CRM Proceedings & Lecture Notes – serie

This volume presents a cross-section of new developments in algebraic topology. The main portion consists of survey articles suitable for advanced graduate students and professionals pursuing research in this area. A great variety of topics are covered, many of which are of interest to researchers working in other areas of mathematics. In addition, some of the articles cover topics in group theory and homological algebra.

The articles in this collection constitute the proceedings of the Canadian Mathematical Society Annual Seminar on Mathematical Quantum Theory, held in Vancouver in August 1993. The meeting was run as a research-level summer school concentrating on two related areas of contemporary mathematical physics. The first area, quantum field theory and many-body theory, is covered in volume 1 of these proceedings. The second area, treated in the present volume, is Schrodinger operators. The meeting featured a series of four-hour mini-courses, designed to introduce students to the state of the art in particular areas, and thirty hour-long expository lectures. With contributions from some of the top experts in the field, this book is an important resource for those interested in activity at the frontiers of mathematical quantum theory.

There is currently considerable mathematical interest and very real potential for applications in using geometry in the design, identification and control of technological processes. Geometry plays the role of a design variable in the shape optimization of mechanical parts. It also appears as a control variable in optimal swimming, shape control of aircraft wings or stabilization of membranes and plates by periodic variations of the boundary. As it is used as a design or control variable, it often undergoes 'mutations' as in the microstructures of materials, crystal growth, image processing or the texture of objects which involve relaxations of classical geometry and geometrical entities. In other areas, such as free and moving boundary problems, the understanding of the underlying phenomena is very much related to the geometric properties of the fronts and the nature of the nonlinearities involved.This book brings together tools that have been developed in a priori distant areas of mathematics, mechanics and physics. It provides coverage of selected contemporary problems in the areas of optimal design, mathematical models in material sciences, hysteresis, superconductivity, phase transition, crystal growth, moving boundary problems, thin shells and some of the associated numerical issues.

This book is devoted to a topic that has undergone rapid and fruitful development over the last few years: symmetries and integrability of difference equations and $q$-difference equations and the theory of special functions that occur as solutions of such equations. Techniques that have been traditionally applied to solve linear and nonlinear differential equations are now being successfully adapted and applied to discrete equations.This volume is based on contributions made by leading experts in the field during the workshop on Symmetries and Integrability of Difference Equations held in Esterel, Quebec, in May 1994. Giving an up-to-date review of the current status of the field, the book treats these specific topics: Lie group and quantum group symmetries of difference and $q$-difference equations, integrable and nonintegrable discretizations of continuous integrable systems, integrability of difference equations, discrete Painleve property and singularity confinement, integrable mappings, applications in statistical mechanics and field theories, Yang-Baxter equations, $q$-special functions and discrete polynomials, and $q$-difference integrable systems.

'This is a very stimulating book!' - N. G. de Bruijn. 'This short book will provide extremely enjoyable reading to anyone with an interest in discrete mathematics and algorithm design' - ""Mathematical Reviews"". 'This book is an excellent (and enjoyable) means of sketching a large area of computer science for specialists in other fields: It requires little previous knowledge, but expects of the reader a degree of mathematical facility and a willingness to participate. It is really neither a survey nor an introduction; rather, it is a paradigm, a fairly complete treatment of a single example used as a synopsis of a larger subject' - ""SIGACT News"". 'Anyone would enjoy reading this book. If one had to learn French first, it would be worth the effort!' - ""Computing Reviews"". The above citations are taken from reviews of the initial French version of this text - a series of seven expository lectures that were given at the University of Montreal in November of 1975.The book uses the appealing theory of stable marriage to introduce and illustrate a variety of important concepts and techniques of computer science and mathematics: data structures, control structures, combinatorics, probability, analysis, algebra, and especially the analysis of algorithms. The presentation is elementary, and the topics are interesting to nonspecialists. The theory is quite beautiful and developing rapidly. Exercises with answers, an annotated bibliography, and research problems are included.The text would be appropriate as supplementary reading for undergraduate research seminars or courses in algorithmic analysis and for graduate courses in combinatorial algorithms, operations research, economics, or analysis of algorithms. Donald E. Knuth is one of the most prominent figures of modern computer science. His works in ""The Art of Computer Programming"" are classic. He is also renowned for his development of TeX and METAFONT. In 1996, Knuth won the prestigious Kyoto Prize, considered to be the nearest equivalent to a Nobel Prize in computer science.

