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Author Bismut, Jean-Michel
Title Hypoelliptic Laplacian and Orbital Integrals (AM-177)
Imprint Princeton : Princeton University Press, 2011
©2011
book jacket
Descript 1 online resource (320 pages)
text txt rdacontent
computer c rdamedia
online resource cr rdacarrier
Series Annals of Mathematics Studies ; v.177
Annals of Mathematics Studies
Note Cover -- Contents -- Introduction -- 0.1 The trace formula as a Lefschetz formula -- 0.2 A short history of the hypoelliptic Laplacian -- 0.3 The hypoelliptic Laplacian on a symmetric space -- 0.4 The hypoelliptic Laplacian and its heat kernel -- 0.5 Elliptic and hypoelliptic orbital integrals -- 0.6 The limit as b → 0 -- 0.7 The limit as b → +∞: an explicit formula for the orbital integrals -- 0.8 The analysis of the hypoelliptic orbital integrals -- 0.9 The heat kernel for bounded b and the Malliavin calculus -- 0.10 The heat kernel for large b, Toponogov, and local index -- 0.11 The hypoelliptic Laplacian and the wave equation -- 0.12 The organization of the book -- Chapter One: Clifford and Heisenberg algebras -- 1.1 The Clifford algebra of a real vector space -- 1.2 The Clifford algebra of V … V* -- 1.3 The Heisenberg algebra -- 1.4 The Heisenberg algebra of V … V* -- 1.5 The Clifford-Heisenberg algebra of V … V* -- 1.6 The Clifford-Heisenberg algebra of V … V* when V is Euclidean -- Chapter Two: The hypoelliptic Laplacian on X = G/K -- 2.1 A pair (G, K) -- 2.2 The flat connection on TX … N -- 2.3 The Clifford algebras of g -- 2.4 The flat connections on … (T * X … N*) -- 2.5 The Casimir operator -- 2.6 The form κg -- 2.7 The Dirac operator of Kostant -- 2.8 The Clifford-Heisenberg algebra of g … g* -- 2.9 The operator Db -- 2.10 The compression of the operator Db -- 2.11 A formula for … -- 2.12 The action of Db on quotients by K -- 2.13 The operators LX and … -- 2.14 The scaling of the form B -- 2.15 The Bianchi identity -- 2.16 A fundamental identity -- 2.17 The canonical vector fields on X -- 2.18 Lie derivatives and the operator … -- Chapter Three: The displacement function and the return map -- 3.1 Convexity, the displacement function, and its critical set -- 3.2 The norm of the canonical vector fields
3.3 The subset X (…) as a symmetric space -- 3.4 The normal coordinate system on X based at X (…) -- 3.5 The return map along the minimizing geodesics in X (…) -- 3.6 The return map on … -- 3.7 The connection form in the parallel transport trivialization -- 3.8 Distances and pseudodistances on X and … -- 3.9 The pseudodistance and Toponogov's theorem -- 3.10 The flat bundle (TX … N) (…) -- Chapter Four: Elliptic and hypoelliptic orbital integrals -- 4.1 An algebra of invariant kernels on X -- 4.2 Orbital integrals -- 4.3 Infinite dimensional orbital integrals -- 4.4 The orbital integrals for the elliptic heat kernel of X -- 4.5 The orbital supertraces for the hypoelliptic heat kernel -- 4.6 A fundamental equality -- 4.7 Another approach to the orbital integrals -- 4.8 The locally symmetric space Z -- Chapter Five: Evaluation of supertraces for a model operator -- 5.1 The operator … and the function … -- 5.2 A conjugate operator -- 5.3 An evaluation of certain infinite dimensional traces -- 5.4 Some formulas of linear algebra -- 5.5 A formula for … -- Chapter Six: A formula for semisimple orbital integrals -- 6.1 Orbital integrals for the heat kernel -- 6.2 A formula for general orbital integrals -- 6.3 The orbital integrals for the wave operator -- Chapter Seven: An application to local index theory -- 7.1 Characteristic forms on X -- 7.2 The vector bundle of spinors on X and the Dirac operator -- 7.3 The McKean-Singer formula on Z -- 7.4 Orbital integrals and the index theorem -- 7.5 A proof of (7.4.4) -- 7.6 The case of complex symmetric spaces -- 7.7 The case of an elliptic element -- 7.8 The de Rham-Hodge operator -- 7.9 The integrand of de Rham torsion -- Chapter Eight: The case where … -- 8.1 The case where G = K -- 8.2 The case … -- 8.3 The case where G = SL2 (R) -- Chapter Nine: A proof of the main identity
