LEADER 08019nam 2201909 a 450 001 9910781084803321 005 20230721005558.0 010 $a1-282-45837-X 010 $a9786612458378 010 $a1-4008-2906-2 024 7 $a10.1515/9781400829064 035 $a(CKB)2550000000003519 035 $a(EBL)483509 035 $a(OCoLC)593214464 035 $a(SSID)ssj0000338850 035 $a(PQKBManifestationID)11230411 035 $a(PQKBTitleCode)TC0000338850 035 $a(PQKBWorkID)10298053 035 $a(PQKB)11597985 035 $a(MiAaPQ)EBC483509 035 $a(DE-B1597)447046 035 $a(OCoLC)979970184 035 $a(DE-B1597)9781400829064 035 $a(Au-PeEL)EBL483509 035 $a(CaPaEBR)ebr10359240 035 $a(CaONFJC)MIL245837 035 $a(EXLCZ)992550000000003519 100 $a20080229d2008 uy 0 101 0 $aeng 135 $aur|n|---||||| 181 $ctxt 182 $cc 183 $acr 200 14$aThe hypoelliptic Laplacian and Ray-Singer metrics$b[electronic resource] /$fJean-Michel Bismut, Gilles Lebeau 205 $aCourse Book 210 $aPrinceton $cPrinceton University Press$d2008 215 $a1 online resource (378 p.) 225 1 $aAnnals of mathematics studies ;$vno. 167 300 $aDescription based upon print version of record. 311 $a0-691-13731-5 311 $a0-691-13732-3 320 $aIncludes bibliographical references (p. [353]-357) and indexes. 327 $tFrontmatter -- $tContents -- $tIntroduction -- $tChapter 1. Elliptic Riemann-Roch-Grothendieck and flat vector bundles -- $tChapter 2. The hypoelliptic Laplacian on the cotangent bundle -- $tChapter 3. Hodge theory, the hypoelliptic Laplacian and its heat kernel -- $tChapter 4. Hypoelliptic Laplacians and odd Chern forms -- $tChapter 5. The limit as t ? +? and b ? 0 of the superconnection forms -- $tChapter 6. Hypoelliptic torsion and the hypoelliptic Ray-Singer metrics -- $tChapter 7. The hypoelliptic torsion forms of a vector bundle -- $tChapter 8. Hypoelliptic and elliptic torsions: a comparison formula -- $tChapter 9. A comparison formula for the Ray-Singer metrics -- $tChapter 10. The harmonic forms for b ? 0 and the formal Hodge theorem -- $tChapter 11. A proof of equation (8.4.6) -- $tChapter 12. A proof of equation (8.4.8) -- $tChapter 13. A proof of equation (8.4.7) -- $tChapter 14. The integration by parts formula -- $tChapter 15. The hypoelliptic estimates -- $tChapter 16. Harmonic oscillator and the J0 function -- $tChapter 17. The limit of A'2?b,ħH as b ? 0 -- $tBibliography -- $tSubject Index -- $tIndex of Notation 330 $aThis book presents the analytic foundations to the theory of the hypoelliptic Laplacian. The hypoelliptic Laplacian, a second-order operator acting on the cotangent bundle of a compact manifold, is supposed to interpolate between the classical Laplacian and the geodesic flow. Jean-Michel Bismut and Gilles Lebeau establish the basic functional analytic properties of this operator, which is also studied from the perspective of local index theory and analytic torsion. The book shows that the hypoelliptic Laplacian provides a geometric version of the Fokker-Planck equations. The authors give the proper functional analytic setting in order to study this operator and develop a pseudodifferential calculus, which provides estimates on the hypoelliptic Laplacian's resolvent. When the deformation parameter tends to zero, the hypoelliptic Laplacian converges to the standard Hodge Laplacian of the base by a collapsing argument in which the fibers of the cotangent bundle collapse to a point. For the local index theory, small time asymptotics for the supertrace of the associated heat kernel are obtained. The Ray-Singer analytic torsion of the hypoelliptic Laplacian as well as the associated Ray-Singer metrics on the determinant of the cohomology are studied in an equivariant setting, resulting in a key comparison formula between the elliptic and hypoelliptic analytic torsions. 