Title:
The Heat Kernel Lefschetz Fixed Point Formula for the Spin-c Dirac Operator by J. J. Duistermaat.
Modern Birkhäuser Classics
Modern Birkhäuser Classics
Author:
Duistermaat, J.J.
SpringerLink (Online service)
General Notes:
1 Introduction -- 2 The Dolbeault-Dirac Operator -- 3 Clifford Modules -- 4 The Spin Group and the Spin-c Group -- 5 The Spin-c Dirac Operator -- 6 Its Square -- 7 The Heat Kernel Method -- 8 The Heat Kernel Expansion -- 9 The Heat Kernel on a Principal Bundle -- 10 The Automorphism -- 11 The Hirzebruch-Riemann-Roch Integrand -- 12 The Local Lefschetz Fixed Point Formula -- 13 Characteristic Case -- 14 The Orbifold Version -- 15 Application to Symplectic Geometry -- 16 Appendix: Equivariant Forms.
<p><p><p>Interest in the spin-c Dirac operator originally came about from the study of complex analytic manifolds, where in the non-Kähler case the Dolbeault operator is no longer suitable for getting local formulas for the Riemann–Roch number or the holomorphic Lefschetz number. However, every symplectic manifold (phase space in classical mechanics) also carries an almost complex structure and hence a corresponding spin-c Dirac operator. Using the heat kernels theory of Berline, Getzler, and Vergne, this work revisits some fundamental concepts of the theory, and presents the application to symplectic geometry.</p><p>J.J. Duistermaat was well known for his beautiful and concise expositions of seemingly familiar concepts, and this classic study is certainly no exception. Reprinted as it was originally published, this work is as an affordable text that will be of interest to a range of researchers in geometric analysis and mathematical physics.</p><p><i>Overall this is a carefully written, highly readable book on a very beautiful subject.<b> </b></i><b>—Mathematical Reviews</b></p><p><i>The book of J.J. Duistermaat is a nice introduction to analysis related [to the] spin-c Dirac operator. ... The book is almost self contained, [is] readable, and will be useful for anybody who is interested in the topic.<b> </b></i><b>—EMS Newsletter</b></p><p><i>The author's book is a marvelous introduction to [these] objects and theories. </i><b>—Zentralblatt MATH</b></p>
Publisher:
Birkhäuser Boston,
Publication Place:
Boston, MA :
ISBN:
9780817682477
Subject:
Mathematics.
Global analysis (Mathematics).
Global analysis.
Operator theory.
Differential equations, Partial.
Global differential geometry.
Mathematics.
Global Analysis and Analysis on Manifolds.
Partial differential equations.
Differential Geometry.
ANALYSIS.
Operator theory.
Mathematical physics.
Series:
Modern Birkhäuser Classics
Modern Birkhäuser Classics
Contents:
1 Introduction -- 2 The Dolbeault-Dirac Operator -- 3 Clifford Modules -- 4 The Spin Group and the Spin-c Group -- 5 The Spin-c Dirac Operator -- 6 Its Square -- 7 The Heat Kernel Method -- 8 The Heat Kernel Expansion -- 9 The Heat Kernel on a Principal Bundle -- 10 The Automorphism -- 11 The Hirzebruch-Riemann-Roch Integrand -- 12 The Local Lefschetz Fixed Point Formula -- 13 Characteristic Case -- 14 The Orbifold Version -- 15 Application to Symplectic Geometry -- 16 Appendix: Equivariant Forms.
Physical Description:
VIII, 247p. digital.
Electronic Location:
http://dx.doi.org/10.1007/978-0-8176-8247-7
Publication Date:
2011.
Title:
Heat Kernel Method and its Applications by Ivan G. Avramidi.
Author:
Avramidi, Ivan G. author.
SpringerLink (Online service)
General Notes:
Part I Analysis -- 1 Background in Analysis -- 2 Introduction to Partial Differential Equations -- Part II Geometry -- 3 Introduction to Differential Geometry -- Part III Perturbations -- 4 Singular Perturbations -- 5 Heat Kernel Asymptotics -- 6 Advanced Topics -- Part IV Applications -- 7 Stochastic Processes -- 8 Applications in Mathematical Finance -- Summary -- References -- Index.
The heart of the book is the development of a short-time asymptotic expansion for the heat kernel. This is explained in detail and explicit examples of some advanced calculations are given. In addition some advanced methods and extensions, including path integrals, jump diffusion and others are presented. The book consists of four parts: Analysis, Geometry, Perturbations and Applications. The first part shortly reviews of some background material and gives an introduction to PDEs. The second part is devoted to a short introduction to various aspects of differential geometry that will be needed later. The third part and heart of the book presents a systematic development of effective methods for various approximation schemes for parabolic differential equations. The last part is devoted to applications in financial mathematics, in particular, stochastic differential equations. Although this book is intended for advanced undergraduate or beginning graduate students in, it should also provide a useful reference for professional physicists, applied mathematicians as well as quantitative analysts with an interest in PDEs.
Publisher:
Springer International Publishing : Imprint: Birkhäuser,
Publication Place:
Cham :
ISBN:
9783319262666
Subject:
Mathematics.
Partial differential equations.
