Holden / Piene The Abel Prize

2003-2007 The First Five Years
1. Auflage 2009
ISBN: 978-3-642-01373-7
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 327 Seiten

Reihe: The Abel Prize

This book presents the first five Abel Prize in Mathematics winners, from 2003 to 2007. Leading experts in the field detail the work of each Abel Prize winner. The book also includes a brief history of the Abel Prize.

Autoren/Hrsg.

Holden, Helge

Piene, Ragni

Inhaltsverzeichnis

1;Preface;5
2;Contents;7
3;The History of the Abel Prize and the Honorary Abel Prize;12
3.1;The History of the Abel Prize;13
3.1.1;References;18
3.2;2002-an Honorary Abel Prize to Atle Selberg;19
3.2.1;References;21
4;2003 Jean-Pierre Serre;22
4.1;Jean-Pierre Serre: Mon premier demi-siècle au Collège de France;23
4.2;Jean-Pierre Serre: An Overview of His Work;42
4.2.1;Introduction;42
4.2.2;The Beginnings;43
4.2.2.1;Homotopy Groups of Spheres;44
4.2.2.2;Hochschild-Serre Spectral Sequence;45
4.2.2.3;Sheaf Cohomology of Complex Manifolds;45
4.2.2.4;The Amsterdam Congress;46
4.2.3;Sheaf Cohomology;46
4.2.3.1;FAC;46
4.2.3.2;GAGA;47
4.2.3.3;Cohomology of Algebraic Varieties;47
4.2.4;Lie Groups and Lie Algebras;49
4.2.5;Local Algebra;50
4.2.6;Projective Modules;50
4.2.7;Algebraic Number Fields;51
4.2.8;Class Field Theory;52
4.2.8.1;Geometric Class Field Theory;52
4.2.8.2;Local Class Field Theory;54
4.2.8.3;A Local Mass Formula;55
4.2.9;p-adic Analysis;55
4.2.10;Group Cohomology;56
4.2.10.1;Cohomology of Profinite Groups and p-adic Lie Groups;56
4.2.10.2;Galois Cohomology;57
4.2.10.3;Galois Cohomology of Linear Algebraic Groups;57
4.2.10.4;Self-dual Normal Basis;59
4.2.10.5;Essential Dimension;59
4.2.11;Discrete Subgroups;60
4.2.11.1;Congruence Subgroups;61
4.2.11.2;Cohomology of Arithmetic Groups;62
4.2.12;Arithmetic of Algebraic Varieties;63
4.2.12.1;Modular Curves;63
4.2.12.2;Varieties Over Finite Fields;65
4.2.12.3;Number of Points of Curves Over Finite Fields;65
4.2.12.4;Diophantine Problems;67
4.2.13;Field Theory;68
4.2.14;Galois Representations;71
4.2.14.1;Hodge-Tate Modules;71
4.2.14.2;Elliptic Curves and l-adic Representations;71
4.2.14.3;Modular Forms and l-adic Representations;74
4.2.14.4;Abelian Varieties and l-adic Representations;78
4.2.14.5;Motives;81
4.2.15;Group Theory;82
4.2.15.1;Representation Theory;82
4.2.15.2;Algebraic Groups;85
4.2.15.3;Finite Subgroups of Lie Groups and of Algebraic Groups;86
4.2.16;Miscellaneous Writings;87
4.2.17;References;89
4.3;List of Publications for Jean-Pierre Serre;90
4.4;Curriculum Vitae for Jean-Pierre Serre;103
5;2004 Sir Michael Atiyah and Isadore M. Singer;105
5.1;Autobiography;106
5.2;Autobiography;115
5.3;The Atiyah-Singer Index Theorem;121
5.3.1;Introduction;121
5.3.2;Background;121
5.3.2.1;The Index;121
5.3.2.2;Riemann-Roch;122
5.3.2.3;The Beginning;123
5.3.2.4;The Signature;124
5.3.2.5;Hirzebruch-Riemann-Roch;127
5.3.2.6;The Dirac Operator;129
5.3.3;The Integer Index;132
