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Hypoelliptic Laplacian and orbital integrals / / Jean-Michel Bismut



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Autore: Bismut Jean-Michel Visualizza persona
Titolo: Hypoelliptic Laplacian and orbital integrals / / Jean-Michel Bismut Visualizza cluster
Pubblicazione: Princeton, : Princeton University Press, 2011
Edizione: Course Book
Descrizione fisica: 1 online resource (320 p.)
Disciplina: 515.7242
Soggetto topico: Differential equations, Hypoelliptic
Laplacian operator
Definite integrals
Orbit method
Soggetto non controllato: Bianchi identity
Brownian motion
Casimir operator
Clifford algebras
Clifford variables
Dirac operator
Euclidean vector space
Feynman-Kac formula
Gaussian integral
Gaussian type estimates
Heisenberg algebras
Kostant
Leftschetz formula
Littlewood-Paley decomposition
Malliavin calculus
Pontryagin maximum principle
Selberg's trace formula
Sobolev spaces
Toponogov's theorem
Witten complex
action functional
complexification
conjugations
convergence
convexity
de Rham complex
displacement function
distance function
elliptic Laplacian
elliptic orbital integrals
fixed point formulas
flat bundle
general kernels
general orbital integrals
geodesic flow
geodesics
harmonic oscillator
heat kernel
heat kernels
heat operators
hypoelliptic Laplacian
hypoelliptic deformation
hypoelliptic heat kernel
hypoelliptic heat kernels
hypoelliptic operators
hypoelliptic orbital integrals
index formulas
index theory
infinite dimensional orbital integrals
keat kernels
local index theory
locally symmetric space
matrix part
model operator
nondegeneracy
orbifolds
orbital integrals
parallel transport trivialization
probabilistic construction
pseudodistances
quantitative estimates
quartic term
real vector space
refined estimates
rescaled heat kernel
resolvents
return map
rough estimates
scalar heat kernel
scalar heat kernels
scalar hypoelliptic Laplacian
scalar hypoelliptic heat kernels
scalar hypoelliptic operator
scalar part
semisimple orbital integrals
smooth kernels
standard elliptic heat kernel
supertraces
symmetric space
symplectic vector space
trace formula
unbounded operators
uniform bounds
uniform estimates
variational problems
vector bundles
wave equation
wave kernel
wave operator
Note generali: Description based upon print version of record.
Nota di bibliografia: Includes bibliographical references and indexes.
Nota di contenuto: Frontmatter -- Contents -- Acknowledgments -- Introduction -- Chapter One. Clifford and Heisenberg algebras -- Chapter Two. The hypoelliptic Laplacian on X = G/K -- Chapter Three. The displacement function and the return map -- Chapter Four. Elliptic and hypoelliptic orbital integrals -- Chapter Five. Evaluation of supertraces for a model operator -- Chapter Six. A formula for semisimple orbital integrals -- Chapter Seven. An application to local index theory -- Chapter Eight. The case where [k (γ) ; p0] = 0 -- Chapter Nine. A proof of the main identity -- Chapter Ten. The action functional and the harmonic oscillator -- Chapter Eleven. The analysis of the hypoelliptic Laplacian -- Chapter Twelve. Rough estimates on the scalar heat kernel -- Chapter Thirteen. Refined estimates on the scalar heat kernel for bounded b -- Chapter Fourteen. The heat kernel qXb;t for bounded b -- Chapter Fifteen. The heat kernel qXb;t for b large -- Bibliography -- Subject Index -- Index of Notation
Sommario/riassunto: This book uses the hypoelliptic Laplacian to evaluate semisimple orbital integrals in a formalism that unifies index theory and the trace formula. The hypoelliptic Laplacian is a family of operators that is supposed to interpolate between the ordinary Laplacian and the geodesic flow. It is essentially the weighted sum of a harmonic oscillator along the fiber of the tangent bundle, and of the generator of the geodesic flow. In this book, semisimple orbital integrals associated with the heat kernel of the Casimir operator are shown to be invariant under a suitable hypoelliptic deformation, which is constructed using the Dirac operator of Kostant. Their explicit evaluation is obtained by localization on geodesics in the symmetric space, in a formula closely related to the Atiyah-Bott fixed point formulas. Orbital integrals associated with the wave kernel are also computed. Estimates on the hypoelliptic heat kernel play a key role in the proofs, and are obtained by combining analytic, geometric, and probabilistic techniques. Analytic techniques emphasize the wavelike aspects of the hypoelliptic heat kernel, while geometrical considerations are needed to obtain proper control of the hypoelliptic heat kernel, especially in the localization process near the geodesics. Probabilistic techniques are especially relevant, because underlying the hypoelliptic deformation is a deformation of dynamical systems on the symmetric space, which interpolates between Brownian motion and the geodesic flow. The Malliavin calculus is used at critical stages of the proof.
Titolo autorizzato: Hypoelliptic Laplacian and orbital integrals  Visualizza cluster
ISBN: 1-283-16387-X
9786613163875
1-4008-4057-0
Formato: Materiale a stampa
Livello bibliografico Monografia
Lingua di pubblicazione: Inglese
Record Nr.: 9910827349603321
Lo trovi qui: Univ. Federico II
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Serie: Annals of mathematics studies ; ; no. 177.