By: Helge Holden (Editor), Ragni Piene (Editor)

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The book presents the winners of the first five Abel Prizes in mathematics: 2003 Jean-Pierre Serre; 2004 Sir Michael Atiyah and Isadore Singer; 2005 Peter D. Lax; 2006 Lennart Carleson; and 2007 S.R. Srinivasa Varadhan. Each laureate provides an autobiography or an interview, a curriculum vitae, and a complete bibliography. This is complemented by a scholarly description of their work written by leading experts in the field and by a brief history of the Abel Prize. Interviews with the laureates are also included on the enclosed DVD.

From the reviews: "The volume under review presents the winners of the first five prizes, together with a brief history of the prize and a very short biographical sketch of Selberg, the first, honorary, laureate. There is a short autobiographical piece by each laureate, followed by a more extensive review of their work. ... an overview suitable for a general mathematical audience." (Anton Cox, The London Mathematical Society, Issue 392, May, 2010) "The book presents the Abel Prize and the laureates of this prize in the first five years. The book is intended as the first volume of a series, each volume covering five years. ... The article is written in such a way as to explain the importance of the results at the level 'of a well-informed and able first year mathematics student' ... . Interesting photographs are included. The articles presenting the results of the recipients of the prizes are written perfectly." (Krzysztof Ciesielski, Mathematical Reviews, Issue 2011 i) "This superbly presented book begins with the history of the prize and is followed by a biographical sketch of Alte Selberg who was presented with an Honorary Abel Prize in Oslo in 2002. ... Each part contains a bibliography and a curriculum vitae. The text is adorned throughout with many photographs. ... This text is an invaluable contribution to the history of twentieth-century mathematics. Save your pennies and buy it." (Graham Hoare, The Mathematical Gazette, Vol. 95 (533), July, 2011) "Created in 2002 and first awarded in 2003, the Abel Prize comes the closest to being a 'Nobel Prize in mathematics.' ... For each winner, the book includes an autobiographical text, curriculum vitae and bibliography, photographs, and an extensive account of the mathematical work. ... This is a very well-made book." (Fernando Q. Gouvea, The Mathematical Association of America, May, 2011)