5.3.3.1;Formulation of the Theorem;132
5.3.3.2;Integrality Theorems;134
5.3.3.3;Positive Scalar Curvature;135
5.3.3.4;Gauge-Theoretic Moduli Spaces;136
5.3.4;The Equivariant Index;138
5.3.4.1;K-Theory;138
5.3.4.2;Fixed Point Theorems;140
5.3.4.3;Rigidity Theorems;143
5.3.5;The mod 2 Index;145
5.3.5.1;Real K-Theory;145
5.3.5.2;Theta Characteristics;146
5.3.5.3;Positive Scalar Curvature;146
5.3.6;The Index for Families;147
5.3.6.1;Fredholm Operators;147
5.3.6.2;Jumping of Dimension;148
5.3.7;The Local Index Theorem;150
5.3.7.1;The Heat Kernel;150
5.3.7.2;The Eta Invariant;152
5.3.7.3;Quantum Field Theory;153
5.3.7.4;The Supersymmetric Proof;154
5.3.8;References;155
5.4;List of Publications for Sir Michael Atiyah;157
5.5;List of Publications for Isadore M. Singer;170
5.6;Curriculum Vitae for Sir Michael Francis Atiyah, OM, FRS, FRSE;177
5.7;Curriculum Vitae for Isadore Manual Singer;181
6;2005 Peter D. Lax;183
6.1;Autobiography;184
6.2;A Survey of Peter D. Lax's Contributions to Mathematics;191
6.2.1;Introduction;191
6.2.2;Partial Differential Equations-General Results;193
6.2.3;Difference Approximations to Partial Differential Equations;197
6.2.4;Hyperbolic Systems of Conservation Laws;198
6.2.5;Integrable Systems;205
6.2.6;Lax-Phillips Scattering Theory;210
6.2.7;References;213
6.3;List of Publications for Peter D. Lax;217
6.4;Curriculum Vitae for Peter David Lax;230
7;2006 Lennart Carleson;232
7.1;Reflections on My Life as a Mathematician;233
7.2;Carleson for Beginners;240
7.2.1;References;267
7.3;List of Publications for Lennart Carleson;268
7.4;Curriculum Vitae for Lennart Axel Edvard Carleson;273
8;2007 S.R. Srinivasa Varadhan;275
8.1;Autobiography;276
8.2;A Personal Perspective on Raghu Varadhan's Role in the Development of Stochastic Analysis;284
8.2.1;A Great Day for the Coin Flippers;284
8.2.2;Stochastic Analysis;284
8.2.3;Varadhan;287
8.2.4;Independent or Uncorrelated;287
8.2.4.1;A Simple Example;288
8.2.5;Diffusion Equations and PDE;291
8.2.5.1;Elliptic PDEs in Non-divergence Form with Continuous Elliptic Coefficients-Uniqueness of Solution;292
8.2.5.2;First-Order PDEs and Large Deviations;294
8.2.5.3;PDEs and Probability in the 60s;295
8.2.6;The Support Theorem-Understanding the Itô Differential Equation;296
8.2.6.1;On Extending the Wong-Zakai Theorem;296
8.2.6.1.1;One-Dimension of Rough Control;299
8.2.6.1.2;Multiple Dimensions of Rough Control;300
8.2.7;The Donsker-Varadhan Theory of Large Deviations;301
8.2.7.1;Short Time Behaviour-Geodesics;302
8.2.7.2;Long Time Behaviour-Occupation Measures;305
8.2.8;Hydrodynamical Limits, Interacting Particles and Other Questions;306
8.2.8.1;Homogenised Hamilton-Jacobi Equations;307
8.2.9;Conclusion;307
8.2.10;References;308
8.3;List of Publications for S.R. Srinivasa Varadhan;310
8.4;Curriculum Vitae for Sathamangalam Rangaiyenga Srinivasa Varadhan;318
9;The Abel Committee;320
10;The Board for the Niels Henrik Abel Memorial Fund;322