Mathematical Sciences Research Institute Publications – serie

When the Mathematical Sciences Research Institute was started in the Fall of 1982, one of the programs was "non-linear partial differential equations". A seminar was organized whose audience consisted of graduate students of the University and mature mathematicians who are not experts in the field. This volume contains 18 of these lectures. An effort is made to have an adequate Bibliography for further information. The Editor wishes to take this opportunity to thank all the speakers and the authors of the articles presented in this volume for their cooperation. S. S. Chern, Editor Table of Contents Geometrical and Analytical Questions Stuart S. Antman 1 in Nonlinear Elasticity An Introduction to Euler's Equations Alexandre J. Chorin 31 for an Incompressible Fluid Linearizing Flows and a Cohomology Phillip Griffiths 37 Interpretation of Lax Equations The Ricci Curvature Equation Richard Hamilton 47 A Walk Through Partial Differential Fritz John 73 Equations Remarks on Zero Viscosity Limit for Tosio Kato 85 Nonstationary Navier-Stokes Flows with Boundary Free Boundary Problems in Mechanics Joseph B. Keller 99 The Method of Partial Regularity as Robert V.

This volume records most of the talks given at the Conference on Infinite-dimensional Groups held at the Mathematical Sciences Research Institute at Berkeley, California, May 10-May 15, 1984, as a part of the special program on Kac-Moody Lie algebras. The purpose of the conference was to review recent developments of the theory of infinite-dimensional groups and its applications. The present collection concentrates on three very active, interrelated directions of the field: general Kac-Moody groups, gauge groups (especially loop groups) and diffeomorphism groups. I would like to express my thanks to the MSRI for sponsoring the meeting, to Ms. Faye Yeager for excellent typing, to the authors for their manuscripts, and to Springer-Verlag for publishing this volume. V. Kac INFINITE DIMENSIONAL GROUPS WITH APPLICATIONS CONTENTS The Lie Group Structure of M. Adams. T. Ratiu 1 Diffeomorphism Groups and & R. Schmid Invertible Fourier Integral Operators with Applications On Landau-Lifshitz Equation and E. Date 71 Infinite Dimensional Groups Flat Manifolds and Infinite D. S. Freed 83 Dimensional Kahler Geometry Positive-Energy Representations R. Goodman 125 of the Group of Diffeomorphisms of the Circle Instantons and Harmonic Maps M. A. Guest 137 A Coxeter Group Approach to Z. Haddad 157 Schubert Varieties Constructing Groups Associated to V. G. Kac 167 Infinite-Dimensional Lie Algebras I. Kaplansky 217 Harish-Chandra Modules Over the Virasoro Algebra & L. J. Santharoubane 233 Rational Homotopy Theory of Flag S.

This book documents the focus on a branch of Riemannian geometry called Comparison Geometry. The simple idea of comparing the geometry of an arbitrary Riemannian manifold with the geometries of constant curvature spaces has seen a tremendous evolution of late. This volume is an up-to-date reflection of the recent development regarding spaces with lower (or two-sided) curvature bounds. The content of the volume reflects some of the most exciting activities in comparison geometry during the year and especially of the Mathematical Sciences Research Institute's workshop devoted to the subject. Both survey and research articles are featured. Complete proofs are often provided, and in one case a new unified strategy is presented and new proofs are offered. This volume will be a valuable source for advanced researchers and those who wish to learn about and contribute to this beautiful subject.

Spaces of holomorphic functions have been a prominent theme in analysis since early in the twentieth century. Of interest to complex analysts, functional analysts, operator theorists and systems theorists, their study is now flourishing. This volume, an outgrowth of a 1995 program at the Mathematical Sciences Research Institute, contains expository articles by programme participants. Here researchers and graduate students will encounter Hardy spaces, Bergman spaces, Dirichlet spaces, Hankel and Toeplitz operators, and a sampling of the role these objects play in modern analysis.

