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Calogero Moser Systems And Representation Theory
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Book Synopsis Calogero-Moser Systems and Representation Theory by : Pavel I. Etingof
Download or read book Calogero-Moser Systems and Representation Theory written by Pavel I. Etingof and published by European Mathematical Society. This book was released on 2007 with total page 108 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Calogero—Moser— Sutherland Models by : Jan F. van Diejen
Download or read book Calogero—Moser— Sutherland Models written by Jan F. van Diejen and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 572 pages. Available in PDF, EPUB and Kindle. Book excerpt: In the 1970s F. Calogero and D. Sutherland discovered that for certain potentials in one-dimensional systems, but for any number of particles, the Schrödinger eigenvalue problem is exactly solvable. Until then, there was only one known nontrivial example of an exactly solvable quantum multi-particle problem. J. Moser subsequently showed that the classical counterparts to these models is also amenable to an exact analytical approach. The last decade has witnessed a true explosion of activities involving Calogero-Moser-Sutherland models, and these now play a role in research areas ranging from theoretical physics (such as soliton theory, quantum field theory, string theory, solvable models of statistical mechanics, condensed matter physics, and quantum chaos) to pure mathematics (such as representation theory, harmonic analysis, theory of special functions, combinatorics of symmetric functions, dynamical systems, random matrix theory, and complex geometry). The aim of this volume is to provide an overview of the many branches into which research on CMS systems has diversified in recent years. The contributions are by leading researchers from various disciplines in whose work CMS systems appear, either as the topic of investigation itself or as a tool for further applications.
Book Synopsis Representation Theory, Mathematical Physics, and Integrable Systems by : Anton Alekseev
Download or read book Representation Theory, Mathematical Physics, and Integrable Systems written by Anton Alekseev and published by Springer Nature. This book was released on 2022-02-05 with total page 652 pages. Available in PDF, EPUB and Kindle. Book excerpt: Over the course of his distinguished career, Nicolai Reshetikhin has made a number of groundbreaking contributions in several fields, including representation theory, integrable systems, and topology. The chapters in this volume – compiled on the occasion of his 60th birthday – are written by distinguished mathematicians and physicists and pay tribute to his many significant and lasting achievements. Covering the latest developments at the interface of noncommutative algebra, differential and algebraic geometry, and perspectives arising from physics, this volume explores topics such as the development of new and powerful knot invariants, new perspectives on enumerative geometry and string theory, and the introduction of cluster algebra and categorification techniques into a broad range of areas. Chapters will also cover novel applications of representation theory to random matrix theory, exactly solvable models in statistical mechanics, and integrable hierarchies. The recent progress in the mathematical and physicals aspects of deformation quantization and tensor categories is also addressed. Representation Theory, Mathematical Physics, and Integrable Systems will be of interest to a wide audience of mathematicians interested in these areas and the connections between them, ranging from graduate students to junior, mid-career, and senior researchers.
Author : Publisher :World Scientific ISBN 13 : Total Pages :1001 pages Book Rating :4./5 ( download)
Download or read book written by and published by World Scientific. This book was released on with total page 1001 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Invariant Manifolds and Dispersive Hamiltonian Evolution Equations by : Kenji Nakanishi
Download or read book Invariant Manifolds and Dispersive Hamiltonian Evolution Equations written by Kenji Nakanishi and published by European Mathematical Society. This book was released on 2011 with total page 264 pages. Available in PDF, EPUB and Kindle. Book excerpt: The notion of an invariant manifold arises naturally in the asymptotic stability analysis of stationary or standing wave solutions of unstable dispersive Hamiltonian evolution equations such as the focusing semilinear Klein-Gordon and Schrodinger equations. This is due to the fact that the linearized operators about such special solutions typically exhibit negative eigenvalues (a single one for the ground state), which lead to exponential instability of the linearized flow and allows for ideas from hyperbolic dynamics to enter. One of the main results proved here for energy subcritical equations is that the center-stable manifold associated with the ground state appears as a hyper-surface which separates a region of finite-time blowup in forward time from one which exhibits global existence and scattering to zero in forward time. The authors' entire analysis takes place in the energy topology, and the conserved energy can exceed the ground state energy only by a small amount. This monograph is based on recent research by the authors. The proofs rely on an interplay between the variational structure of the ground states and the nonlinear hyperbolic dynamics near these states. A key element in the proof is a virial-type argument excluding almost homoclinic orbits originating near the ground states, and returning to them, possibly after a long excursion. These lectures are suitable for graduate students and researchers in partial differential equations and mathematical physics. For the cubic Klein-Gordon equation in three dimensions all details are provided, including the derivation of Strichartz estimates for the free equation and the concentration-compactness argument leading to scattering due to Kenig and Merle.
