Bücher der Reihe Progress in Mathematics

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 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.

This book contains selected papers based on talks given at the "Representation Theory, Number Theory, and Invariant Theory" conference held at Yale University from June 1 to June 5, 2015.

This book studies some of the groundbreaking advances that have been made regarding analytic capacity and its relationship to rectifiability in the decade 1995-2005.

Deals with 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. This book is suitable for researchers and postgraduates interested in pseudodifferential analysis.

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

Dynamical zeta functions are associated to dynamical systems with a countable set of periodic orbits. The dynamical zeta functions of the geodesic flow of lo cally symmetric spaces of rank one are known also as the generalized Selberg zeta functions.

The decomposition of the space L2 (G(Q)/G(/A), where G is a reductive group defined over (Q and /A is the ring of adeles of (Q, is a deep problem at the intersection of number and group theory. This book describes this proof in detail.

The framework of 'symmetry' provides an important route between the abstract theory and experimental observations. From the reviews:"[The] rich collection of examples makes the book...extremely useful for motivation and for spreading the ideas to a large Community."--MATHEMATICAL REVIEWS

Includes: Plenary Lectures; The Vassiliev Theory of Discriminants and Knots; Transparent Proofs and Limits to Approximation; Poisson Algebraic Groups and Representations of Quantum Groups at Roots of 1; Gauge Theory and Four-Manifold Topology; Spectral Theory and Geometry; and, Pattern Theory: A Unifying Perspective.

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.

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This volume presents original research articles and extended surveys related to the mathematical interest and work of Jean-Michel Bismut.

This open access book provides an extensive treatment of Hardy inequalities and closely related topics from the point of view of Folland and Stein's homogeneous (Lie) groups.

A wide range of mathematical disciplines play a significant role in this work: as well as various aspects of infinite group theory, these include finite simple groups and permutation groups, profinite groups, arithmetic groups and Strong Approximation, algebraic and analytic number theory, probability, and p-adic model theory.

The distri bution of prime numbers in arithmetic progressions is intimately connected with non-vanishing properties of various L-functions.

Combining algebraic groups and number theory, this volume gathers material from the representation theory of this group for the first time, doing so for both local (Archimedean and non-Archimedean) cases as well as for the global number field case.

The various contributions to this volume cover a broad range of topics in metric and differential geometry, including metric spaces, Ricci flow, Einstein manifolds, Kahler geometry, index theory, hypoelliptic Laplacian and analytic torsion.

This book is an introduction to the subject of mean curvature flow of hypersurfaces with special emphasis on the analysis of singularities. This flow occurs in the description of the evolution of numerous physical models where the energy is given by the area of the interfaces. These notes provide a detailed discussion of the classical parametric approach (mainly developed by R. Hamilton and G. Huisken). They are well suited for a course at PhD/PostDoc level and can be useful for any researcher interested in a solid introduction to the technical issues of the field. All the proofs are carefully written, often simplified, and contain several comments. Moreover, the author revisited and organized a large amount of material scattered around in literature in the last 25 years.

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.