Recent Developments in Integrable Systems and Riemann-Hilbert Problems


Book Description

This volume is a collection of papers presented at a special session on integrable systems and Riemann-Hilbert problems. The goal of the meeting was to foster new research by bringing together experts from different areas. Their contributions to the volume provide a useful portrait of the breadth and depth of integrable systems. Topics covered include discrete Painleve equations, integrable nonlinear partial differential equations, random matrix theory, Bose-Einstein condensation, spectral and inverse spectral theory, and last passage percolation models. In most of these articles, the Riemann-Hilbert problem approach plays a central role, which is powerful both analytically and algebraically. The book is intended for graduate students and researchers interested in integrable systems and its applications.




Riemann-Hilbert Problems, Their Numerical Solution, and the Computation of Nonlinear Special Functions


Book Description

Riemann?Hilbert problems are fundamental objects of study within complex analysis. Many problems in differential equations and integrable systems, probability and random matrix theory, and asymptotic analysis can be solved by reformulation as a Riemann?Hilbert problem.This book, the most comprehensive one to date on the applied and computational theory of Riemann?Hilbert problems, includes an introduction to computational complex analysis, an introduction to the applied theory of Riemann?Hilbert problems from an analytical and numerical perspective, and a discussion of applications to integrable systems, differential equations, and special function theory. It also includes six fundamental examples and five more sophisticated examples of the analytical and numerical Riemann?Hilbert method, each of mathematical or physical significance or both.?




Recent Advances in Riemannian and Lorentzian Geometries


Book Description

This volume covers material presented by invited speakers at the AMS special session on Riemannian and Lorentzian geometries held at the annual Joint Mathematics Meetings in Baltimore. Topics covered include classification of curvature-related operators, curvature-homogeneous Einstein 4-manifolds, linear stability/instability singularity and hyperbolic operators of spacetimes, spectral geometry of holomorphic manifolds, cut loci of nilpotent Lie groups, conformal geometry of almost Hermitian manifolds, and also submanifolds of complex and contact spaces. This volume can serve as a good reference source and provide indications for further research. It is suitable for graduate students and research mathematicians interested in differential geometry.




Twelfth Marcel Grossmann Meeting, The: On Recent Developments In Theoretical And Experimental General Relativity, Astrophysics And Relativistic Field Theories (In 3 Volumes) - Proceedings Of The Mg12 Meeting On General Relativity


Book Description

Marcel Grossmann Meetings are formed to further the development of General Relativity by promoting theoretical understanding in the fields of physics, mathematics, astronomy and astrophysics and to direct future technological, observational, and experimental efforts. In these meetings are discussed recent developments in classical and quantum gravity, general relativity and relativistic astrophysics, with major emphasis on mathematical foundations and physical predictions, with the main objective of gathering scientists from diverse backgrounds for deepening the understanding of spacetime structure and reviewing the status of test-experiments for Einstein's theory of gravitation. The range of topics is broad, going from the more abstract classical theory, quantum gravity and strings, to the more concrete relativistic astrophysics observations and modeling.The three volumes of the proceedings of MG12 give a broad view of all aspects of gravitational physics and astrophysics, from mathematical issues to recent observations and experiments. The scientific program of the meeting includes 29 plenary talks stretched over 6 mornings, and 74 parallel sessions over 5 afternoons. Volume A contains plenary and review talks ranging from the mathematical foundations of classical and quantum gravitational theories including recent developments in string theories, to precision tests of general relativity including progress towards the detection of gravitational waves, to relativistic astrophysics including such topics as gamma ray bursts, black hole physics both in our galaxy, in active galactic nuclei and in other galaxies, neutron stars, pulsar astrophysics, gravitational lensing effects, neutrino physics and ultra high energy cosmic rays. The rest of the volumes include parallel sessions on dark matter, neutrinos, X-ray sources, astrophysical black holes, neutron stars, binary systems, radiative transfer, accretion disks, alternative gravitational theories, perturbations of collapsed objects, analog models, black hole thermodynamics, cosmic background radiation & observational cosmology, numerical relativity & algebraic computing, gravitational lensing, variable ';constants'; of nature, large scale structure, topology of the universe, brane-world cosmology, early universe models & cosmic microwave background anisotropies, inhomogeneous cosmology, inflation, gamma ray burst modeling, supernovas, global structure, singularities, cosmic censorship, chaos, Einstein-Maxwell systems, inertial forces, gravitomagnetism, wormholes & time machines, exact solutions of Einstein's equations, gravitational waves, gravitational wave detectors & data analysis, precision gravitational measurements, history of relativity, quantum gravity & loop quantum gravity, Casimir effect, quantum cosmology, strings & branes, self-gravitating systems, gamma ray astronomy, cosmic rays, gamma ray bursts and quasars.




