The Abel Prize 2013-2017

Preface......Page 7
Contents......Page 9
Part I 2013 Pierre Deligne......Page 12
 Mathematical Autobiography......Page 14
  Definition of Mixed Hodge Structures......Page 18
  Conjectures on Critical Zeta Values......Page 19
  Relations Between Multizeta Values......Page 20
 Pierre Deligne: A Poet of Arithmetic Geometry......Page 23
   1.1 General Topology......Page 24
    Degeneration and Decomposition in the Derived Category......Page 25
    Décalage of Filtrations......Page 26
   1.3 Cohomological Descent......Page 28
    Global Duality......Page 29
    The Functors Rf! and Rf!......Page 30
    Picard Stacks and Geometric Class Field Theory......Page 31
    Symmetric Künneth Formula......Page 33
    Finiteness......Page 34
   2.1 Deligne–Mumford Stacks......Page 35
   2.2 Moduli of Curves of Genus ≥2......Page 36
    Generalized Elliptic Curves and Compactifications......Page 37
    The Curve Case......Page 38
    Higher Dimension: The Riemann–Hilbert Correspondence......Page 39
   3.2 Betti–de Rham Comparison Theorems......Page 41
    Discontinuous Crystals......Page 43
    Liftings of K3 Surfaces, Canonical Coordinates......Page 44
    The de Rham–Witt Complex......Page 45
   3.4 Irregular Connections......Page 46
   3.5 Monodromy of the Hypergeometric Equation, Lattices......Page 47
   4.1 Hodge I......Page 50
    Homological Algebra Infrastructure......Page 51
    Mixed Hodge Theory......Page 54
    The Fixed Part and Semisimplicity Theorems......Page 58
    1-Motives......Page 60
    The du Bois Complex......Page 61
    Hodge Theory and Rational Homotopy......Page 62
    Axiomatization of Shimura Varieties......Page 63
    Canonical Models......Page 65
   4.4 Absolute Hodge Cycles......Page 66
    Link with the Hodge Conjecture and Intermediate Jacobians......Page 69
    Link with the Tame Symbol......Page 70
   4.6 Liftings mod p2 and Hodge Degeneration......Page 71
   4.7 The Hodge Locus......Page 72
    Basic Definitions......Page 73
    Statement of the Weil Conjectures......Page 74
    Grothendieck\'s Trace Formula......Page 76
   5.2 A False Good Plan: Grothendieck\'s Standard Conjectures......Page 77
    K3 Surfaces......Page 79
    Complete Intersections of Hodge Level ≤1......Page 80
    Independence of l......Page 81
   5.5 Weil I......Page 84
    Mixed Sheaves, Statement of the Main Theorem......Page 87
    Ingredients of the Proof......Page 89
    First Applications......Page 93
    Exponential Sums......Page 98
    Definition and First Properties......Page 102
    Laumon\'s Contribution and Applications......Page 103
    t-Structures......Page 104
    Perverse Sheaves in the Étale Setting......Page 107
    The Purity and Decomposition Theorems......Page 108
    Consequences Over C......Page 110
    A Finiteness Theorem......Page 111
    Counting Lisse -Adic Sheaves......Page 112
   6.1 Construction of -Adic Representations......Page 113
   6.2 The Weil–Deligne Group......Page 114
    Construction of Local Constants......Page 116
    The Case of Function Fields......Page 117
    Additional Results......Page 119
   6.4 Abelian L-Functions and Hilbert–Blumenthal Moduli Spaces......Page 120
   7.1 The Picard–Lefschetz Formula......Page 121
   7.2 The Milnor Number......Page 122
   7.3 Wild Ramification and Euler–Poincaré Characteristics......Page 123
    Local Behavior......Page 124
    Surfaces......Page 125
   7.4 Nearby Cycles Over General Bases......Page 126
    Deligne–Lusztig Varieties......Page 127
    A Fixed Point Formula......Page 129
    Properties of the RθT......Page 130
    Duality......Page 131
   8.2 Central Extensions......Page 132
   8.3 Braid Groups......Page 133
   8.4 Reductive Groups Over Local Fields......Page 134
    The Tannakian Formalism......Page 136
    Further Results......Page 138
