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Lectures on the Geometry of Poisson Manifolds

Author : Izu Vaisman
Publisher : Birkhäuser
Release : 2012-12-06
Category : Mathematics
ISBN : 9783034884952

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Book Lectures on the Geometry of Poisson Manifolds Description/Summary:

This book is addressed to graduate students and researchers in the fields of mathematics and physics who are interested in mathematical and theoretical physics, differential geometry, mechanics, quantization theories and quantum physics, quantum groups etc., and who are familiar with differentiable and symplectic manifolds. The aim of the book is to provide the reader with a monograph that enables him to study systematically basic and advanced material on the recently developed theory of Poisson manifolds, and that also offers ready access to bibliographical references for the continuation of his study. Until now, most of this material was dispersed in research papers published in many journals and languages. The main subjects treated are the Schouten-Nijenhuis bracket; the generalized Frobenius theorem; the basics of Poisson manifolds; Poisson calculus and cohomology; quantization; Poisson morphisms and reduction; realizations of Poisson manifolds by symplectic manifolds and by symplectic groupoids and Poisson-Lie groups. The book unifies terminology and notation. It also reports on some original developments stemming from the author's work, including new results on Poisson cohomology and geometric quantization, cofoliations and biinvariant Poisson structures on Lie groups.

Lectures on Poisson Geometry

Author : Marius Crainic,Rui Loja Fernandes,Ioan Mărcuţ
Publisher : American Mathematical Soc.
Release : 2021-10-14
Category : Education
ISBN : 9781470466671

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Book Lectures on Poisson Geometry Description/Summary:

This excellent book will be very useful for students and researchers wishing to learn the basics of Poisson geometry, as well as for those who know something about the subject but wish to update and deepen their knowledge. The authors' philosophy that Poisson geometry is an amalgam of foliation theory, symplectic geometry, and Lie theory enables them to organize the book in a very coherent way. —Alan Weinstein, University of California at Berkeley This well-written book is an excellent starting point for students and researchers who want to learn about the basics of Poisson geometry. The topics covered are fundamental to the theory and avoid any drift into specialized questions; they are illustrated through a large collection of instructive and interesting exercises. The book is ideal as a graduate textbook on the subject, but also for self-study. —Eckhard Meinrenken, University of Toronto

Lectures on Symplectic Geometry

Author : Ana Cannas da Silva
Publisher : Springer
Release : 2004-10-27
Category : Mathematics
ISBN : 9783540453307

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Book Lectures on Symplectic Geometry Description/Summary:

The goal of these notes is to provide a fast introduction to symplectic geometry for graduate students with some knowledge of differential geometry, de Rham theory and classical Lie groups. This text addresses symplectomorphisms, local forms, contact manifolds, compatible almost complex structures, Kaehler manifolds, hamiltonian mechanics, moment maps, symplectic reduction and symplectic toric manifolds. It contains guided problems, called homework, designed to complement the exposition or extend the reader's understanding. There are by now excellent references on symplectic geometry, a subset of which is in the bibliography of this book. However, the most efficient introduction to a subject is often a short elementary treatment, and these notes attempt to serve that purpose. This text provides a taste of areas of current research and will prepare the reader to explore recent papers and extensive books on symplectic geometry where the pace is much faster. For this reprint numerous corrections and clarifications have been made, and the layout has been improved.

Poisson Geometry in Mathematics and Physics

Author : Giuseppe Dito
Publisher : American Mathematical Soc.
Release : 2008
Category : Mathematics
ISBN : 9780821844236

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Book Poisson Geometry in Mathematics and Physics Description/Summary:

This volume is a collection of articles by speakers at the conference ""Poisson 2006: Poisson Geometry in Mathematics and Physics"", which was held June 5-9, 2006, in Tokyo, Japan. Poisson 2006 was the fifth in a series of international conferences on Poisson geometry that are held once every two years. The aim of these conferences is to bring together mathematicians and mathematical physicists who work in diverse areas but have common interests in Poisson geometry. The program for Poisson 2006 was remarkable for the overlap of topics that included deformation quantization, generalized complex structures, differentiable stacks, normal forms, and group-valued moment maps and reduction. The articles represent current research in Poisson geometry and should be valuable to anyone interested in Poisson geometry, symplectic geometry, and mathematical physics. This volume also contains lectures by the principal speakers of the three-day school held at Keio University that preceded Poisson 2006.

