Verfügbarkeit: From probability to geometry

From probability to geometry: volume in honor of the 60th birthday of Jean-Michel Bismut 1 From probability to geometry

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1. Verfasser:	Dai, Xianzhe (VerfasserIn)
Format:	Buch
Sprache:	English
Veröffentlicht:	Paris Soc. Math. de France 2009
Schriftenreihe:	Astérisque 327
Schlagworte:	Wahrscheinlichkeitstheorie Festschrift
Online-Zugang:	Inhaltsverzeichnis
Beschreibung:	XXXVII, 424 S. Ill.
ISBN:	9782856292884

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adam_text	TABLE OF CONTENTS Preface by Paul Malliavin .............................................. xv Preface by Sir Michael Atiyah ......................................... xvii A letter from a friend ................................................... xix Curriculum vitae of Jean-Michel Bismut .............................. xxi The mathematical work of Jean-Michel Bismut: a brief summary xxv 1. From probability theory ................................................. xxv 2. ... to Index Theory ...................................................... xxvi 2.1. Superconnections, Quillen metrics and 77-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 — Semi-classical limit of the lowest eigenvalue of a Schrödinger 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 — Infinite dimensional oscillatory integrals with polynomial phase function and the trace formula for the heat semigroup ......................... 17 1. Introduction ............................................................. 17 2. Infinite dimensional oscillatory integrals ................................. 19 3. The asymptotic expansion ............................................... 27 4. A degenerate case ....................................................... 30 Appendix. Abstract Wiener spaces ......................................... 41 References .................................................................. 43 vi TABLE OF CONTENTS Richard F. Bass fc Edwin Perkins — A new technique for proving uniqueness for martingale problems ........................... 47 1. Introduction ............................................................. 47 2. Some estimates .......................................................... 49 3. Proof of Theorem 1.1 .................................................... 51 References .................................................................. 53 Martin Grothaus, Ludwig Streit L· Anna Vogel — Feynman integrals as Hida distributions: the case of non-perturbative potentials ......................................................... 55 1. Introduction ............................................................. 55 2. White Noise Analysis .................................................... 56 3. Hida distributions as candidates for Feynman Integrands ................ 57 4. Solution to time-dependent Schrödinger equation ........................ 59 5. General construction of the Feynman integrand ......................... 62 6. Examples ................................................................ 63 6.1. The Feynman integrand for polynomial potentials .................. 64 6.2. Non-perturbative accessible potentials .............................. 65 References .................................................................. 67 Hiroshi Kunita — Smooth Density of Canonical Stochastic Differential Equation with Jumps ............................. 69 1. Introduction and main results ........................................... 69 2. Malliavin calculus for canonical SDE .................................... 73 3. SDE s for derivatives of stochastic flow .................................. 76 4. Alternative criterion for the smooth density ............................. 80 5. Relation with Lie algebra ................................................ 83 6. Appendix. An analogue of Norris estimate .............................. 87 References .................................................................. 90 James R. Norris — Two-parameter stochastic calculus and Malliavin s integration-by-parts formula on Wiener space . 93 1. Introduction ............................................................. 93 2. Integration-by-parts formula ............................................. 94 3. Review of two-parameter stochastic calculus ............................. 96 4. A regularity result for two-parameter stochastic differential equations ... 100 5. Derivation of the formula ................................................ 109 References .................................................................. 113 Ichiro Shigekawa — Witten Laplacian on a lattice spin system ... 115 1. Introduction ............................................................. 115 2. Witten Laplacian in finite dimension ................................... 116 ASTÉRISQUE 327 TABLE OF CONTENTS vii 3. Witten Laplacian acting on differential forms ............................ 118 4. Witten Laplacian in one-dimension ...................................... 121 5. Positivity of the lowest eigenvalue for the Witten Laplacian ............. 124 References .................................................................. 129 Anton Alekseev, Henrique Bursztyn & Eckhard Meinrenken — Pure Spinora on Lie groupa .................................... 131 0. Introduction ............................................................. 131 1. Linear Dirac geometry ................................................... 134 1.1. Clifford algebras .................................................... 134 1.2. Pure spinors ........................................................ 136 1.3. The bilinear pairing of spinors ...................................... 136 1.4. Contravariant spinors ............................................... 