This volume commemorates the 25th anniversary of the Centre de Recherches Mathematiques (CRM) - a national institute for research in the mathematical sciences in Canada. It includes contributions by eminent scientists who have been closely involved with the CRM. Various topics in pure and applied mathematics, statistics, theoretical physics, and mathematical biology are covered. Original research papers and reviews, as well as historical notes and reminiscences, are included in the volume.

This volume presents lectures given at the 1995 Annual Seminar of the Canadian Mathematical Society on Partial Differential Equations and Their Applications held at the University of Toronto in June 1995. The conference consisted of a combination of minicourses, invited presentations, and contributed talks. In this volume readers will find contributions on a variety of topics related to PDE, such as spectral asymptotics, harmonic analysis, differential operators in hyperbolic manifolds, applications to geometry, mathematical physics, hydrodynamics, and the interaction between theory and numerical methods in PDE.

This work is based on a series of thematic workshops on the theory of wavelets and the theory of splines. Important applications are included. The volume is divided into four parts: Spline Functions, Theory of Wavelets, Wavelets in Physics, and Splines and Wavelets in Statistics. Part one presents the broad spectrum of current research in the theory and applications of spline functions. Theory ranges from classical univariate spline approximation to an abstract framework for multivariate spline interpolation. Applications include scattered-data interpolation, differential equations and various techniques in CAGD. Part two considers two developments in subdivision schemes; one for uniform regularity and the other for irregular situations. The latter includes construction of multidimensional wavelet bases and determination of bases with a given time frequency localization. In part three, the multifractal formalism is extended to fractal functions involving oscillating singularities.There is a review of a method of quantization of classical systems based on the theory of coherent states. Wavelets are applied in the domains of atomic, molecular and condensed-matter physics. In part four, ways in which wavelets can be used to solve important function estimation problems in statistics are shown. Different wavelet estimators are proposed in the following distinct cases: functions with discontinuities, errors that are no longer Gaussian, wavelet estimation with robustness, and error distribution that is no longer stationary. Some of the contributions in this volume are current research results not previously available in monograph form. The volume features many applications and interesting new theoretical developments. Readers will find powerful methods for studying irregularities in mathematics, physics, and statistics.

The lectures collected for this volume were given during a workshop entitled, 'Semi-analytic Methods for the Navier Stokes Equations' held at the CRM in Montreal. The title reflects the current reality in fluid dynamics: Navier-Stokes equations (NSE) describe the behavior of fluid in a wide range of physical situations, the solutions of these equations are sufficiently complicated, so that another level of analysis is clearly needed. The fundamental problem is not just to solve the NSE, but also to understand what the solutions mean. One of the goals of the workshop was to bring together people who, while working in different fields, share a common perspective on the nature of the problem to be solved. The lectures present a diverse set of techniques for modelling, computing, and understanding phenomena such as instabilities, turbulence and spatiotemporal chaos in fluids.

The third annual CRM Summer School took place in Banff (Alberta, Canada) and was aimed toward advanced students and recent PhDs. This volume presents surveys from the group theory part of the theme year and examines different approaches to the topic: a geometric approach, an approach using methods from logic, and an approach with roots in the Bass-Serre theory of groups acting on trees. The work offers a concise introduction to current directions of research in combinatorial group theory. Surveys in the text are by leading researchers in the field who are experienced expositors. The text is suitable for use in a graduate course in geometric and combinatorial group theory.

