Lectures on Integrable Systems
Lectures on Integrable Systems

Author: Jens Hoppe

Publisher: Springer Science & Business Media

Published: 2008-09-15

Total Pages: 109

ISBN-13: 3540472746

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Mainly drawing on explicit examples, the author introduces the reader to themost recent techniques to study finite and infinite dynamical systems. Without any knowledge of differential geometry or lie groups theory the student can follow in a series of case studies the most recent developments. r-matrices for Calogero-Moser systems and Toda lattices are derived. Lax pairs for nontrivial infinite dimensionalsystems are constructed as limits of classical matrix algebras. The reader will find explanations of the approach to integrable field theories, to spectral transform methods and to solitons. New methods are proposed, thus helping students not only to understand established techniques but also to interest them in modern research on dynamical systems.

Calogero-Moser Systems and Representation Theory
Calogero-Moser Systems and Representation Theory

Author: Pavel I. Etingof

Publisher: European Mathematical Society

Published: 2007

Total Pages: 108

ISBN-13: 9783037190340

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Calogero-Moser systems, which were originally discovered by specialists in integrable systems, are currently at the crossroads of many areas of mathematics and within the scope of interests of many mathematicians. More specifically, these systems and their generalizations turned out to have intrinsic connections with such fields as algebraic geometry (Hilbert schemes of surfaces), representation theory (double affine Hecke algebras, Lie groups, quantum groups), deformation theory (symplectic reflection algebras), homological algebra (Koszul algebras), Poisson geometry, etc. The goal of the present lecture notes is to give an introduction to the theory of Calogero-Moser systems, highlighting their interplay with these fields. Since these lectures are designed for non-experts, the author gives short introductions to each of the subjects involved and provides a number of exercises.

Symmetries, Integrable Systems and Representations
Symmetries, Integrable Systems and Representations

Author: Kenji Iohara

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 633

ISBN-13: 1447148630

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This volume is the result of two international workshops; Infinite Analysis 11 – Frontier of Integrability – held at University of Tokyo, Japan in July 25th to 29th, 2011, and Symmetries, Integrable Systems and Representations held at Université Claude Bernard Lyon 1, France in December 13th to 16th, 2011. Included are research articles based on the talks presented at the workshops, latest results obtained thereafter, and some review articles. The subjects discussed range across diverse areas such as algebraic geometry, combinatorics, differential equations, integrable systems, representation theory, solvable lattice models and special functions. Through these topics, the reader will find some recent developments in the field of mathematical physics and their interactions with several other domains.

Integrable Systems
Integrable Systems

Author: N.J. Hitchin

Publisher: Oxford University Press, USA

Published: 2013-03-14

Total Pages: 148

ISBN-13: 0199676771

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Designed to give graduate students an understanding of integrable systems via the study of Riemann surfaces, loop groups, and twistors, this book has its origins in a lecture series given by the internationally renowned authors. Written in an accessible, informal style, it fills a gap in the existing literature.

Integrable Systems and Quantum Groups
Integrable Systems and Quantum Groups

Author: Ron Donagi

Publisher: Springer

Published: 2006-11-14

Total Pages: 496

ISBN-13: 3540477063

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The aim of this CIME Session was to review the state of the art in the recent development of the theory of integrable systems and their relations with quantum groups. The purpose was to gather geometers and mathematical physicists to allow a broader and more complete view of these attractive and rapidly developing fields. The papers contained in this volume have at the same time the character of survey articles and of research papers, since they contain both a survey of current problems and a number of original contributions to the subject.

Lectures on the Orbit Method
Lectures on the Orbit Method

Author: Aleksandr Aleksandrovich Kirillov

Publisher: American Mathematical Soc.

Published: 2004

Total Pages: 434

ISBN-13: 0821835300

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Describes the essence of the orbit method for non-experts and gives a detailed exposition of the method. This work can be used as a text for a graduate course, as well as a handbook for non-experts and a reference book for research mathematicians and mathematical physicists.

Lectures on Mechanics
Lectures on Mechanics

Author: Jerrold E. Marsden

Publisher: Cambridge University Press

Published: 1992-04-30

Total Pages: 272

ISBN-13: 9780521428446

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Based on the 1991 LMS Invited Lectures given by Professor Marsden, this book discusses and applies symmetry methods to such areas as bifurcations and chaos in mechanical systems.

Discrete Systems and Integrability
Discrete Systems and Integrability

Author: J. Hietarinta

Publisher: Cambridge University Press

Published: 2016-09

Total Pages: 461

ISBN-13: 1107042720

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A first introduction to the theory of discrete integrable systems at a level suitable for students and non-experts.

Seiberg-Witten Theory and Integrable Systems
Seiberg-Witten Theory and Integrable Systems

Author: Andrei Marshakov

Publisher: World Scientific

Published: 1999

Total Pages: 268

ISBN-13: 9789810236366

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In the past few decades many attempts have been made to search for a consistent formulation of quantum field theory beyond perturbation theory. One of the most interesting examples is the Seiberg-Witten ansatz for the N=2 SUSY supersymmetric Yang-Mills gauge theories in four dimensions. The aim of this book is to present in a clear form the main ideas of the relation between the exact solutions to the supersymmetric (SUSY) Yang-Mills theories and integrable systems. This relation is a beautiful example of reformulation of close-to-realistic physical theory in terms widely known in mathematical physics ? systems of integrable nonlinear differential equations and their algebro-geometric solutions.First, the book reviews what is known about the physical problem: the construction of low-energy effective actions for the N=2 Yang-Mills theories from the traditional viewpoint of quantum field theory. Then the necessary background information from the theory of integrable systems is presented. In particular the author considers the definition of the algebro-geometric solutions to integrable systems in terms of complex curves or Riemann surfaces and the generating meromorphic 1-form. These definitions are illustrated in detail on the basic example of the periodic Toda chain.Several ?toy-model? examples of string theory solutions where the structures of integrable systems appear are briefly discussed. Then the author proceeds to the Seiberg-Witten solutions and show that they are indeed defined by the same data as finite-gap solutions to integrable systems. The complete formulation requires the introduction of certain deformations of the finite-gap solutions described in terms of quasiclassical or Whitham hierarchies. The explicit differential equations and direct computations of the prepotential of the effective theory are presented and compared when possible with the well-known computations from supersymmetric quantum gauge theories.Finally, the book discusses the properties of the exact solutions to SUSY Yang-Mills theories and their relation to integrable systems in the general context of the modern approach to nonperturbative string or M-theory.