Hypoelliptic Laplacian and Orbital Integrals (eBook)

Jean-Michel Bismut (Autor)

eBook Download: EPUB

2011
344 Seiten
Princeton University Press (Verlag)
978-1-4008-4057-1 (ISBN)

Infos

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.

Autor

Jean-Michel Bismut is professor of mathematics at the Université Paris-Sud, Orsay.

Produkt-Details

• Reihe/Serie: Annals of Mathematics Studies
• Reihe/Serie: Annals of Mathematics Studies
• Zusatzinfo: 2 line illus.
• Verlagsort: Princeton
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Technik
• Schlagworte: Action functional • adjoint representation • analytic torsion • Asymptote • Atiyah–Singer index theorem • automorphism • Bianchi identity • bilinear form • Boundary value problem • Brownian motion • Casimir Operator • Change of variables • Clifford Algebra • Clifford Algebras • Clifford variables • coefficient • commutator • complexification • Computation • conjugations • Connection form • continuous function • convergence • Convexity • coordinate system • cotangent bundle • covariant derivative • de Rham cohomology • de Rham complex • Derivative • Determinant • diffeomorphism • differential equation • differential form • Differential operator • Dimension (vector space) • Dirac Operator • Direct proof • displacement function • distance function • Division by zero • dot product • Eigenvalues and Eigenvectors • elliptic Laplacian • elliptic operator • elliptic orbital integrals • Endomorphism • Equation • estimation • Euclidean space • Euclidean vector space • existential quantification • Explicit formula • Explicit formulae (L-function) • exponential function • Feynman-Kac formula • Feynman–Kac formula • fiber bundle • fixed point formulas • flat bundle • Fourier transform • Gaussian integral • Gaussian process • Gaussian type estimates • general kernels • general orbital integrals • geodesic • Geodesic flow • geodesics • Harmonic oscillator • heat equation • Heat kernel • heat kernels • heat operators • Heisenberg algebras • Hermitian matrix • hilbert space • hypoelliptic deformation • hypoelliptic heat kernel • hypoelliptic heat kernels • hypoelliptic Laplacian • hypoelliptic operator • hypoelliptic operators • hypoelliptic orbital integrals • index formulas • Index Theory • infinite dimensional orbital integrals • Integration by parts • keat kernels • Kostant • Leftschetz formula • Levi-Civita connection • Lie algebra • Lie derivative • Linear map • Littlewood-Paley decomposition • local index theory • locally symmetric space • Malliavin calculus • matrix part • model operator • nondegeneracy • orbifold • orbifolds • Orbital integral • Orbital Integrals • Orthonormal basis • Parallel Transport • parallel transport trivialization • Parameter • Pointwise • polynomial • Pontryagin Maximum Principle • Pontryagin's Maximum Principle • principal bundle • probabilistic construction • Probabilistic method • Probability • probability measure • Projection (linear algebra) • pseudo-differential operator • pseudodistances • Quantitative Estimates • quartic term • Random Variable • real vector space • refined estimates • rescaled heat kernel • resolvents • return map • Riemannian manifold • rough estimates • scalar heat kernel • scalar heat kernels • scalar hypoelliptic heat kernels • scalar hypoelliptic Laplacian • scalar hypoelliptic operator • scalar part • scientific notation • Selberg's trace formula • Self-adjoint • Semigroup • semisimple orbital integrals • smooth kernels • Smoothness • Sobolev Space • Sobolev spaces • Special case • Spinor • Square-integrable function • square root • standard elliptic heat kernel • stochastic differential equation • submanifold • Summation • Supertrace • supertraces • Support (mathematics) • Symmetric bilinear form • symmetric space • symplectic vector space • tangent bundle • Theorem • Toponogov's theorem • trace formula • unbounded operator • unbounded operators • uniform bounds • uniform estimates • Variable (mathematics) • Variational Problems • vector bundle • Vector Bundles • Vector field • Vector Space • Volume element • wave equation • wave kernel • wave operator • Witten complex
• ISBN-10: 1-4008-4057-0 / 1400840570
• ISBN-13: 978-1-4008-4057-1 / 9781400840571