Although originally posed in the context of mathematical problems related to medical imaging, the bispectral problem is now closely related to other topics and has connections to many areas of pure and applied mathematics. The central theme of this book is the search for solutions to eigenvalue problems that satisfy additional equations in the spectral parameter, for example, pairs of eigenvalue equations. This problem, which looks very simple at first, has turned out to be both deep and difficult. Moreover, this concept of bispectrality has been shown to be useful in many active areas of current research in mathematics and physics.Following several years of exciting new results on the subject, in March 1997 the Centre de Recherches Mathematiques held the first scientific meeting devoted exclusively to the bispectral problem. Collected in this volume are contributions from the speakers at this meeting. The participants at this workshop included a majority of those researchers who have made significant contributions to the subject and many others working on related problems.

This volume features the proceedings from the Summer Seminar of the Canadian Mathematical Society held at Universite Laval. The purpose of the seminar was to gather both mathematicians and engineers interested in the theory or application of plates and shells, or more generally, in the modelisation of thin structures. From this, it was hoped that a better understanding of the problem would emerge for both groups of professionals. New aspects from the mathematical point of view and new applications posing new challenges are reported. This volume offers a snapshot of the state of the art of this rapidly evolving topic.

This book contains papers presented at the fifth Canadian Number Theory Association (CNTA) conference held at Carleton University (Ottawa, ON). The invited speakers focused on arithmetic algebraic geometry and elliptic curves, diophantine problems, analytic number theory, and algebraic and computational number theory. The contributed talks represented a wide variety of areas in number theory. David Boyd gave an hour-long talk on ""Mahler's Measure and Elliptic Curves"". This lecture was open to the public and attracted a large audience from outside the conference.

The NATO ASI/CRM Summer School at Banff offered a unique, full, and in-depth account of the topic, ranging from introductory courses by leading experts to discussions of the latest developments by all participants. The papers have been organized into three categories: cohomological methods; Chow groups and motives; and arithmetic methods. As a subfield of algebraic geometry, the theory of algebraic cycles has gone through various interactions with algebraic $K$-theory, Hodge theory, arithmetic algebraic geometry, number theory, and topology.These interactions have led to developments such as a description of Chow groups in terms of algebraic $K$-theory; the application of the Merkurjev-Suslin theorem to the arithmetic Abel-Jacobi mapping; progress on the celebrated conjectures of Hodge, and of Tate, which compute cycles class groups respectively in terms of Hodge theory or as the invariants of a Galois group action on etale cohomology; and, the conjectures of Bloch and Beilinson, which explain the zero or pole of the $L$-function of a variety and interpret the leading non-zero coefficient of its Taylor expansion at a critical point, in terms of arithmetic and geometric invariant of the variety and its cycle class groups. The immense recent progress in the theory of algebraic cycles is based on its many interactions with several other areas of mathematics. This conference was the first to focus on both arithmetic and geometric aspects of algebraic cycles. It brought together leading experts to speak from their various points of view. A unique opportunity was created to explore and view the depth and the breadth of the subject. This volume presents the intriguing results.

This volume presents the papers based upon lectures given at the 1999 Seminaire de Mathematiques Superieurs 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 surprisingly 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.

This book contains the proceedings from the workshop, Nonlinear Dynamics and Renormalization Group, held at the Centre de recherches mathematiques (CRM) in Montreal (Canada), as part of the year-long program devoted to mathematical physics. In the book, active researchers in the fields of nonlinear partial differential equations and renormalization group contribute recent results on topics such as Ginzburg-Landau equations and blow-up of solutions of the nonlinear Schroedinger equations, quantum resonances, and renormalization group analysis in constructive quantum field theory. This volume offers the latest research in the rapidly developing fields of nonlinear equations and renormalization group.

