European Congress of Mathematics, Amsterdam, 14-18 July, 2008

Contents......Page 8
Committees......Page 10
Sponsors......Page 12
Opening Ceremony......Page 14
Prize Ceremony......Page 20
The PrizeWinners......Page 21
Congress Programme......Page 28
Invited Lectures......Page 38
Introduction......Page 40
Notion of solution......Page 42
Ingredients about optimal transport......Page 44
Uniqueness for bounded weak solutions......Page 46
Uniqueness for aggregation/diffusion competition......Page 49
References......Page 51
Introduction......Page 54
Galois representations......Page 56
Johan Bosman\'s examples......Page 59
Modular forms of level 1 and arbitrary weight......Page 60
Lattices......Page 64
Perspectives......Page 65
References......Page 66
Model cases of equidistribution problems......Page 68
Equidistribution of unipotent and closed orbits on homogeneous spaces......Page 73
Effective decay of matrix coefficients and spectral gap......Page 76
An effective pointwise ergodic theorem......Page 80
Effective equidistribution for semisimple subgroups......Page 83
Transportation of spectral gap......Page 86
References......Page 87
Survey on aspherical manifolds by\rWolfgang Lück......Page 90
Homotopy theory of aspherical manifolds......Page 91
Examples of aspherical manifolds......Page 92
Non-aspherical closed manifolds......Page 96
The Borel Conjecture......Page 97
Poincaré duality groups......Page 101
Product decompositions......Page 104
Novikov Conjecture......Page 107
Boundaries of hyperbolic groups......Page 109
L2-invariants......Page 110
The universe of closed manifolds......Page 112
References......Page 114
Introduction......Page 120
Directed graphs......Page 122
Decorated directed graphs......Page 123
Remark......Page 125
Props, properads, operads, etc. as ===G-algebras......Page 126
Morphisms and resolutions of ===G-algebras......Page 127
Claim......Page 128
Remark......Page 129
Claim......Page 130
Theorem......Page 131
Reminder on L_infinity-algebras and their homotopy classification......Page 132
Theorem......Page 133
Lie 1-bialgebras and Poisson geometry......Page 134
Theorem......Page 135
Pre-Lie algebras, Nijenhuis geometry and contractible dg manifolds......Page 136
Theorem......Page 137
Gerstenhaber algebras and Hertling–Manin geometry......Page 138
Theorem......Page 139
Theorem (MV)......Page 140
Deformation theory......Page 142
Koszul duality......Page 143
Definition......Page 144
Example: unimodular Lie 1-bialgebras versus quantum BV manifolds......Page 145
Formal quantum BV manifolds......Page 146
Definition......Page 147
Homotopy transfer of quantum BV-structures via Feynman integral......Page 148
References......Page 149
Introduction......Page 152
Delsarte\'s method......Page 154
The kissing problem......Page 156
Spherical two-distance sets......Page 158
Sylvester\'s theorem and SDP bounds for codes......Page 161
Multivariate positive definite functions......Page 166
References......Page 169
Introduction......Page 172
Classification by shallow minors......Page 174
Classification by edge densities......Page 175
Classification by topological resolution......Page 176
Classification by independence......Page 177
Classification by decomposition......Page 178
Classification by vertex ordering......Page 180
Simplicial graphs......Page 182
Bounded expansion classes......Page 183
Algebra of classes......Page 186
Distance between graph sets......Page 187
The left distance......Page 188
The right distance......Page 189
Full distance......Page 190
Dualities......Page 192
Duality of directed graphs......Page 194
Restricted dualities......Page 195
Planar graphs......Page 196
Characterization of graph classes with restricted dualities......Page 197
Concluding remarks......Page 198
References......Page 199
Introduction......Page 204
Hermitian line bundles and holonomy......Page 205
Descent of bundles......Page 207
Bundle gerbes......Page 209
Surface holonomy......Page 213
Wess–Zumino terms......Page 214
Sigma models......Page 215
Rational conformal field theory......Page 216
The TFT construction of full RCFT......Page 217
Jandl gerbes: Holonomy for unoriented surfaces......Page 220
D-branes: Holonomy for surfaces with boundary......Page 224
Gerbe bimodules and bi-branes......Page 226
Holonomy and Wess–Zumino term for defects......Page 227
Fusion of defects......Page 228
References......Page 231
Introduction......Page 234
Two dimensions......Page 235
Frobenius structure on quantum cohomology......Page 236
Givental\'s reconstruction conjecture......Page 238
Dubrovin\'s conjecture......Page 239
Cohomological field theory......Page 240
Moral interpretation of R......Page 242
Construction of the theory......Page 243
Some open questions......Page 244
References......Page 246
Lecture on Invitation\rby the KWG......Page 248
Introduction......Page 250
The ur-intuition......Page 253
Choice sequences......Page 257
Brouwer on logic......Page 259
Life and career......Page 262
References......Page 266
Plenary Lectures......Page 268
