Fields Institute Communications – serie

This volume presents new research in classical Hamiltonian and quantum systems from the Workshop on Conservative Systems and Quantum Chaos held during The Fields Institute Program Year on Dynamical Systems and Bifurcation Theory in October 1992 (Waterloo, Canada). The workshop was organized so that there were presentations that formed a bridge between classical and quantum mechanical systems. Four of these papers appear in this collection, with the remaining six papers concentrating on classical Hamiltonian dynamics. It includes applications ranging from numerical exploration of a coupled mechanical system to a Hamiltonian approach to the problem of spatio-temporal chaos in biological models from population genetics and ecology.

This volume brings together ideas from several areas of mathematics that have traditionally been rather disparate. The conference at The Fields Institute which gave rise to these proceedings was intended to encourage such connections. One of the key interactions occurs between dynamical systems and algorithms, one example being the by now classic observation that the QR algorithm for diagonalizing matrices may be viewed as the time-1 map of the Toda lattice flow.Another link occurs with interior point methods for linear programming, where certain smooth flows associated with such programming problems have proved valuable in the analysis of the corresponding discrete problems. More recently, other smooth flows have been introduced which carry out discrete computations (such as sorting sets of numbers) and which solve certain least squares problems. Another interesting facet of the flows described here is that they often have a dual Hamiltonian and gradient structure, both of which turn out to be useful in analyzing and designing algorithms for solving optimization problems. This volume explores many of these interactions, as well as related work in optimal control and partial differential equations.

This book is the Proceedings of The Fields Institute Conference/NATO Advanced Research Workshop held in June 1993. The articles review the diverse recent progress in the theory and development of lattice-gas and lattice Boltzmann methods and their applications to hydrodynamics, multi-phase flows, flows through porous media, reaction-diffusion systems, pattern formation phenomena, and phase separation processes. Discussed here are various aspects of the statistical analysis of these methods, with emphasis on fluctuations and correlations, as well as computational prospects including development of dedicated hardware. Features include up-to-date articles covering theory and applications interdisciplinary approach, which includes mathematics, physics, chemistry, and geophysics abstracts of papers published from 1992 through 1995.

Dedicated to the late Juan Carlos Simo, this volume contains the proceedings of a workshop held at the Fields Institute in October 1993. The articles focus on current algorithms for the integration of mechanical systems, from systems in celestial mechanics to coupled rigid bodies to fluid mechanics. The scope of the articles ranges from symplectic integration methods to energy-momentum methods and related themes.

This volume presents new research on normal forms, symmetry, homoclinic cycles, and chaos, from the Workshop on Normal Forms and Homoclinic Chaos held during The Fields Institute Program Year on Dynamical Systems and Bifurcation Theory in November 1992, in Waterloo, Canada. The workshop bridged the local and global analysis of dynamical systems with emphasis on normal forms and the recently discovered homoclinic cycles which may arise in normal forms.Specific topics covered in this volume include...normal forms for dissipative, conservative, and reversible vector fields, and for symplectic maps; the effects of symmetry on normal forms; the persistence of homoclinic cycles; symmetry-breaking, both spontaneous and induced; mode interactions; resonances; intermittency; numerical computation of orbits in phase space; applications to flow-induced vibrations and to mechanical and structural systems; general methods for calculation of normal forms; chaotic dynamics arising from normal forms. Of the 32 presentations given at this workshop, 14 of them are represented by papers in this volume.

This volume is the refereed proceedings of the Sixth Canadian Conference on General Relativity and Relativistic Astrophysics held in May 1995 at the University of New Brunswick. The book includes invited talks and contributed talks and posters including state-of-the-art reviews of many of the most recent important developments in gravitational physics. This book would serve as a good supplement to standard texts on the topic. It features: review articles in key areas - black holes, numerical relativity, etc.; contributions covering most of gravitational physics; useful articles for students who wish to begin exploring the issues discussed; and, invited talks given by researchers known for their ability to communicate their expertise.

This book contains contributions from the proceedings at The Fields Institute workshop on Special Functions, $q$-Series and Related Topics that was held in June 1995. The articles cover areas from quantum groups and their representations, multivariate special functions, $q$-series, and symbolic algebra techniques as well as the traditional areas of single-variable special functions. The book contains both pure and applied topics and reflects recent trends of research in the various areas of special functions.

