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Poisson Structures and Their Normal Forms

Author : Jean-Paul Dufour,Nguyen Tien Zung
Publisher : Birkhäuser
Release : 2005-09-16
Category : Mathematics
ISBN : 3764373342

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Book Poisson Structures and Their Normal Forms Description/Summary:

The aim of this book is twofold. On the one hand, it gives a quick, self-contained introduction to Poisson geometry and related subjects. On the other hand, it presents a comprehensive treatment of the normal form problem in Poisson geometry. Even when it comes to classical results, the book gives new insights. It contains results obtained over the past 10 years which are not available in other books.

Poisson Structures and Their Normal Forms

Author : Jean-Paul Dufour,Nguyen Tien Zung
Publisher : Springer Science & Business Media
Release : 2006-01-17
Category : Mathematics
ISBN : 9783764373351

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Book Poisson Structures and Their Normal Forms Description/Summary:

The aim of this book is twofold. On the one hand, it gives a quick, self-contained introduction to Poisson geometry and related subjects. On the other hand, it presents a comprehensive treatment of the normal form problem in Poisson geometry. Even when it comes to classical results, the book gives new insights. It contains results obtained over the past 10 years which are not available in other books.

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.​

Council for African American Researchers in the Mathematical Sciences: Volume V

Author : Alfred G. Noël
Publisher : American Mathematical Soc.
Release : 2008
Category : Mathematics
ISBN : 9780821844571

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Book Council for African American Researchers in the Mathematical Sciences: Volume V Description/Summary:

This volume contains articles based on talks presented at the Thirteenth Conference of African American Researchers in the Mathematical Sciences (CAARMS), held at Northeastern University and the University of Massachusetts, Boston on June 19-22, 2007. The representation theory of Lie groups and its applications were a major focus of the talks. An overview is included of the recent achievements of the Atlas of Lie Groups Project and the work that remains to be done by the Atlas Project to fully understand the unitary representations of reductive groups. Other articles highlight the significance of scientific computing in Lie Theory and applications of the representation theory of Lie groups to the physics of black holes.This volume contains two research papers not related to Lie Theory: one on the geometry of coisotropic submanifolds of Poisson manifolds and one on recent results on the structure of the set of ultrafilters on the collection of finite subsets of an infinite set. It concludes with a survey of CAARMS12 participants. For volumes based on previous CAARMS proceedings, see: ""African Americans in Mathematics"" (volume 34 in the ""AMS"" series, DIMACS), ""African Americans in Mathematics: Volume II"" (volume 252 in the ""AMS"" series ""Contemporary Mathematics""), ""African Americans in the Mathematical Sciences: Volume III"" (volume 275 in the ""Contemporary Mathematics"" series) and Volume IV (volume 284 in the ""Contemporary Mathematics"" series).

Hamiltonian Dynamical Systems and Applications

Author : Walter Craig
Publisher : Springer Science & Business Media
Release : 2008-02-17
Category : Mathematics
ISBN : 9781402069642

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Book Hamiltonian Dynamical Systems and Applications Description/Summary:

This volume is the collected and extended notes from the lectures on Hamiltonian dynamical systems and their applications that were given at the NATO Advanced Study Institute in Montreal in 2007. Many aspects of the modern theory of the subject were covered at this event, including low dimensional problems. Applications are also presented to several important areas of research, including problems in classical mechanics, continuum mechanics, and partial differential equations.

Geometry and Physics: Volume 2

Author : Jørgen Ellegaard Andersen,Andrew Dancer,Oscar García-Prada
Publisher : Oxford University Press
Release : 2018-10-18
Category : Mathematics
ISBN : 9780192522375

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Book Geometry and Physics: Volume 2 Description/Summary:

Nigel Hitchin is one of the world's foremost figures in the fields of differential and algebraic geometry and their relations with mathematical physics, and he has been Savilian Professor of Geometry at Oxford since 1997. Geometry and Physics: A Festschrift in honour of Nigel Hitchin contain the proceedings of the conferences held in September 2016 in Aarhus, Oxford, and Madrid to mark Nigel Hitchin's 70th birthday, and to honour his far-reaching contributions to geometry and mathematical physics. These texts contain 29 articles by contributors to the conference and other distinguished mathematicians working in related areas, including three Fields Medallists. The articles cover a broad range of topics in differential, algebraic and symplectic geometry, and also in mathematical physics. These volumes will be of interest to researchers and graduate students in geometry and mathematical physics.

