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The Index Theorem And The Heat Equation Method
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Book Synopsis The Index Theorem And The Heat Equation Method by : Yanlin Yu
Download or read book The Index Theorem And The Heat Equation Method written by Yanlin Yu and published by World Scientific. This book was released on 2001-07-02 with total page 309 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides a self-contained representation of the local version of the Atiyah-Singer index theorem. It contains proofs of the Hodge theorem, the local index theorems for the Dirac operator and some first order geometric elliptic operators by using the heat equation method. The proofs are up to the standard of pure mathematics. In addition, a Chern root algorithm is introduced for proving the local index theorems, and it seems to be as efficient as other methods.
Book Synopsis Invariance Theory by : Peter B. Gilkey
Download or read book Invariance Theory written by Peter B. Gilkey and published by CRC Press. This book was released on 1994-12-22 with total page 534 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book treats the Atiyah-Singer index theorem using the heat equation, which gives a local formula for the index of any elliptic complex. Heat equation methods are also used to discuss Lefschetz fixed point formulas, the Gauss-Bonnet theorem for a manifold with smooth boundary, and the geometrical theorem for a manifold with smooth boundary. The author uses invariance theory to identify the integrand of the index theorem for classical elliptic complexes with the invariants of the heat equation.
Book Synopsis The Laplacian on a Riemannian Manifold by : Steven Rosenberg
Download or read book The Laplacian on a Riemannian Manifold written by Steven Rosenberg and published by Cambridge University Press. This book was released on 1997-01-09 with total page 190 pages. Available in PDF, EPUB and Kindle. Book excerpt: This text on analysis of Riemannian manifolds is aimed at students who have had a first course in differentiable manifolds.
Book Synopsis Elliptic Operators, Topology, and Asymptotic Methods by : John Roe
Download or read book Elliptic Operators, Topology, and Asymptotic Methods written by John Roe and published by Longman Scientific and Technical. This book was released on 1988 with total page 208 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis The Atiyah-Patodi-Singer Index Theorem by : Richard Melrose
Download or read book The Atiyah-Patodi-Singer Index Theorem written by Richard Melrose and published by CRC Press. This book was released on 1993-03-31 with total page 392 pages. Available in PDF, EPUB and Kindle. Book excerpt: Based on the lecture notes of a graduate course given at MIT, this sophisticated treatment leads to a variety of current research topics and will undoubtedly serve as a guide to further studies.
Book Synopsis The Index Theorem and the Heat Equation by : Peter B. Gilkey
Download or read book The Index Theorem and the Heat Equation written by Peter B. Gilkey and published by Publish or Perish. This book was released on 1974 with total page 134 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Heat Kernel and Analysis on Manifolds by : Alexander Grigoryan
Download or read book Heat Kernel and Analysis on Manifolds written by Alexander Grigoryan and published by American Mathematical Soc.. This book was released on 2009 with total page 504 pages. Available in PDF, EPUB and Kindle. Book excerpt: The heat kernel has long been an essential tool in both classical and modern mathematics but has become especially important in geometric analysis as a result of major innovations beginning in the 1970s. The methods based on heat kernels have been used in areas as diverse as analysis, geometry, and probability, as well as in physics. This book is a comprehensive introduction to heat kernel techniques in the setting of Riemannian manifolds, which inevitably involves analysis of the Laplace-Beltrami operator and the associated heat equation. The first ten chapters cover the foundations of the subject, while later chapters deal with more advanced results involving the heat kernel in a variety of settings. The exposition starts with an elementary introduction to Riemannian geometry, proceeds with a thorough study of the spectral-theoretic, Markovian, and smoothness properties of the Laplace and heat equations on Riemannian manifolds, and concludes with Gaussian estimates of heat kernels. Grigor'yan has written this book with the student in mind, in particular by including over 400 exercises. The text will serve as a bridge between basic results and current research.Titles in this series are co-published with International Press, Cambridge, MA, USA.
