Symplectic Geometry, Groupoids, and Integrable Systems
Symplectic Geometry, Groupoids, and Integrable Systems

Author: Pierre Dazord

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 318

ISBN-13: 1461397197

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The papers, some of which are in English, the rest in French, in this volume are based on lectures given during the meeting of the Seminare Sud Rhodanien de Geometrie (SSRG) organized at the Mathematical Sciences Research Institute in 1989. The SSRG was established in 1982 by geometers and mathematical physicists with the aim of developing and coordinating research in symplectic geometry and its applications to analysis and mathematical physics. Among the subjects discussed at the meeting, a special role was given to the theory of symplectic groupoids, the subject of fruitful collaboration involving geometers from Berkeley, Lyon, and Montpellier.

Geometric Models for Noncommutative Algebras
Geometric Models for Noncommutative Algebras

Author: Ana Cannas da Silva

Publisher: American Mathematical Soc.

Published: 1999

Total Pages: 202

ISBN-13: 9780821809525

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The volume is based on a course, ``Geometric Models for Noncommutative Algebras'' taught by Professor Weinstein at Berkeley. Noncommutative geometry is the study of noncommutative algebras as if they were algebras of functions on spaces, for example, the commutative algebras associated to affine algebraic varieties, differentiable manifolds, topological spaces, and measure spaces. In this work, the authors discuss several types of geometric objects (in the usual sense of sets with structure) that are closely related to noncommutative algebras. Central to the discussion are symplectic and Poisson manifolds, which arise when noncommutative algebras are obtained by deforming commutative algebras. The authors also give a detailed study of groupoids (whose role in noncommutative geometry has been stressed by Connes) as well as of Lie algebroids, the infinitesimal approximations to differentiable groupoids. Featured are many interesting examples, applications, and exercises. The book starts with basic definitions and builds to (still) open questions. It is suitable for use as a graduate text. An extensive bibliography and index are included.

The Breadth of Symplectic and Poisson Geometry
The Breadth of Symplectic and Poisson Geometry

Author: Jerrold E. Marsden

Publisher: Springer Science & Business Media

Published: 2007-07-03

Total Pages: 666

ISBN-13: 0817644199

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* The invited papers in this volume are written in honor of Alan Weinstein, one of the world’s foremost geometers * Contributions cover a broad range of topics in symplectic and differential geometry, Lie theory, mechanics, and related fields * Intended for graduate students and working mathematicians, this text is a distillation of prominent research and an indication of future trends in geometry, mechanics, and mathematical physics

Lectures on the Geometry of Quantization
Lectures on the Geometry of Quantization

Author: Sean Bates

Publisher: American Mathematical Soc.

Published: 1997

Total Pages: 150

ISBN-13: 9780821807989

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These notes are based on a course entitled ``Symplectic Geometry and Geometric Quantization'' taught by Alan Weinstein at the University of California, Berkeley (fall 1992) and at the Centre Emile Borel (spring 1994). The only prerequisite for the course needed is a knowledge of the basic notions from the theory of differentiable manifolds (differential forms, vector fields, transversality, etc.). The aim is to give students an introduction to the ideas of microlocal analysis and the related symplectic geometry, with an emphasis on the role these ideas play in formalizing the transition between the mathematics of classical dynamics (hamiltonian flows on symplectic manifolds) and quantum mechanics (unitary flows on Hilbert spaces). These notes are meant to function as a guide to the literature. The authors refer to other sources for many details that are omitted and can be bypassed on a first reading.

Geometric, Algebraic and Topological Methods for Quantum Field Theory
Geometric, Algebraic and Topological Methods for Quantum Field Theory

Author: Sylvie Payche

Publisher: World Scientific

Published: 2014

Total Pages: 378

ISBN-13: 9814460052

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Based on lectures held at the 7th Villa de Leyva summer school, this book presents an introduction to topics of current interest in the interface of geometry, topology and physics. It is aimed at graduate students in physics or mathematics with interests in geometric, algebraic as well as topological methods and their applications to quantum field theory. This volume contains the written notes corresponding to lectures given by experts in the field. They cover current topics of research in a way that is suitable for graduate students of mathematics or physics interested in the recent developments and interactions between geometry, topology and physics. The book also contains contributions by younger participants, displaying the ample range of topics treated in the school. A key feature of the present volume is the provision of a pedagogical presentation of rather advanced topics, in a way which is suitable for both mathematicians and physicists.

Hamiltonian Reduction by Stages
Hamiltonian Reduction by Stages

Author: Jerrold E. Marsden

Publisher: Springer

Published: 2007-06-05

Total Pages: 527

ISBN-13: 3540724702

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This volume provides a detailed account of the theory of symplectic reduction by stages, along with numerous illustrations of the theory. It gives special emphasis to group extensions, including a detailed discussion of the Euclidean group, the oscillator group, the Bott-Virasoro group and other groups of matrices. The volume also provides ample background theory on symplectic reduction and cotangent bundle reduction.

Poisson Geometry in Mathematics and Physics
Poisson Geometry in Mathematics and Physics

Author: Giuseppe Dito

Publisher: American Mathematical Soc.

Published: 2008

Total Pages: 330

ISBN-13: 0821844237

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This volume is a collection of articles by speakers at the Poisson 2006 conference. The program for Poisson 2006 was an overlap of topics that included deformation quantization, generalized complex structures, differentiable stacks, normal forms, and group-valued moment maps and reduction.

New Spaces in Mathematics
New Spaces in Mathematics

Author: Mathieu Anel

Publisher: Cambridge University Press

Published: 2021-04

Total Pages: 601

ISBN-13: 1108490638

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In this graduate-level book, leading researchers explore various new notions of 'space' in mathematics.

Elements of Classical and Quantum Integrable Systems
Elements of Classical and Quantum Integrable Systems

Author: Gleb Arutyunov

Publisher: Springer

Published: 2019-07-23

Total Pages: 420

ISBN-13: 303024198X

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Integrable models have a fascinating history with many important discoveries that dates back to the famous Kepler problem of planetary motion. Nowadays it is well recognised that integrable systems play a ubiquitous role in many research areas ranging from quantum field theory, string theory, solvable models of statistical mechanics, black hole physics, quantum chaos and the AdS/CFT correspondence, to pure mathematics, such as representation theory, harmonic analysis, random matrix theory and complex geometry. Starting with the Liouville theorem and finite-dimensional integrable models, this book covers the basic concepts of integrability including elements of the modern geometric approach based on Poisson reduction, classical and quantum factorised scattering and various incarnations of the Bethe Ansatz. Applications of integrability methods are illustrated in vast detail on the concrete examples of the Calogero-Moser-Sutherland and Ruijsenaars-Schneider models, the Heisenberg spin chain and the one-dimensional Bose gas interacting via a delta-function potential. This book has intermediate and advanced topics with details to make them clearly comprehensible.

Current Developments in Algebraic Geometry
Current Developments in Algebraic Geometry

Author: Lucia Caporaso

Publisher: Cambridge University Press

Published: 2012-03-19

Total Pages: 437

ISBN-13: 052176825X

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This volume, based on a workshop by the MSRI, offers an overview of the state of the art in many areas of algebraic geometry.