Introduction to Symplectic Dirac Operators

Author: Katharina Habermann,Lutz Habermann

Publisher: Springer

ISBN: 3540334211

Category: Mathematics

Page: 125

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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.
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Geometric Methods in Physics

XXXI Workshop, Białowieża, Poland, June 24–30, 2012

Author: Piotr Kielanowski,S. Twareque Ali,Alexander Odesskii,Anatol Odzijewicz,Martin Schlichenmaier,Theodore Voronov

Publisher: Springer Science & Business Media

ISBN: 3034806450

Category: Mathematics

Page: 237

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The Białowieża workshops on Geometric Methods in Physics, taking place in the unique environment of the Białowieża natural forest in Poland, are among the important meetings in the field. Every year some 80 to 100 participants both from mathematics and physics join to discuss new developments and to interchange ideas. The current volume was produced on the occasion of the XXXI meeting in 2012. For the first time the workshop was followed by a School on Geometry and Physics, which consisted of advanced lectures for graduate students and young researchers. Selected speakers of the workshop were asked to contribute, and additional review articles were added. The selection shows that despite its now long tradition the workshop remains always at the cutting edge of ongoing research. The XXXI workshop had as a special topic the works of the late Boris Vasilievich Fedosov (1938–2011) who is best known for a simple and very natural construction of a deformation quantization for any symplectic manifold, and for his contributions to index theory.​
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A Course in Algebra

Author: Ėrnest Borisovich Vinberg

Publisher: American Mathematical Soc.

ISBN: 9780821834138

Category: Mathematics

Page: 511

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Great book! The author's teaching experinece shows in every chapter. --Efim Zelmanov, University of California, San Diego Vinberg has written an algebra book that is excellent, both as a classroom text or for self-study. It is plain that years of teaching abstract algebra have enabled him to say the right thing at the right time. --Irving Kaplansky, MSRI This is a comprehensive text on modern algebra written for advanced undergraduate and basic graduate algebra classes. The book is based on courses taught by the author at the Mechanics and Mathematics Department of Moscow State University and at the Mathematical College of the Independent University of Moscow. The unique feature of the book is that it contains almost no technically difficult proofs. Following his point of view on mathematics, the author tried, whenever possible, to replace calculations and difficult deductions with conceptual proofs and to associate geometric images to algebraic objects. Another important feature is that the book presents most of the topics on several levels, allowing the student to move smoothly from initial acquaintance to thorough study and deeper understanding of the subject. Presented are basic topics in algebra such as algebraic structures, linear algebra, polynomials, groups, as well as more advanced topics like affine and projective spaces, tensor algebra, Galois theory, Lie groups, associative algebras and their representations. Some applications of linear algebra and group theory to physics are discussed. Written with extreme care and supplied with more than 200 exercises and 70 figures, the book is also an excellent text for independent study.
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Twenty Years of Bialowieza: A Mathematical Anthology

Aspects of Differential Geometric Methods in Physics

Author: S Twareque Ali,Gerard G Emch,Anatol Odzijewicz,Martin Schlichenmaier,Stanislaw L Woronowicz