Since the publication of ""Theory of Games and Economic Behavior"" by von Neumann and Morgenstern, the concept of games has played an increasing role in economics. It also plays a role of growing importance in other sciences, including biology, political science, and psychology. Many scientists have made seminal advances and continue to be leaders in the field, including Harsanyi, Shapley, Shubik, and Selten. Professor Robert Aumann, in addition to his important contributions to game theory and economics, made a number of significant contributions to mathematics.This volume provides a collection of essays in mathematical economics and game theory, including cutting-edge research on noncooperative game theory and its foundations, bargaining theory, and general equilibrium theory. Also included is a reprint of Aumann's classic paper, 'Acceptable Points in General Cooperative $n$-Person Games' and of the oft-cited, yet hard to find, paper by Maschler, 'The Worth of a Cooperative Enterprise to Each Member'. This book illustrates the wide range of applications of mathematics to economics, game theory, and social choice. The volume is dedicated to Professor Robert J. Aumann, Hebrew University, Jerusalem, Israel, for his contributions in mathematics and social sciences.

This conference proceedings volume is produced in connection with the second Great Lakes K-theory Conference that was held at The Fields Institute for Research in Mathematical Sciences in March 1996. The volume is dedicated to the late Bob Thomason, one of the leading research mathematicians specializing in algebraic K-theory. In addition to research papers treated directly in the lectures at the conference, this volume contains the following: several timely articles inspired by those lectures (particularly by that of V. Voevodsky), an extensive exposition by Steve Mitchell of Thomason's famous result concerning the relationship between algebraic K-theory and etale cohomology, and a definitive exposition by J. L. Colliot-Thelene, R. Hoobler, and B. Kahn (explaining and elaborating upon unpublished work of O. Gabber) of Bloch-Ogus-Gersten type resolutions in K-theory and algebraic geometry.This volume will be important both for researchers who want access to details of recent development in K-theory and also to graduate students and researchers seeking good advanced exposition. It features: invaluable access to new, previously unavailable research results; an ideal reference for some important results in K-theory; techniques in algebraic geometry; material for advanced seminars for graduate students; and a number of papers connecting number theory and K-theory in a way that complements ""The Fields Institute Monograph Series"" from the 1993-1994 emphasis year on $L$-functions.

This volume presents the proceedings of workshops on stable homotopy theory and on unstable homotopy theory held at The Fields Institute as part of the homotopy program during the year 1996. The papers in the volume describe current research in the subject, and all included works were refereed. Rather than being a summary of work to be published elsewhere, each paper is the unique source for the new material it contains. The book contains current research from international experts in the subject area, and presents open problems with directions for future research.

This book presents the proceedings from the International Conference held in Halifax, NS in July 1997. Funded by The Fields Institute and Le Centre de Recherches Mathematiques, the conference was held in honor of the retirement of Professors Lynn Erbe and Herb I. Freedman (University of Alberta). Featured topics include ordinary, partial, functional, and stochastic differential equations and their applications to biology, epidemiology, neurobiology, physiology and other related areas. The 41 papers included in this volume represent the recent work of leading researchers over a wide range of subjects, including bifurcation theory, chaos, stability theory, boundary value problems, persistence theory, neural networks, disease transmission, population dynamics, pattern formation and more.The text would be suitable for a graduate or advanced undergraduate course study in mathematical biology. It's features include: an overview of current developments in differential equations and mathematical biology; authoritative contributions from over 60 leading worldwide researchers; and, original, refereed contributions.

This volume presents the proceedings of a workshop on geometry, topology, and markets held at The Fields Institute. The workshop was attended by eminent mathematicians and financial and economic theorists. Using a topological approach, the volume discusses new mathematics and its applications to social sciences and financial markets. Topics addressed at the workshop included new topological invariants for existence, characterization and computation of market equilibria and their relation to social choice and to other forms of resource allocation, competitive and cooperative systems, algebraic geometry and markets with increasing returns, computational complexity, and stochastic processes and financial markets.

This volume contains papers written by participants at the Conference on Functional Differential and Difference Equations held at the Instituto Superior Tecnico in Lisbon, Portugal. The conference brought together mathematicians working in a wide range of topics, including qualitative properties of solutions, bifurcation and stability theory, oscillatory behavior, control theory and feedback systems, biological models, state-dependent delay equations, Lyapunov methods, etc. The articles are written by leading experts in the field. A comprehensive overview is given of these active areas of current research. The book will be of interest to both theoretical and applied mathematical scientists.

