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The Hypoelliptic Laplacian And Ray Singer Metrics
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Book Synopsis The Hypoelliptic Laplacian and Ray-Singer Metrics. (AM-167) by : Jean-Michel Bismut
Download or read book The Hypoelliptic Laplacian and Ray-Singer Metrics. (AM-167) written by Jean-Michel Bismut and published by Princeton University Press. This book was released on 2008-09-07 with total page 377 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis The Hypoelliptic Laplacian and Ray-Singer Metrics by : Jean-Michel Bismut
Download or read book The Hypoelliptic Laplacian and Ray-Singer Metrics written by Jean-Michel Bismut and published by Princeton University Press. This book was released on 2008-08-18 with total page 378 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Hypoelliptic Laplacian and Orbital Integrals by : Jean-Michel Bismut
Download or read book Hypoelliptic Laplacian and Orbital Integrals written by Jean-Michel Bismut and published by Princeton University Press. This book was released on 2011-08-08 with total page 343 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Hypoelliptic Laplacian and Bott–Chern Cohomology by : Jean-Michel Bismut
Download or read book Hypoelliptic Laplacian and Bott–Chern Cohomology written by Jean-Michel Bismut and published by Springer Science & Business Media. This book was released on 2013-05-23 with total page 211 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book provides the proof of a complex geometric version of a well-known result in algebraic geometry: the theorem of Riemann–Roch–Grothendieck for proper submersions. It gives an equality of cohomology classes in Bott–Chern cohomology, which is a refinement for complex manifolds of de Rham cohomology. When the manifolds are Kähler, our main result is known. A proof can be given using the elliptic Hodge theory of the fibres, its deformation via Quillen's superconnections, and a version in families of the 'fantastic cancellations' of McKean–Singer in local index theory. In the general case, this approach breaks down because the cancellations do not occur any more. One tool used in the book is a deformation of the Hodge theory of the fibres to a hypoelliptic Hodge theory, in such a way that the relevant cohomological information is preserved, and 'fantastic cancellations' do occur for the deformation. The deformed hypoelliptic Laplacian acts on the total space of the relative tangent bundle of the fibres. While the original hypoelliptic Laplacian discovered by the author can be described in terms of the harmonic oscillator along the tangent bundle and of the geodesic flow, here, the harmonic oscillator has to be replaced by a quartic oscillator. Another idea developed in the book is that while classical elliptic Hodge theory is based on the Hermitian product on forms, the hypoelliptic theory involves a Hermitian pairing which is a mild modification of intersection pairing. Probabilistic considerations play an important role, either as a motivation of some constructions, or in the proofs themselves.
Book Synopsis Metric and Differential Geometry by : Xianzhe Dai
Download or read book Metric and Differential Geometry written by Xianzhe Dai and published by Springer Science & Business Media. This book was released on 2012-06-01 with total page 401 pages. Available in PDF, EPUB and Kindle. Book excerpt: Metric and Differential Geometry grew out of a similarly named conference held at Chern Institute of Mathematics, Tianjin and Capital Normal University, Beijing. The various contributions to this volume cover a broad range of topics in metric and differential geometry, including metric spaces, Ricci flow, Einstein manifolds, Kähler geometry, index theory, hypoelliptic Laplacian and analytic torsion. It offers the most recent advances as well as surveys the new developments. Contributors: M.T. Anderson J.-M. Bismut X. Chen X. Dai R. Harvey P. Koskela B. Lawson X. Ma R. Melrose W. Müller A. Naor J. Simons C. Sormani D. Sullivan S. Sun G. Tian K. Wildrick W. Zhang
Book Synopsis Algebraic Analysis of Differential Equations by : T. Aoki
Download or read book Algebraic Analysis of Differential Equations written by T. Aoki and published by Springer Science & Business Media. This book was released on 2009-03-15 with total page 349 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains 23 articles on algebraic analysis of differential equations and related topics, most of which were presented as papers at the conference "Algebraic Analysis of Differential Equations – from Microlocal Analysis to Exponential Asymptotics" at Kyoto University in 2005. This volume is dedicated to Professor Takahiro Kawai, who is one of the creators of microlocal analysis and who introduced the technique of microlocal analysis into exponential asymptotics.