    Values of L-Functions at Critical Integers......Page 139
    Mixed Tate Motives......Page 142
  10 Deligne\'s Conjectures......Page 147
  11 Expository Articles......Page 151
  References......Page 152
 List of Publications for Pierre R. Deligne......Page 163
 Curriculum Vitae for Pierre R. Deligne......Page 173
Part II 2014 Yakov G. Sinai......Page 175
 Autobiography......Page 178
  1 Introduction......Page 183
  2 A Geometric Trapping Method for NSS......Page 184
  3 Power Series and Diagrams......Page 188
  4 Complex Valued Solutions and Renormalization Group......Page 193
  5 Bifurcations of Solutions to Two-Dimensional Navier–Stokes Systems......Page 197
  6 Stochastic Hydrodynamics......Page 198
  References......Page 201
 Billiards......Page 203
  1 Mathematical Billiards......Page 204
  3 Hyperbolicity of Sinai Billiards......Page 206
  4 Sinai\'s Fundamental Theorem of the Theory of Hyperbolic Billiards......Page 209
  5 Mechanism of Defocusing and Hyperbolic Focusing Billiards......Page 210
  References......Page 212
 Ya.G. Sinai\'s Work on Number Theory......Page 214
  References......Page 225
  1 Prehistory of Dynamical Entropy (Shannon, Khinchin, Kolmogorov)......Page 228
  2 Emergence of Dynamical Entropy......Page 229
  3 Early Development of Entropy Theory......Page 232
   3.2 Isomorphism Problem: First Results......Page 234
   3.3 First Examples of Smooth K-Systems......Page 236
   3.5 Entropy and Periodic Orbits......Page 238
  4 Topological Entropy and Emergence of Thermodynamic Formalism......Page 239
  5 Ornstein\'s Theory......Page 242
  6 Kakutani Equivalence......Page 243
  7 Other Entropy Type Characteristics of Dynamics......Page 244
  8 Entropy for Actions of General Groups......Page 245
  References......Page 247
  1 Introduction......Page 250
  2 Statistical Mechanics......Page 251
  3 Spectral Theory of Schrödinger Operators......Page 253
  4 Random Walks in Random Environment......Page 256
  5 Renormalization Methods in Dynamical Systems......Page 257
  6 Random Burgers Equation......Page 259
  References......Page 261
  1 Introduction......Page 263
   1.1 From Scientific Determinism to Deterministic Chaos......Page 264
   1.2 Markov Partitions and Symbolic Representations of Chaotic Dynamics......Page 265
   1.3 Entropy......Page 266
   1.4 Hyperbolicity......Page 267
   2.1 Uniform Hyperbolicity......Page 268
   2.2 Nonuniform Hyperbolicity......Page 270
   2.3 Partial Hyperbolicity......Page 271
   3.1 Definition of Markov Partitions......Page 272
   3.2 Symbolic Models......Page 273
   3.3 Sinai\'s Construction of Markov Partitions......Page 274
   4.2 Natural and Physical Measures......Page 276
   4.3 SRB Measures......Page 277
   4.4 Uniformly Hyperbolic Attractors......Page 279
   4.5 First Proof of Theorem 4 (Sinai [60])......Page 280
   4.6 Second Proof of Theorem 4 (Pesin and Sinai, [53])......Page 281
   5.1 Chaotic Attractors: The Concept......Page 282
   5.3 The Hénon Attractor......Page 283
   5.5 Chaotic Attractors: Constructing SRB-Measures......Page 284
   6.1 Partially Hyperbolic Attractors......Page 285
   6.2 SRB-Measures on Partially Hyperbolic Attractors......Page 286
   6.3 Dominated Splitting and SRB-Measures......Page 287
  References......Page 288
  2 Posing the Problem: The Investigated Models......Page 292
   2.2 Transitive Cylindric Billiards......Page 294
   2.4 Transverseness......Page 295
   3.2 Step II......Page 296
   3.3 Step III......Page 297
  4 The Conditional Proof......Page 298
   5.1 Formulation of Theorem......Page 301
  References......Page 302
  1 Introduction......Page 304
  2 Markov Partitions for Anosov Maps (and for Axiom A Maps)......Page 306
  3 Sinai Billiard and Lorentz Process......Page 309
    Handling the Singularities......Page 312
  4 Statistical Properties of 2D Periodic Lorentz Processes......Page 313