The beauty and the mystery surrounding the interplay between mathematics and physics is captured by E. Wigner's famous expression, ""The unreasonable effectiveness of mathematics"". We don't know why, but physical laws are described by mathematics, and good mathematics sooner or later finds applications in physics, often in a surprising way. In this sense, mathematical physics is a very old subject - as Egyptian, Phoenician, or Greek history tells us. But mathematical physics is a very modern subject, as any working mathematician or physicist can witness. It is a challenging discipline that has to provide results of interest for both mathematics and physics.Ideas and motivations from both these sciences give it a vitality and freshness that is difficult to find anywhere else. One of the big physical revolutions in the twentieth century, quantum physics, opened a new magnificent era for this interplay. With the appearance of noncommutative analysis, the role of classical calculus has been taken by commutation relations, a subject still growing in an astonishing way. A good example where mathematical physics showed its power, beauty, and interdisciplinary character is the Doplicher-Haag-Roberts analysis of superselection sectors in the late 1960s.Not only did this theory explain the origin of statistics and classify it, but year after year, new connections have merged, for example with Tomita-Takesaki modular theory, Jones theory of subfactors, and Doplicher-Roberts abstract duality for compact groups. This volume contains the proceedings of the conference, ""Mathematical Physics in Mathematics and Physics"", dedicated to Sergio Doplicher and John E. Roberts held in Siena, Italy. The articles offer current research in various fields of mathematical physics, primarily concerning quantum aspects of operator algebras.

This book is the second of two proceedings volumes stemming from the International Conference and Workshop on Valuation Theory held at the University of Saskatchewan (Saskatoon, SK, Canada). It contains the most recent applications of valuation theory to a broad range of mathematical ideas. Valuation theory arose in the early part of the twentieth century in connection with number theory and continues to have many important applications to algebra, geometry, and analysis.The research and survey papers in this volume cover a variety of topics, including Galois theory, the Grunwald-Wang Theorem, algebraic geometry, resolution of singularities, curves over Prufer domains, model theory of valued fields and the Frobenius, Hardy fields, Hensel's Lemma, fixed point theorems, and computations in valued fields. It is suitable for graduate students and research mathematicians interested in algebra, algebraic geometry, number theory, and mathematical logic. Also available from the AMS is ""Valuation Theory and Its Applications"", Volume I.

During the last decade, many novel approaches have been considered for dealing with computationally difficult discrete optimization problems. Such approaches include interior point methods, semidefinite programming techniques, and global optimization. More efficient computational algorithms have been developed and larger problem instances of hard discrete problems have been solved. This progress is due in part to these novel approaches, but also to new computing facilities and massive parallelism. This volume contains the papers presented at the workshop on 'Novel Approaches to Hard Discrete Optimization'. The articles cover a spectrum of issues regarding computationally hard discrete problems.

This volume is based on talks given at the Workshop on Categorical Structures for Descent and Galois Theory, Hopf Algebras, and Semiabelian Categories held at The Fields Institute for Research in Mathematical Sciences (Toronto, ON, Canada). The meeting brought together researchers working in these interrelated areas. This collection of survey and research papers gives an up-to-date account of the many current connections among Galois theories, Hopf algebras, and semiabelian categories. The book features articles by leading researchers on a wide range of themes, specifically, abstract Galois theory, Hopf algebras, and categorical structures, in particular quantum categories and higher-dimensional structures. Articles are suitable for graduate students and researchers, specifically those interested in Galois theory and Hopf algebras and their categorical unification.

This volume contains papers from the 7th International Conference on Difference Equations held at Hunan University (Changsa, China), a satellite conference of ICM2002 Beijing. The volume captures the spirit of the meeting and includes peer-reviewed survey papers, research papers, and open problems and conjectures. Articles of this title cover stability, oscillation, chaos, symmetries, boundary value problems and bifurcations for discrete dynamical systems, difference-differential equations, and discretization of continuous systems. The book presents state-of-the-art research in these important areas. It is suitable for graduate students and researchers in difference equations and related topics.

