This Seminar began in Moscow in November 1943 and has continued without interruption up to the present. We are happy that with this vol ume, Birkhiiuser has begun to publish papers of talks from the Seminar. It was, unfortunately, difficult to organize their publication before 1990. Since 1990, most of the talks have taken place at Rutgers University in New Brunswick, New Jersey. Parallel seminars were also held in Moscow, and during July, 1992, at IRES in Bures-sur-Yvette, France. Speakers were invited to submit papers in their own style, and to elaborate on what they discussed in the Seminar. We hope that readers will find the diversity of styles appealing, and recognize that to some extent this reflects the diversity of styles in a mathematical society. The principal aim was to have interesting talks, even if the topic was not especially popular at the time. The papers listed in the Table of Contents reflect some of the rich variety of ideas presented in the Seminar. Not all the speakers submit ted papers. Among the interesting talks that influenced the seminar in an important way, let us mention, for example, that of R. Langlands on per colation theory and those of J. Conway and J. McKay on sporadic groups. In addition, there were many extemporaneous talks as well as short discus sions.

The Seminar has taken place at Rutgers University in New Brunswick, New Jersey, since 1990 and it has become a tradition, starting in 1992, that the Seminar be held during July at IHES in Bures-sur-Yvette, France. This is the second Gelfand Seminar volume published by Birkhauser, the first having covered the years 1990-1992. Most of the papers in this volume result from Seminar talks at Rutgers, and some from talks at IHES. In the case of a few of the papers the authors did not attend, but the papers are in the spirit of the Seminar. This is true in particular of V. Arnold's paper. He has been connected with the Seminar for so many years that his paper is very natural in this volume, and we are happy to have it included here. We hope that many people will find something of interest to them in the special diversity of topics and the uniqueness of spirit represented here. The publication of this volume would be impossible without the devoted attention of Ann Kostant. We are extremely grateful to her. I. Gelfand J. Lepowsky M. Smirnov Questions and Answers About Geometric Evolution Processes and Crystal Growth Fred Almgren We discuss evolutions of solids driven by boundary curvatures and crystal growth with Gibbs-Thomson curvature effects. Geometric measure theo retic techniques apply both to smooth elliptic surface energies and to non differentiable crystalline surface energies.

The Seminar has taken place at Rutgers University in New Brunswick, New Jersey, since 1990 and it has become a tradition, starting in 1992, that the Seminar be held during July at IHES in Bures-sur-Yvette, France. This is the second Gelfand Seminar volume published by Birkhauser, the first having covered the years 1990-1992. Most of the papers in this volume result from Seminar talks at Rutgers, and some from talks at IHES. In the case of a few of the papers the authors did not attend, but the papers are in the spirit of the Seminar. This is true in particular of V. Arnold's paper. He has been connected with the Seminar for so many years that his paper is very natural in this volume, and we are happy to have it included here. We hope that many people will find something of interest to them in the special diversity of topics and the uniqueness of spirit represented here. The publication of this volume would be impossible without the devoted attention of Ann Kostant. We are extremely grateful to her. I. Gelfand J. Lepowsky M. Smirnov Questions and Answers About Geometric Evolution Processes and Crystal Growth Fred Almgren We discuss evolutions of solids driven by boundary curvatures and crystal growth with Gibbs-Thomson curvature effects. Geometric measure theo retic techniques apply both to smooth elliptic surface energies and to non differentiable crystalline surface energies.

Dedicated to the memory of Chih-Han Sah, this volume continues a long tradition of one of the most influential mathematical seminars of this century. A number of topics are covered, including combinatorial geometry, connections between logic and geometry, Lie groups, algebras and their representations. An additional area of importance is noncommutative algebra and geometry, and its relations to modern physics. Distinguished mathematicians contributing to this work: T.V. Alekseevskaya V. Kac

Singularities arise naturally in a huge number of different areas of mathematics and science. As a consequence, singularity theory lies at the crossroads of paths that connect many of the most important areas of applications of mathematics with some of its most abstract regions. The main goal in most problems of singularity theory is to understand the dependence of some objects of analysis, geometry, physics, or other science (functions, varieties, mappings, vector or tensor fields, differential equations, models, etc.) on parameters. The articles collected here can be grouped under three headings. (A) Singularities of real maps; (B) Singular complex variables; and (C) Singularities of homomorphic maps.

These two volumes contain eighteen invited papers by distinguished mathematicians in honor of the eightieth birthday of Israel M. Gelfand, one of the most remarkable mathematicians of our time. Gelfand has played a crucial role in the development of functional analysis during the last half-century. His work and his philosophy have in fact helped shape our understanding of the term 'functional analysis'. The papers in these volumes largely concern areas in which Gelfand has a very strong interest today, including geometric quantum field theory, representation theory, combinatorial structures underlying various 'continuous' constructions, quantum groups and geometry.

The Mathematics of Chip-firing is a solid introduction and overview of the growing field of chip-firing. It offers an appreciation for the richness and diversity of the subject. Chip-firing refers to a discrete dynamical system — a commodity is exchanged between sites of a network according to very simple local rules. Although governed by local rules, the long-term global behavior of the system reveals fascinating properties. The Fundamental properties of chip-firing are covered from a variety of perspectives. This gives the reader both a broad context of the field and concrete entry points from different backgrounds. Broken into two sections, the first examines the fundamentals of chip-firing, while the second half presents more general frameworks for chip-firing. Instructors and students will discover that this book provides a comprehensive background to approaching original sources. Features: Provides a broad introduction for researchers interested in the subject of chip-firing The text includes historical and current perspectives Exercises included at the end of each chapter About the Author: Caroline J. Klivans received a BA degree in mathematics from Cornell University and a PhD in applied mathematics from MIT. Currently, she is an Associate Professor in the Division of Applied Mathematics at Brown University. She is also an Associate Director of ICERM (Institute for Computational and Experimental Research in Mathematics). Before coming to Brown she held positions at MSRI, Cornell and the University of Chicago. Her research is in algebraic, geometric and topological combinatorics.