Book Synopsis Geometric Numerical Integration and Schrödinger Equations by : Erwan Faou
Download or read book Geometric Numerical Integration and Schrödinger Equations written by Erwan Faou and published by European Mathematical Society. This book was released on 2012 with total page 152 pages. Available in PDF, EPUB and Kindle. Book excerpt: The goal of geometric numerical integration is the simulation of evolution equations possessing geometric properties over long periods of time. Of particular importance are Hamiltonian partial differential equations typically arising in application fields such as quantum mechanics or wave propagation phenomena. They exhibit many important dynamical features such as energy preservation and conservation of adiabatic invariants over long periods of time. In this setting, a natural question is how and to which extent the reproduction of such long-time qualitative behavior can be ensured by numerical schemes. Starting from numerical examples, these notes provide a detailed analysis of the Schrodinger equation in a simple setting (periodic boundary conditions, polynomial nonlinearities) approximated by symplectic splitting methods. Analysis of stability and instability phenomena induced by space and time discretization are given, and rigorous mathematical explanations are provided for them. The book grew out of a graduate-level course and is of interest to researchers and students seeking an introduction to the subject matter.
Book Synopsis Topics in Occupation Times and Gaussian Free Fields by : Alain-Sol Sznitman
Download or read book Topics in Occupation Times and Gaussian Free Fields written by Alain-Sol Sznitman and published by European Mathematical Society. This book was released on 2012 with total page 128 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book grew out of a graduate course at ETH Zurich during the spring 2011 term. It explores various links between such notions as occupation times of Markov chains, Gaussian free fields, Poisson point processes of Markovian loops, and random interlacements, which have been the object of intensive research over the last few years. These notions are developed in the convenient setup of finite weighted graphs endowed with killing measures. This book first discusses elements of continuous-time Markov chains, Dirichlet forms, potential theory, together with some consequences for Gaussian free fields. Next, isomorphism theorems and generalized Ray-Knight theorems, which relate occupation times of Markov chains to Gaussian free fields, are presented. Markovian loops are constructed and some of their key properties derived. The field of occupation times of Poisson point processes of Markovian loops is investigated. Of special interest are its connection to the Gaussian free field, and a formula of Symanzik. Finally, links between random interlacements and Markovian loops are discussed, and some further connections with Gaussian free fields are mentioned.
Book Synopsis Lectures on Representation Theory by : Jing-Song Huang
Download or read book Lectures on Representation Theory written by Jing-Song Huang and published by World Scientific. This book was released on 1999 with total page 206 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is an expanded version of the lectures given at the Nankai Mathematical Summer School in 1997. It provides an introduction to Lie groups, Lie algebras and their representations as well as introduces some directions of current research for graduate students who have little specialized knowledge in representation theory. It only assumes that the reader has a good knowledge of linear algebra and some basic knowledge of abstract algebra.Parts I-III of the book cover the relatively elementary material of representation theory of finite groups, simple Lie algebras and compact Lie groups. These theories are natural continuation of linear algebra. The last chapter of Part III includes some recent results on extension of Weyl's construction to exceptional groups. Part IV covers some advanced material on infinite-dimensional representations of non-compact groups such as the orbit method, minimal representations and dual pair correspondences, which introduces some directions of the current research in representation theory.
Book Synopsis Noncommutative Geometry and Representation Theory in Mathematical Physics by : Jürgen Fuchs
Download or read book Noncommutative Geometry and Representation Theory in Mathematical Physics written by Jürgen Fuchs and published by American Mathematical Soc.. This book was released on 2005 with total page 402 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 influentialin 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 fromthe 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 suitablefor graduate students and researchers interested in mathematical physics.
