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



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Autore: Bismut Jean-Michel Visualizza persona
Titolo: The hypoelliptic Laplacian and Ray-Singer metrics [[electronic resource] /] / Jean-Michel Bismut, Gilles Lebeau Visualizza cluster
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  Visualizza cluster
ISBN: 1-282-45837-X
9786612458378
1-4008-2906-2
Formato: Materiale a stampa
Livello bibliografico Monografia
Lingua di pubblicazione: Inglese
Record Nr.: 9910781084803321
Lo trovi qui: Univ. Federico II
Opac: Controlla la disponibilità qui
Serie: Annals of mathematics studies ; ; no. 167.