137 1.5. Action of the orthogonal group ...................................... 138 1.6. Morphisms .......................................................... 139 1.7. Dirac spaces ......................................................... 141 1.8. Lagrangian splittings ................................................ 142 2. Pure spinors on manifolds ............................................... 146 2.1. Dirac structures ..................................................... 146 2.2. Dirac morphisms .................................................... 148 2.3. Bivector fields ....................................................... 150 2.4. Dirac cohomology ................................................... 152 2.5. Classical dynamical Yang-Baxter equation .......................... 154 3. Dirac structures on Lie groups ........................................... 155 3.1. The isomorphism TG = G χ (g φ g) ................................. 155 3.2. η -twisted Dirac structures on G ..................................... 156 3.3. The Cartan-Dirac structure ......................................... 157 3.4. Group multiplication ................................................ 159 3.5. Exponential map .................................................... 161 3.6. The Gauss-Dirac structure .......................................... 164 4. Pure spinors on Lie groups .............................................. 167 4.1. Cl(g) as a spinor module over Cl(g θ g) ............................. 167 4.2. The isomorphism KTG 9 G χ Cl(g) ................................ 170 4.3. Group multiplication ................................................ 174 4.4. Exponential map .................................................... 175 4.5. The Gauss-Dirac spinor ............................................. 178 5. q-Hamiltonian G-manifolds .............................................. 182 5.1. Dirac morphisms and group-valued moment maps .................. 182 5.2. Volume forms ....................................................... 184 5.3. The volume form in terms of the Gauss-Dirac spinor ................ 187 5.4. q-Hamiltonian q-Poisson g-manifolds ................................ 188 5.5. t -valued moment maps ............................................. 191 6. if-valued moment maps ................................................ 192 SOCIÉTÉ MATHÉMATIQUE DE FRANCE 2009 viii TABLE OF CONTENTS 6.1. Review of JC -valued moment maps ................................. 193 6.2. P-valued moment maps ............................................. 194 6.3. Equivalence between K -valueâ and P-valued moment maps ....... 195 6.4. Equivalence between P-valued and t -valued moment maps ......... 196 References .................................................................. 196 Moulay-Tahar Benameur L· Paolo Piazza — Index, eta and rho invarianta on foliated bundles .................................. 201 Introduction and main results .............................................. 202 1. Group actions ........................................................... 208 1.1. The discrete groupoid @ ............................................ 208 1.2. C-algebras associated to the discrete groupoid $ .................. 209 1.3. von Neumann algebras associated to the discrete groupoid í^ ....... 209 1.4. Traces ............................................................... 211 2. Foliated spaces .......................................................... 213 2.1. Foliated spaces ...................................................... 213 2.2. The monodramy groupoid and the C-algebra of the foliation ...... 215 2.3. von Neumann Algebras of foliations ................................. 216 2.4. Traces ............................................................... 218 2.5. Compatibility with Morita isomorphisms ............................ 221 3. Hilbert modules and Dirac operators .................................... 226 3.1. Connes-Skandalis Hilbert module ................................... 226 3.2. Г -equivariant pseudodifferential operators ........................... 231 3.3. Functional calculus for Dirac operators .............................. 235 4. Index theory ............................................................. 242 4.1. The numeric index .................................................. 242 4.2. The index class in the maximal C-algebra ......................... 244 4.3. The signature operator for odd foliations ............................ 246 5. Foliated rho invariants ................................................... 246 5.1. Foliated eta and rho invariants ...................................... 247 5.2. Eta invariants and determinants of paths ........................... 250 6. Stability properties of p for the signature operator ..................... 255 6.1. Leafwise homotopies ................................................ 255 6.2. pv{V, 9) is metric independent ...................................... 258 7. Loops, determinants and Bott periodicity ............................... 261 8. On the homotopy invariance of rho on foliated bundles .................. 263 8.1. The Baum-Connes map for the discrete groupoid Τ χ Γ ............ 264 8.2. Homotopy invariance of (r{V, Í7) for special homotopy equivalences 266 9. Proof of the homotopy invariance for special homotopy equivalences: details .......................................... 268 9.1. Consequences of surjectivity I: equality of determinants ............. 268 9.2. Consequences of surjectivity II: the large time path ................. 269 9.3. The determinants of the large time path ............................ 271 ASTÉRISQUE 327 TABLE OF CONTENTS ix 9.4. Consequences of injectivity: the small time path .................... 