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The present monograph further develops the study, via the techniques of combinatorial anabelian geometry, of the profinite fundamental groups of configuration spaces associated to hyperbolic curves over algebraically closed fields of characteristic zero.The starting point of the theory of the present monograph is a combinatorial anabelian result which allows one to reduce issues concerning the anabelian geometry of configuration spaces to issues concerning the anabelian geometry of hyperbolic curves, as well as to give purely group-theoretic characterizations of the cuspidal inertia subgroups of one-dimensional subquotients of the profinite fundamental group of a configuration space.We then turn to the study of tripod synchronization, i.e., of the phenomenon that an outer automorphism of the profinite fundamental group of a log configuration space associated to a stable log curve induces the same outer automorphism on certain subquotients of such a fundamental group determined by tripods [i.e., copies of the projective line minus three points]. The theory of tripod synchronization shows that such outer automorphisms exhibit somewhat different behavior from the behavior that occurs in the case of discrete fundamental groups and, moreover, may be applied to obtain various strong results concerning profinite Dehn multi-twists.In the final portion of the monograph, we develop a theory of localizability, on the dual graph of a stable log curve, for the condition that an outer automorphism of the profinite fundamental group of the stable log curve lift to an outer automorphism of the profinite fundamental group of a corresponding log configuration space. This localizability is combined with the theory of tripod synchronization to construct a purely combinatorial analogue of the natural outer surjection from the etale fundamental group of the moduli stack of hyperbolic curves over the field of rational numbers to the absolute Galois group of the field of rational numbers.

This book gives a proof of Cherlin¿s conjecture for finite binary primitive permutation groups. Motivated by the part of model theory concerned with Lachlan¿s theory of finite homogeneous relational structures, this conjecture proposes a classification of those finite primitive permutation groups that have relational complexity equal to 2. The first part gives a full introduction to Cherlin¿s conjecture, including all the key ideas that have been used in the literature to prove some of its special cases. The second part completes the proof by dealing with primitive permutation groups that are almost simple with socle a group of Lie type. A great deal of material concerning properties of primitive permutation groups and almost simple groups is included, and new ideas are introduced. Addressing a hot topic which cuts across the disciplines of group theory, model theory and logic, this book will be of interest toa wide range of readers. It will be particularly useful for graduate students and researchers who need to work with simple groups of Lie type.

The papers in this volume reflect the richness and diversity of the subject of dynamics. Some are lectures given at the three conferences (Ergodic Theory and Topological Dynamics, Symbolic Dynamics and Coding Theory and Smooth Dynamics, Dynamics and Applied Dynamics) held in Maryland between October 1986 and March 1987; some are work which was in progress during the Special Year, and some are work which was done because of questions and problems raised at the conferences. In addition, a paper of John Milnor and William Thurston, versions of which had been available as notes but not yet published, is included.

The Taniguchi Symposium on global analysis on manifolds focused mainly on the relationships between some geometric structures of manifolds and analysis, especially spectral analysis on noncompact manifolds. Included in the present volume are expanded versions of most of the invited lectures. In these original research articles, the reader will find up-to date accounts of the subject.

The Latin American School of Mathematics (ELAM) is one of the most important mathematical events in Latin America. It has been held every other year since 1968 in a different country of the region, and its theme varies according to the areas of interest of local research groups. The subject of the 1986 school was Partial Differential Equations with emphasis on Microlocal Analysis, Scattering Theory and the applications of Nonlinear Analysis to Elliptic Equations and Hamiltonian Systems.

This volume contains original research papers on topics central to Dynamical Systems, such as fractional dimensions (Hausdorff dimension, limity capacity) and limit cycles of polynomial vector fields concerning the well-known Dulac and Hilbert's 16th problems. Stability and bifurcations, intermittency, normal forms, Anosov flows and foliations are also themes treated in the papers. Many of the authors are renowned for their important contributions to the field. This volume should be of much interest to people working in dynamical systems, including, physicists, biologists and engineers.

The main theme of the meeting was to illustrate the use of stochastic processes in the study of topological problems in quantum physics and statistical mechanics. Much discussion of current problems was generated and there was a considerable amount of interaction between mathematicians and physicists. The papers presented in the proceedings are essentially of a research nature but some (Lewis, Hudson) are introductions or surveys.

At the Summer School Saint Petersburg 2001, the main lecture courses bore on recent progress in asymptotic representation theory: those written up for this volume deal with the theory of representations of infinite symmetric groups, and groups of infinite matrices over finite fields; Riemann-Hilbert problem techniques applied to the study of spectra of random matrices and asymptotics of Young diagrams with Plancherel measure; the corresponding central limit theorems; the combinatorics of modular curves and random trees with application to QFT; free probability and random matrices, and Hecke algebras.

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Spin glass theory is going through a stunning period of progress while finding exciting new applications in areas beyond theoretical physics, in particular in combinatorics and computer science. This collection of state-of-the-art review papers written by leading experts in the field covers the topic from a wide variety of angles. The topics covered are mean field spin glasses, including a pedagogical account of Talagrand's proof of the Parisi solution, short range spin glasses, emphasizing the open problem of the relevance of the mean-field theory for lattice models, and the dynamics of spin glasses, in particular the problem of ageing in mean field models.The book will serve as a concise introduction to the state of the art of spin glass theory, useful to both graduate students and young researchers, as well as to anyone curious to know what is going on in this exciting area of mathematical physics.

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