This book is devoted to a classical topic that has undergone rapid and fruitful development over the past 25 years, namely Backlund and Darboux transformations and their applications in the theory of integrable systems, also known as soliton theory. The book consists of two parts. The first is a series of introductory pedagogical lectures presented by leading experts in the field. They are devoted respectively to Backlund transformations of Painleve equations, to the dressing method and Backlund and Darboux transformations, and to the classical geometry of Backlund transformations and their applications to soliton theory.The second part contains original contributions that represent new developments in the theory and applications of these transformations. Both the introductory lectures and the original talks were presented at an International Workshop that took place in Halifax, Nova Scotia (Canada). This volume covers virtually all recent developments in the theory and applications of Backlund and Darboux transformations.

The area of inverse scattering transform method or soliton theory has evolved over the past two decades in a vast variety of exciting new algebraic and analytic directions and has found numerous new applications. Methods and applications range from quantum group theory and exactly solvable statistical models to random matrices, random permutations, and number theory. The theory of isomonodromic deformations of systems of differential equations with rational coefficents, and most notably, the related apparatus of the Riemann-Hilbert problem, underlie the analytic side of this striking development.The contributions in this volume are based on lectures given by leading experts at the CRM workshop (Montreal, Canada). Included are both survey articles and more detailed expositions relating to the theory of isomonodromic deformations, the Riemann-Hilbert problem, and modern applications. The first part of the book represents the mathematical aspects of isomonodromic deformations; the second part deals mostly with the various appearances of isomonodromic deformations and Riemann-Hilbert methods in the theory of exactly solvable quantum field theory and statistical mechanical models, and related issues. The book elucidates for the first time in the current literature the important role that isomonodromic deformations play in the theory of integrable systems and their applications to physics.

This volume contains the proceedings of the Moonshine workshop held at the Centre de Recherches Mathematiques (CRM) in Montreal. A glance at the contents will reveal that the connection of some papers to Moonshine is not immediate; however, Moonshine has proved to be a very fertile area, and it does not stretch the imagination to believe that many more threads will be drawn together before we understand what is really going on. In this volume, all the classical Moonshine themes are presented, namely the Monster simple group and other finite groups, automorphic functions and forms and related congruence groups, and vertex algebras and their representations. These topics appear in either a pure form or in a blend of algebraic geometry dealing with algebraic surfaces, Picard-Fuchs equations, and hypergeometric functions.

This volume includes the proceedings of a workshop on Invariant Theory held at Queen's University (Ontario). The workshop was part of the theme year held under the auspices of the Centre de recherches mathematiques (CRM) in Montreal. The gathering brought together two communities of researchers: those working in characteristic 0 and those working in positive characteristic. The book contains three types of papers: survey articles providing introductions to computational invariant theory, modular invariant theory of finite groups, and the invariant theory of Lie groups; expository works recounting recent research in these three areas and beyond; and open problems of current interest. The book is suitable for graduate students and researchers working in invariant theory.

An entirely new branch of science now known as Laser Control of Molecular Processes is steadily making an impact on the experimental and technological worlds, with internationally distinguished scientists making many outstanding contributions. In parallel, mathematicians from control theory and numerical simulation are getting progressively involved and making their contributions to this scientific endeavor. This volume presents the proceedings of the workshop, Quantum Control: Mathematical and Numerical Challenges, held at the Centre de recherches mathematiques of the Universite de Montreal (CRM).The workshop concentrated on advanced numerical methods and new mathematical control and optimization approaches and tools for the quantum control of matter at the molecular level using current laser technology. It brought together mathematicians, theoretical chemists, and physicists working in the area of control and optimization of systems to address the outstanding numerical and mathematical problems. The volume is suitable for graduate students and research mathematicians interested in mathematical methods of control of molecular processes. It will also be useful to chemical engineers and chemists working in control and optimization of systems.

This volume contains a collection of articles from the meeting of the Canadian Number Theory Association held at the Centre de Recherches Mathematiques (CRM) at the University of Montreal. The book represents a cross section of current research and new results in number theory. Topics covered in this book include algebraic number theory, analytic number theory, arithmetic algebraic geometry, computational number theory, and Diophantine analysis and approximation. The volume contains both research and expository papers suitable for graduate students and researchers interested in number theory.