273 9.5. The determinants of the small time path ............................ 278 References .................................................................. 284 Alain Berthomieu — Direct image for some secondary К -theories 289 1. Introduction ............................................................. 289 2. Various K-theorìes ....................................................... 293 2.1. Preliminaries ........................................................ 293 2.1.1. Connections and vector bundle morphisms ..................... 293 2.1.2. Chern-Simons transgression forms .............................. 294 2.2. Definitions of the considered if-groups .............................. 295 2.2.1. Topological АГ -theory ........................................... 295 2.2.2. К°-іЬеоту of the category of flat bundles ....................... 295 2.2.3. Relative ѓГ -theory .............................................. 296 2.2.4. Free multiplicative or non hermitian smooth .ЌГ -theory ...... 297 2.3. Chern-Simons class on relative ir-theory ............................ 297 2.4. Relations between the preceding if-groups .......................... 298 2.5. Symmetries associated to hermitian metrics ......................... 299 2.6. Borel-Kamber-Tondeur class on Кљ ................................ 301 3. Direct images for if-groups .............................................. 303 3.1. The case of topological Jff-theory .....................,.............. 303 3.1.1. Preliminary: construction of family index bundles .............. 303 3.1.2. Definition of the direct image morphism for K^op and Jf¿,p ..... 304 3.2. The case of the ir°-theory of flat bundles ........................... 306 3.3. The case of relative if-theory ....................................... 307 3.3.1. The notion of link ............................................ 307 3.3.2. Definition of the direct image for K°el .......................... 307 3.4. The case of multiplicative, or smooth, Ä^-theory ................... 309 3.4.1. Transgression of the family index theorem ...................... 309 3.4.2. Direct image for multiplicative/smooth K°-theory .............. 310 3.5. Hermitian symmetry and functoriality results ....................... 311 3.5.1. Direct images and symmetries .................................. 311 3.5.2. Double fibrations ............................................... 312 4. Proof of Theorems 25 and 27 ............................................ 312 4.1. Proof of Theorem 25 ................................................ 312 4.1.1. Links and exact sequences of vector bundles .................... 312 4.1.2. Link with positive kernel family index bundles ................ 313 4.1.3. Deformation of ψ, h* and Vţ ................................... 314 4.1.4. General construction (and proof of Theorem 25) ............... 315 4.2. Proof of Theorem 27 ................................................ 316 4.2.1. Reduction of the problem ....................................... 316 4.2.2. Sheaf theoretic direct images and short exact sequences ........ 317 SOCIÉTÉ MATHÉMATIQUE DE FRANCE 3009 TABLE OF CONTENTS 4.2.3. Adiabatic limit of harmonie forms ............................ 318 4.2.4. End of proof of Proposition 43 ................................. 319 5. JT-forms .................................................................. 320 5.1. Zi-graded theory .................................................... 320 5.1.1. Za-graded bundles and superconnections ....................... 320 5.1.2. Special adjunction .............................................. 321 5.2. Adaptation of Bismut s superconnection ............................ 322 5.2.1. Definition of Bismut and Lott s Levi-Civita superconnection ... 322 5.2.2. Properties and asymptotics of the Chern character of Ct ....... 323 5.2.3. Calculating Ct for the product with the real line ............... 324 5.2.4. t —► 0 asymptotics of the infinitesimal transgression form ..... 325 5.2.5. Adapting Ct to some suitable triple ............................ 326 5.2.6. t —> +0O asymptotics of the infinitesimal transgression form . . 327 5.3. Proof of the first part of Theorem 28 ................................ 328 5.3.1. Chern-Simons transgression and links .......................... 328 5.3.2. Definition of the 77-form and check of its properties ............. 329 5.3.3. Invariance properties of 77 ....................................... 330 5.4. Anomaly formulae and their consequences .......................... 332 5.4.1. Anomaly formulae .............................................. 332 5.4.2. End of proof of Theorem 28 .................................... 334 5.4.3. Proof of Theorem 29 ........................................... 334 5.4.4. Proof of Theorem 31 ........................................... 334 5.4.5. Influence of the vertical metric and the horizontal distribution . 336 6. Fiberwise Hodge symmetry .............................................. 337 6.1. Symmetries induced on family index bundles ........................ 337 6.1.1. The fiberwise Hodge * operator ................................ 337 6.1.2. Symmetry induced by *z on fiberwise twisted Euler operators . 338 6.1.3. Odd dimensional fibre case ..................................... 338 6.1.4. Symmetry on canonical links ................................... 339 6.1.5. Symmetry on connections on the infinite rank bundle S ........ 341 6.2. Proof of results about Ä^at and K°el ................................ 342 6.2.1. End of proof of Theorem 32 .................................... 342 6.2.2. Results on τιν- .................................................. 342 6.3. End of proof of Theorem 33 ......................................... 344 7. Double fibrations ........................................................ 