   4.1 Bunimovich–Sinai, 1980......Page 314
   4.2 Bunimovich–Chernov–Sinai, 1990–1991......Page 315
   5.1 Markov Towers......Page 316
   5.2 Standard Pairs......Page 318
    Applications of the Method of Standard Pairs......Page 319
  References......Page 321
 List of Publications for Yakov G. Sinai......Page 325
 Curriculum Vitae for Yakov Grigorevich Sinai......Page 348
Part III 2015 John F. Nash, Jr. and Louis Nirenberg......Page 350
 Autobiography......Page 354
 John Nash, His Life......Page 359
  References......Page 378
 Autobiography......Page 380
  1 Introduction......Page 391
   2.1 Introduction......Page 392
   2.2 Real Algebraic Structures and Main Statements......Page 394
   2.3 Technical Preliminaries......Page 397
   2.4 The Algebraic Description of Nash Manifolds and the Characterization of Representations as Sheets......Page 400
   2.5 Proof of the Existence of Representations and of the Approximation Theorem......Page 403
   2.6 Proof of the Uniqueness of the Nash Ring......Page 409
   3.1 Introduction......Page 410
   3.2 Main Iteration......Page 413
   3.3 Decomposition in Primitive Metrics......Page 418
   3.4 Proof of the Main Iterative Statement......Page 419
   4.1 Introduction......Page 424
   4.2 The Perturbation Theorem......Page 426
   4.3 Proof of the Smooth Isometric Embedding Theorem......Page 428
   4.4 Smoothing Operator......Page 433
   4.5 A Smooth Path to Prove the Perturbation Theorem......Page 437
   4.6 A Priori Estimates for Solutions of Nash\'s Regularized Flow Equations......Page 440
   4.7 Global Existence and Convergence to an Isometry......Page 447
   4.8 Higher Regularity of the Map u......Page 450
   4.9 The Nonclosed Case......Page 453
   5.1 Introduction......Page 454
   5.2 Preliminaries and Main Statements......Page 458
   5.3 Three Elementary Inequalities......Page 461
   5.4 Energy, Entropy and Moment Bounds......Page 465
   5.5 G Bound......Page 467
   5.6 Overlap Estimate......Page 473
   5.7 Proof of the a Priori Estimate......Page 479
   5.8 Proof of Nash\'s Parabolic Regularity Theorem......Page 480
   5.9 Proof of the De Giorgi–Nash Theorem......Page 486
   6.1 A Path Space and Stiefel–Whitney Classes......Page 489
   6.2 Le problème de Cauchy pour les équations différentielles d\'un fluide général......Page 490
   6.4 Arc Structure of Singularities......Page 492
   6.5 The Nash Blow-Up......Page 494
  References......Page 495
  1 Introduction......Page 500
  2 The Weyl Problem, the Minkowski Problem, and Fully Nonlinear PDE in Two Space Dimensions......Page 502
  3 Elliptic Regularity for Boundary Value Problems: The Agmon–Douglis–Nirenberg Estimates......Page 505
  4 Functions with Bounded Mean Oscillation......Page 508
  5 Partial Regularity for the 3D Navier–Stokes Equations......Page 511
  6 Nonlinear Elliptic Equations Involving Critical Exponents......Page 516
  7 The Method of Moving Planes and the Sliding Method......Page 519
  8 Conclusion......Page 523
  References......Page 524
 List of Publications for John Forbes Nash, Jr.......Page 528
 List of Publications for Louis Nirenberg......Page 531
 Curriculum Vitae for John Forbes Nash, Jr.......Page 544
 Curriculum Vitae for Louis Nirenberg......Page 546
Part IV 2016 Sir Andrew J. Wiles......Page 548
 First Steps......Page 551
  1 Introduction......Page 557
  2 Explicit Reciprocity Laws......Page 558
   2.1 The Hilbert Symbol......Page 559
   2.2 The Explicit Reciprocity Laws of Artin, Hasse, and Iwasawa......Page 560
   2.3 Wiles\'s Higher Explicit Reciprocity Laws......Page 561
   2.4 Example: The Formal Multiplicative Group Over Zp......Page 563
   2.5 Further Developments......Page 564
   3.1 The Coates–Wiles Homomorphisms......Page 565
   3.2 Groups of Local Units......Page 567
   3.3 Example: Cyclotomic Units......Page 568