Geometry and Physics: Volume 2

Author : Andrew Dancer,Jørgen Ellegaard Andersen,Oscar García-Prada
Publisher : Unknown
Release : 2018-10-25
Category : Uncategorized
ISBN : 9780198802020

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Book Geometry and Physics: Volume 2 Description/Summary:

These texts contain 29 articles that cover a broad range of topics in differential, algebraic and symplectic geometry, and also in mathematical physics. These volumes will be of interest to researchers and graduate students in geometry and mathematical physics

Formality Theory

Author : Chiara Esposito
Publisher : Springer
Release : 2014-09-04
Category : Science
ISBN : 9783319092904

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Book Formality Theory Description/Summary:

This book is a survey of the theory of formal deformation quantization of Poisson manifolds, in the formalism developed by Kontsevich. It is intended as an educational introduction for mathematical physicists who are dealing with the subject for the first time. The main topics covered are the theory of Poisson manifolds, star products and their classification, deformations of associative algebras and the formality theorem. Readers will also be familiarized with the relevant physical motivations underlying the purely mathematical construction.

Geometry, Mechanics, and Dynamics

Author : Dong Eui Chang,Darryl D. Holm,George Patrick,Tudor Ratiu
Publisher : Springer
Release : 2015-04-16
Category : Mathematics
ISBN : 9781493924417

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Book Geometry, Mechanics, and Dynamics Description/Summary:

This book illustrates the broad range of Jerry Marsden’s mathematical legacy in areas of geometry, mechanics, and dynamics, from very pure mathematics to very applied, but always with a geometric perspective. Each contribution develops its material from the viewpoint of geometric mechanics beginning at the very foundations, introducing readers to modern issues via illustrations in a wide range of topics. The twenty refereed papers contained in this volume are based on lectures and research performed during the month of July 2012 at the Fields Institute for Research in Mathematical Sciences, in a program in honor of Marsden's legacy. The unified treatment of the wide breadth of topics treated in this book will be of interest to both experts and novices in geometric mechanics. Experts will recognize applications of their own familiar concepts and methods in a wide variety of fields, some of which they may never have approached from a geometric viewpoint. Novices may choose topics that interest them among the various fields and learn about geometric approaches and perspectives toward those topics that will be new for them as well.

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

Geometric Structures of Statistical Physics, Information Geometry, and Learning

Author : Frédéric Barbaresco,Frank Nielsen
Publisher : Springer Nature
Release : 2021
Category : Artificial intelligence
ISBN : 9783030779573

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Book Geometric Structures of Statistical Physics, Information Geometry, and Learning Description/Summary:

Machine learning and artificial intelligence increasingly use methodological tools rooted in statistical physics. Conversely, limitations and pitfalls encountered in AI question the very foundations of statistical physics. This interplay between AI and statistical physics has been attested since the birth of AI, and principles underpinning statistical physics can shed new light on the conceptual basis of AI. During the last fifty years, statistical physics has been investigated through new geometric structures allowing covariant formalization of the thermodynamics. Inference methods in machine learning have begun to adapt these new geometric structures to process data in more abstract representation spaces. This volume collects selected contributions on the interplay of statistical physics and artificial intelligence. The aim is to provide a constructive dialogue around a common foundation to allow the establishment of new principles and laws governing these two disciplines in a unified manner. The contributions were presented at the workshop on the Joint Structures and Common Foundation of Statistical Physics, Information Geometry and Inference for Learning which was held in Les Houches in July 2020. The various theoretical approaches are discussed in the context of potential applications in cognitive systems, machine learning, signal processing.

Geometric and Topological Methods for Quantum Field Theory

Author : Alexander Cardona,Iván Contreras,Andrés F. Reyes-Lega
Publisher : Cambridge University Press
Release : 2013-05-09
Category : Science
ISBN : 9781107026834

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Book Geometric and Topological Methods for Quantum Field Theory Description/Summary:

"Based on lectures given at the renowed Villa de Leyva summer school, this book provides a unique presentation of modern geometric methods in quantum field theory. Written by experts, it enables readers to enter some of the most fascinating research topics in this subject. Covering a series of topics on geometry, topology, algebra, number theory methods and their applications to quantum field theory, the book covers topics such as Dirac structures, holomorphic bundles and stability, Feynman integrals, geometric aspects of quantum field theory and the standard model, spectral and Riemannian geometry and index theory. This is a valuable guide for graduate students and researchers in physics and mathematics wanting to enter this interesting research field at the borderline between mathematics and physics"--