Book Synopsis Rectifiable Sets, Densities and Tangent Measures by : Camillo De Lellis
Download or read book Rectifiable Sets, Densities and Tangent Measures written by Camillo De Lellis and published by European Mathematical Society. This book was released on 2008 with total page 140 pages. Available in PDF, EPUB and Kindle. Book excerpt: The characterization of rectifiable sets through the existence of densities is a pearl of geometric measure theory. The difficult proof, due to Preiss, relies on many beautiful and deep ideas and novel techniques. Some of them have already proven useful in other contexts, whereas others have not yet been exploited. These notes give a simple and short presentation of the former and provide some perspective of the latter. This text emerged from a course on rectifiability given at the University of Zurich. It is addressed both to researchers and students; the only prerequisite is a solid knowledge in standard measure theory. The first four chapters give an introduction to rectifiable sets and measures in Euclidean spaces, covering classical topics such as the area formula, the theorem of Marstrand and the most elementary rectifiability criterions. The fifth chapter is dedicated to a subtle rectifiability criterion due to Marstrand and generalized by Mattila, and the last three focus on Preiss' result. The aim is to provide a self-contained reference for anyone interested in an overview of this fascinating topic.
Book Synopsis Invitation to Topological Robotics by : Michael Farber
Download or read book Invitation to Topological Robotics written by Michael Farber and published by European Mathematical Society. This book was released on 2008 with total page 148 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book discusses several selected topics of a new emerging area of research on the interface between topology and engineering. The first main topic is topology of configuration spaces of mechanical linkages. These manifolds arise in various fields of mathematics and in other sciences, e.g., engineering, statistics, molecular biology. To compute Betti numbers of these configuration spaces the author applies a new technique of Morse theory in the presence of an involution. A significant result of topology of linkages presented in this book is a solution of a conjecture of Kevin Walker which states that the relative sizes of bars of a linkage are determined, up to certain equivalence, by the cohomology algebra of the linkage configuration space. This book also describes a new probabilistic approach to topology of linkages which treats the bar lengths as random variables and studies mathematical expectations of Betti numbers. The second main topic is topology of configuration spaces associated to polyhedra. The author gives an account of a beautiful work of S. R. Gal, suggesting an explicit formula for the generating function encoding Euler characteristics of these spaces. Next the author studies the knot theory of a robot arm, focusing on a recent important result of R. Connelly, E. Demain, and G. Rote. Finally, he investigates topological problems arising in the theory of robot motion planning algorithms and studies the homotopy invariant TC(X) measuring navigational complexity of configuration spaces. This book is intended as an appetizer and will introduce the reader to many fascinating topological problems motivated by engineering.
Book Synopsis Hypergeometry, Integrability and Lie Theory by : Erik Koelink
Download or read book Hypergeometry, Integrability and Lie Theory written by Erik Koelink and published by American Mathematical Soc.. This book was released on 2022-08-30 with total page 362 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains the proceedings of the virtual conference on Hypergeometry, Integrability and Lie Theory, held from December 7–11, 2020, which was dedicated to the 50th birthday of Jasper Stokman. The papers represent recent developments in the areas of representation theory, quantum integrable systems and special functions of hypergeometric type.
Book Synopsis First International Congress of Chinese Mathematicians by : Stephen Shing-Toung Yau
Download or read book First International Congress of Chinese Mathematicians written by Stephen Shing-Toung Yau and published by American Mathematical Soc.. This book was released on 2001 with total page 596 pages. Available in PDF, EPUB and Kindle. Book excerpt: The International Congress of Mathematicians was an historical event that was held at the Morningside Center of Mathematics of the Chinese Academy of Sciences (Beijing). It was the first occasion where Chinese mathematicians from all over the world gathered to present their research. The Morningside Mathematics lectures were given by R. Borcherds, J. Coates, R. Graham, and D. Stroock. Other distinguished speakers included J.-P. Bourguignon, J. Jöst, M. Taylor, and S. L. Lee. Topics covered in the volume include algebra and representation theory, algebraic geometry, number theory and automorphic forms, Riemannian geometry and geometric analysis, mathematical physics, topology, complex analysis and complex geometry, computational mathematics, and combinatorics. Titles in this series are copublished with International Press, Cambridge, MA.