The Breadth of Symplectic and Poisson Geometry

Author : Jerrold E. Marsden,Tudor S. Ratiu
Publisher : Springer Science & Business Media
Release : 2007-07-03
Category : Mathematics
ISBN : 9780817644192

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Book The Breadth of Symplectic and Poisson Geometry Description/Summary:

* 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

Quantum Algebras and Poisson Geometry in Mathematical Physics

Author : Mikhail Vladimirovich Karasev,Elena M. Novikova,Yurii Mikhailovich Vorobjev
Publisher : American Mathematical Soc.
Release : 2005
Category : Mathematics
ISBN : 0821840401

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Book Quantum Algebras and Poisson Geometry in Mathematical Physics Description/Summary:

This collection presents new and interesting applications of Poisson geometry to some fundamental well-known problems in mathematical physics. The methods used by the authors include, in addition to advanced Poisson geometry, unexpected algebras with non-Lie commutation relations, nontrivial (quantum) Kahlerian structures of hypergeometric type, dynamical systems theory, semiclassical asymptotics, etc.

Poisson Structures

Author : Camille Laurent-Gengoux,Anne Pichereau,Pol Vanhaecke
Publisher : Springer Science & Business Media
Release : 2012-08-27
Category : Mathematics
ISBN : 9783642310904

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Book Poisson Structures Description/Summary:

Poisson structures appear in a large variety of contexts, ranging from string theory, classical/quantum mechanics and differential geometry to abstract algebra, algebraic geometry and representation theory. In each one of these contexts, it turns out that the Poisson structure is not a theoretical artifact, but a key element which, unsolicited, comes along with the problem that is investigated, and its delicate properties are decisive for the solution to the problem in nearly all cases. Poisson Structures is the first book that offers a comprehensive introduction to the theory, as well as an overview of the different aspects of Poisson structures. The first part covers solid foundations, the central part consists of a detailed exposition of the different known types of Poisson structures and of the (usually mathematical) contexts in which they appear, and the final part is devoted to the two main applications of Poisson structures (integrable systems and deformation quantization). The clear structure of the book makes it adequate for readers who come across Poisson structures in their research or for graduate students or advanced researchers who are interested in an introduction to the many facets and applications of Poisson structures.​

Introduction to Symplectic Geometry

Author : Jean-Louis Koszul,Yi Ming Zou
Publisher : Springer
Release : 2019-04-15
Category : Science
ISBN : 9789811339875

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Book Introduction to Symplectic Geometry Description/Summary:

This introductory book offers a unique and unified overview of symplectic geometry, highlighting the differential properties of symplectic manifolds. It consists of six chapters: Some Algebra Basics, Symplectic Manifolds, Cotangent Bundles, Symplectic G-spaces, Poisson Manifolds, and A Graded Case, concluding with a discussion of the differential properties of graded symplectic manifolds of dimensions (0,n). It is a useful reference resource for students and researchers interested in geometry, group theory, analysis and differential equations.This book is also inspiring in the emerging field of Geometric Science of Information, in particular the chapter on Symplectic G-spaces, where Jean-Louis Koszul develops Jean-Marie Souriau's tools related to the non-equivariant case of co-adjoint action on Souriau’s moment map through Souriau’s Cocycle, opening the door to Lie Group Machine Learning with Souriau-Fisher metric.

Geometric Models for Noncommutative Algebras

Author : Ana Cannas da Silva,Alan Weinstein
Publisher : American Mathematical Soc.
Release : 1999
Category : Mathematics
ISBN : 0821809520

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Book Geometric Models for Noncommutative Algebras Description/Summary:

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.

Geometric Methods in Physics

Author : Piotr Kielanowski,Pierre Bieliavsky,Alexander Odesskii,Anatol Odzijewicz,Martin Schlichenmaier,Theodore Voronov
Publisher : Springer
Release : 2014-08-19
Category : Mathematics
ISBN : 9783319062488

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Book Geometric Methods in Physics Description/Summary:

The Białowieża Workshops on Geometric Methods in Physics, which are hosted in the unique setting of the Białowieża natural forest in Poland, are among the most important meetings in the field. Every year some 80 to 100 participants from both the mathematics and physics world join to discuss new developments and to exchange ideas. The current volume was produced on the occasion of the 32nd meeting in 2013. It is now becoming a tradition that the Workshop is followed by a School on Geometry and Physics, which consists of advanced lectures for graduate students and young researchers. Selected speakers at the 2013 Workshop were asked to contribute to this book, and their work was supplemented by additional review articles. The selection shows that, despite its now long tradition, the workshop remains at the cutting edge of research. The 2013 Workshop also celebrated the 75th birthday of Daniel Sternheimer, and on this occasion the discussion mainly focused on his contributions to mathematical physics such as deformation quantization, Poisson geometry, symplectic geometry and non-commutative differential geometry.