Inverse Problems: Theory and Applications


Book Description

This volume presents the proceedings of a workshop on Inverse Problems and Applications and a special session on Inverse Boundary Problems and Applications. Inverse problems arise in practical situations, such as medical imaging, exploration geophysics, and non-destructive evaluation where measurements made in the exterior of a body are used to deduce properties of the hidden interior. A large class of inverse problems arise from a physical situation modeled by partial differential equations. The inverse problem is to determine some coefficients of the equation given some information about solutions. Analysis of such problems is a fertile area for interaction between pure and applied mathematics. This interplay is well represented in this volume where several theoretical and applied aspects of inverse problems are considered. The book includes articles on a broad range of inverse problems including the inverse conductivity problem, inverse problems for Maxwell's equations, time reversal mirrors, ultrasound using elastic pressure waves, inverse problems arising in the environment, inverse scattering for the three-body problem, and optical tomography. Also included are several articles on unique continuation and on the study of propagation of singularities for hyperbolic equations in anisotropic media. This volume is suitable for graduate students and research mathematicians interested in inverse problems and applications.




Integrable Systems: From Classical to Quantum


Book Description

This volume presents the papers based upon lectures given at the 1999 Séminaire de Mathémathiques Supérieurs held in Montreal. It includes contributions from many of the most active researchers in the field. This subject has been in a remarkably active state of development throughout the past three decades, resulting in new motivation for study in r s3risingly different directions. Beyond the intrinsic interest in the study of integrable models of many-particle systems, spin chains, lattice and field theory models at both the classical and the quantum level, and completely solvable models in statistical mechanics, there have been new applications in relation to a number of other fields of current interest. These fields include theoretical physics and pure mathematics, for example the Seiberg-Witten approach to supersymmetric Yang-Mills theory, the spectral theory of random matrices, topological models of quantum gravity, conformal field theory, mirror symmetry, quantum cohomology, etc. This collection gives a nice cross-section of the current state of the work in the area of integrable systems which is presented by some of the leading active researchers in this field. The scope and quality of the articles in this volume make this a valuable resource for those interested in an up-to-date introduction and an overview of many of the main areas of study in the theory of integral systems.




Current Trends in Scientific Computing


Book Description

This volume contains 36 research papers written by prominent researchers. The papers are based on a large satellite conference on scientific computing held at the International Congress of Mathematics (ICM) in Xi'an, China. Topics covered include a variety of subjects in modern scientific computing and its applications, such as numerical discretization methods, linear solvers, parallel computing, high performance computing, and applications to solid and fluid mechanics, energy, environment, and semiconductors. The book will serve as an excellent reference work for graduate students and researchers working with scientific computing for problems in science and engineering.




Homotopy Theory: Relations with Algebraic Geometry, Group Cohomology, and Algebraic $K$-Theory


Book Description

As part of its series of Emphasis Years in Mathematics, Northwestern University hosted an International Conference on Algebraic Topology. The purpose of the conference was to develop new connections between homotopy theory and other areas of mathematics. This proceedings volume grew out of that event. Topics discussed include algebraic geometry, cohomology of groups, algebraic $K$-theory, and $\mathbb{A 1$ homotopy theory. Among the contributors to the volume were Alejandro Adem,Ralph L. Cohen, Jean-Louis Loday, and many others. The book is suitable for graduate students and research mathematicians interested in homotopy theory and its relationship to other areas of mathematics.




Integrable Systems and Random Matrices


Book Description

This volume contains the proceedings of a conference held at the Courant Institute in 2006 to celebrate the 60th birthday of Percy A. Deift. The program reflected the wide-ranging contributions of Professor Deift to analysis with emphasis on recent developments in Random Matrix Theory and integrable systems. The articles in this volume present a broad view on the state of the art in these fields. Topics on random matrices include the distributions and stochastic processes associated with local eigenvalue statistics, as well as their appearance in combinatorial models such as TASEP, last passage percolation and tilings. The contributions in integrable systems mostly deal with focusing NLS, the Camassa-Holm equation and the Toda lattice. A number of papers are devoted to techniques that are used in both fields. These techniques are related to orthogonal polynomials, operator determinants, special functions, Riemann-Hilbert problems, direct and inverse spectral theory. Of special interest is the article of Percy Deift in which he discusses some open problems of Random Matrix Theory and the theory of integrable systems.




Noncompact Problems at the Intersection of Geometry, Analysis, and Topology


Book Description

This proceedings volume contains articles from the conference held at Rutgers University in honor of Haim Brezis and Felix Browder, two mathematicians who have had a profound impact on partial differential equations, functional analysis, and geometry. Mathematicians attending the conference had interests in noncompact variational problems, pseudo-holomorphic curves, singular and smooth solutions to problems admitting a conformal (or some group) invariance, Sobolev spaces on manifolds, and configuration spaces. One day of the proceedings was devoted to Einstein equations and related topics. Contributors to the volume include, among others, Sun-Yung A. Chang, Luis A. Caffarelli, Carlos E. Kenig, and Gang Tian. The material is suitable for graduate students and researchers interested in problems in analysis and differential equations on noncompact manifolds.