410 0$aAnnals of mathematics studies ;$vno. 167. 606 $aDifferential equations, Hypoelliptic 606 $aLaplacian operator 606 $aMetric spaces 610 $aAlexander Grothendieck. 610 $aAnalytic function. 610 $aAsymptote. 610 $aAsymptotic expansion. 610 $aBerezin integral. 610 $aBijection. 610 $aBrownian dynamics. 610 $aBrownian motion. 610 $aChaos theory. 610 $aChern class. 610 $aClassical Wiener space. 610 $aClifford algebra. 610 $aCohomology. 610 $aCombination. 610 $aCommutator. 610 $aComputation. 610 $aConnection form. 610 $aCoordinate system. 610 $aCotangent bundle. 610 $aCovariance matrix. 610 $aCurvature tensor. 610 $aCurvature. 610 $aDe Rham cohomology. 610 $aDerivative. 610 $aDeterminant. 610 $aDifferentiable manifold. 610 $aDifferential operator. 610 $aDirac operator. 610 $aDirect proof. 610 $aEigenform. 610 $aEigenvalues and eigenvectors. 610 $aEllipse. 610 $aEmbedding. 610 $aEquation. 610 $aEstimation. 610 $aEuclidean space. 610 $aExplicit formula. 610 $aExplicit formulae (L-function). 610 $aFeynman?Kac formula. 610 $aFiber bundle. 610 $aFokker?Planck equation. 610 $aFormal power series. 610 $aFourier series. 610 $aFourier transform. 610 $aFredholm determinant. 610 $aFunction space. 610 $aGirsanov theorem. 610 $aGround state. 610 $aHeat kernel. 610 $aHilbert space. 610 $aHodge theory. 610 $aHolomorphic function. 610 $aHolomorphic vector bundle. 610 $aHypoelliptic operator. 610 $aIntegration by parts. 610 $aInvertible matrix. 610 $aLogarithm. 610 $aMalliavin calculus. 610 $aMartingale (probability theory). 610 $aMatrix calculus. 610 $aMellin transform. 610 $aMorse theory. 610 $aNotation. 610 $aParameter. 610 $aParametrix. 610 $aParity (mathematics). 610 $aPolynomial. 610 $aPrincipal bundle. 610 $aProbabilistic method. 610 $aProjection (linear algebra). 610 $aRectangle. 610 $aResolvent set. 610 $aRicci curvature. 610 $aRiemann?Roch theorem. 610 $aScientific notation. 610 $aSelf-adjoint operator. 610 $aSelf-adjoint. 610 $aSign convention. 610 $aSmoothness. 610 $aSobolev space. 610 $aSpectral theory. 610 $aSquare root. 610 $aStochastic calculus. 610 $aStochastic process. 610 $aSummation. 610 $aSupertrace. 610 $aSymmetric space. 610 $aTangent space. 610 $aTaylor series. 610 $aTheorem. 610 $aTheory. 610 $aTorus. 610 $aTrace class. 610 $aTranslational symmetry. 610 $aTransversality (mathematics). 610 $aUniform convergence. 610 $aVariable (mathematics). 610 $aVector bundle. 610 $aVector space. 610 $aWave equation. 615 0$aDifferential equations, Hypoelliptic. 615 0$aLaplacian operator. 615 0$aMetric spaces. 676 $a515/.7242 686 $aSK 620$2rvk 700 $aBismut$b Jean-Michel$044924 701 $aLebeau$b Gilles$063020 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910781084803321 996 $aHypoelliptic laplacian and ray-singer metrics$91013782 997 $aUNINA