Hypoelliptic Laplacian and Orbital Integrals (AM-177)

Nonfiction, Science & Nature, Mathematics, Matrices, Geometry
Cover of the book Hypoelliptic Laplacian and Orbital Integrals (AM-177) by Jean-Michel Bismut, Princeton University Press
View on Amazon View on AbeBooks View on Kobo View on B.Depository View on eBay View on Walmart
Author: Jean-Michel Bismut ISBN: 9781400840571
Publisher: Princeton University Press Publication: August 8, 2011
Imprint: Princeton University Press Language: English
Author: Jean-Michel Bismut
ISBN: 9781400840571
Publisher: Princeton University Press
Publication: August 8, 2011
Imprint: Princeton University Press
Language: English

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.

View on Amazon View on AbeBooks View on Kobo View on B.Depository View on eBay View on Walmart

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.

More books from Princeton University Press

Cover of the book Kazantzakis, Volume 2 by Jean-Michel Bismut
Cover of the book Capitalism and the Jews by Jean-Michel Bismut
Cover of the book Is Democracy Possible Here? by Jean-Michel Bismut
Cover of the book The Rise and Fall of American Growth by Jean-Michel Bismut
Cover of the book The Concise Princeton Encyclopedia of American Political History by Jean-Michel Bismut
Cover of the book Reputation and International Cooperation by Jean-Michel Bismut
Cover of the book Legitimacy and Power Politics by Jean-Michel Bismut
Cover of the book The Lost History of Liberalism by Jean-Michel Bismut
Cover of the book The New American Judaism by Jean-Michel Bismut
Cover of the book Tobacco Culture by Jean-Michel Bismut
Cover of the book Reluctant Accomplice by Jean-Michel Bismut
Cover of the book The Microtheory of Innovative Entrepreneurship by Jean-Michel Bismut
Cover of the book Creating the Market University by Jean-Michel Bismut
Cover of the book Jung on Christianity by Jean-Michel Bismut
Cover of the book Life on a Young Planet by Jean-Michel Bismut
We use our own "cookies" and third party cookies to improve services and to see statistical information. By using this website, you agree to our Privacy Policy