Introduction To Symplectic Dirac Operators
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Introduction To Symplectic Dirac Operators
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Author : Katharina Habermann
language : en
Publisher:
Release Date : 2006-01-01
Introduction To Symplectic Dirac Operators written by Katharina Habermann and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-01-01 with Dirac equation categories.
One of the basic ideas in differential geometry is that the study of analytic properties of certain differential operators acting on sections of vector bundles yields geometric and topological properties of the underlying base manifold. Symplectic spinor fields are sections in an L2-Hilbert space bundle over a symplectic manifold and symplectic Dirac operators, acting on symplectic spinor fields, are associated to the symplectic manifold in a very natural way. They may be expected to give interesting applications in symplectic geometry and symplectic topology. These symplectic Dirac operators are called Dirac operators, since they are defined in an analogous way as the classical Riemannian Dirac operator known from Riemannian spin geometry. They are called symplectic because they are constructed by use of the symplectic setting of the underlying symplectic manifold. This volume is the first one that gives a systematic and self-contained introduction to the theory of symplectic Dirac operators and reflects the current state of the subject. At the same time, it is intended to establish the idea that symplectic spin geometry and symplectic Dirac operators may give valuable tools in symplectic geometry and symplectic topology, which have become important fields and very active areas of mathematical research.
Introduction To Symplectic Dirac Operators
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Author : Katharina Habermann
language : en
Publisher: Springer
Release Date : 2006-10-28
Introduction To Symplectic Dirac Operators written by Katharina Habermann and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-10-28 with Mathematics categories.
This volume is the first one that gives a systematic and self-contained introduction to the theory of symplectic Dirac operators and reflects the current state of the subject. At the same time, it is intended to establish the idea that symplectic spin geometry and symplectic Dirac operators may give valuable tools in symplectic geometry and symplectic topology, which have become important fields and very active areas of mathematical research.
Introduction To Symplectic Dirac Operators
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Author :
language : en
Publisher:
Release Date : 2006
Introduction To Symplectic Dirac Operators written by and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006 with Clifford algebras categories.
Dirac Operators In Representation Theory
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Author : Jing-Song Huang
language : en
Publisher: Springer Science & Business Media
Release Date : 2007-05-27
Dirac Operators In Representation Theory written by Jing-Song Huang and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007-05-27 with Mathematics categories.
This book presents a comprehensive treatment of important new ideas on Dirac operators and Dirac cohomology. Using Dirac operators as a unifying theme, the authors demonstrate how some of the most important results in representation theory fit together when viewed from this perspective. The book is an excellent contribution to the mathematical literature of representation theory, and this self-contained exposition offers a systematic examination and panoramic view of the subject. The material will be of interest to researchers and graduate students in representation theory, differential geometry, and physics.
Dirac Operators In Riemannian Geometry
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Author : Thomas Friedrich
language : en
Publisher: American Mathematical Soc.
Release Date : 2000
Dirac Operators In Riemannian Geometry written by Thomas Friedrich and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2000 with Mathematics categories.
For a Riemannian manifold M, the geometry, topology and analysis are interrelated in ways that have become widely explored in modern mathematics. Bounds on the curvature can have significant implications for the topology of the manifold. The eigenvalues of the Laplacian are naturally linked to the geometry of the manifold. For manifolds that admit spin structures, one obtains further information from equations involving Dirac operators and spinor fields. In the case of four-manifolds, for example, one has the remarkable Seiberg-Witten invariants. In this text, Friedrich examines the Dirac operator on Riemannian manifolds, especially its connection with the underlying geometry and topology of the manifold. The presentation includes a review of Clifford algebras, spin groups and the spin representation, as well as a review of spin structures and $\textrm{spin}mathbb{C}$ structures. With this foundation established, the Dirac operator is defined and studied, with special attention to the cases of Hermitian manifolds and symmetric spaces. Then, certain analytic properties are established, including self-adjointness and the Fredholm property. An important link between the geometry and the analysis is provided by estimates for the eigenvalues of the Dirac operator in terms of the scalar curvature and the sectional curvature. Considerations of Killing spinors and solutions of the twistor equation on M lead to results about whether M is an Einstein manifold or conformally equivalent to one. Finally, in an appendix, Friedrich gives a concise introduction to the Seiberg-Witten invariants, which are a powerful tool for the study of four-manifolds. There is also an appendix reviewing principal bundles and connections. This detailed book with elegant proofs is suitable as a text for courses in advanced differential geometry and global analysis, and can serve as an introduction for further study in these areas. This edition is translated from the German edition published by Vieweg Verlag.