Book Synopsis Random Walk and the Heat Equation by : Gregory F. Lawler
Download or read book Random Walk and the Heat Equation written by Gregory F. Lawler and published by American Mathematical Soc.. This book was released on 2010-11-22 with total page 170 pages. Available in PDF, EPUB and Kindle. Book excerpt: The heat equation can be derived by averaging over a very large number of particles. Traditionally, the resulting PDE is studied as a deterministic equation, an approach that has brought many significant results and a deep understanding of the equation and its solutions. By studying the heat equation and considering the individual random particles, however, one gains further intuition into the problem. While this is now standard for many researchers, this approach is generally not presented at the undergraduate level. In this book, Lawler introduces the heat equations and the closely related notion of harmonic functions from a probabilistic perspective. The theme of the first two chapters of the book is the relationship between random walks and the heat equation. This first chapter discusses the discrete case, random walk and the heat equation on the integer lattice; and the second chapter discusses the continuous case, Brownian motion and the usual heat equation. Relationships are shown between the two. For example, solving the heat equation in the discrete setting becomes a problem of diagonalization of symmetric matrices, which becomes a problem in Fourier series in the continuous case. Random walk and Brownian motion are introduced and developed from first principles. The latter two chapters discuss different topics: martingales and fractal dimension, with the chapters tied together by one example, a random Cantor set. The idea of this book is to merge probabilistic and deterministic approaches to heat flow. It is also intended as a bridge from undergraduate analysis to graduate and research perspectives. The book is suitable for advanced undergraduates, particularly those considering graduate work in mathematics or related areas.
Download or read book The Heat Equation written by D. V. Widder and published by Academic Press. This book was released on 1976-01-22 with total page 285 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Heat Equation
Book Synopsis Modern Group Theoretical Methods in Physics by : J. Bertrand
Download or read book Modern Group Theoretical Methods in Physics written by J. Bertrand and published by Springer Science & Business Media. This book was released on 2013-06-29 with total page 329 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book contains the proceedings of a meeting that brought together friends and colleagues of Guy Rideau at the Université Denis Diderot (Paris, France) in January 1995. It contains original results as well as review papers covering important domains of mathematical physics, such as modern statistical mechanics, field theory, and quantum groups. The emphasis is on geometrical approaches. Several papers are devoted to the study of symmetry groups, including applications to nonlinear differential equations, and deformation of structures, in particular deformation-quantization and quantum groups. The richness of the field of mathematical physics is demonstrated with topics ranging from pure mathematics to up-to-date applications such as imaging and neuronal models. Audience: Researchers in mathematical physics.
Book Synopsis Elliptic Operators, Topology, and Asymptotic Methods, Second Edition by : John Roe
Download or read book Elliptic Operators, Topology, and Asymptotic Methods, Second Edition written by John Roe and published by CRC Press. This book was released on 1999-01-06 with total page 222 pages. Available in PDF, EPUB and Kindle. Book excerpt: Ten years after publication of the popular first edition of this volume, the index theorem continues to stand as a central result of modern mathematics-one of the most important foci for the interaction of topology, geometry, and analysis. Retaining its concise presentation but offering streamlined analyses and expanded coverage of important examples and applications, Elliptic Operators, Topology, and Asymptotic Methods, Second Edition introduces the ideas surrounding the heat equation proof of the Atiyah-Singer index theorem. The author builds towards proof of the Lefschetz formula and the full index theorem with four chapters of geometry, five chapters of analysis, and four chapters of topology. The topics addressed include Hodge theory, Weyl's theorem on the distribution of the eigenvalues of the Laplacian, the asymptotic expansion for the heat kernel, and the index theorem for Dirac-type operators using Getzler's direct method. As a "dessert," the final two chapters offer discussion of Witten's analytic approach to the Morse inequalities and the L2-index theorem of Atiyah for Galois coverings. The text assumes some background in differential geometry and functional analysis. With the partial differential equation theory developed within the text and the exercises in each chapter, Elliptic Operators, Topology, and Asymptotic Methods becomes the ideal vehicle for self-study or coursework. Mathematicians, researchers, and physicists working with index theory or supersymmetry will find it a concise but wide-ranging introduction to this important and intriguing field.