The History of the Abel Prize	p. 1
References	p. 6
2002-an Honorary Abel Prize to Atle Selberg	p. 7
References	p. 9
2003: Jean-Pierre Serre
Jean-Pierre Serre: Mon premier demi-siècie au Collège de France	p. 13
My First Fifty Years at the Collège de France	p. 13
Jean-Pierre Serre: An Overview of His Work	p. 33
Introduction	p. 33
The Beginnings	p. 34
Homotopy Groups of Spheres	p. 35
Hochschild-Serre Spectral Sequence	p. 36
Sheaf Cohomology of Complex Manifolds	p. 36
The Amsterdam Congress	p. 37
Sheaf Cohomology	p. 37
FAC	p. 37
GAGA	p. 38
Cohomology of Algebraic Varieties	p. 38
Lie Groups and Lie Algebras	p. 40
Local Algebra	p. 41
Projective Modules	p. 41
Algebraic Number Fields	p. 42
Class Field Theory	p. 43
Geometric Class Field Theory	p. 43
Local Class Field Theory	p. 45
A Local Mass Formula	p. 46
p-adic Analysis	p. 46
Group Cohomology	p. 47
Cohomology of Profinite Groups and p-adic Lie Groups	p. 47
Galois Cohomology	p. 48
Galois Cohomology of Linear Algebraic Groups	p. 48
Self-dual Normal Basis	p. 50
Essential Dimension	p. 50
Discrete Subgroups	p. 51
Congruence Subgroups	p. 52
Cohomology of Arithmetic Groups	p. 53
Arithmetic of Algebraic Varieties	p. 54
Modular Curves	p. 54
Varieties Over Finite Fields	p. 56
Number of Points of Curves Over Finite Fields	p. 56
Diophantine Problems	p. 58
Field Theory	p. 59
Galois Representations	p. 62
Hodge-Tate Modules	p. 62
Elliptic Curves and l-adic Representations	p. 62
Modular Forms and l-adic Representations	p. 65
Abehan Varieties and l-adic Representations	p. 69
Motives	p. 72
Group Theory	p. 73
Representation Theory	p. 73
Algebraic Groups	p. 76
Finite Subgroups of Lie Groups and of Algebraic Groups	p. 77
Miscellaneous Writings	p. 78
References	p. 80
List of Publications for Jean-Pierre Serre	p. 81
Curriculum Vitae for Jean-Pierre Serre	p. 95
2004: Sir Michael Atiyah and Isadore M. Singer
Autobiography	p. 99
Autobiography	p. 109
The Atiyah-Singer Index Theorem	p. 115
Introduction	p. 115
Background	p. 115
The Index	p. 115
Riemann-Roch	p. 116
The Beginning	p. 117
The Signature	p. 118
Hirzebruch-Riemann-Roch	p. 123
The Dirac Operator	p. 123
The Integer Index	p. 126
Formulation of the Theorem	p. 126
Integrality Theorems	p. 128
Positive Scalar Curvature	p. 129
Gauge-Theoretic Moduli Spaces	p. 130
The Equivariant Index	p. 132
K-Theory	p. 132
Fixed Point Theorems	p. 134
Rigidity Theorems	p. 137
The mod 2 Index	p. 139
Real K-Theory	p. 139
Theta Characteristics	p. 140
Positive Scalar Curvature	p. 140
The Index for Families	p. 141
Fredholm Operators	p. 141
Jumping of Dimension	p. 142
The Local Index Theorem	p. 144
The Heat Kernel	p. 144
The Eta Invariant	p. 146
Quantum Field Theory	p. 147
The Supersymmetric Proof	p. 148
References	p. 149
List of Publications for Sir Michael Atiyah	p. 151
List of Publications for Isadore M. Singer	p. 165
Curriculum Vitae for Sir Michael Francis Atiyah, OM, FRS, FRSE	p. 173
Curriculum Vitae for Isadore Manual Singer	p. 177
2005: Peter D. Lax
Autobiography	p. 181
A Survey of Peter D. Lax's Contributions to Mathematics	p. 189
Introduction	p. 189
Partial Differential Equations-General Results	p. 191
Difference Approximations to Partial Differential Equations	p. 195
Hyperbolic Systems of Conservation Laws	p. 196
Integrable Systems	p. 203
Lax-Phillips Scattering Theory	p. 208
References	p. 211
List of Publications for Peter D. Lax	p. 215
Curriculum Vitae for Peter David Lax	p. 229
2006: Lennart Carleson
Reflections on My Life as a Mathematician	p. 233
Carleson for Beginners	p. 241
References	p. 268
List of Publications for Lennart Carleson	p. 269
Curriculum Vitae for Lennart Axel Edvard Carleson	p. 275
2007: S.R. Srinivasa Varadhan
Autobiography	p. 279
A Personal Perspective on Raghu Varadhan's Role in the Development of Stochastic Analysis	p. 287
A Great Day for the Coin Flippers	p. 287
Stochastic Analysis	p. 287
Varadhan	p. 290
Independent or Uncorrelated	p. 290
A Simple Example	p. 291
Diffusion Equations and PDE	p. 294
Elliptic PDEs in Non-divergence Form with Continuous Elliptic Coefficients-Uniqueness of Solution	p. 295
First-Order PDEs and Large Deviations	p. 297
PDEs and Probability in the 60s	p. 298
The Support Theorem-Understanding the It6 Differential Equation	p. 299
On Extending the Wong-Zakai Theorem	p. 299
The Donsker-Varadhan Theory of Large Deviations	p. 304
Short Time Behaviour-Geodesies	p. 305
Long Time Behaviour-Occupation Measures	p. 308
Hydrodynamical Limits, Interacting Particles and Other Questions	p. 309
Homogenised Hamilton-Jacobi Equations	p. 310
Conclusion	p. 310
References	p. 311
List of Publications for S.R. Srinivasa Varadhan	p. 313
Curriculum Vitae for Sathamangalam Rangaiyenga Srinivasa Varadhan	p. 321
The Abel Committee	p. 323
The Board for the Niels Henrik Abel Memorial Fund	p. 325
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9783642013720
ISBN-10: 3642013724
Audience: Professional
Format: Book with CD or DVD
Language: English
Number Of Pages: 338
Published: 1st January 2010
Dimensions (cm): 23.5 x 15.5  x 1.8
Weight (kg): 0.612