2002 an Honorary Abel Prize to Atle Selberg... 7 Nils A. Baas References... 9

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1 Contents The History of the Abel Prize... 1 Arild Stubhaug References an Honorary Abel Prize to Atle Selberg... 7 Nils A. Baas References : Jean-Pierre Serre Jean-Pierre Serre: Mon premier demi-siècle au Collège de France My First Fifty Years at the Collège de France Marc Kirsch Jean-Pierre Serre: An Overview of His Work Pilar Bayer Introduction TheBeginnings Homotopy Groups of Spheres Hochschild Serre Spectral Sequence Sheaf Cohomology of Complex Manifolds The Amsterdam Congress Sheaf Cohomology FAC GAGA Cohomology of Algebraic Varieties Lie Groups and Lie Algebras Local Algebra Projective Modules Algebraic Number Fields Class Field Theory Geometric Class Field Theory vii

2 viii Contents 7.2 Local Class Field Theory A Local Mass Formula p-adicanalysis Group Cohomology Cohomology of Profinite Groups and p-adic Lie Groups Galois Cohomology Galois Cohomology of Linear Algebraic Groups Self-dual Normal Basis EssentialDimension Discrete Subgroups Congruence Subgroups Cohomology of Arithmetic Groups Arithmetic of Algebraic Varieties Modular Curves VarietiesOverFiniteFields NumberofPointsofCurvesOverFiniteFields Diophantine Problems Field Theory GaloisRepresentations Hodge Tate Modules Elliptic Curves and l-adicrepresentations Modular Forms and l-adicrepresentations Abelian Varieties and l-adicrepresentations Motives Group Theory Representation Theory Algebraic Groups Finite Subgroups of Lie Groups and of Algebraic Groups Miscellaneous Writings References List of Publications for Jean-Pierre Serre Curriculum Vitae for Jean-Pierre Serre : Sir Michael Atiyah and Isadore M. Singer Autobiography Sir Michael Atiyah Autobiography Isadore M. Singer The Atiyah Singer Index Theorem Nigel Hitchin 1 Introduction

3 Contents ix 2 Background TheIndex Riemann Roch TheBeginning The Signature Hirzebruch Riemann Roch TheDiracOperator TheIntegerIndex Formulation of the Theorem Integrality Theorems Positive Scalar Curvature Gauge-Theoretic Moduli Spaces TheEquivariantIndex K-Theory Fixed Point Theorems Rigidity Theorems Themod2Index Real K-Theory ThetaCharacteristics Positive Scalar Curvature The Index for Families Fredholm Operators JumpingofDimension The Local Index Theorem TheHeatKernel TheEtaInvariant Quantum Field Theory The Supersymmetric Proof References List of Publications for Sir Michael Atiyah List of Publications for Isadore M. Singer Curriculum Vitae for Sir Michael Francis Atiyah, OM, FRS, FRSE Curriculum Vitae for Isadore Manual Singer : Peter D. Lax Autobiography Peter D. Lax A Survey of Peter D. Lax s Contributions to Mathematics Helge Holden and Peter Sarnak 1 Introduction

4 x Contents 2 Partial Differential Equations General Results Difference Approximations to Partial Differential Equations Hyperbolic Systems of Conservation Laws IntegrableSystems Lax Phillips Scattering Theory References List of Publications for Peter D. Lax Curriculum Vitae for Peter David Lax : Lennart Carleson Reflections on My Life as a Mathematician Lennart Carleson Carleson for Beginners Tom Körner References List of Publications for Lennart Carleson Curriculum Vitae for Lennart Axel Edvard Carleson : S.R. Srinivasa Varadhan Autobiography S.R. Srinivasa Varadhan A Personal Perspective on Raghu Varadhan s Role in the Development of Stochastic Analysis Terry Lyons 1 A Great Day for the Coin Flippers Stochastic Analysis Varadhan Independent or Uncorrelated ASimpleExample DiffusionEquationsandPDE Elliptic PDEs in Non-divergence Form with Continuous Elliptic Coefficients Uniqueness of Solution First-OrderPDEsandLargeDeviations PDEs and Probability in the 60s The Support Theorem Understanding the Itô Differential Equation On Extending the Wong Zakai Theorem The Donsker Varadhan Theory of Large Deviations Short Time Behaviour Geodesics Long Time Behaviour Occupation Measures...308

5 Contents xi 8 Hydrodynamical Limits, Interacting Particles and Other Questions Homogenised Hamilton Jacobi Equations Conclusion References List of Publications for S.R. Srinivasa Varadhan Curriculum Vitae for Sathamangalam Rangaiyenga Srinivasa Varadhan 321 The Abel Committee The Board for the Niels Henrik Abel Memorial Fund...325

Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem

PETER B. GILKEY Department of Mathematics, University of Oregon Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem Second Edition CRC PRESS Boca Raton Ann Arbor London Tokyo Contents