VOLUME IN HONOR OF THE 60th BIRTHDAY OF JEAN-MICHEL BISMUT

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1 FROM PROBABILITY TO GEOMETRY (I) VOLUME IN HONOR OF THE 60th BIRTHDAY OF JEAN-MICHEL BISMUT Xianzhe DAI, Remi LEANDRE, Xiaonan MA and Weiping ZHANG, editors

2 Semi-classical Infinite Preface by Paul Malliavin Preface by Sir Michael Atiyah A letter from a friend Curriculum vitae of Jean-Michel Bismut xv xvii xix xxi The mathematical work of Jean-Michel Bismut: a brief summary xxv 1. Prom probability theory xxv to Index Theory xxvi 2.1. Superconnections, Quillen metrics and ^-invariants xxvi 2.2. Analytic torsion and complex geometry xxvii 2.3. Prom loop spaces to the hypoelliptic Laplacian xxviii 3. Conclusion xxix References xxix Shigeki Aida limit of the lowest eigenvalue of a Schrodinger operator on a Wiener space: I. Unbounded one particle Hamiltonians 1 1. Introduction 1 2. Preliminaries 2 3. Results 8 References 15 Sergio Albeverio & Sonia Mazzucchi dimensional oscillatory integrals with polynomial phase function and the trace formula for the heat semigroup Introduction Infinite dimensional oscillatory integrals The asymptotic expansion A degenerate case 30 Appendix. Abstract Wiener spaces 41 References 43

3 Smooth Witten Two-parameter A Richard F. Bass & Edwin Perkins new technique for proving uniqueness for martingale problems Introduction Some estimates Proof of Theorem References 53 Martin Grothaus, Ludwig Streit &: Anna Vogel Feynman integrals as Hida distributions: the case of non-perturbative potentials Introduction White Noise Analysis Hida distributions as candidates for Feynman Integrands Solution to time-dependent Schrodinger equation General construction of the Feynman integrand Examples The Feynman integrand for polynomial potentials Non-perturbative accessible potentials 65 References 67 Hiroshi Kunita Density of Canonical Stochastic Differential Equation with Jumps Introduction and main results Malliavin calculus for canonical SDE SDE's for derivatives of stochastic flow Alternative criterion for the smooth density Relation with Lie algebra Appendix. An analogue of Norris' estimate 87 References 90 James R. Norris - stochastic calculus and Malliavin''s integration-by-parts formula on Wiener space Introduction Integration-by-parts formula Review of two-parameter stochastic calculus A regularity result for two-parameter stochastic differential equations Derivation of the formula 109 References 113 Ichiro Shigekawa Laplacian on a lattice spin system Introduction Witten Laplacian in finite dimension 116 AST6RISQUB 327

4 ' vii 3. Witten Laplacian acting on differential forms Witten Laplacian in one-dimension Positivity of the lowest eigenvalue for the Witten Laplacian 124 References 129 Anton Alekseev, Henrique Bursztyn & Eckhard Meinrenken Pure Spinors on Lie groups Introduction Linear Dirac geometry Clifford algebras Pure spinors The bilinear pairing of spinors Contravariant spinors Action of the orthogonal group Morphisms Dirac spaces Lagrangian splittings Pure spinors on manifolds Dirac structures Dirac morphisms Bivector fields Dirac cohomology Classical dynamical Yang-Baxter equation Dirac structures on Lie groups The isomorphism TG = G x (g g) twisted Dirac structures on G The Cartan-Dirac structure Group multiplication Exponential map The Gauss-Dirac structure Pure spinors on Lie groups Cl(cj) as a spinor module over Cl(g ffig) The isomorphism AT*G = G x Cl(fl) Group multiplication Exponential map The Gauss-Dirac spinor q-hamiltonian G-manifolds Dirac morphisms and group-valued moment maps Volume forms The volume form in terms of the Gauss-Dirac spinor q-hamiltonian q-poisson g-manifolds *-valued moment maps it*-valued moment maps 192 societe mathematique de France 2009

5 Index, viii 6.1. Review of if*-valued moment maps P-valued moment maps Equivalence between if "-valued and P-valued moment maps Equivalence between P-valued and 6*-valued moment maps 196 References 196 Moulay-Tahar Benameur & Paolo Piazza eta and rho invariants on foliated bundles 201 Introduction and main results Group actions The discrete groupoid $ C*-algebras associated to the discrete groupoid $ von Neumann algebras associated to the discrete groupoid $ Traces Foliated spaces Foliated spaces The monodromy groupoid and the C*-algebra of the foliation von Neumann Algebras of foliations Traces Compatibility with Morita isomorphisms Hilbert modules and Dirac operators Connes-Skandalis Hilbert module T-equivariant pseudodifferential operators Functional calculus for Dirac operators Index theory The numeric index The index class in the maximal C*-algebra The signature operator for odd foliations Foliated rho invariants Foliated eta and rho invariants , Eta invariants and determinants of paths Stability properties of pv for the signature operator Leafwise homotopies pv(y,9) is metric independent Loops, determinants and Bott periodicity On the homotopy invariance of rho on foliated bundles The Baum-Connes map for the discrete groupoid TxT Homotopy invariance of pu(v, 57) for special homotopy equivalences Proof of the homotopy invariance for special homotopy equivalences: details Consequences of surjectivity I: equality of determinants Consequences of surjectivity II: the large time path The determinants of the large time path 271 ASTfiRISQUE 327