Publisher: World Scientific

ISBN: 9814481033

Category: Science

Page: 276

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This volume marks the twentieth anniversary of the Bialowieza series of meetings on Differential Geometric Methods in Physics; the anniversary meeting was held during July 1-7, 2001. The Bialowieza meetings, held every year during the first week of July, have now grown into an annual pilgrimage for an international group of physicists and mathematicians. The topics discussed at the meetings, while within the broad area of differential geometric methods in physics, have focused around quantization, coherent states, infinite dimensional systems, symplectic geometry, spectral theory and harmonic analysis. The present volume brings together a set of specially invited papers from leading experts in the various fields, who have contributed to these meetings and whose work represents a cross-section of the topics discussed. Consequently, rather than a proceedings volume, this book embodies the spirit of the Bialowieza workshops and reflects their scientific tenor, as a tribute to the completion of two decades of a shared scientific experience. This book will be of interest to researchers and graduate students working in the area of differential geometric methods in physics, as it gives interesting glimpses into the present state of the art from different points of view. Contents:Aspects of Quantization:Diffeomorphism Groups and Quantum Configurations (G A Goldin)Functorial Quantization and the Guillemin–Sternberg Conjecture (N P Landsman)Coherent State Method in Geometric Quantization (A Odzijewicz)The Group of Volume Preserving Diffeomorphisms and the Lie Algebra of Unimodular Vector Fields: Survey of Some Classical and Not-So-Classical Results (C Roger)Symplectic and Poisson Geometry:Moduli Space of Germs of Symplectic Connections of Ricci Type (M Cahen)Banach Lie–Poisson Spaces (A Odzijewicz & T S Ratiu)Other Mathematical Methods:Spectra of Operators Associated with Dynamical Systems: From Ergodicity to the Duality Principle (A B Antonevich et al.)An Ergodic Arnold-Liouville Theorem for Locally Symmetric Spaces (J Hilgert)The Renormalization Fixed Point as a Mathematical Object (R P Langlands)A Cohomological Description of Abelian Bundles and Gerbes (R Picken)On a Quantum Group of Unitary Operators: The Quantum az + b Group (W Pusz & S L Woronowicz) Readership: Physicists and mathematicians in the area of differential geometric methods in physics. Keywords:Quantization;Symplectic Geometry;Coherent States;Diffeomorphism Groups;Quantum Groups;Ergodicity;Renormalization;Symmetric SpacesKey Features:This volume marks the twentieth anniversary of the Bialowieza series of meetings on Differential Geometric Methods in PhysicsThe articles collected here are written by leading experts in the field
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Moment Maps, Cobordisms, and Hamiltonian Group Actions

Author: Victor Guillemin,Yael Karshon,Viktor L. Ginzburg,T L Ohsawa

Publisher: American Mathematical Soc.

ISBN: 0821805029

Category: Mathematics

Page: 350

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This research monograph presents many new results in a rapidly developing area of great current interest. Guillemin, Ginzburg, and Karshon show that the underlying topological thread in the computation of invariants of G-manifolds is a consequence of a linearization theorem involving equivariant cobordisms. The book incorporates a novel approach and showcases exciting new research. During the last 20 years, 'localization' has been one of the dominant themes in the area of equivariant differential geometry. Typical results are the Duistermaat-Heckman theory, the Berline-Vergne-Atiyah-Bott localization theorem in equivariant de Rham theory, and the 'quantization commutes with reduction' theorem and its various corollaries. To formulate the idea that these theorems are all consequences of a single result involving equivariant cobordisms, the authors have developed a cobordism theory that allows the objects to be non-compact manifolds.A key ingredient in this non-compact cobordism is an equivariant-geometrical object which they call an 'abstract moment map'. This is a natural and important generalization of the notion of a moment map occurring in the theory of Hamiltonian dynamics. The book contains a number of appendices that include introductions to proper group-actions on manifolds, equivariant cohomology, Spin${^\mathrm{c}}$-structures, and stable complex structures. It is geared toward graduate students and research mathematicians interested in differential geometry. It is also suitable for topologists, Lie theorists, combinatorists, and theoretical physicists. Prerequisite is some expertise in calculus on manifolds and basic graduate-level differential geometry.
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An Introduction to Noncommutative Geometry

Author: Joseph C. Várilly

Publisher: European Mathematical Society

ISBN: 9783037190241

Category: Mathematics

Page: 113

View: 5742

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Noncommutative geometry, inspired by quantum physics, describes singular spaces by their noncommutative coordinate algebras and metric structures by Dirac-like operators. Such metric geometries are described mathematically by Connes' theory of spectral triples. These lectures, delivered at an EMS Summer School on noncommutative geometry and its applications, provide an overview of spectral triples based on examples. This introduction is aimed at graduate students of both mathematics and theoretical physics. It deals with Dirac operators on spin manifolds, noncommutative tori, Moyal quantization and tangent groupoids, action functionals, and isospectral deformations. The structural framework is the concept of a noncommutative spin geometry; the conditions on spectral triples which determine this concept are developed in detail. The emphasis throughout is on gaining understanding by computing the details of specific examples. The book provides a middle ground between a comprehensive text and a narrowly focused research monograph. It is intended for self-study, enabling the reader to gain access to the essentials of noncommutative geometry. New features since the original course are an expanded bibliography and a survey of more recent examples and applications of spectral triples.
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