Papers in this volume are based on the Workshop on Symmetries in Physics held at the Center de recherches mathematiques (University of Montreal) in memory of Robert T. Sharp. Contributed articles are on a variety of topics revolving around the theme of symmetry in physics. The preface presents a biographical and scientific retrospect of the life and work of Robert Sharp. Other articles in the volume represent his diverse range of interests, including representation theoretic methods for Lie algebras, quantization techniques and foundational considerations, modular group invariants and applications to conformal models, various physical models and equations, geometric calculations with symmetries, and pedagogical methods for developing spatio-temporal intuition. The book is suitable for graduate students and researchers interested in group theoretic methods, symmetries, and mathematical physics.

The Workshop on Group Theory and Numerical Analysis brought together scientists working in several different but related areas. The unifying theme was the application of group theory and geometrical methods to the solution of differential and difference equations. The emphasis was on the combination of analytical and numerical methods and also the use of symbolic computation. This meeting was organized under the auspices of the Centre de Recherches Mathematiques, Universite de Montreal (Canada). This volume has the character of a monograph and should represent a useful reference book for scientists working in this highly topical field.

This book contains recent and exciting developments on the structure of moduli spaces, with an emphasis on the algebraic structures that underlie this structure. Topics covered in this title include Hilbert schemes of points, moduli of instantons, coherent sheaves and their derived categories, moduli of flat connections, Hodge structures, and the topology of affine varieties. Two beautiful series of lectures are a particularly fine feature of the book. One is an introductory series by Manfred Lehn on the topology and geometry of Hilbert schemes of points on surfaces, and the other, by Hiraku Nakajima and Kota Yoshioka, explains their recent work on the moduli space of instantons over ${\mathbb R}^4$. The material is suitable for graduate students and researchers interested in moduli spaces in algebraic geometry, topology, and mathematical physics.

Special geometries as well as the relation between curvature and topology have always been of interest to differential geometers. More recently, these topics have turned out to be of use in physical problems related to string theory as well. This volume provides a unique and thorough survey on the latest developments on Riemannian geometry, special geometrical structures on manifolds, and their interactions with other fields such as mathematical physics, complex analysis, and algebraic geometry. This volume presents ten papers written by participants of the 'Short Program on Riemannian Geometry', a workshop held at the CRM in Montreal in 2004. It will be a valuable reference for graduate students and research mathematicians alike.

High-dimensional spatio-temporal partial differential equations are a major challenge to scientific computing of the future. Up to now deemed prohibitive, they have recently become manageable by combining recent developments in numerical techniques, appropriate computer implementations, and the use of computers with parallel and even massively parallel architectures. This opens new perspectives in many fields of applications. Kinetic plasma physics equations, the many body Schrodinger equation, Dirac and Maxwell equations for molecular electronic structures and nuclear dynamic computations, options pricing equations in mathematical finance, as well as Fokker-Planck and fluid dynamics equations for complex fluids, are examples of equations that can now be handled. The objective of this volume is to bring together contributions by experts of international stature in that broad spectrum of areas to confront their approaches and possibly bring out common problem formulations and research directions in the numerical solutions of high-dimensional partial differential equations in various fields of science and engineering with special emphasis on chemistry and physics.Information for our distributors: Titles in this series are co-published with the Centre de Recherches Mathematiques.

This volume is based on talks given at a conference celebrating Stanislav Molchanov's 65th birthday held in June of 2005 at the Centre de Recherches Mathematiques (Montreal, QC, Canada). The meeting brought together researchers working in an exceptionally wide range of topics reflecting the quality and breadth of Molchanov's past and present research accomplishments. This collection of survey and research papers gives a glance of the profound consequences of Molchanov's contributions in stochastic differential equations, spectral theory for deterministic and random operators, localization and intermittency, mathematical physics and optics, and other topics. Information for our distributors: Titles in this series are co-published with the Centre de Recherches Mathematiques.