6 Direct ix 9.4. Consequences of injectivity: the small time path The determinants of the small time path 278 References 284 Alain Berthomieu image for some secondary K-theories Introduction Various if-theories Preliminaries Connections and vector bundle morphisms Chern-Simons transgression forms Definitions of the considered if-groups Topological if-theory if -theory of the category of flat bundles Relative if-theory "Free multiplicative" or "non hermitian smooth" if-theory Chern-Simons class on relative if-theory Relations between the preceding if-groups Symmetries associated to hermitian metrics Borel-Kamber-Tondeur class on ifch Direct images for if-groups The case of topological if-theory Preliminary: construction of family 303 index bundles Definition of the direct image morphism for ift Qp and K\ov The case of the if -theory of fiat bundles The case of relative if-theory The notion of "link" Definition of the direct image for if el The case of multiplicative, or smooth, if -theory Transgression of the family index theorem Direct image for multiplicative/smooth if -theory Hermitian symmetry and functoriality results Direct images and symmetries Double fibrations Proof of Theorems 25 and Proof of Theorem Links and exact sequences of vector bundles Link with "positive kernel" family index bundles Deformation of ip, h and General construction (and proof of Theorem 25) Proof of Theorem Reduction of the problem Sheaf theoretic direct images and short exact sequences 317 soci te mathematique de prance 2009

7 x "Adiabatic" limit of harmonic forms End of proof of Proposition yj-forms Z2-graded theory Z2-graded bundles and superconnections Special adjunction Adaptation of Bismut's superconnection Definition of Bismut and Lott's Levi-Civita superconnection Properties and asymptotics of the Chern character of Ct Calculating Ct for the product with the real line t > 0 asymptotics of the infinitesimal transgression form Adapting Ct to some suitable triple t > +co asymptotics of the infinitesimal transgression form 5.3. Proof of the first part of Theorem Chern-Simons transgression and links Definition of the»7-form and check of its properties Invariance properties of rj Anomaly formulae and their consequences Anomaly formulae End of proof of Theorem Proof of Theorem Proof of Theorem Influence of the vertical metric and the horizontal distribution 6. Fiberwise Hodge symmetry Symmetries induced on family index bundles The fiberwise Hodge * operator Symmetry induced by *z on fiberwise twisted Euler operators Odd dimensional fibre case Symmetry on canonical links Symmetry on connections on the infinite rank bundle & Proof of results about JTgat and K el End of proof of Theorem Results on tt<_ End of proof of Theorem Double fibrations Topological K-theory Fiberwise exterior differentials: Fiberwise Euler operators Introducing some intermediate suitable triple Estimates on the operator A\ Spectral convergence of Euler operators Construction of the canonical link (proof of Theorem 61) Flat and relative K-theory 352 AST&RISQUE 327

8 Hermitian xi Leray spectral sequence Compatibility of topological and sheaf theoretic links Proof of Theorem Multiplicative and smooth if-theory Calculation of tt u o 7if,u - (tt2 o 7n)fu Proof of Theorem References 358 Jean-Benoit Bost & Klaus Kunnemann vector bundles and extension groups on arithmetic schemes II. The arithmetic Atiyah extension Introduction Atiyah extensions in algebraic and analytic geometry Definition and basic properties Cotangent complex and Atiyah class <S -connections compatible with the holomorphic structure The arithmetic Atiyah class of a vector bundle with connection Definition and basic properties The first Chern class in arithmetic Hodge cohomology Hermitian line bundles with vanishing arithmetic Atiyah class Transcendence and line bundles with connections on abelian varieties Line bundles with connections on abelian varieties The complex case An application of the Theorem of Schneider-Lang Reality I Reality II Conclusion of the proof of Theorem Hermitian line bundles with vanishing arithmetic Atiyah class on smooth projective varieties over number fields Finiteness results on the kernel of cf A geometric analogue Line bundles with vanishing relative Atiyah class on fibered projective varieties Notation Variants and complements Hodge cohomology and first Chern class Hodge cohomology groups The first Chern class in Hodge cohomology An application of the Hodge Index Theorem The Hodge Index Theorem in Hodge cohomology An application to projective varieties fibered over curves The equivalence of VA1 and VA The Picard variety of a variety over a function field 410 SOClfiTl!: MATHEMATIQUE DB PRANCE 2009

9 xii 4.7. The equivalence of VA2 and VA3 412 Appendix A. Arithmetic extensions and Cech cohomology 414 Appendix B. The universal vector extension of a Picard variety 416 References 422 ASTtSRISQUE 327

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