Geometric Science of Information

Author : Frank Nielsen,Frédéric Barbaresco
Publisher : Springer Nature
Release : 2021-07-14
Category : Computers
ISBN : 9783030802097

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Book Geometric Science of Information Description/Summary:

This book constitutes the proceedings of the 5th International Conference on Geometric Science of Information, GSI 2021, held in Paris, France, in July 2021. The 98 papers presented in this volume were carefully reviewed and selected from 125 submissions. They cover all the main topics and highlights in the domain of geometric science of information, including information geometry manifolds of structured data/information and their advanced applications. The papers are organized in the following topics: Probability and statistics on Riemannian Manifolds; sub-Riemannian geometry and neuromathematics; shapes spaces; geometry of quantum states; geometric and structure preserving discretizations; information geometry in physics; Lie group machine learning; geometric and symplectic methods for hydrodynamical models; harmonic analysis on Lie groups; statistical manifold and Hessian information geometry; geometric mechanics; deformed entropy, cross-entropy, and relative entropy; transformation information geometry; statistics, information and topology; geometric deep learning; topological and geometrical structures in neurosciences; computational information geometry; manifold and optimization; divergence statistics; optimal transport and learning; and geometric structures in thermodynamics and statistical physics.

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.

Quantum Mathematical Physics

Author : Felix Finster,Johannes Kleiner,Christian Röken,Jürgen Tolksdorf
Publisher : Birkhäuser
Release : 2016-02-24
Category : Science
ISBN : 9783319269023

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Book Quantum Mathematical Physics Description/Summary:

Quantum physics has been highly successful for more than 90 years. Nevertheless, a rigorous construction of interacting quantum field theory is still missing. Moreover, it is still unclear how to combine quantum physics and general relativity in a unified physical theory. Attacking these challenging problems of contemporary physics requires highly advanced mathematical methods as well as radically new physical concepts. This book presents different physical ideas and mathematical approaches in this direction. It contains a carefully selected cross-section of lectures which took place in autumn 2014 at the sixth conference ``Quantum Mathematical Physics - A Bridge between Mathematics and Physics'' in Regensburg, Germany. In the tradition of the other proceedings covering this series of conferences, a special feature of this book is the exposition of a wide variety of approaches, with the intention to facilitate a comparison. The book is mainly addressed to mathematicians and physicists who are interested in fundamental questions of mathematical physics. It allows the reader to obtain a broad and up-to-date overview of a fascinating active research area.

Material Geometry: Groupoids In Continuum Mechanics

Author : Manuel De Leon,Marcelo Epstein,Victor Manuel Jimenez
Publisher : World Scientific
Release : 2021-04-23
Category : Mathematics
ISBN : 9789811232565

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Book Material Geometry: Groupoids In Continuum Mechanics Description/Summary:

This monograph is the first in which the theory of groupoids and algebroids is applied to the study of the properties of uniformity and homogeneity of continuous media. It is a further step in the application of differential geometry to the mechanics of continua, initiated years ago with the introduction of the theory of G-structures, in which the group G denotes the group of material symmetries, to study smoothly uniform materials.The new approach presented in this book goes much further by being much more general. It is not a generalization per se, but rather a natural way of considering the algebraic-geometric structure induced by the so-called material isomorphisms. This approach has allowed us to encompass non-uniform materials and discover new properties of uniformity and homogeneity that certain material bodies can possess, thus opening a new area in the discipline.

D-Modules, Perverse Sheaves, and Representation Theory

Author : Ryoshi Hotta,Toshiyuki Tanisaki
Publisher : Springer Science & Business Media
Release : 2007-11-07
Category : Mathematics
ISBN : 9780817643638

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Book D-Modules, Perverse Sheaves, and Representation Theory Description/Summary:

D-modules continues to be an active area of stimulating research in such mathematical areas as algebraic, analysis, differential equations, and representation theory. Key to D-modules, Perverse Sheaves, and Representation Theory is the authors' essential algebraic-analytic approach to the theory, which connects D-modules to representation theory and other areas of mathematics. To further aid the reader, and to make the work as self-contained as possible, appendices are provided as background for the theory of derived categories and algebraic varieties. The book is intended to serve graduate students in a classroom setting and as self-study for researchers in algebraic geometry, representation theory.

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.