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This volume introduces equivariant homotopy, homology, and cohomology theory, along with various related topics in modern algebraic topology. It explains the main ideas behind some of the most striking recent advances in the subject. The book begins with a development of the equivariant algebraic topology of spaces culminating in a discussion of the Sullivan conjecture that emphasizes its relationship with classical Smith theory. It then introduces equivariant stable homotopy theory, the equivariant stable homotopy category, and the most important examples of equivariant cohomology theories. The basic machinery that is needed to make serious use of equivariant stable homotopy theory is presented next, along with discussions of the Segal conjecture and generalized Tate cohomology.Finally, the book gives an introduction to 'brave new algebra', the study of point-set level algebraic structures on spectra and its equivariant applications. Emphasis is placed on equivariant complex cobordism, and related results on that topic are presented in detail. It introduces many of the fundamental ideas and concepts of modern algebraic topology. It presents comprehensive material not found in any other book on the subject. It provides a coherent overview of many areas of current interest in algebraic topology. It surveys a great deal of material, explaining main ideas without getting bogged down in details.

In a very broad sense, 'spaces' are objects of study in geometry, and 'functions' are objects of study in analysis. There are, however, deep relations between functions defined on a space and the shape of the space, and the study of these relations is the main theme of Morse theory. In particular, its feature is to look at the critical points of a function, and to derive information on the shape of the space from the information about the critical points. Morse theory deals with both finite-dimensional and infinite-dimensional spaces. In particular, it is believed that Morse theory on infinite-dimensional spaces will become more and more important in the future as mathematics advances.This book describes Morse theory for finite dimensions. Finite-dimensional Morse theory has an advantage in that it is easier to present fundamental ideas than in infinite-dimensional Morse theory, which is theoretically more involved. Therefore, finite-dimensional Morse theory is more suitable for beginners to study. On the other hand, finite-dimensional Morse theory has its own significance, not just as a bridge to infinite dimensions. It is an indispensable tool in the topological study of manifolds. That is, one can decompose manifolds into fundamental blocks such as cells and handles by Morse theory, and thereby compute a variety of topological invariants and discuss the shapes of manifolds. These aspects of Morse theory will continue to be a treasure in geometry for years to come. This textbook aims at introducing Morse theory to advanced undergraduates and graduate students. It is the English translation of a book originally published in Japanese.

This book contains the collected works of A. Adrian Albert, a leading algebraist of the twentieth century. Albert made many important contributions to the theory of the Brauer group and central simple algebras, Riemann matrices, nonassociative algebras, and other topics. Part 1 focuses on associative algebras and Riemann matrices, and Part 2 on nonassociative algebras and miscellany. Because much of Albert's work remains of vital interest in contemporary research, this volume will interest mathematicians in a variety of areas.

This book contains papers of the outstanding and versatile mathematician, Witold Hurewicz. Preceding the collection are introductory articles describing Hurewicz's contributions to Borel sets, dimension theory, and algebraic topology. Hurewicz first studied set theory and dimension, and his papers on this topic are especially clear and precise, making them accessible to beginning mathematicians. His work in algebraic topology is marked by five fundamental papers which provide an introduction to homotopy groups and the Hurewicz Theorem concerning the relation between homotopy and singular homology. These papers are included here in their original form along with English translations. Each paper in the collection is followed by a review from one of the major reviewing journals. These reviews were written by eminent mathematicians and serve as excellent abstracts for the papers.

A memorial to the late John von Neumann.

These lectures emphasized specific areas of operations research and the mathematics used in modeling and solving the related problems. The lecturers were: Seth Bonder, Ralph L. Disney, Frederick C. Johnson, William P. Pierskalla, Robert B. Rovinsky, Warren E. Walker. Each lecturer attempted to make his presentation self-contained in terms of defining the application areas and mathematics employed.

The theory of networks is a very lively one, both in terms of developments in the theory itself and of the variety of its applications. This book, based on the 1981 AMS Short Course on the Mathematics of Networks, introduces most of the basic ideas of network theory and develops some of these ideas considerably. A number of more specialized topics are introduced, including areas of active research and a wide variety of applications. Frank Boesch gives the basic definitions in the mathematics of networks and in the closely-related topic of graph theory. He discusses two of the most fundamental network problems - the shortest path problem and the minimum spanning tree problem as well as some of their variants.Boesch also gives an interesting presentation in the area of network reliability. Frances Yao considers maximum flows in networks, the problem most often thought of in connection with the mathematics of networks. Richard Karp gives an account of the computational complexity of network problems. Using the case study method, Shen Lin demonstrates the effective use of heuristic algorithms in network design. Four applications of the mathematics of networks are presented by Daniel Kleitman. These include: the design of irrigation systems, the theory of electrical networks, the scheduling of delivery trucks, and the physics of ice. Finally, Nicholas Pippenger presents a chapter on telephone switching networks, an area of network theory that leads to difficult mathematics drawn from such apparently unrelated fields as harmonic analysis.