Book Synopsis From Quantum to Classical Molecular Dynamics by : Christian Lubich
Download or read book From Quantum to Classical Molecular Dynamics written by Christian Lubich and published by European Mathematical Society. This book was released on 2008 with total page 164 pages. Available in PDF, EPUB and Kindle. Book excerpt: Quantum dynamics of molecules poses a variety of computational challenges that are presently at the forefront of research efforts in numerical analysis in a number of application areas: high-dimensional partial differential equations, multiple scales, highly oscillatory solutions, and geometric structures such as symplecticity and reversibility that are favourably preserved in discretizations. This text addresses such problems in quantum mechanics from the viewpoint of numerical analysis, illustrating them to a large extent on intermediate models between the Schrodinger equation of full many-body quantum dynamics and the Newtonian equations of classical molecular dynamics. The fruitful interplay between quantum dynamics and numerical analysis is emphasized.
Book Synopsis The Dynamical Yang-Baxter Equation, Representation Theory, and Quantum Integrable Systems by : Pavel I. Etingof
Download or read book The Dynamical Yang-Baxter Equation, Representation Theory, and Quantum Integrable Systems written by Pavel I. Etingof and published by Oxford University Press, USA. This book was released on 2005 with total page 151 pages. Available in PDF, EPUB and Kindle. Book excerpt: The text is based on an established graduate course given at MIT that provides an introduction to the theory of the dynamical Yang-Baxter equation and its applications, which is an important area in representation theory and quantum groups. The book, which contains many detailed proofs and explicit calculations, will be accessible to graduate students of mathematics, who are familiar with the basics of representation theory of semisimple Lie algebras.
Book Synopsis Mathematical Problems of General Relativity I by : Demetrios Christodoulou
Download or read book Mathematical Problems of General Relativity I written by Demetrios Christodoulou and published by European Mathematical Society. This book was released on 2008 with total page 164 pages. Available in PDF, EPUB and Kindle. Book excerpt: General relativity is a theory proposed by Einstein in 1915 as a unified theory of space, time and gravitation. It is based on and extends Newton's theory of gravitation as well as Newton's equations of motion. It is thus fundamentally rooted in classical mechanics. The theory can be seen as a development of Riemannian geometry, itself an extension of Gauss' intrinsic theory of curved surfaces in Euclidean space. The domain of application of the theory is astronomical systems. One of the mathematical methods analyzed and exploited in the present volume is an extension of Noether's fundamental principle connecting symmetries to conserved quantities. This is involved at a most elementary level in the very definition of the notion of hyperbolicity for an Euler-Lagrange system of partial differential equations. Another method, the study and systematic use of foliations by characteristic (null) hypersurfaces, is in the spirit of Roger Penrose's approach in his incompleteness theorem. The methods have applications beyond general relativity to problems in fluid mechanics and, more generally, to the mechanics and electrodynamics of continuous media. The book is intended for advanced students and researchers seeking an introduction to the methods and applications of general relativity.
Book Synopsis Special Functions, KZ Type Equations, and Representation Theory by : Aleksandr Nikolaevich Varchenko
Download or read book Special Functions, KZ Type Equations, and Representation Theory written by Aleksandr Nikolaevich Varchenko and published by American Mathematical Soc.. This book was released on 2003 with total page 130 pages. Available in PDF, EPUB and Kindle. Book excerpt: The last twenty years have seen an active interaction between mathematics and physics. This book is devoted to one of the new areas which deals with mathematical structures related to conformal field theory and its sqs-deformations. In the book, the author discusses the interplay between Knizhnik-Zamolodchikov type equations, the Bethe ansatz method, representation theory, and geometry of multi-dimensional hypergeometric functions. This book aims to provide an introduction to the area and expose different facets of the subject. It contains constructions, discussions of notions, statements of main results, and illustrative examples. The exposition is restricted to the simplest case of the theory associated with the Lie algebra s\mathfrak{sl 2s. This book is intended for researchers and graduate students in mathematics and in mathematical physics, in particular to those interested in applications of special functions.
