The Abel Prize (eBook)

Preface 5
Contents 7
The History of the Abel Prize and the Honorary Abel Prize 12
The History of the Abel Prize 13
References 18
2002-an Honorary Abel Prize to Atle Selberg 19
References 21
2003 Jean-Pierre Serre 22
Jean-Pierre Serre: Mon premier demi-siècle au Collège de France 23
Jean-Pierre Serre: An Overview of His Work 42
Introduction 42
The Beginnings 43
Homotopy Groups of Spheres 44
Hochschild-Serre Spectral Sequence 45
Sheaf Cohomology of Complex Manifolds 45
The Amsterdam Congress 46
Sheaf Cohomology 46
FAC 46
GAGA 47
Cohomology of Algebraic Varieties 47
Lie Groups and Lie Algebras 49
Local Algebra 50
Projective Modules 50
Algebraic Number Fields 51
Class Field Theory 52
Geometric Class Field Theory 52
Local Class Field Theory 54
A Local Mass Formula 55
p-adic Analysis 55
Group Cohomology 56
Cohomology of Profinite Groups and p-adic Lie Groups 56
Galois Cohomology 57
Galois Cohomology of Linear Algebraic Groups 57
Self-dual Normal Basis 59
Essential Dimension 59
Discrete Subgroups 60
Congruence Subgroups 61
Cohomology of Arithmetic Groups 62
Arithmetic of Algebraic Varieties 63
Modular Curves 63
Varieties Over Finite Fields 65
Number of Points of Curves Over Finite Fields 65
Diophantine Problems 67
Field Theory 68
Galois Representations 71
Hodge-Tate Modules 71
Elliptic Curves and l-adic Representations 71
Modular Forms and l-adic Representations 74
Abelian Varieties and l-adic Representations 78
Motives 81
Group Theory 82
Representation Theory 82
Algebraic Groups 85
Finite Subgroups of Lie Groups and of Algebraic Groups 86
Miscellaneous Writings 87
References 89
List of Publications for Jean-Pierre Serre 90
Curriculum Vitae for Jean-Pierre Serre 103
2004 Sir Michael Atiyah and Isadore M. Singer 105
Autobiography 106
Autobiography 115
The Atiyah-Singer Index Theorem 121
Introduction 121
Background 121
The Index 121
Riemann-Roch 122
The Beginning 123
The Signature 124
Hirzebruch-Riemann-Roch 127
The Dirac Operator 129
The Integer Index 132
Formulation of the Theorem 132
Integrality Theorems 134
Positive Scalar Curvature 135
Gauge-Theoretic Moduli Spaces 136
The Equivariant Index 138
K-Theory 138
Fixed Point Theorems 140
Rigidity Theorems 143
The mod 2 Index 145
Real K-Theory 145
Theta Characteristics 146
Positive Scalar Curvature 146
The Index for Families 147
Fredholm Operators 147
Jumping of Dimension 148
The Local Index Theorem 150
The Heat Kernel 150
The Eta Invariant 152
Quantum Field Theory 153
The Supersymmetric Proof 154
References 155
List of Publications for Sir Michael Atiyah 157
List of Publications for Isadore M. Singer 170
Curriculum Vitae for Sir Michael Francis Atiyah, OM, FRS, FRSE 177
Curriculum Vitae for Isadore Manual Singer 181
2005 Peter D. Lax 183
Autobiography 184
A Survey of Peter D. Lax's Contributions to Mathematics 191
Introduction 191
Partial Differential Equations-General Results 193
Difference Approximations to Partial Differential Equations 197
Hyperbolic Systems of Conservation Laws 198
Integrable Systems 205
Lax-Phillips Scattering Theory 210
References 213
List of Publications for Peter D. Lax 217
Curriculum Vitae for Peter David Lax 230
2006 Lennart Carleson 232
Reflections on My Life as a Mathematician 233
Carleson for Beginners 240
References 267
List of Publications for Lennart Carleson 268
Curriculum Vitae for Lennart Axel Edvard Carleson 273
2007 S.R. Srinivasa Varadhan 275
Autobiography 276
A Personal Perspective on Raghu Varadhan's Role in the Development of Stochastic Analysis 284
A Great Day for the Coin Flippers 284
Stochastic Analysis 284
Varadhan 287
Independent or Uncorrelated 287
A Simple Example 288
Diffusion Equations and PDE 291
Elliptic PDEs in Non-divergence Form with Continuous Elliptic Coefficients-Uniqueness of Solution 292
First-Order PDEs and Large Deviations 294
PDEs and Probability in the 60s 295
The Support Theorem-Understanding the Itô Differential Equation 296
On Extending the Wong-Zakai Theorem 296
One-Dimension of Rough Control 299
Multiple Dimensions of Rough Control 300
The Donsker-Varadhan Theory of Large Deviations 301
Short Time Behaviour-Geodesics 302
Long Time Behaviour-Occupation Measures 305
Hydrodynamical Limits, Interacting Particles and Other Questions 306
Homogenised Hamilton-Jacobi Equations 307
Conclusion 307
References 308
List of Publications for S.R. Srinivasa Varadhan 310
Curriculum Vitae for Sathamangalam Rangaiyenga Srinivasa Varadhan 318
The Abel Committee 320
The Board for the Niels Henrik Abel Memorial Fund 322