Mathematics.
Partial differential equations.
Edition:
1st ed. 2015.
Contents:
Part I Analysis -- 1 Background in Analysis -- 2 Introduction to Partial Differential Equations -- Part II Geometry -- 3 Introduction to Differential Geometry -- Part III Perturbations -- 4 Singular Perturbations -- 5 Heat Kernel Asymptotics -- 6 Advanced Topics -- Part IV Applications -- 7 Stochastic Processes -- 8 Applications in Mathematical Finance -- Summary -- References -- Index.
Physical Description:
XIX, 390 p. 17 illus. in color. online resource.
Electronic Location:
http://dx.doi.org/10.1007/978-3-319-26266-6
Publication Date:
2015.
Title:
Heat Kernels for Elliptic and Sub-elliptic Operators Methods and Techniques / by Ovidiu Calin, Der-Chen Chang, Kenro Furutani, Chisato Iwasaki.
Applied and Numerical Harmonic Analysis
Applied and numerical harmonic analysis
Author:
Calin, Ovidiu.
Chang, Der-Chen.
Furutani, Kenro.
Iwasaki, Chisato.
SpringerLink (Online service)
General Notes:
Part I. Traditional Methods for Computing Heat Kernels -- Introduction -- Stochastic Analysis Method -- A Brief Introduction to Calculus of Variations -- The Path Integral Approach -- The Geometric Method -- Commuting Operators -- Fourier Transform Method -- The Eigenfunctions Expansion Method -- Part II. Heat Kernel on Nilpotent Lie Groups and Nilmanifolds -- Laplacians and Sub-Laplacians -- Heat Kernels for Laplacians and Step 2 Sub-Laplacians -- Heat Kernel for Sub-Laplacian on the Sphere S^3 -- Part III. Laguerre Calculus and Fourier Method -- Finding Heat Kernels by Using Laguerre Calculus -- Constructing Heat Kernel for Degenerate Elliptic Operators -- Heat Kernel for the Kohn Laplacian on the Heisenberg Group -- Part IV. Pseudo-Differential Operators -- The Psuedo-Differential Operators Technique -- Bibliography -- Index.
This monograph is a unified presentation of several theories of finding explicit formulas for heat kernels for both elliptic and sub-elliptic operators. These kernels are important in the theory of parabolic operators because they describe the distribution of heat on a given manifold as well as evolution phenomena and diffusion processes. <br> <br> The work is divided into four main parts: Part I treats the heat kernel by traditional methods, such as the Fourier transform method, paths integrals, variational calculus, and eigenvalue expansion; Part II deals with the heat kernel on nilpotent Lie groups and nilmanifolds; Part III examines Laguerre calculus applications; Part IV uses the method of pseudo-differential operators to describe heat kernels.<br> <br> Topics and features:<br> <br> •comprehensive treatment from the point of view of distinct branches of mathematics, such as stochastic processes, differential geometry, special functions, quantum mechanics, and PDEs;<br> <br> •novelty of the work is in the diverse methods used to compute heat kernels for elliptic and sub-elliptic operators;<br> <br> •most of the heat kernels computable by means of elementary functions are covered in the work;<br> <br> •self-contained material on stochastic processes and variational methods is included.<br> <br> <em>Heat Kernels for Elliptic and Sub-elliptic Operators</em> is an ideal reference for graduate students, researchers in pure and applied mathematics, and theoretical physicists interested in understanding different ways of approaching evolution operators.<br> <br />
Publisher:
Birkhäuser Boston,
Publication Place:
Boston :
ISBN:
9780817649951
Subject:
Mathematics.
Harmonic analysis.
Operator theory.
Differential equations, Partial.
Global differential geometry.
Distribution (Probability theory).
Mathematical physics.
Mathematics.
Partial differential equations.
Mathematical Methods in Physics.
Operator theory.
Differential Geometry.
Probability Theory and Stochastic Processes.
Abstract Harmonic Analysis.
Series:
Applied and Numerical Harmonic Analysis
Applied and numerical harmonic analysis
Edition:
1.
Contents:
Part I. Traditional Methods for Computing Heat Kernels -- Introduction -- Stochastic Analysis Method -- A Brief Introduction to Calculus of Variations -- The Path Integral Approach -- The Geometric Method -- Commuting Operators -- Fourier Transform Method -- The Eigenfunctions Expansion Method -- Part II. Heat Kernel on Nilpotent Lie Groups and Nilmanifolds -- Laplacians and Sub-Laplacians -- Heat Kernels for Laplacians and Step 2 Sub-Laplacians -- Heat Kernel for Sub-Laplacian on the Sphere S^3 -- Part III. Laguerre Calculus and Fourier Method -- Finding Heat Kernels by Using Laguerre Calculus -- Constructing Heat Kernel for Degenerate Elliptic Operators -- Heat Kernel for the Kohn Laplacian on the Heisenberg Group -- Part IV. Pseudo-Differential Operators -- The Psuedo-Differential Operators Technique -- Bibliography -- Index.
Physical Description:
XVIII, 436p. 25 illus. digital.
Electronic Location:
http://dx.doi.org/10.1007/978-0-8176-4995-1
Publication Date:
2011.
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