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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.

Differential equations, Hypoelliptic. --- Laplacian operator. --- Definite integrals. --- Orbit method. --- 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.

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The arithmetic Riemann-Roch Theorem has been shown recently by Bismut-Gillet-Soul. The proof mixes algebra, arithmetic, and analysis. The purpose of this book is to give a concise introduction to the necessary techniques, and to present a simplified and extended version of the proof. It should enable mathematicians with a background in arithmetic algebraic geometry to understand some basic techniques in the rapidly evolving field of Arakelov-theory.

Algebraic geometry --- Algebraïsche meetkunde --- Geometry [Algebraic ] --- Géométrie algébrique --- Meetkunde [Algebraïsche ] --- Riemann-Roch theorema's --- Riemann-Roch thoerems --- Theoremes de Riemann-Roch --- Geometry, Algebraic. --- Riemann-Roch theorems. --- Theorems, Riemann-Roch --- Algebraic functions --- Geometry, Algebraic --- Geometry --- Addition. --- Adjoint. --- Alexander Grothendieck. --- Algebraic geometry. --- Analytic torsion. --- Arakelov theory. --- Asymptote. --- Asymptotic expansion. --- Asymptotic formula. --- Big O notation. --- Cartesian coordinate system. --- Characteristic class. --- Chern class. --- Chow group. --- Closed immersion. --- Codimension. --- Coherent sheaf. --- Cohomology. --- Combination. --- Commutator. --- Computation. --- Covariant derivative. --- Curvature. --- Derivative. --- Determinant. --- Diagonal. --- Differentiable manifold. --- Differential form. --- Dimension (vector space). --- Divisor. --- Domain of a function. --- Dual basis. --- E6 (mathematics). --- Eigenvalues and eigenvectors. --- Embedding. --- Endomorphism. --- Exact sequence. --- Exponential function. --- Generic point. --- Heat kernel. --- Injective function. --- Intersection theory. --- K-group. --- Levi-Civita connection. --- Line bundle. --- Linear algebra. --- Local coordinates. --- Mathematical induction. --- Morphism. --- Natural number. --- Neighbourhood (mathematics). --- Parameter. --- Projective space. --- Pullback (category theory). --- Pullback (differential geometry). --- Pullback. --- Riemannian manifold. --- Riemann–Roch theorem. --- Self-adjoint operator. --- Smoothness. --- Sobolev space. --- Stochastic calculus. --- Summation. --- Supertrace. --- Theorem. --- Transition function. --- Upper half-plane. --- Vector bundle. --- Volume form.

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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.

Differential equations, Hypoelliptic. --- Laplacian operator. --- Metric spaces. --- Spaces, Metric --- Operator, Laplacian --- Hypoelliptic differential equations --- Generalized spaces --- Set theory --- Topology --- Differential equations, Partial --- 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.