Resulting from the Tenth International Conference on Representations of Algebras and Related Topics held at The Fields Institute (Toronto, ON, Canada), this collection of research and survey articles, honoring Vlastimil Dlab's seventieth birthday, reflects state-of-the-art research on the topic. Leading experts contributed papers, demonstrating the interaction between representation theory of finite dimensional algebras and neighboring subjects. A wide range of topics are covered, including quantum groups, the theory of Lie algebras, the geometry and combinatorics of tilting theory, commutative algebra, algebraic geometry, homology theories, and derived and triangulated categories. The book is suitable for graduate students and researchers interested in the theory of algebras.

This book covers a variety of topics related to kinetic theory in neutral gases and magnetized plasmas, with extensions to other systems such as quantum plasmas and granular flows. A comprehensive presentation is given for the Boltzmann equations and other kinetic equations for a neutral gas, together with the derivations of compressible and incompressible fluid dynamical systems, and their rigorous justification. Several contributions are devoted to collisionless magnetized plasmas. Rigorous results concerning the well-posedness of the Vlasov-Maxwell system are presented. Special interest is devoted to asymptotic regimes where the scales of variation of the electromagnetic field are clearly separated from those associated with the gyromotion of the particles. This volume collects lectures given at the Short Course and Workshop on Kinetic Theory organized at the Fields Institute of Mathematical Sciences in Toronto during the Spring of 2004.

This book is a collection of research and expository papers reflecting the interfacing of two fields: nonlinear dynamics (in the physiological and biological sciences) and statistics. It presents the proceedings of a four-day workshop entitled ""Nonlinear Dynamics and Time Series: Building a Bridge Between the Natural and Statistical Sciences"" held at the Centre de Recherches Mathematiques (CRM) in Montreal in July 1995. The goal of the workshop was to provide an exchange forum and to create a link between two diverse groups with a common interest in the analysis of nonlinear time series data. The editors and peer reviewers of this work have attempted to minimize the problems of maintaining communication between the different scientific fields. The result is a collection of interrelated papers that highlight current areas of research in statistics that might have particular applicability to nonlinear dynamics and new methodology and open data analysis problems in nonlinear dynamics that might find their way into the toolkits and research interests of statisticians.Features: A survey of state-of-the-art developments in nonlinear dynamics time series analysis with open statistical problems and areas for further research. Contributions by statisticians to understanding and improving modern techniques commonly associated with nonlinear time series analysis, such as surrogate data methods and estimation of local Lyapunov exponents. Starting point for both scientists and statisticians who want to explore the field. Expositions that are readable to scientists outside the featured fields of specialization. Information for our distributors: Titles in this series are co-published with the Fields Institute for Research in Mathematical Sciences (Toronto, Ontario, Canada).

This book covers a wide range of phenomena in the natural sciences dominated by notions of universality and renormalization. The contributions in this volume are equally broad in their approach to these phenomena, offering the mathematical as well as the perspective of the applied sciences. They explore renormalization theory in quantum field theory and statistical physics, and its connections to modern mathematics as well as physics on scales from the microscopic to the macroscopic. Information for our distributors: Titles in this series are co-published with the Fields Institute for Research in Mathematical Sciences (Toronto, Ontario, Canada).

It is by now a well-known paradigm that public-key cryptosystems can be built using finite Abelian groups and that algebraic geometry provides a supply of such groups through Abelian varieties over finite fields. Of special interest are the Abelian varieties that are Jacobians of algebraic curves. All of the articles in this volume are centered on the theme of point counting and explicit arithmetic on the Jacobians of curves over finite fields. The topics covered include Schoof's $\ell$-adic point counting algorithm, the $p$-adic algorithms of Kedlaya and Denef-Vercauteren, explicit arithmetic on the Jacobians of $C_{ab}$ curves and zeta functions. This volume is based on seminars on algebraic curves and cryptography held at the GANITA Lab of the University of Toronto during 2001-2008. The articles are mostly suitable for independent study by graduate students who wish to enter the field, both in terms of introducing basic material as well as guiding them in the literature. The literature in cryptography seems to be growing at an exponential rate. For a new entrant into the subject, navigating through this ocean can seem quite daunting. In this volume, the reader is steered toward a discussion of a few key ideas of the subject, together with some brief guidance for further reading. It is hoped that this approach may render the subject more approachable. Titles in this series are co-published with the Fields Institute for Research in Mathematical Sciences (Toronto, Ontario, Canada).|It is by now a well-known paradigm that public-key cryptosystems can be built using finite Abelian groups and that algebraic geometry provides a supply of such groups through Abelian varieties over finite fields. Of special interest are the Abelian varieties that are Jacobians of algebraic curves. All of the articles in this volume are centered on the theme of point counting and explicit arithmetic on the Jacobians of curves over finite fields. The topics covered include Schoof's $\ell$-adic point counting algorithm, the $p$-adic algorithms of Kedlaya and Denef-Vercauteren, explicit arithmetic on the Jacobians of $C_{ab}$ curves and zeta functions. This volume is based on seminars on algebraic curves and cryptography held at the GANITA Lab of the University of Toronto during 2001-2008. The articles are mostly suitable for independent study by graduate students who wish to enter the field, both in terms of introducing basic material as well as guiding them in the literature. The literature in cryptography seems to be growing at an exponential rate. For a new entrant into the subject, navigating through this ocean can seem quite daunting. In this volume, the reader is steered toward a discussion of a few key ideas of the subject, together with some brief guidance for further reading. It is hoped that this approach may render the subject more approachable. Titles in this series are co-published with the Fields Institute for Research in Mathematical Sciences (Toronto, Ontario, Canada).