   3.4 Example: Elliptic Units......Page 570
   3.5 Further Developments......Page 573
   4.1 Elliptic Curves and the BSD Conjecture......Page 574
   4.2 The Coates–Wiles Theorem......Page 575
   4.3 Idea of the Proof: Bird\'s Eye View......Page 576
   4.4 A Variation on the Proof in CoatesWiles-BSD......Page 577
   4.5 Further Developments......Page 578
   5.1 Iwasawa\'s Main Conjecture......Page 579
    The Main Conjecture of Iwasawa Theory for Totally Real Fields......Page 580
    The Main Conjecture with μ-Invariant......Page 582
    A Refined Class Number Formula......Page 583
    Other Formulations......Page 584
   5.2 The Proof by Mazur and Wiles......Page 585
    The Good Quotient......Page 586
    The Extension......Page 587
   5.3 Wiles\'s Proof for Totally Real Fields......Page 588
   5.4 Further Developments......Page 590
  6 Galois Representations......Page 591
   6.1 Galois Representations of Modular Forms......Page 592
   6.2 Pseudorepresentations......Page 593
   6.3 Ordinarity of Galois Representations......Page 594
   6.4 Galois Representations of Hilbert Modular Forms......Page 595
   6.5 Further Developments......Page 596
  7 Modularity of Elliptic Curves and Fermat\'s Last Theorem......Page 597
   7.1 Elliptic Curves and Modular Forms......Page 598
   7.2 Modularity in Terms of Galois Representations......Page 599
   7.4 Modularity of Semistable Elliptic Curves and FLT......Page 601
    A Ring-Theoretic Reformulation......Page 602
    RD = TD: the minimal case......Page 605
    RD= TD: Relaxing Restrictions......Page 606
   7.7 Modularity of Residually Reducible Representations......Page 607
   7.8 Further Developments......Page 610
  8 Solvable Points on Genus One Curves......Page 611
    The Case of Analytic Rank at Most One......Page 612
  9 Class Groups of Quadratic Fields......Page 613
  References......Page 615
  Other References......Page 616
 List of Publications for Sir Andrew J. Wiles......Page 619
 Curriculum Vitae for Sir Andrew John Wiles, KBE, FRS......Page 622
Part V 2017 Yves Meyer......Page 624
 For My Mother......Page 627
  References......Page 637
  Foreword......Page 638
  1 Harmonic Analysis, Quasicrystals and Number Theory......Page 639
   1.1 Spectral Synthesis......Page 640
   1.2 Coherent Sets of Frequencies......Page 643
   1.3 Mathematical Models of Quasicrystals......Page 645
   1.4 Universal Sampling Sets......Page 648
   1.5 New Poisson Summation Formulas......Page 650
  2 The Calderón Program......Page 652
   2.1 Calderón–Zygmund Operators and Calderón\'s Conjectures......Page 653
   2.2 Commutators and the Cauchy Integral......Page 656
   2.3 Offsprings and Related Results......Page 659
  3 Wavelets and Time-Frequency Analysis......Page 662
   3.1 From Fourier Transforms to Wavelet Transforms......Page 663
   3.2 Orthonormal Wavelet Bases......Page 665
   3.3 Multiresolution Analysis......Page 668
   3.4 Smoothness Spaces and Sparsity......Page 671
   3.5 Chirps, Time-Frequency Bases and Gravitational Waves......Page 674
   3.6 Pointwise Smoothness and Multifractal Analysis......Page 676
   4.1 The Wave Equation and Control Problems......Page 678
   4.2 Compensated Compactness......Page 681
   4.3 Improved Sobolev Embeddings and the Space BV......Page 684
   4.4 Oscillations in Nonlinear Evolution PDEs......Page 687
  References......Page 690
 List of Publications for Yves F. Meyer......Page 695
 Curriculum Vitae for Yves F. Meyer......Page 710
Part VI Abel Activities......Page 712
 Abel Prize Citations 2003–2012......Page 729
The Abel Committee......Page 742
The Niels Henrik Abel Board......Page 744
The Abel Lectures 2013–2017......Page 746
The Abel Laureate Presenters 2013–2017......Page 748
The Interviews with the Abel Laureates......Page 749
Addenda, Errata, and Updates......Page 751