Book Synopsis Differential Geometry and Physics by : Mo-Lin Ge
Download or read book Differential Geometry and Physics written by Mo-Lin Ge and published by World Scientific. This book was released on 2006 with total page 542 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volumes provides a comprehensive review of interactions between differential geometry and theoretical physics, contributed by many leading scholars in these fields. The contributions promise to play an important role in promoting the developments in these exciting areas. Besides the plenary talks, the coverage includes: models and related topics in statistical physics; quantum fields, strings and M-theory; Yang-Mills fields, knot theory and related topics; K-theory, including index theory and non-commutative geometry; mirror symmetry, conformal and topological quantum field theory; development of integrable systems; and random matrix theory. Sample Chapter(s). Chapter 1: Yangian and Applications (787 KB). Contents: Yangian and Applications (C-M Bai et al.); The Hypoelliptic Laplacian and the ChernOCoGaussOCoBonnet (J-M Bismut); S S Chern and ChernOCoSimos Terms (R Jackiw); Localization and Conjectures from String Duality (K F Liu); Topologization of Electron Liquids with ChernOCoSimons Theory and Quantum Computation (Z H Wang); Topology and Quantum Information (L H Kauffman); Toeplitz Quantization and Symplectic Reduction (X N Ma & W P Zhang); Murphy Operators in Knot Theory (H R Morton); Separation Between Spin and Charge in SU(2) YangOCoMills Theory (A J Niemi); LAwner Equations and Dispersionless Hierarchies (K Takasaki & T Takebe); and other papers. Readership: Graduate students and professional researchers in geometry and physics."
Book Synopsis Differential Geometry And Physics - Proceedings Of The 23th International Conference Of Differential Geometric Methods In Theoretical Physics by : Weiping Zhang
Download or read book Differential Geometry And Physics - Proceedings Of The 23th International Conference Of Differential Geometric Methods In Theoretical Physics written by Weiping Zhang and published by World Scientific. This book was released on 2006-12-11 with total page 542 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volumes provides a comprehensive review of interactions between differential geometry and theoretical physics, contributed by many leading scholars in these fields. The contributions promise to play an important role in promoting the developments in these exciting areas. Besides the plenary talks, the coverage includes: models and related topics in statistical physics; quantum fields, strings and M-theory; Yang-Mills fields, knot theory and related topics; K-theory, including index theory and non-commutative geometry; mirror symmetry, conformal and topological quantum field theory; development of integrable systems; and random matrix theory.
Book Synopsis Advances in Partial Differential Equations and Control by : Kaïs Ammari
Download or read book Advances in Partial Differential Equations and Control written by Kaïs Ammari and published by Springer Nature. This book was released on with total page 250 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Markov Processes, Semigroups and Generators by : Vassili N. Kolokoltsov
Download or read book Markov Processes, Semigroups and Generators written by Vassili N. Kolokoltsov and published by Walter de Gruyter. This book was released on 2011-03-29 with total page 449 pages. Available in PDF, EPUB and Kindle. Book excerpt: Markov processes represent a universal model for a large variety of real life random evolutions. The wide flow of new ideas, tools, methods and applications constantly pours into the ever-growing stream of research on Markov processes that rapidly spreads over new fields of natural and social sciences, creating new streamlined logical paths to its turbulent boundary. Even if a given process is not Markov, it can be often inserted into a larger Markov one (Markovianization procedure) by including the key historic parameters into the state space. This monograph gives a concise, but systematic and self-contained, exposition of the essentials of Markov processes, together with recent achievements, working from the "physical picture" - a formal pre-generator, and stressing the interplay between probabilistic (stochastic differential equations) and analytic (semigroups) tools. The book will be useful to students and researchers. Part I can be used for a one-semester course on Brownian motion, Lévy and Markov processes, or on probabilistic methods for PDE. Part II mainly contains the author's research on Markov processes. From the contents: Tools from Probability and Analysis Brownian motion Markov processes and martingales SDE, ψDE and martingale problems Processes in Euclidean spaces Processes in domains with a boundary Heat kernels for stable-like processes Continuous-time random walks and fractional dynamics Complex chains and Feynman integral
Book Synopsis Higher Topos Theory (AM-170) by : Jacob Lurie
Download or read book Higher Topos Theory (AM-170) written by Jacob Lurie and published by Princeton University Press. This book was released on 2009-07-26 with total page 948 pages. Available in PDF, EPUB and Kindle. Book excerpt: In 'Higher Topos Theory', Jacob Lurie presents the foundations of this theory using the language of weak Kan complexes introduced by Boardman and Vogt, and shows how existing theorems in algebraic topology can be reformulated and generalized in the theory's new language.