Book Synopsis Partial Differential Equations and Spectral Theory by : Michael Demuth
Download or read book Partial Differential Equations and Spectral Theory written by Michael Demuth and published by Birkhäuser. This book was released on 2012-12-06 with total page 346 pages. Available in PDF, EPUB and Kindle. Book excerpt: The intention of the international conference PDE2000 was to bring together specialists from different areas of modern analysis, mathematical physics and geometry, to discuss not only the recent progress in their own fields but also the interaction between these fields. The special topics of the conference were spectral and scattering theory, semiclassical and asymptotic analysis, pseudodifferential operators and their relation to geometry, as well as partial differential operators and their connection to stochastic analysis and to the theory of semigroups. The scientific advisory board of the conference in Clausthal consisted of M. Ben-Artzi (Jerusalem), Chen Hua (Peking), M. Demuth (Clausthal), T. Ichinose (Kanazawa), L. Rodino (Turin), B.-W. Schulze (Potsdam) and J. Sjöstrand (Paris). The book is aimed at researchers in mathematics and mathematical physics with interests in partial differential equations and all its related fields.
Book Synopsis Operator Algebras and Applications, Part 1 by : Richard V. Kadison
Download or read book Operator Algebras and Applications, Part 1 written by Richard V. Kadison and published by American Mathematical Soc.. This book was released on 1982 with total page 654 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Michael Atiyah Collected Works by : Michael Atiyah
Download or read book Michael Atiyah Collected Works written by Michael Atiyah and published by Oxford University Press. This book was released on 1988-04-28 with total page 656 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is a collection of the works of Michael Atiyah, a well-established mathematician and winner of the Fields Medal. It is thematically divided into volumes; this one discusses index theory.
Book Synopsis Differential Geometry For Physicists by : Bo-yu Hou
Download or read book Differential Geometry For Physicists written by Bo-yu Hou and published by World Scientific Publishing Company. This book was released on 1997-10-31 with total page 561 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is divided into fourteen chapters, with 18 appendices as introduction to prerequisite topological and algebraic knowledge, etc. The first seven chapters focus on local analysis. This part can be used as a fundamental textbook for graduate students of theoretical physics. Chapters 8-10 discuss geometry on fibre bundles, which facilitates further reference for researchers. The last four chapters deal with the Atiyah-Singer index theorem, its generalization and its application, quantum anomaly, cohomology field theory and noncommutative geometry, giving the reader a glimpse of the frontier of current research in theoretical physics.
Book Synopsis Collected Papers of V.K. Patodi by : Vijay Kumar Patodi
Download or read book Collected Papers of V.K. Patodi written by Vijay Kumar Patodi and published by World Scientific. This book was released on 1996 with total page 318 pages. Available in PDF, EPUB and Kindle. Book excerpt: Vijay Kumar Patodi was a brilliant Indian mathematicians who made, during his short life, fundamental contributions to the analytic proof of the index theorem and to the study of differential geometric invariants of manifolds. This set of collected papers edited by Prof M Atiyah and Prof Narasimhan includes his path-breaking papers on the McKean-Singer conjecture and the analytic proof of Riemann-Roch-Hirzebruch theorem for Kähler manifolds. It also contains his celebrated joint papers on the index theorem and the Atiyah-Patodi-Singer invariant.
Book Synopsis Partial Differential Equations and Boundary-Value Problems with Applications by : Mark A. Pinsky
Download or read book Partial Differential Equations and Boundary-Value Problems with Applications written by Mark A. Pinsky and published by American Mathematical Soc.. This book was released on 2011 with total page 545 pages. Available in PDF, EPUB and Kindle. Book excerpt: Building on the basic techniques of separation of variables and Fourier series, the book presents the solution of boundary-value problems for basic partial differential equations: the heat equation, wave equation, and Laplace equation, considered in various standard coordinate systems--rectangular, cylindrical, and spherical. Each of the equations is derived in the three-dimensional context; the solutions are organized according to the geometry of the coordinate system, which makes the mathematics especially transparent. Bessel and Legendre functions are studied and used whenever appropriate throughout the text. The notions of steady-state solution of closely related stationary solutions are developed for the heat equation; applications to the study of heat flow in the earth are presented. The problem of the vibrating string is studied in detail both in the Fourier transform setting and from the viewpoint of the explicit representation (d'Alembert formula). Additional chapters include the numerical analysis of solutions and the method of Green's functions for solutions of partial differential equations. The exposition also includes asymptotic methods (Laplace transform and stationary phase). With more than 200 working examples and 700 exercises (more than 450 with answers), the book is suitable for an undergraduate course in partial differential equations.