Geometric and Algebraic Topological Methods in Quantum Mechanics

Author : Giovanni Giachetta,Luigi Mangiarotti,Gennadi Sardanashvily
Publisher : World Scientific
Release : 2005-01-27
Category : Science
ISBN : 9789814481144

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Book Geometric and Algebraic Topological Methods in Quantum Mechanics Description/Summary:

' In the last decade, the development of new ideas in quantum theory, including geometric and deformation quantization, the non-Abelian Berry's geometric factor, super- and BRST symmetries, non-commutativity, has called into play the geometric techniques based on the deep interplay between algebra, differential geometry and topology. The book aims at being a guide to advanced differential geometric and topological methods in quantum mechanics. Their main peculiarity lies in the fact that geometry in quantum theory speaks mainly the algebraic language of rings, modules, sheaves and categories. Geometry is by no means the primary scope of the book, but it underlies many ideas in modern quantum physics and provides the most advanced schemes of quantization. Contents:Commutative GeometryClassical Hamiltonian SystemsAlgebraic QuantizationGeometry of Algebraic QuantizationGeometric QuantizationSupergeometryDeformation QuantizationNon-Commutative GeometryGeometry of Quantum Groups Readership: Theoreticians and mathematicians of postgraduate and research level. Keywords:Algebraic Quantum Theory;Poisson Manifold;Hilbert Manifold;Geometric Quantization;Deformation Quantization;Supergeometry;Noncommutative Geometry;Constraint System;Quantum GroupKey Features:The book collects all the advanced methods of quantization in the last decadeIt presents in a compact way all the necessary up to date mathematical tools to be used in studying quantum problems.Reviews:“This book is well-written and I am convinced that it will be useful to all those interested in quantum theory.”Zentralblatt MATH “With respect to a propsective reader having a reasonably good background in mathematics, the notions, concepts, etc, are introduced in a self-contained but condensed manner … The book gives a very helpful supply of mathematical tools needed by a theoretical or mathematical physicist to effect entry into some of the new directions in theoretical physics. Also, a mathematician might appreciate the condensed presentation of definitions and results in one of the modern fields of mathematics for which one may be seeking an overview.”Mathematical Reviews '

Quantization of Singular Symplectic Quotients

Author : N.P. Landsman,Markus Pflaum,Martin Schlichenmaier
Publisher : Birkhäuser
Release : 2012-12-06
Category : Mathematics
ISBN : 9783034883641

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Book Quantization of Singular Symplectic Quotients Description/Summary:

This is the first exposition of the quantization theory of singular symplectic (Marsden-Weinstein) quotients and their applications to physics. The reader will acquire an introduction to the various techniques used in this area, as well as an overview of the latest research approaches. These involve classical differential and algebraic geometry, as well as operator algebras and noncommutative geometry. Thus one will be amply prepared to follow future developments in this field.

Geometric Mechanics and Symmetry

Author : James Montaldi,Tudor Ratiu,J. W. S. Cassels
Publisher : Cambridge University Press
Release : 2005-05-05
Category : Mathematics
ISBN : 0521539579

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Book Geometric Mechanics and Symmetry Description/Summary:

The lectures in this 2005 book are intended to bring young researchers to the current frontier of knowledge in geometrical mechanics and dynamical systems.

Geometric and Algebraic Structures in Differential Equations

Author : P.H. Kersten,I.S. Krasil'shchik
Publisher : Springer Science & Business Media
Release : 2012-12-06
Category : Mathematics
ISBN : 9789400901797

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Book Geometric and Algebraic Structures in Differential Equations Description/Summary:

The geometrical theory of nonlinear differential equations originates from classical works by S. Lie and A. Bäcklund. It obtained a new impulse in the sixties when the complete integrability of the Korteweg-de Vries equation was found and it became clear that some basic and quite general geometrical and algebraic structures govern this property of integrability. Nowadays the geometrical and algebraic approach to partial differential equations constitutes a special branch of modern mathematics. In 1993, a workshop on algebra and geometry of differential equations took place at the University of Twente (The Netherlands), where the state-of-the-art of the main problems was fixed. This book contains a collection of invited lectures presented at this workshop. The material presented is of interest to those who work in pure and applied mathematics and especially in mathematical physics.