An Introduction To Symplectic Geometry
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Author : Rolf Berndt
language : en
Publisher: American Mathematical Society
Release Date : 2024-04-15
An Introduction To Symplectic Geometry written by Rolf Berndt and has been published by American Mathematical Society this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-04-15 with Mathematics categories.
Symplectic geometry is a central topic of current research in mathematics. Indeed, symplectic methods are key ingredients in the study of dynamical systems, differential equations, algebraic geometry, topology, mathematical physics and representations of Lie groups. This book is a true introduction to symplectic geometry, assuming only a general background in analysis and familiarity with linear algebra. It starts with the basics of the geometry of symplectic vector spaces. Then, symplectic manifolds are defined and explored. In addition to the essential classic results, such as Darboux's theorem, more recent results and ideas are also included here, such as symplectic capacity and pseudoholomorphic curves. These ideas have revolutionized the subject. The main examples of symplectic manifolds are given, including the cotangent bundle, Kähler manifolds, and coadjoint orbits. Further principal ideas are carefully examined, such as Hamiltonian vector fields, the Poisson bracket, and connections with contact manifolds. Berndt describes some of the close connections between symplectic geometry and mathematical physics in the last two chapters of the book. In particular, the moment map is defined and explored, both mathematically and in its relation to physics. He also introduces symplectic reduction, which is an important tool for reducing the number of variables in a physical system and for constructing new symplectic manifolds from old. The final chapter is on quantization, which uses symplectic methods to take classical mechanics to quantum mechanics. This section includes a discussion of the Heisenberg group and the Weil (or metaplectic) representation of the symplectic group. Several appendices provide background material on vector bundles, on cohomology, and on Lie groups and Lie algebras and their representations. Berndt's presentation of symplectic geometry is a clear and concise introduction to the major methods and applications of the subject, and requires only a minimum of prerequisites. This book would be an excellent text for a graduate course or as a source for anyone who wishes to learn about symplectic geometry.
Operator And Matrix Theory Function Spaces And Applications
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Author : Marek Ptak
language : en
Publisher: Springer Nature
Release Date : 2024-04-02
Operator And Matrix Theory Function Spaces And Applications written by Marek Ptak and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-04-02 with Mathematics categories.
This volume features presentations from the International Workshop on Operator Theory and its Applications that was held in Kraków, Poland, September 6-10, 2022. The volume reflects the wide interests of the participants and contains original research papers in the active areas of Operator Theory. These interests include weighted Hardy spaces, geometry of Banach spaces, dilations of the tetrablock contractions, Toeplitz and Hankel operators, symplectic Dirac operator, pseudodifferential and differential operators, singular integral operators, non-commutative probability, quasi multipliers, Hilbert transform, small rank perturbations, spectral constants, Banach-Lie groupoids, reproducing kernels, and the Kippenhahn curve. The volume includes contributions by a number of the world's leading experts and can therefore be used as an introduction to the currently active research areas in operator theory.
Beyond Partial Differential Equations
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Author : Horst Reinhard Beyer
language : en
Publisher: Springer
Release Date : 2007-04-10
Beyond Partial Differential Equations written by Horst Reinhard Beyer and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007-04-10 with Mathematics categories.
This book introduces the treatment of linear and nonlinear (quasi-linear) abstract evolution equations by methods from the theory of strongly continuous semigroups. The theoretical part is accessible to graduate students with basic knowledge in functional analysis, with only some examples requiring more specialized knowledge from the spectral theory of linear, self-adjoint operators in Hilbert spaces. Emphasis is placed on equations of the hyperbolic type which are less often treated in the literature.
Fluctuation Theory For L Vy Processes
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Author : Ronald A. Doney
language : en
Publisher: Springer
Release Date : 2007-04-25
Fluctuation Theory For L Vy Processes written by Ronald A. Doney and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007-04-25 with Mathematics categories.
Lévy processes, that is, processes in continuous time with stationary and independent increments, form a flexible class of models, which have been applied to the study of storage processes, insurance risk, queues, turbulence, laser cooling, and of course finance, where they include particularly important examples having "heavy tails." Their sample path behaviour poses a variety of challenging and fascinating problems, which are addressed in detail.
Construction Of Global Lyapunov Functions Using Radial Basis Functions
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Author : Peter Giesl
language : en
Publisher: Springer
Release Date : 2007-04-11
Construction Of Global Lyapunov Functions Using Radial Basis Functions written by Peter Giesl and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007-04-11 with Mathematics categories.
The basin of attraction of an equilibrium of an ordinary differential equation can be determined using a Lyapunov function. A new method to construct such a Lyapunov function using radial basis functions is presented in this volume intended for researchers and advanced students from both dynamical systems and radial basis functions. Besides an introduction to both areas and a detailed description of the method, it contains error estimates and many examples.