Introduction......Page 270
Sum-product theorem in finite fields......Page 271
Exponential sums over multiplicative subgroups......Page 272
Extensions to ``almost groups\'\'......Page 273
More on exponential sums......Page 274
Hyperelliptic curves......Page 277
Exponential sums over general sets and the theory of extractors in computer science......Page 278
SL_2(p) results and generalizations......Page 279
SU(2)......Page 282
Actions of linear groups on tori [19]......Page 283
References......Page 285
Introduction......Page 290
Bijections between maps and trees......Page 296
Plane trees......Page 299
Labeled trees and mobiles......Page 301
Convergence towards the Brownian map......Page 303
Two theorems about the Brownian map......Page 307
Geodesics in the Brownian map......Page 308
References......Page 312
An example: counting subgroups of finite index in nilpotent groups......Page 314
p-adic integrals......Page 315
Universal additive invariants......Page 317
Euler characteristics versus counting......Page 318
Euler characteristics......Page 319
The original construction......Page 320
Second application: finite group actions......Page 321
Third application: motivic Milnor fiber......Page 322
General motivic measure......Page 323
Generalization to definable sets......Page 324
An example......Page 325
Recent developments......Page 326
References......Page 327
Introduction: quantum fields and motives, an unlikely match......Page 330
A bottom-up approach to Feynman integrals and motives......Page 343
The Connes–Kreimer theory......Page 353
A top-down approach via Galois theory......Page 356
The geometry of Dim Reg......Page 361
References......Page 366
Introduction......Page 370
Space-time categories......Page 372
The framework of a local quantum field theory......Page 373
Local Lagrangian classical field theory......Page 374
Pure Euclidean d-dimensional Yang–Mills......Page 375
3-dimensional Chern–Simons theory......Page 376
Hamiltonian classical field theory......Page 377
Asymptotical expansion of non-degenerate oscillatory integrals and Feynman diagrams......Page 379
Feynman diagrams in G-invariant oscillatory integrals......Page 380
Path integrals......Page 383
Digression: renormalization......Page 384
The Yang–Mills theory......Page 386
The Chern–Simons theory......Page 388
Digression: quantized universal enveloping algebras at roots of unity......Page 392
Specialization at roots of unity......Page 393
Representations at roots of unity......Page 394
Braiding......Page 395
Invariants of framed tangled graphs with flat connections in the compliment......Page 396
Invariants of 3-manifolds and TQFT......Page 398
Relating combinatorial and semiclassical pictures......Page 399
Chern–Simons TQFT and geometric quantization......Page 403
The volume conjecture......Page 404
Complex Chern–Simons theory......Page 406
Other developments......Page 407
References......Page 408
Prize Lectures......Page 416
Computational complexity of linear problems......Page 418
Algebraic complexity of matrix multiplication......Page 420
Asymptotic bilinear complexity via tensor ranks......Page 421
Group-theoretic methods of fast matrix multiplication......Page 424
Asymptotic complexity of other linear problems......Page 425
Numerical stability of linear problems......Page 426
Recursive matrix multiplication: Strassen and beyond......Page 427
Group-theoretic matrix multiplication......Page 430
Matrix decompositions and other linear problems......Page 434
References......Page 436
Introduction......Page 438
An example: The sphere......Page 439
Sudakov\'s theorem......Page 441
Convexity......Page 442
Basic volumetric properties of convex sets......Page 443
Spectral gap......Page 444
Strong uniform convexity assumptions......Page 445
A central limit theorem for convex bodies......Page 446
Rate of convergence......Page 449
References......Page 452
The sixth problem of Hilbert......Page 456
From microscopic to mesoscopic and macroscopic models......Page 457
The Boltzmann equation......Page 458
The compressible Euler limit......Page 459
Mathematical setting......Page 460
Fluctuations around a global equilibrium......Page 461
Local thermodynamic equilibrium......Page 462
Motion and heat equations......Page 463
Mathematical difficulties......Page 464
Time regularity......Page 465
Spatial regularity......Page 466
Renormalized solutions to the Boltzmann equation......Page 467
The incompressible Navier–Stokes limit......Page 468
From Boltzmann to Navier–Stokes......Page 469
Strategy of the proof: the moment method......Page 470
The incompressible Euler limit......Page 471
From Boltzmann to Euler......Page 472
Strategy of the proof: the modulated entropy method......Page 473
References......Page 475
Introduction......Page 478
Algebras generated by elements a, b, c......Page 479
Algebras generated by elements x, y, z......Page 480
Some results from other papers......Page 482
The main result......Page 484
Conclusions......Page 485
References......Page 486
Author index......Page 488