Hopf algebras have important connections to quantum theory, Lie algebras, knot and braid theory, operator algebras and other areas of physics and mathematics. They have been intensely studied in the past; in particular, the solution of a number of conjectures of Kaplansky from the 1970s has led to progress on the classification of semisimple Hopf algebras and on the structure of pointed Hopf algebras. Among the topics covered are results toward the classification of finite-dimensional Hopf algebras (semisimple and non-semisimple), as well as what is known about the extension theory of Hopf algebras. Some papers consider Hopf versions of classical topics, such as the Brauer group, while others are closer to work in quantum groups. The book also explores the connections and applications of Hopf algebras to other fields.

Model theory has made substantial contributions to semialgebraic, subanalytic, p-adic, rigid and diophantine geometry. These applications range from a proof of the rationality of certain Poincare series associated to varieties over p-adic fields, to a proof of the Mordell-Lang conjecture for function fields in positive characteristic. In some cases (such as the latter) it is the most abstract aspects of model theory which are relevant. This book, originally published in 2000, arising from a series of introductory lectures for graduate students, provides the necessary background to understanding both the model theory and the mathematics behind these applications. The book is unique in that the whole spectrum of contemporary model theory (stability, simplicity, o-minimality and variations) is covered and diverse areas of geometry (algebraic, diophantine, real analytic, p-adic, and rigid) are introduced and discussed, all by leading experts in their fields.

Several Complex Variables is a central area of mathematics with strong interactions with partial differential equations, algebraic geometry, number theory, and differential geometry. The 1995-1996 MSRI program on Several Complex Variables emphasized these interactions and concentrated on developments and problems of interest that capitalize on this interplay of ideas and techniques. This collection, first published in 2000, provides a remarkably clear and complete picture of the status of research in these overlapping areas and will provide a basis for significant continued contributions from researchers. Several of the articles are expository or have extensive expository sections, making this an excellent introduction for students to the use of techniques from these other areas in several complex variables. Thanks to its distinguished list of contributors this volume provides a representative sample of the work done in Several Complex Variables.

This 2003 book provides an analysis of combinatorial games - games not involving chance or hidden information. It contains a fascinating collection of articles by some well-known names in the field, such as Elwyn Berlekamp and John Conway, plus other researchers in mathematics and computer science, together with some top game players. The articles run the gamut from theoretical approaches (infinite games, generalizations of game values, 2-player cellular automata, Alpha-Beta pruning under partial orders) to other games (Amazons, Chomp, Dot-and-Boxes, Go, Chess, Hex). Many of these advances reflect the interplay of the computer science and the mathematics. The book ends with a bibliography by A. Fraenkel and a list of combinatorial game theory problems by R. K. Guy. Like its predecessor, Games of No Chance, this should be on the shelf of all serious combinatorial games enthusiasts.

Convex geometry is at once simple and amazingly rich. While the classical results go back many decades, during that previous to this book's publication in 1999, the integral geometry of convex bodies had undergone a dramatic revitalization, brought about by the introduction of methods, results and, most importantly, new viewpoints, from probability theory, harmonic analysis and the geometry of finite-dimensional normed spaces. This book is a collection of research and expository articles on convex geometry and probability, suitable for researchers and graduate students in several branches of mathematics coming under the broad heading of 'Geometric Functional Analysis'. It continues the Israel GAFA Seminar series, which is widely recognized as the most useful research source in the area. The collection reflects the work done at the program in Convex Geometry and Geometric Analysis that took place at MSRI in 1996.

The seminal formula of Gross and Zagier relating heights of Heegner points to derivatives of the associated Rankin L-series has led to many generalisations and extensions in a variety of different directions, spawning a fertile area of study that remains active to this day. This volume, based on a workshop on Special Values of Rankin L-series held at the MSRI in December 2001, is a collection of thirteen articles written by many of the leading contributors in the field, having the Gross-Zagier formula and its avatars as a common unifying theme. It serves as a valuable reference for mathematicians wishing to become further acquainted with the theory of complex multiplication, automorphic forms, the Rankin-Selberg method, arithmetic intersection theory, Iwasawa theory, and other topics related to the Gross-Zagier formula.

Signal processing is everywhere in modern technology. Its mathematical basis and many areas of application are the subject of this book, based on a series of graduate-level lectures held at the Mathematical Sciences Research Institute. Emphasis is on challenges in the subject, particular techniques adapted to particular technologies, and certain advances in algorithms and theory. The book covers two main areas: computational harmonic analysis, envisioned as a technology for efficiently analysing real data using inherent symmetries; and the challenges inherent in the acquisition, processing and analysis of images and sensing data in general [EMDASH] ranging from sonar on a submarine to a neuroscientist's fMRI study.

In 2002, an introductory workshop was held at the Mathematical Sciences Research Institute in Berkeley to survey some of the many directions of the commutative algebra field. Six principal speakers each gave three lectures, accompanied by a help session, describing the interaction of commutative algebra with other areas of mathematics for a broad audience of graduate students and researchers. This book is based on those lectures, together with papers from contributing researchers. David Benson and Srikanth Iyengar present an introduction to the uses and concepts of commutative algebra in the cohomology of groups. Mark Haiman considers the commutative algebra of n points in the plane. Ezra Miller presents an introduction to the Hilbert scheme of points to complement Professor Haiman's paper. Further contributors include David Eisenbud and Jessica Sidman; Melvin Hochster; Graham Leuschke; Rob Lazarsfeld and Manuel Blickle; Bernard Teissier; and Ana Bravo.

Finsler geometry generalises Riemannian geometry in the same sense that Banach spaces generalise Hilbert spaces. This book presents an expository account of seven important topics in Riemann-Finsler geometry, ones which have undergone significant development but have not had a detailed pedagogical treatment elsewhere. Each article will open the door to an active area of research, and is suitable for a special topics course in graduate-level differential geometry. The contributors consider issues related to volume, geodesics, curvature, complex differential geometry and parametrised jet bundles, and include a variety of instructive examples.

Inverse problems arise in practical situations such as medical imaging, geophysical exploration, and non-destructive evaluation where measurements made on the exterior of a body are used to determine properties of the inaccessible interior. There have been substantial developments in the mathematical theory of inverse problems, and applications have expanded greatly. In this volume, leading experts in the theoretical and applied aspects of inverse problems offer extended surveys on several important topics in modern inverse problems, such as microlocal analysis, reflection seismology, tomography, inverse scattering, and X-ray transforms. Each article covers a particular topic or topics with an emphasis on accessibility and integration with the whole volume. Thus the collection can be at the same time stimulating to researchers and accessible to graduate students.

This book contains eight expository articles by well-known authors of the theory of Galois groups and fundamental groups. They focus on presenting developments, avoiding classical aspects which have already been described at length in the standard literature. The volume grew from the special semester held at the MSRI in Berkeley in 1999 and many of the results are due to work accomplished during that program. Among the subjects covered are elliptic surfaces, Grothendieck's anabelian conjecture, fundamental groups of curves and differential Galois theory in positive characteristic. Although the articles contain fresh results, the authors have striven to make them as introductory as possible, making them accessible to graduate students as well as researchers in algebraic geometry and number theory. The volume also contains a lengthy overview by Leila Schneps that sets the individual articles into the broader context of contemporary research in Galois groups.

First published in 1997, this book contains six in-depth articles on various aspects of the field of tight and taut submanifolds and concludes with an extensive bibliography of the entire field. The book is dedicated to the memory of Nicolaas H. Kuiper; the first paper is an unfinished but insightful survey of the field of tight immersions and maps written by Kuiper himself. Other papers by leading researchers in the field treat topics such as the smooth and polyhedral portions of the theory of tight immersions, taut, Dupin and isoparametric submanifolds of Euclidean space, taut submanifolds of arbitrary complete Riemannian manifolds, and real hypersurfaces in complex space forms with special curvature properties. Taken together these articles provide a comprehensive survey of the field and point toward several directions for future research.

Random matrices arise from, and have important applications to, number theory, probability, combinatorics, representation theory, quantum mechanics, solid state physics, quantum field theory, quantum gravity, and many other areas of physics and mathematics. This 2001 volume of surveys and research results, based largely on lectures given at the Spring 1999 MSRI program of the same name, covers broad areas such as topologic and combinatorial aspects of random matrix theory; scaling limits, universalities and phase transitions in matrix models; universalities for random polynomials; and applications to integrable systems. Its stress on the interaction between physics and mathematics will make it a welcome addition to the shelves of graduate students and researchers in both fields, as will its expository emphasis.

Although topology was recognized by Gauss and Maxwell to play a pivotal role in the formulation of electromagnetic boundary value problems, it is a largely unexploited tool for field computation. The development of algebraic topology since Maxwell provides a framework for linking data structures, algorithms, and computation to topological aspects of three-dimensional electromagnetic boundary value problems. This book attempts to expose the link between Maxwell and a modern approach to algorithms. The first chapters lay out the relevant facts about homology and cohomology, stressing their interpretations in electromagnetism. These topological structures are subsequently tied to variational formulations in electromagnetics, the finite element method, algorithms, and certain aspects of numerical linear algebra. A recurring theme is the formulation of and algorithms for the problem of making branch cuts for computing magnetic scalar potentials and eddy currents.

The three main themes of this book, probability theory, differential geometry, and the theory of integrable systems, reflect the broad range of mathematical interests of Henry McKean, to whom it is dedicated. Written by experts in probability, geometry, integrable systems, turbulence, and percolation, the seventeen papers included here demonstrate a wide variety of techniques that have been developed to solve various mathematical problems in these areas. The topics are often combined in an unusual and interesting fashion to give solutions outside of the standard methods. The papers contain some exciting results and offer a guide to the contemporary literature on these subjects.

Based on the subjects from the Clay Mathematics Institute/Mathematical Sciences Research Institute Workshop titled 'Recent Progress in Dynamics' in September and October 2004, this volume contains surveys and research articles by leading experts in several areas of dynamical systems that have experienced substantial progress. One of the major surveys is on symplectic geometry, which is closely related to classical mechanics and an exciting addition to modern geometry. The survey on local rigidity of group actions gives a broad and up-to-date account of another flourishing subject. Other papers cover hyperbolic, parabolic, and symbolic dynamics as well as ergodic theory. Students and researchers in dynamical systems, geometry, and related areas will find this book fascinating. The book also includes a fifty-page commented problem list that takes the reader beyond the areas covered by the surveys, to inspire and guide further research.

During the past few decades, the gradual merger of Discrete Geometry and the newer discipline of Computational Geometry has provided enormous impetus to mathematicians and computer scientists interested in geometric problems. This 2005 volume, which contains 32 papers on a broad range of topics of interest in the field, is an outgrowth of that synergism. It includes surveys and research articles exploring geometric arrangements, polytopes, packing, covering, discrete convexity, geometric algorithms and their complexity, and the combinatorial complexity of geometric objects, particularly in low dimension. There are points of contact with many applied areas such as mathematical programming, visibility problems, kinetic data structures, and biochemistry, as well as with algebraic topology, geometric probability, real algebraic geometry, and combinatorics.

During 1996-7 MSRI held a full academic year program on Combinatorics, with special emphasis on the connections with other branches of mathematics, such as algebraic geometry, topology, commutative algebra, representation theory, and convex geometry. The rich combinatorial problems arising from the study of various algebraic structures are the subject of this book, which represents work done or presented at seminars during the program. It contains contributions on matroid bundles, combinatorial representation theory, lattice points in polyhedra, bilinear forms, combinatorial differential topology and geometry, Macdonald polynomials and geometry, enumeration of matchings, the generalized Baues problem, and Littlewood-Richardson semigroups. These expository articles, written by some of the most respected researchers in the field, will continue to be of use to graduate students and researchers in combinatorics as well as algebra, geometry, and topology.

Appearance of singularities is pervasive in many problems in topology, differential geometry and algebraic geometry. This book concerns the study of singular spaces using techniques from a variety of areas of geometry and topology and the interactions among them. Expository chapters by well-known experts cover intersection homology, L2 cohomology and differential operators, topology of algebraic varieties, signatures and characteristic classes, mixed Hodge theory and elliptic genera of singular complex and real algebraic varieties. The book concludes with a list of open problems.

This book provides an introduction to (1) various zeta functions (for example, Riemann, Hurwitz, Barnes, Epstein, Selberg, and Ruelle), including graph zeta functions; (2) modular forms (Eisenstein series, Hecke and Dirichlet L-functions, Ramanujan's tau function, and cusp forms); and (3) vertex operator algebras (correlation functions, quasimodular forms, modular invariance, rationality, and some current research topics including higher genus conformal field theory). Various concrete applications of the material to physics are presented. These include Kaluza-Klein extra dimensional gravity, Bosonic string calculations, an abstract Cardy formula for black hole entropy, Patterson-Selberg zeta function expression of one-loop quantum field and gravity partition functions, Casimir energy calculations, atomic Schrödinger operators, Bose-Einstein condensation, heat kernel asymptotics, random matrices, quantum chaos, elliptic and theta function solutions of Einstein's equations, a soliton-black hole connection in two-dimensional gravity, and conformal field theory.

Number theory is one of the oldest and most appealing areas of mathematics. Computation has always played a role in number theory, a role which has increased dramatically in the last 20 or 30 years, both because of the advent of modern computers, and because of the discovery of surprising and powerful algorithms. As a consequence, algorithmic number theory has gradually emerged as an important and distinct field with connections to computer science and cryptography as well as other areas of mathematics. This text provides a comprehensive introduction to algorithmic number theory for beginning graduate students, written by the leading experts in the field. It includes several articles that cover the essential topics in this area, and in addition, there are contributions pointing in broader directions, including cryptography, computational class field theory, zeta functions and L-series, discrete logarithm algorithms, and quantum computing.

The 1992/3 academic year at the Mathematical Sciences Research Institute was devoted to complex algebraic geometry. This 1996 volume collects survey articles that arose from this event, which took place at a time when algebraic geometry was undergoing a major change. To put it succinctly, algebraic geometry has opened up to ideas and connections from other fields that have traditionally been far away. The editors of the volume, Herbert Clemens and Janos Kollar, chaired the organizing committee. Activities were centered around themes, one per month, under the guidance of experts in each area. There were also four short workshops, which attracted many participants and helped considerably in communicating the new directions in algebraic geometry to a large audience. This book gives a good idea of the intellectual content of the special year and of the workshops.

Is Nine-Men Morris, in the hands of perfect players, a win for white or for black - or a draw? Can king, rook, and knight always defeat king and two knights in chess? What can Go players learn from economists? What are nimbers, tinies, switches and minies? This book deals with combinatorial games, that is, games not involving chance or hidden information. Their study is at once old and young: though some games, such as chess, have been analyzed for centuries, the first full analysis of a nontrivial combinatorial game (Nim) only appeared in 1902. The first part of this book will be accessible to anyone, regardless of background: it contains introductory expositions, reports of unusual tournaments, and a fascinating article by John H. Conway on the possibly everlasting contest between an angel and a devil. For those who want to delve more deeply, the book also contains combinatorial studies of chess and Go; reports on computer advances such as the solution of Nine-Men Morris and Pentominoes; and theoretical approaches to such problems as games with many players. If you have read and enjoyed Martin Gardner, or if you like to learn and analyze new games, this book is for you.

This book documents the focus on a branch of Riemannian geometry called Comparison Geometry. The simple idea of comparing the geometry of an arbitrary Riemannian manifold with the geometries of constant curvature spaces has seen a tremendous evolution of late. This volume is an up-to-date reflection of the recent development regarding spaces with lower (or two-sided) curvature bounds. The content of the volume reflects some of the most exciting activities in comparison geometry during the year and especially of the Mathematical Sciences Research Institute's workshop devoted to the subject. Both survey and research articles are featured. Complete proofs are often provided, and in one case a new unified strategy is presented and new proofs are offered. This volume will be a valuable source for advanced researchers and those who wish to learn about and contribute to this beautiful subject.

Foliated spaces look locally like products, but their global structure is generally not a product, and tangential differential operators are correspondingly more complex. In the 1980s, Alain Connes founded what is now known as noncommutative geometry and topology. One of the first results was his generalization of the Atiyah-Singer index theorem to compute the analytic index associated with a tangential (pseudo) - differential operator and an invariant transverse measure on a foliated manifold, in terms of topological data on the manifold and the operator. This second edition presents a complete proof of this beautiful result, generalized to foliated spaces (not just manifolds). It includes the necessary background from analysis, geometry, and topology. The present edition has improved exposition, an updated bibliography, an index, and additional material covering developments and applications since the first edition came out, including the confirmation of the Gap Labeling Conjecture of Jean Bellissard.