The hypoelliptic Laplacian and Ray-Singer metrics [[electronic resource] /] / Jean-Michel Bismut, Gilles Lebeau

• Autore: Bismut Jean-Michel
• Pubblicazione: Princeton, : Princeton University Press, 2008
• Edizione: Course Book
• Descrizione fisica: 1 online resource (378 p.)
• Disciplina: 515/.7242
• Soggetto topico: Differential equations, Hypoelliptic; Laplacian operator; Metric spaces
• Soggetto non controllato: Alexander Grothendieck; Analytic function; Asymptote; Asymptotic expansion; Berezin integral; Bijection; Brownian dynamics; Brownian motion; Chaos theory; Chern class; Classical Wiener space; Clifford algebra; Cohomology; Combination; Commutator; Computation; Connection form; Coordinate system; Cotangent bundle; Covariance matrix; Curvature tensor; Curvature; De Rham cohomology; Derivative; Determinant; Differentiable manifold; Differential operator; Dirac operator; Direct proof; Eigenform; Eigenvalues and eigenvectors; Ellipse; Embedding; Equation; Estimation; Euclidean space; Explicit formula; Explicit formulae (L-function); Feynman–Kac formula; Fiber bundle; Fokker–Planck equation; Formal power series; Fourier series; Fourier transform; Fredholm determinant; Function space; Girsanov theorem; Ground state; Heat kernel; Hilbert space; Hodge theory; Holomorphic function; Holomorphic vector bundle; Hypoelliptic operator; Integration by parts; Invertible matrix; Logarithm; Malliavin calculus; Martingale (probability theory); Matrix calculus; Mellin transform; Morse theory; Notation; Parameter; Parametrix; Parity (mathematics); Polynomial; Principal bundle; Probabilistic method; Projection (linear algebra); Rectangle; Resolvent set; Ricci curvature; Riemann–Roch theorem; Scientific notation; Self-adjoint operator; Self-adjoint; Sign convention; Smoothness; Sobolev space; Spectral theory; Square root; Stochastic calculus; Stochastic process; Summation; Supertrace; Symmetric space; Tangent space; Taylor series; Theorem; Theory; Torus; Trace class; Translational symmetry; Transversality (mathematics); Uniform convergence; Variable (mathematics); Vector bundle; Vector space; Wave equation
• Classificazione: SK 620
• Altri autori: LebeauGilles
• Note generali: Description based upon print version of record.
• Nota di bibliografia: Includes bibliographical references (p. [353]-357) and indexes.
• Nota di contenuto: Frontmatter -- Contents -- Introduction -- Chapter 1. Elliptic Riemann-Roch-Grothendieck and flat vector bundles -- Chapter 2. The hypoelliptic Laplacian on the cotangent bundle -- Chapter 3. Hodge theory, the hypoelliptic Laplacian and its heat kernel -- Chapter 4. Hypoelliptic Laplacians and odd Chern forms -- Chapter 5. The limit as t → +∞ and b → 0 of the superconnection forms -- Chapter 6. Hypoelliptic torsion and the hypoelliptic Ray-Singer metrics -- Chapter 7. The hypoelliptic torsion forms of a vector bundle -- Chapter 8. Hypoelliptic and elliptic torsions: a comparison formula -- Chapter 9. A comparison formula for the Ray-Singer metrics -- Chapter 10. The harmonic forms for b → 0 and the formal Hodge theorem -- Chapter 11. A proof of equation (8.4.6) -- Chapter 12. A proof of equation (8.4.8) -- Chapter 13. A proof of equation (8.4.7) -- Chapter 14. The integration by parts formula -- Chapter 15. The hypoelliptic estimates -- Chapter 16. Harmonic oscillator and the J0 function -- Chapter 17. The limit of A'2φb,±H as b → 0 -- Bibliography -- Subject Index -- Index of Notation
• Sommario/riassunto: This 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.
• Titolo autorizzato: Hypoelliptic laplacian and ray-singer metrics
• ISBN: 1-282-45837-X; 9786612458378; 1-4008-2906-2
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9910823892903321
• Lo trovi qui: Univ. Federico II
• Serie: Annals of mathematics studies ; ; no. 167.