Noncommutative Geometry and Representation Theory in Mathematical Physics

Author : Jürgen Fuchs
Publisher : American Mathematical Soc.
Release : 2005
Category : Mathematics
ISBN : 9780821837184

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Book Noncommutative Geometry and Representation Theory in Mathematical Physics Description/Summary:

Mathematics provides a language in which to formulate the laws that govern nature. It is a language proven to be both powerful and effective. In the quest for a deeper understanding of the fundamental laws of physics, one is led to theories that are increasingly difficult to put to the test. In recent years, many novel questions have emerged in mathematical physics, particularly in quantum field theory. Indeed, several areas of mathematics have lately become increasingly influential in physics and, in turn, have become influenced by developments in physics. Over the last two decades, interactions between mathematicians and physicists have increased enormously and have resulted in a fruitful cross-fertilization of the two communities.This volume contains the plenary talks from the international symposium on Noncommutative Geometry and Representation Theory in Mathematical Physics held at Karlstad University (Sweden) as a satellite conference to the Fourth European Congress of Mathematics. The scope of the volume is large and its content is relevant to various scientific communities interested in noncommutative geometry and representation theory. It offers a comprehensive view of the state of affairs for these two branches of mathematical physics. The book is suitable for graduate students and researchers interested in mathematical physics.

Dynamical Systems

Author : Albert Fathi,J.-C. Yoccoz
Publisher : Cambridge University Press
Release : 2006-02-02
Category : Mathematics
ISBN : 9780521860680

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Book Dynamical Systems Description/Summary:

A collection of up-to-date research and classic papers reflecting the work of Michael Herman.

Geometric Formulation of Classical and Quantum Mechanics

Author : G. Giachetta,L. G. Magiaradze,Gennadi? Aleksandrovich Sardanashvili
Publisher : World Scientific
Release : 2011
Category : Science
ISBN : 9789814313728

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Book Geometric Formulation of Classical and Quantum Mechanics Description/Summary:

The geometric formulation of autonomous Hamiltonian mechanics in the terms of symplectic and Poisson manifolds is generally accepted. This book provides the geometric formulation of non-autonomous mechanics in a general setting of time-dependent coordinate and reference frame transformations.

Discrete Mechanics, Geometric Integration and Lie–Butcher Series

Author : Kurusch Ebrahimi-Fard,María Barbero Liñán
Publisher : Springer
Release : 2018-11-05
Category : Mathematics
ISBN : 9783030013974

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Book Discrete Mechanics, Geometric Integration and Lie–Butcher Series Description/Summary:

This volume resulted from presentations given at the international “Brainstorming Workshop on New Developments in Discrete Mechanics, Geometric Integration and Lie–Butcher Series”, that took place at the Instituto de Ciencias Matemáticas (ICMAT) in Madrid, Spain. It combines overview and research articles on recent and ongoing developments, as well as new research directions. Why geometric numerical integration? In their article of the same title Arieh Iserles and Reinout Quispel, two renowned experts in numerical analysis of differential equations, provide a compelling answer to this question. After this introductory chapter a collection of high-quality research articles aim at exploring recent and ongoing developments, as well as new research directions in the areas of geometric integration methods for differential equations, nonlinear systems interconnections, and discrete mechanics. One of the highlights is the unfolding of modern algebraic and combinatorial structures common to those topics, which give rise to fruitful interactions between theoretical as well as applied and computational perspectives. The volume is aimed at researchers and graduate students interested in theoretical and computational problems in geometric integration theory, nonlinear control theory, and discrete mechanics.

Differential Geometry and Its Applications

Author : Anonim
Publisher : World Scientific
Release : 2008
Category : Electronic books
ISBN : 9789812790613

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Book Differential Geometry and Its Applications Description/Summary:

This volume contains invited lectures and selected research papers in the fields of classical and modern differential geometry, global analysis, and geometric methods in physics, presented at the 10th International Conference on Differential Geometry and its Applications (DGA2007), held in Olomouc, Czech Republic.The book covers recent developments and the latest results in the following fields: Riemannian geometry, connections, jets, differential invariants, the calculus of variations on manifolds, differential equations, Finsler structures, and geometric methods in physics. It is also a celebration of the 300th anniversary of the birth of one of the greatest mathematicians, Leonhard Euler, and includes the Euler lecture OC Leonhard Euler OCo 300 years onOCO by R Wilson. Notable contributors include J F Cariena, M Castrilln Lpez, J Erichhorn, J-H Eschenburg, I KoliO, A P Kopylov, J Korbai, O Kowalski, B Kruglikov, D Krupka, O Krupkovi, R L(r)andre, Haizhong Li, S Maeda, M A Malakhaltsev, O I Mokhov, J Muoz Masqu(r), S Preston, V Rovenski, D J Saunders, M Sekizawa, J Slovik, J Szilasi, L Tamissy, P Walczak, and others."