This volume contains the proceedings of the NSF-CBMS Regional Conference on Algebraic Geometry, held in Sundance, Utah in July 1988. The conference focused on algebraic curves and related varieties. Some of the papers collected here represent lectures delivered at the conference, some report on research done during the conference, while others describe related work carried out elsewhere.

Symbolic dynamics originated as a tool for analyzing dynamical systems and flows by discretizing space as well as time. The development of information theory gave impetus to the study of symbol sequences as objects in their own right. Today, symbolic dynamics has expanded to encompass multi-dimensional arrays of symbols and has found diverse applications both within and beyond mathematics. This volume is based on the AMS Short Course on Symbolic Dynamics and its Applications. It contains introductory articles on the fundamental ideas of the field and on some of its applications. Topics include the use of symbolic dynamics techniques in coding theory and in complex dynamics, the relation between the theory of multi-dimensional systems and the dynamics of tilings, and strong shift equivalence theory. Contributors to the volume are experts in the field and are clear expositors. The book is suitable for graduate students and research mathematicians interested in symbolic dynamics and its applications.

Table of Contents: D. Duffie: Martingales, Arbitrage, and Portfolio Choice • J. Fröhlich: Mathematical Aspects of the Quantum Hall Effect • M. Giaquinta: Analytic and Geometric Aspects of Variational Problems for Vector Valued Mappings • U. Hamenstädt: Harmonic Measures for Leafwise Elliptic Operators Along Foliations • M. Kontsevich: Feynman Diagrams and Low-Dimensional Topology • S.B. Kuksin: KAM-Theory for Partial Differential Equations • M. Laczkovich: Paradoxical Decompositions: A Survey of Recent Results • J.-F. Le Gall: A Path-Valued Markov Process and its Connections with Partial Differential Equations • I. Madsen: The Cyclotomic Trace in Algebraic K-Theory • A.S. Merkurjev: Algebraic K-Theory and Galois Cohomology • J. Nekovár: Values of L-Functions and p-Adic Cohomology • Y.A. Neretin: Mantles, Trains and Representations of Infinite Dimensional Groups • M.A. Nowak: The Evolutionary Dynamics of HIV Infections • R. Piene: On the Enumeration of Algebraic Curves - from Circles to Instantons • A. Quarteroni: Mathematical Aspects of Domain Decomposition Methods • A. Schrijver: Paths in Graphs and Curves on Surfaces • B. Silverman: Function Estimation and Functional Data Analysis • V. Strassen: Algebra and Complexity • P. Tukia: Generalizations of Fuchsian and Kleinian Groups • C. Viterbo: Properties of Embedded Lagrange Manifolds • D. Voiculescu: Alternative Entropies in Operator Algebras • M. Wodzicki : Algebraic K-Theory and Functional Analysis • D. Zagier: Values of Zeta Functions and Their Applications

1989 marked the 150th anniversary of the birth of the great Danish mathematician Hieronymus George Zeuthen. Zeuthen's name is known to every algebraic geometer because of his discovery of a basic invariant of surfaces. However, he also did fundamental research in intersection theory, enumerative geometry, and the projective geometry of curves and surfaces. Zeuthen's extraordinary devotion to his subject, his characteristic depth, thoroughness, and clarity of thought, and his precise and succinct writing style are truly inspiring. During the past ten years or so, algebraic geometers have reexamined Zeuthen's work, drawing from it inspiration and new directions for development in the field. The 1989 Zeuthen Symposium, held in the summer of 1989 at the Mathematical Institute of the University of Copenhagen, provided a historic opportunity for mathematicians to gather and examine those areas in contemporary mathematical research which have evolved from Zeuthen's fruitful ideas. This volume, containing papers presented during the symposium, as well as others inspired by it, illuminates some currently active areas of research in enumerative algebraic geometry.

Two meetings of the AMS in the autumn of 1989 - one at the Stevens Institute of Technology and the other at Ball State University - included Special Sessions on the role of p-adic methods in number theory and algebraic geometry. This volume grew out of these Special Sessions. Drawn from a wide area of mathematics, the articles presented here provide an excellent sampling of the broad range of trends and applications in p-adic methods.

In the mid-1960's, several Italian mathematicians began to study the connections between classical arguments in commutative algebra and algebraic geometry, and the contemporaneous development of algebraic K-theory in the US. These connections were exemplified by the work of Andreotti-Bombieri, Salmon, and Traverso on seminormality, and by Bass-Murthy on the Picard groups of polynomial rings. Interactions proceeded far beyond this initial point to encompass Chow groups of singular varieties, complete intersections, and applications of K-theory to arithmetic and real geometry. This volume contains the proceedings from a US-Italy Joint Summer Seminar, which focused on this circle of ideas. The conference, held in June 1989 in Santa Margherita Ligure, Italy, was supported jointly by the Consiglio Nazionale delle Ricerche and the National Science Foundation. The book contains contributions from some of the leading experts in this area.

Introduces some of the main ideas of modern intersection theory, traces their origins in classical geometry and sketches a few typical applications. Suitable for graduate students in mathematics, this book describes the construction and computation of intersection products by means of the geometry of normal cones.

The papers collected here present an up-to-date record of the current research developments in the fields of real algebraic geometry and quadratic forms. Articles range from the technical to the expository and there are also indications to new research directions.

The Geometry of Algebraic Fermi Curves deals with the geometry of algebraic Fermi curves, with emphasis on the inverse spectral problem. Topics covered include the periodic Schrödinger operator and electrons in a crystal; one-dimensional algebraic Bloch varieties; separable Bloch varieties; and monodromy for separable and generic Bloch varieties. Compactification, the potential zero, and density of states are also discussed. This book consists of 13 chapters and begins by recalling the static lattice approximation for electronic motion at low temperature in a pure, finite sample of a d-dimensional crystal. The position of the Fermi energy and the geometry of the Fermi hypersurface in relation to the metallic properties of the crystal are described. The following chapters focus on the Bloch variety associated with a discrete two-dimensional periodic Schrödinger operator; algebraic Bloch varieties in one dimension; compactification of the Bloch variety; and the potential zero. The geometry of the Bloch variety of a separable potential is also considered, along with the topology of the family of Fermi curves. The final chapter demonstrates how the Bloch variety is determined by the density of states. This monograph will be a useful resource for students and teachers of mathematics.