Spencer J. Bloch has, and continues to have, a profound influence on the subject of Algebraic $K$-Theory, Cycles and Motives. This book, which is comprised of a number of independent research articles written by leading experts in the field, is dedicated in his honour, and gives a snapshot of the current and evolving nature of the subject. Some of the articles are written in an expository style, providing perspective on the current state of the subject to those wishing to learn more about it. Others are more technical, representing new developments and making them especially interesting to researchers for keeping abreast of recent progress.

This volume represents the proceedings of the Noncommutative Geometry Workshop that was held as part of the thematic program on operator algebras at the Fields Institute in May 2008. Pioneered by Alain Connes starting in the late 1970s, noncommutative geometry was originally inspired by global analysis, topology, operator algebras, and quantum physics. Its main applications were to settle some long-standing conjectures, such as the Novikov conjecture and the Baum-Connes conjecture. Next came the impact of spectral geometry and the way the spectrum of a geometric operator, like the Laplacian, holds information about the geometry and topology of a manifold, as in the celebrated Weyl law. This has now been vastly generalized through Connes' notion of spectral triples. Finally, recent years have witnessed the impact of number theory, algebraic geometry and the theory of motives, and quantum field theory on noncommutative geometry. Almost all of these aspects are touched upon with new results in the papers of this volume. This book is intended for graduate students and researchers in both mathematics and theoretical physics who are interested in noncommutative geometry and its applications.

This volume is a collection of papers on number theory which evolved out of the workshop WIN—Women In Numbers, held November 2–7, 2008, at the Banff International Research Station (BIRS) in Banff, Alberta, Canada. It includes articles showcasing outcomes from collaborative research initiated during the workshop as well as survey papers aimed at introducing graduate students and recent PhDs to important research topics in number theory. The contributions in this volume span a wide range of topics in arithmetic geometry and algebraic, algorithmic, and analytic number theory. Clusters of papers center around the four topics of moduli spaces and Shimura curves, curves and Jacobians over finite fields, Galois covers of function fields in positive characteristic, and zeta functions of graphs, with a fifth group of three individual articles on modular forms, Iwasawa theory, and Galois representations, respectively. The workshop and this volume are part of a broader WIN initiative, whose goals are to highlight and increase the research activities of women in number theory and to train female graduate students in number theory and related fields.

This book contains a collection of papers presented at the Fields Institute workshop, 'The Falling Cat and Related Problems', held in March 1992. The theme of the workshop was the application of methods from geometric mechanics and mathematical control theory to problems in the dynamics and control of freely rotating systems of coupled rigid bodies and related nonholonomic mechanical systems. This book will prove useful in providing insight into this new and exciting area of research.

Flexible structures arise in significant important areas of application, such as robotics, large space structures, and antenna control. Difficulties related to sensing and identification hamper control of such systems. These problems require collaboration between mathematicians and engineers. To promote such collaboration, the Fields Institute sponsored a three-day workshop entitled 'Problems in Sensing, Identification, and Control of Flexible Structures' in June 1992. This volume contains papers presented at the workshop. Topics range from theoretical research on the well-posedness of systems, to experimental implementations of various controllers. A number of controller design techniques are discussed and compared, and there are several papers on modelling the complex dynamics of flexible structures. This book is a useful resource to control theorists, engineers, and mathematicians interested in this important field of research.