Book Synopsis The Structure of Affine Buildings. (AM-168) by : Richard Mark Weiss
Download or read book The Structure of Affine Buildings. (AM-168) written by Richard Mark Weiss and published by Princeton University Press. This book was released on 2009 with total page 388 pages. Available in PDF, EPUB and Kindle. Book excerpt: Richard Weiss gives a detailed presentation of the complete proof of the classification of Bruhat-Tits buildings first completed by Jacques Tits in 1986. The book includes numerous results about automorphisms, completions and residues of these buildings.
Book Synopsis Annals of Mathematics Studies by : Jean-Michel Bismut
Download or read book Annals of Mathematics Studies written by Jean-Michel Bismut and published by . This book was released on 1940 with total page 350 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Arithmetic L-Functions and Differential Geometric Methods by : Pierre Charollois
Download or read book Arithmetic L-Functions and Differential Geometric Methods written by Pierre Charollois and published by Springer Nature. This book was released on 2021-05-17 with total page 324 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is an outgrowth of the conference “Regulators IV: An International Conference on Arithmetic L-functions and Differential Geometric Methods” that was held in Paris in May 2016. Gathering contributions by leading experts in the field ranging from original surveys to pure research articles, this volume provides comprehensive coverage of the front most developments in the field of regulator maps. Key topics covered are: • Additive polylogarithms • Analytic torsions • Chabauty-Kim theory • Local Grothendieck-Riemann-Roch theorems • Periods • Syntomic regulator The book contains contributions by M. Asakura, J. Balakrishnan, A. Besser, A. Best, F. Bianchi, O. Gregory, A. Langer, B. Lawrence, X. Ma, S. Müller, N. Otsubo, J. Raimbault, W. Raskin, D. Rössler, S. Shen, N. Triantafi llou, S. Ünver and J. Vonk.
Book Synopsis Geometry and Dynamics of Groups and Spaces by : Mikhail Kapranov
Download or read book Geometry and Dynamics of Groups and Spaces written by Mikhail Kapranov and published by Springer Science & Business Media. This book was released on 2008-03-05 with total page 759 pages. Available in PDF, EPUB and Kindle. Book excerpt: Alexander Reznikov (1960-2003) was a brilliant and highly original mathematician. This book presents 18 articles by prominent mathematicians and is dedicated to his memory. In addition it contains an influential, so far unpublished manuscript by Reznikov of book length. The book further provides an extensive survey on Kleinian groups in higher dimensions and some articles centering on Reznikov as a person.
Book Synopsis Convolution and Equidistribution by : Nicholas M. Katz
Download or read book Convolution and Equidistribution written by Nicholas M. Katz and published by . This book was released on 1940 with total page 216 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Some Problems of Unlikely Intersections in Arithmetic and Geometry by : Umberto Zannier
Download or read book Some Problems of Unlikely Intersections in Arithmetic and Geometry written by Umberto Zannier and published by . This book was released on 1940 with total page 178 pages. Available in PDF, EPUB and Kindle. Book excerpt:
