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Everybody having even the slightest interest in analytical mechanics remembers having met there the Poisson bracket of two functions of 2n variables (pi, qi) f g ~(8f8g 8 8 ) (0.1) {f,g} = L... ~[ji - [ji~ , ;=1 p, q q p, and the fundamental role it plays in that field. In modern works, this bracket is derived from a symplectic structure, and it appears as one of the main in­ gredients of symplectic manifolds. In fact, it can even be taken as the defining clement of the structure (e.g., [TIl]). But, the study of some mechanical sys­ tems, particularly systems with symmetry groups or constraints, may lead to more general Poisson brackets. Therefore, it was natural to define a mathematical structure where the notion of a Poisson bracket would be the primary notion of the theory, and, from this viewpoint, such a theory has been developed since the early 19708, by A. Lichnerowicz, A. Weinstein, and many other authors (see the references at the end of the book). But, it has been remarked by Weinstein [We3] that, in fact, the theory can be traced back to S. Lie himself [Lie].

This book presents topics in module theory and ring theory: some, such as Goldie dimension and semiperfect rings are now considered classical and others more specialized, such as dual Goldie dimension, semilocal endomorphism rings, serial rings and modules.

Table of contents: Plenary Lectures ¿ V.I. Arnold: The Vassiliev Theory of Discriminants and Knots ¿ L. Babai: Transparent Proofs and Limits to Approximation ¿ C. De Concini: Poisson Algebraic Groups and Representations of Quantum Groups at Roots of 1 ¿ S.K. Donaldson: Gauge Theory and Four-Manifold Topology ¿ W. Müller: Spectral Theory and Geometry ¿ D. Mumford: Pattern Theory: A Unifying Perspective ¿ A.-S. Sznitman: Brownian Motion and Obstacles ¿ M. Vergne: Geometric Quantization and Equivariant Cohomology ¿ Parallel Lectures ¿ Z. Adamowicz: The Power of Exponentiation in Arithmetic ¿ A. Björner: Subspace Arrangements ¿ B. Bojanov: Optimal Recovery of Functions and Integrals ¿ J.-M. Bony: Existence globale et diffusion pour les modèles discrets ¿ R.E. Borcherds: Sporadic Groups and String Theory ¿ J. Bourgain: A Harmonic Analysis Approach to Problems in Nonlinear Partial Differatial Equations ¿ F. Catanese: (Some) Old and New Results on Algebraic Surfaces ¿ Ch. Deninger: Evidence for a Cohomological Approach to Analytic Number Theory ¿ S. Dostoglou and D.A. Salamon: Cauchy-Riemann Operators, Self-Duality, and the Spectral Flow

A joint research project of algebraists from the universities of Antwerp, Biele­ feld, Essen, Leeds, Paris VI and Trondheim on "Invariants and Representations of Algebras" has been supported from 1991 to 1997 by the European Union programmes "Science" and "Human Capital and Mobility", it was coordinated by Mme M. -P. Malliavin (Paris VI). Later, algebraists from the universities of Edinburgh, Ioannina, Murcia and Torun joined the collaboration. This network is now coordinated by C. M. Ringel (Bielefeld). It has received funds from the European Commission in order to organize four conferences as part of the pro­ gramme "Training and Mobility of Researchers", to be held during the period 1997-1999 at Essen, Murcia, Bielefeld and Ioannina. The first Euroconference of this series took place at the University of Essen, April 1-4, 1997. It was devoted to "Computational Methods for Representations of Groups and Algebras" . The organizers were P. Draxler (Bielefeld) and G. Michler (Essen). This volume collects most of the material presented at the conference. There had been an additional introductory lecture by H. Gollan; it is not included here, since its contents is available in the lecture notes: P. Fleischmann, G. O. Michler, P. Roelse, J. Rosenboom, R. Staszewski, C. Wagner, M. Weller, "Linear algebra over small finite fields on parallel machines", Vorlesungen Fachbereich Math. Univ. Essen, 23 (1995). Together with these notes, this volume will provide a survey on the present state of art.

It would be difficult to overestimate the influence and importance of modular forms, modular curves, and modular abelian varieties in the development of num­ ber theory and arithmetic geometry during the last fifty years. These subjects lie at the heart of many past achievements and future challenges. For example, the theory of complex multiplication, the classification of rational torsion on el­ liptic curves, the proof of Fermat's Last Theorem, and many results towards the Birch and Swinnerton-Dyer conjecture all make crucial use of modular forms and modular curves. A conference was held from July 15 to 18, 2002, at the Centre de Recerca Matematica (Bellaterra, Barcelona) under the title "Modular Curves and Abelian Varieties". Our conference presented some of the latest achievements in the theory to a diverse audience that included both specialists and young researchers. We emphasized especially the conjectural generalization of the Shimura-Taniyama conjecture to elliptic curves over number fields other than the field of rational numbers (elliptic Q-curves) and abelian varieties of dimension larger than one (abelian varieties of GL2-type).

This book puts the modern theory of complex linear convexity on a solid footing, and gives a thorough and up-to-date survey of its current status. Applications include the Fantappie transformation of analytic functionals, integral representation formulas, polynomial interpolation, and solutions to linear partial differential equations.

Daniel Quillen's definition of the higher algebraic K-groups of a ring emphasized the importance of computing the homology of groups of matrices. It presents the stability theorems and low-dimensional results of A. Coverage also examines the Friedlander-Milnor-conjecture concerning the homology of algebraic groups made discrete.

This book contains a collection of articles summarizing together the state of knowl­ edge in a broad portion of modern homotopy theory. These articles were assembled during 1998 and 1999, on the occasion of an emphasis semester organized by the Centre de Recerca Matematica (CRM) and its highlight, the 1998 Barcelona Con­ ference on Algebraic Topology (BCAT). First of all, we are indebted to all the authors for submitting their work, and to the referees for their help in the selec­ tion and for their generous contribution to the content of the articles. Many talks given during the CRM semester or at the conference focused on aspects of the following topics: abstract stable homotopy, model categories, homotopical localizations and cellular approximations, p-compact groups, mod­ ules over the Steenrod algebra, classifying spaces for proper actions of discrete groups, K-theory and other generalized cohomology theories, cohomology of fi­ nite and profinite groups, Hochschild homology, configuration spaces, Lusternik­ Schnirelmann category, stable and unstable splittings. Other talks treated multi­ disciplinary subjects related to quantum field theory, differential geometry, homo­ topical dynamics, tilings, and various aspects of group theory. In addition, an advanced course on Classifying Spaces and Cohomology of Groups was organized by the CRM in the days preceding the conference. Lecture notes from this course will be published by Birkhauser Verlag as the first volume of a newly created CRM Advanced Course series.

Abelian varieties and their moduli are a topic of increasing importance in today`s mathematics, applications ranging from algebraic geometry and number theory to mathematical physics.

= {~k' k = 0, 1, ... } with transition probability function (t.pJ.) P(x, B), i.e., P(x, B) := Prob (~k+1 E B I ~k = x) for each x E X, B E B, and k = 0,1, .... is said to be stable if there exists a probability measure (p.m.) /.l on B such that (*) VB EB. /.l(B) = Ix /.l(dx) P(x, B) If (*) holds then /.l is called an invariant p.m. for the Me ~.

The first European Congress of Mathematics was held in Paris from July 6 to July 10, 1992, at the Sorbonne and Pantheon-Sorbonne universities. Moreover, a Junior Mathematical Congress was organized, in parallel with the Congress, which brought together two hundred high school students.

This book grew out of the work developed at the University of Warwick, under the supervision of Ian Stewart, which formed the core of my Ph.D. More importantly, the work should assume as little as possible, and it should be brought to a form which is pleasurable, not painful, to read.

This is the second volume of the procedings of the second European Congress of Mathematics. Together with volume II it contains a collection of contributions by the invited lecturers. Finally, volume II also presents reports on some of the Round Table discussions. II: J.

Harmonic analysis and probability have long enjoyed a mutually beneficial relationship that has been rich and fruitful. This monograph, aimed at researchers and students in these fields, explores several aspects of this relationship. The primary focus of the text is the nontangential maximal function and the area function of a harmonic function and their probabilistic analogues in martingale theory. The text first gives the requisite background material from harmonic analysis and discusses known results concerning the nontangential maximal function and area function, as well as the central and essential role these have played in the development of the field.The book next discusses further refinements of traditional results: among these are sharp good-lambda inequalities and laws of the iterated logarithm involving nontangential maximal functions and area functions. Many applications of these results are given. Throughout, the constant interplay between probability and harmonic analysis is emphasized and explained. The text contains some new and many recent results combined in a coherent presentation.

From April 1, 1984 until March 31, 1991 the Deutsche Forschungsgemeinschaft has sponsored the project "Representation Theory of Finite Groups and Finite Di mensional Algebras".

The school, the book This book is based on lectures given by the authors of the various chapters in a three week long CIMPA summer school, held in Sophia-Antipolis (near Nice) in July 1992. The first week was devoted to the basics of symplectic and Riemannian geometry (Banyaga, Audin, Lafontaine, Gauduchon), the second was the technical one (Pansu, Muller, Duval, Lalonde and Sikorav). The final week saw the conclusion ofthe school (mainly McDuffand Polterovich, with complementary lectures by Lafontaine, Audin and Sikorav). Globally, the chapters here reflect what happened there. Locally, we have tried to reorganise some ofthe material to make the book more coherent. Hence, for instance, the collective (Audin, Lalonde, Polterovich) chapter on Lagrangian submanifolds and the appendices added to some of the chapters. Duval was not able to write up his lectures, so that genuine complex analysis will not appear in the book, although it is a very current tool in symplectic and contact geometry (and conversely). Hamiltonian systems and variational methods were the subject of some of Sikorav's talks, which he also was not able to write up. On the other hand, F. Labourie, who could not be at the school, wrote a chapter on pseudo-holomorphic curves in Riemannian geometry.

Award-winning monograph of the Ferran Sunyer i Balaguer Prize 2002. The subject of this book is the study of automorphic distributions, by which is meant distributions on R2 invariant under the linear action of SL(2,Z), and of the operators associated with such distributions under the Weyl rule of symbolic calculus. Researchers and postgraduates interested in pseudodifferential analyis, the theory of non-holomorphic modular forms, and symbolic calculi will benefit from the clear exposition and new results and insights.

The material and references in this extended second edition of "The Topology of Torus Actions on Symplectic Manifolds", published as Volume 93 in this series in 1991, have been updated. Symplectic manifolds and torus actions are investigated, with numerous examples of torus actions, for instance on some moduli spaces.

Award-winning monograph of the Ferran Sunyer i Balaguer Prize 2003.This book contains a detailed mathematical analysis of the variational approach to image restoration based on the minimization of the total variation submitted to the constraints given by the image acquisition model. This model, initially introduced by Rudin, Osher, and Fatemi, had a strong influence in the development of variational methods for image denoising and restoration, and pioneered the use of the BV model in image processing. After a full analysis of the model, the minimizing total variation flow is studied under different boundary conditions, and its main qualitative properties are exhibited. In particular, several explicit solutions of the denoising problem are computed.

The book consists of articles at the frontier of current research in Algebraic Topology.

Based on the courses given at the Working Week in Obergurgl, Austria, September 7-14, 1997

The Third European Congress of Mathematics (3ecm) took place from July 10th to July 14th, 2000 in Barcelona. It was organised by the Societat Catalana de Matematiques (Catalan Mathematical Society), under the auspices of the Euro­ pean Mathematical Society (EMS). As foreseen by the EMS and the International Mathematical Union, this Congress was a major event in World Mathematical Year 2000. In addition to reviewing outstanding research achievements, important aspects of the life of European mathematics were discussed. Mathematics is undergoing a period of rapid changes. Effective computation and applications in science and technology go ever more hand in hand with con­ ceptual developments. It was one of the aims of 3ecm to reflect this mutual enrich­ ment, while steering present and future trends of mathematical sciences. In fact, the motto of the Congress, Shaping the 21st Century, was meant to capture these views. Nearly 1400 people from 87 countries gathered together in the Palau de Con­ gressos of Barcelona in order to take part in the activities of the 3ecm scientific programme: Nine plenary lectures, thirty invited lectures in parallel sessions, lec­ tures given by EMS prize winners, ten mini-symposia on special topics, seven round tables, poster sessions, presentations of mathematical software and video exhibitions. Twenty events were satellites of 3ecm in various countries.

The second volume collects articles by prize winners and speakers of the mini-symposia. This two-volume set thus gives an overview of the state of the art in many fields of mathematics and is therefore of interest to every professional mathematician.

This volume arises from the contributions presented at the MEGA 94 Con­ ference (Metodos Efectivos en Geomctria Algebraica = Effective Methods in Algebraic Geometry), held at the University of Cantabria (Santander, Spain) April 59, 1994. Previous sessions of this biannual conference had taken place in Castiglioncello (Livorno, Italy, 1990) and in Nice (France, 1992) and the cor­ responding proceedings have been published in the Birkhauser series Progress in Mathematics. volumes no. 94 and 109, respectively. The present collection consists of twenty articles involvillg miscellaneous topics concerning algorithms in algebra, algebraic geometry and related appli­ cations. Fourteen of these papers correspond to the contents of the Conference's regular scientific program and have been selected, by the MEGA Committee, from the submitted contributions after a very rigorous refereeing procedure entailing an average of three independent reports per paper and two Program Committee panel discussions before and after the Conference. The remaining six papers (by S. Beck & M. Kreuzer, M. Bronstein, E. V. Flvnn. 1. Itenberg, J.-P. Merlet and 1\1. Seppala) correspond to invited talks and have also been subject to a post-conference refereeing procedure.

This is the first volume of the procedings of the second European Congress of Mathematics. Volume I presents the speeches delivered at the Congress, the list of lectures, and short summaries of the achievements of the prize winners. Together with volume II it contains a collection of contributions by the invited lecturers.

Lagrangian systems constitute a very important and old class in dynamics. Their origin dates back to the end of the eighteenth century, with Joseph-Louis Lagrange's reformulation of classical mechanics. The main feature of Lagrangian dynamics is its variational flavor: orbits are extremal points of an action functional. The development of critical point theory in the twentieth century provided a powerful machinery to investigateexistence and multiplicity questions for orbits of Lagrangian systems. This monograph gives a modern account of the application of critical point theory, and more specifically Morse theory, to Lagrangian dynamics, with particular emphasis toward existence and multiplicity of periodic orbits of non-autonomous and time-periodic systems.

<i>This book presents a functional calculus for <i>n</i>-tuples of not necessarily commuting linear operators. In particular, a functional calculus for quaternionic linear operators is developed. These calculi are based on a new theory of hyperholomorphicity for functions with values in a Clifford algebra: the so-called slice monogenic functions which are carefully described in the book. In the case of functions with values in the algebra of quaternions these functions are named slice regular functions.</i><br> <p>Except for the appendix and the introduction all results are new and appear for the first time organized in a monograph. The material has been carefully prepared to be as self-contained as possible. The intended audience consists of researchers, graduate and postgraduate students interested in operator theory, spectral theory, hypercomplex analysis, and mathematical physics.</p>

The relevance of commutator methods in spectral and scattering theory has been known for a long time, and numerous interesting results have been ob tained by such methods.

The aim of this monograph is to present a self-contained introduction to some geometric and analytic aspects of the Yamabe problem. The book also describes a wide range of methods and techniques that can be successfully applied to nonlinear differential equations in particularly challenging situations. Such situations occur where the lack of compactness, symmetry and homogeneity prevents the use of more standard tools typically used in compact situations or for the Euclidean setting. The work is written in an easy style that makes it accessible even to non-specialists.After a self-contained treatment of the geometric tools used in the book, readers are introduced to the main subject by means of a concise but clear study of some aspects of the Yamabe problem on compact manifolds. This study provides the motivation and geometrical feeling for the subsequent part of the work. In the main body of the book, it is shown how the geometry and the analysis of nonlinear partial differential equations blend together to give up-to-date results on existence, nonexistence, uniqueness and a priori estimates for solutions of general Yamabe-type equations and inequalities on complete, non-compact Riemannian manifolds.