 2. The descent method and applications......Page 1293 -functions for orthogonal groups; non-generic representations......Page 1298 References......Page 1303 1. Introduction......Page 1305 and automorphic representations......Page 1306 3. Locally symmetric spaces in the adelic language......Page 1309 4. Modular symbols and automorphic representations......Page 1310 5. A conjecture......Page 1311 References......Page 1312 1. Introduction......Page 1314 -action on a compactificatio......Page 1316 -action......Page 1318 4. Character sheaves on......Page 1322 References......Page 1323 1. Introduction......Page 1326 2. Quasiconformal and quasisymmetric maps......Page 1327 3. The quasisymmetric uniformization problem......Page 1330 4. Gromov hyperbolic spaces and quasisymmetric maps......Page 1332 5. Cannon’s conjecture and fractal 2-spheres......Page 1334 6. Post-critically finit rational maps......Page 1336 7. Sierpinski ́ carpets......Page 1340 8. Rigidity of square carpets......Page 1344 9. Conclusion......Page 1347 References......Page 1348 1. Introduction......Page 1351 2. Local Tb theorems for square functions and applications......Page 1355 3. Local Tb theorems for singular integrals and applications......Page 1360 References......Page 1366 1. Introduction......Page 1369 2. Convergence of the sequence of all partial sums......Page 1371 3. Convergence of subsequences of the sequence of partial sums......Page 1372 4. Ul’yanov’s problem......Page 1374 5. Strong summability......Page 1375 References......Page 1377 1. Introduction and notation......Page 1380 2. Iterated Segre mappings......Page 1382 3. Nondegeneracy conditions for generic submanifolds......Page 1384 4. Transversality of mappings......Page 1386 5. Finite jet determination......Page 1388 6. Stability groups......Page 1389 7. Algebraicity of mappings......Page 1391 References......Page 1392 1. Introduction......Page 1395 2. Lattice models......Page 1398 3. Schramm–Loewner evolution......Page 1405 4. SLE as a scaling limit......Page 1409 5. Ising model and beyond......Page 1414 6. Conclusion......Page 1422 References......Page 1423 1. Introduction......Page 1426 2. Compactness......Page 1430 3. Global regularity......Page 1439 References......Page 1446 1. Introduction. Historical remarks......Page 1452 2. Greedy algorithms with regard to bases......Page 1457 3. Optimal methods in nonlinear approximation......Page 1463 4. The TGA with regard to the trigonometric system......Page 1465 5. Convergence of the TGA with regard to the trigonometric system......Page 1466 6. General greedy algorithms......Page 1469 References......Page 1475 1. Introduction......Page 1478 2. Analytic capacity and the Cauchy transform......Page 1480 3. Principal values for the Cauchy integral and related results......Page 1489 4. Lipschitz harmonic capacity and Riesz transforms......Page 1492 5. Some open problems......Page 1493 References......Page 1497 1. Introduction......Page 1501 2. Functional extensions, functional tools......Page 1503 3. Multilinear inequalities......Page 1506 4. Geometry in Gauss space......Page 1509 5. Shannon entropy......Page 1511 References......Page 1515 1. Introduction......Page 1519 2. Symmetrization of convex bodies......Page 1521 3. Volume distribution in convex bodies......Page 1525 4. Beyond Brunn–Minkowski and Santaló inequalities......Page 1529 References......Page 1531 1. Introduction......Page 1535 2. Amenable actions and exactness......Page 1536 3. Amenable compactification which are small......Page 1544 4. Application to von Neumann algebra theory......Page 1545 References......Page 1549 1. Introduction......Page 1553 -algebras......Page 1555 3. Elliott’s classificatio conjecture......Page 1560 4. Almost commuting self-adjoint matrices: an application of real rank zero and stable rank one......Page 1562 -algebras......Page 1564 References......Page 1569 1. Introduction......Page 1571 2. Metric entropy and its duality......Page 1573 3. Geometric complexity of convex bodies and their diversity......Page 1577 4. Algorithmic complexity and derandomization, pseudorandom matrices......Page 1581 References......Page 1587 1. Introduction......Page 1594 2. Higher index theory of elliptic operators......Page 1596 3. Geometry of groups and metric spaces......Page 1601 4. Main results......Page 1605 References......Page 1606 1. Introduction......Page 1611 2. General classical problems of the spectral ergodic theory......Page 1613 3. Weak operator convergence......Page 1614 4. The homogeneous spectrum problem of arbitrary multiplicity......Page 1615 5. Spectral rigidity of group actions......Page 1617 6. Spectral invariants in natural subclasses of dynamical systems......Page 1618 7. Some more problems......Page 1621 References......Page 1622 Introduction......Page 1624 1. A brief survey......Page 1625 2. Ramsey theory and multiple recurrence along polynomials......Page 1633 3. Ergodic Ramsey theory in a noncommutative setting......Page 1637 4. Generalized polynomials and dynamical systems on nilmanifolds......Page 1639 5. Amenable groups and ergodic Ramsey theory......Page 1642 References......Page 1645 1. Introduction......Page 1648 2. Dispersing billiards......Page 1650 3. Slow mixing and non-standard limit theorems......Page 1652 4. Transport coefficient......Page 1655 5. Interacting particles......Page 1657 6. Infinit measure systems......Page 1665 References......Page 1669 1. Introduction......Page 1674 2. A mathematical formulation of the instability problem......Page 1675 3. The example of [Arn64] and the large gap problem......Page 1679 4. The role of normally hyperbolic manifolds......Page 1680 5. Overcoming the large gap problem by the method of [DdlLS03b]......Page 1683 6. Perturbations of geodesic flws and of superlinear oscillators......Page 1685 7. The method of correctly aligned windows......Page 1687 8. The method of normally hyperbolic laminations......Page 1689 9. The scattering map and the obstruction mechanism......Page 1691 References......Page 1692 1. Introduction......Page 1699 2. Entropy and classificatio of invariant measures......Page 1701 3. Brief review of some elements of entropy theory......Page 1708 4. Entropy and the set of values obtained by products of linear forms......Page 1712 5. Entropy and arithmetic quantum unique ergodicity......Page 1715 6. Entropy and distribution of periodic orbits......Page 1719 References......Page 1723