346 7.1. Topological К-Љеоту ............................................... 346 7.1.1. Fiberwise exterior differentials: ................................. 347 7.1.2. Fiberwise Euler operators ...................................... 347 7.1.3. Introducing some intermediate suitable triple ................... 348 7.1.4. Estimates on the operator A .................................. 349 7.1.5. Spectral convergence of Euler operators ........................ 350 7.1.6. Construction of the canonical link (proof of Theorem 61) ....... 351 7.2. Flat and relative Jř-theory .......................................... 352 ASTÉRISQUE 32? TABLE OF CONTENTS xi 7.2.1. Leray spectral sequence ......................................... 353 7.2.2. Compatibility of topologicai and sheaf theoretic links .......... 353 7.2.3. Proof of Theorem 34 ........................................... 355 7.3. Multiplicative and smooth liľ-theory ................................ 356 7.3.1. Calculation of тг\|и о irf? - (тг2 о Wl)fu .......................... 356 7.3.2. Proof of Theorem 35 ........................................... 357 References .................................................................. 358 Jean-Benoît Bost & Klaus Künnemann — Hermitian vector bundles and extension groups on arithmetic schemes II. The arithmetic Atiyah extension .............................. 361 0. Introduction ............................................................. 362 1. Atiyah extensions in algebraic and analytic geometry .................... 370 1.1. Definition and basic properties ...................................... 370 1.2. Cotangent complex and Atiyah class ................................ 375 1.3. ţ?00 -connections compatible with the holomorphic structure ........ 377 2. The arithmetic Atiyah class of a vector bundle with connection ......... 378 2.1. Definition and basic properties ...................................... 378 2.2. The first Chern class in arithmetic Hodge cohomology .............. 383 3. Hermitian line bundles with vanishing arithmetic Atiyah class ........... 386 3.1. Transcendence and line bundles with connections on abelian varieties 386 3.1.3. Line bundles with connections on abelian varieties ............. 387 3.1.4. The complex case ............................................... 389 3.1.5. An application of the Theorem of Schneider-Lang .............. 390 3.1.8. Reality I ........................................................ 391 3.1.10. Reality II ...................................................... 392 3.1.12. Conclusion of the proof of Theorem 3.1.1 ...................... 393 3.2. Hermitian line bundles with vanishing arithmetic Atiyah class on smooth projective varieties over number fields ...................... 394 3.3. Finiteness results on the kernel of őf ............................... 398 4. A geometric analogue .................................................... 399 4.1. Line bundles with vanishing relative Atiyah class on fibered projective varieties ............................................................ 399 4.1.1. Notation ........................................................ 399 4.2. Variants and complements .......................................... 402 4.3. Hodge cohomology and first Chern class ............................ 404 4.3.1. Hodge cohomology groups ...................................... 404 4.3.2. The first Chern class in Hodge cohomology ..................... 406 4.4. An application of the Hodge Index Theorem ........................ 407 4.4.1. The Hodge Index Theorem in Hodge cohomology .............. 407 4.4.3. An application to projective varieties fibered over curves ....... 407 4.5. The equivalence of VAI and VA2 .................................. 409 4.6. The Picard variety of a variety over a function field ................. 410 SOCIÉTÉ MATHÉMATIQUE DE FRANCE 3009 xii TABLE OF CONTENTS 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 ASTÉRISQUE 327
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spellingShingle	Dai, Xianzhe From probability to geometry volume in honor of the 60th birthday of Jean-Michel Bismut Astérisque Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd
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title	From probability to geometry volume in honor of the 60th birthday of Jean-Michel Bismut
title_auth	From probability to geometry volume in honor of the 60th birthday of Jean-Michel Bismut
title_exact_search	From probability to geometry volume in honor of the 60th birthday of Jean-Michel Bismut
title_full	From probability to geometry volume in honor of the 60th birthday of Jean-Michel Bismut 1 From probability to geometry Xianzhe Dai ... éds.
title_fullStr	From probability to geometry volume in honor of the 60th birthday of Jean-Michel Bismut 1 From probability to geometry Xianzhe Dai ... éds.
title_full_unstemmed	From probability to geometry volume in honor of the 60th birthday of Jean-Michel Bismut 1 From probability to geometry Xianzhe Dai ... éds.
title_short	From probability to geometry
title_sort	from probability to geometry volume in honor of the 60th birthday of jean michel bismut from probability to geometry
title_sub	volume in honor of the 60th birthday of Jean-Michel Bismut
topic	Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd
topic_facet	Wahrscheinlichkeitstheorie Festschrift
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020476639&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
volume_link	(DE-604)BV036555176 (DE-604)BV002579439
work_keys_str_mv	AT daixianzhe fromprobabilitytogeometryvolumeinhonorofthe60thbirthdayofjeanmichelbismut1 AT bismutjeanmichel fromprobabilitytogeometryvolumeinhonorofthe60thbirthdayofjeanmichelbismut1

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