9.1 Estimates on the heat kernel … away from … -- 9.2 A rescaling on the coordinates (f, Y) -- 9.3 A conjugation of the Clifford variables -- 9.4 The norm of … -- 9.5 A conjugation of the hypoelliptic Laplacian -- 9.6 The limit of the rescaled heat kernel -- 9.7 A proof of Theorem 6.1.1 -- 9.8 A translation on the variable YTX -- 9.9 A coordinate system and a trivialization of the vector bundles -- 9.10 The asymptotics of the operator … as b → + ∞ -- 9.11 A proof of Theorem 9.6.1 -- Chapter Ten: The action functional and the harmonic oscillator -- 10.1 A variational problem -- 10.2 The Pontryagin maximum principle -- 10.3 The variational problem on an Euclidean vector space -- 10.4 Mehler's formula -- 10.5 The hypoelliptic heat kernel on an Euclidean vector space -- 10.6 Orbital integrals on an Euclidean vector space -- 10.7 Some computations involving Mehler's formula -- 10.8 The probabilistic interpretation of the harmonic oscillator -- Chapter Eleven: The analysis of the hypoelliptic Laplacian -- 11.1 The scalar operators … -- … on X -- 11.2 The Littlewood-Paley decomposition along the fibres TX -- 11.3 The Littlewood-Paley decomposition on X -- 11.4 The Littlewood Paley decomposition on X -- 11.5 The heat kernels for … -- … -- 11.6 The scalar hypoelliptic operators on … -- 11.7 The scalar hypoelliptic operator on … with a quartic term -- 11.8 The heat kernel associated with the operator … -- Chapter Twelve: Rough estimates on the scalar heat kernel -- 12.1 The Malliavin calculus for the Brownian motion on X -- 12.2 The probabilistic construction of exp … over X -- 12.3 The operator … and the wave equation -- 12.4 The Malliavin calculus for the operator … -- 12.5 The tangent variational problem and integration by parts -- 12.6 A uniform control of the integration by parts formula as b → 0 -- 12.7 Uniform rough estimates on … for bounded b
12.8 The limit as b → 0 -- 12.9 The rough estimates as b → +∞ -- 12.10 The heat kernel … on … -- 12.11 The heat kernel … on … -- Chapter Thirteen: Refined estimates on the scalar heat kernel for bounded b -- 13.1 The Hessian of the distance function -- 13.2 Bounds on the scalar heat kernel on X for bounded b -- 13.3 Bounds on the scalar heat kernel on … for bounded b -- Chapter Fourteen: The heat kernel … for bounded b -- 14.1 A probabilistic construction of exp … -- 14.2 The operator … and the wave equation -- 14.3 Changing Y into −Y -- 14.4 A probabilistic construction of exp … -- 14.5 Estimating V. -- 14.6 Estimating W. -- 14.7 A proof of (4.5.3) when E is trivial -- 14.8 A proof of the estimate (4.5.3) in the general case -- 14.9 Rough estimates on the derivatives of … for bounded b -- 14.10 The behavior of V. as b → 0 -- 14.11 The limit of … as b → 0 -- Chapter Fifteen: The heat kernel … for b large -- 15.1 Uniform estimates on the kernel … over X -- 15.2 The deviation from the geodesic flow for large b -- 15.3 The scalar heat kernel on X away from … -- 15.4 Gaussian estimates for … near iaX … -- 15.5 The scalar heat kernel on … away from … -- 15.6 Estimates on the scalar heat kernel on … near …a N (k−1) -- 15.7 A proof of Theorem 9.1.1 -- 15.8 A proof of Theorem 9.1.3 -- 15.9 A proof of Theorem 9.5.6 -- 15.10 A proof of Theorem 9.11.1 -- Bibliography -- Subject Index -- Index of Notation
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
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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2020. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries
Link Print version: Bismut, Jean-Michel Hypoelliptic Laplacian and Orbital Integrals (AM-177) Princeton : Princeton University Press,c2011 9780691151304
Subject Differential equations, Hypoelliptic.;Laplacian operator.;Definite integrals.;Orbit method
Electronic books
Alt Author Bismut, Jean-Michel
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