This book consists of two main parts. The first part, ""Hyperbolic and Parabolic Equations"", written by F. John, contains a well-chosen assortment of material intended to give an understanding of some problems and techniques involving hyperbolic and parabolic equations. The emphasis is on illustrating the subject without attempting to survey it. The point of view is classical, and this serves well in furnishing insight into the subject; it also makes it possible for the lectures to be read by someone familiar with only the fundamentals of real and complex analysis.The second part, ""Elliptic Equations"", written by L. Bers and M. Schechter, contains a very readable account of the results and methods of the theory of linear elliptic equations, including the maximum principle, Hilbert-space methods, and potential-theoretic methods. It also contains a brief discussion of some quasi-linear elliptic equations. The book is suitable for graduate students and researchers interested in partial differential equations.

This is the proceedings of the 1983 AMS Short Course given at Denver, Colorado. Computer communications is characterized by rapid technological advances presenting problems of a theoretical nature that are often very difficult to solve. They range from those that arise on a single chip, where communication among thousands of elements on a chip is influenced by electrical properties, to those that arise when human beings communicate with databases where the logical aspects of communications play a more important role. A variety of mathematical methods is needed to attempt to solve such problems; they vary from partial differential equations to temporal or modal logic. The article on ""Diffusion Approximation"" is probably one of the best reviews of this field. The book is unique in having contributors from a variety of different fields of computer communications. All of the articles are of high research value and are self-contained.

The lecture notes contained in this volume were presented at the AMS Short Course on Population Biology, held August 6-7, 1983, in Albany, New York in conjunction with the summer meeting of the American Mathematical Society. These notes will acquaint the reader with the mathematical ideas that pervade almost every level of thinking in population biology and provide an introduction to the many applications of mathematics in the field.Research mathematicians, college teachers of mathematics, and graduate students all should find this book of interest. Population biology is probably the oldest area in mathematical biology, but remains a constant source of new mathematical problems and the area of biology best integrated with mathematical theory. The need for mathematical approaches has never been greater, as evolutionary theory is challenged by new interpretations of the paleontological record and new discoveries at the molecular level, as world resources for feeding populations become limiting, as the problems of pollution increase, and as both animal and plant epidemiological problems receive closer scrutiny. A background of advanced calculus, introduction to ordinary and partial differential equations, and linear algebra will make the book accessible. All of the papers included have high research value. A list of the contents follows.

These introductory survey lectures, the result of a 1984 AMS Short Course, focus on the algorithmic problems arising in the construction and utilization of large-scale information systems. Addressed to both mathematicians and computer scientists, the lectures require a background in the methodologies of discrete mathematics, in particular the elements of algebra, combinatorics and graph theory, discrete probability, logic and the theory of computation. All of the articles either are of high research value or survey profound themes in current research. They cover the two fundamental aspects of the field, i.e., database systems and communication networks.An overview of database architectures, the theory of data dependencies, and transaction management are provided, respectively, by the articles of Jacobs, Fagin and Vardi, and Garcia-Molina. Chung evaluates problems in the design of communication networks. Miller's discussion of data compression algorithms links current research to classical information theory. Finally, Tuzhilin describes a general framework evolved in the Soviet Union for modelling problems of information processing.

This collection of six papers provides a valuable source of material on the real-world problem of allocating objects among competing claimants. The examples given show how mathematics, particularly the axiomatic method, can be applied to give insight into complex social problems. Originally presented as an AMS Short Course, these papers could serve as a suitable text for courses touching on game theory, decision sciences, economics, or quantitative political science. Most of the material is accessible to the mathematically mature undergraduate with a background in advanced calculus and algebra. Each article surveys the recent literature and includes statements and sketches of proofs, as well as unsolved problems which should excite student curiosity.The articles analyze the question of fair allocation via six examples: the apportionment of political representation, the measurement of income inequality, the allocation of joint costs, the levying of taxes, the design of voting laws, and the framing of auction procedures. In each of these examples fairness has a somewhat different significance, but common axiomatic threads reveal broad underlying principles. Each of the topics is concerned with norms of comparative equity for evaluating allocations or with standards of procedures for effecting them; it is this focus on normative properties which suggests that a mathematical analysis is appropriate. Though game theory provides a useful tool in many of these allocation problems, the emphasis here is on standards rather than strategy and equity rather than rationality, an approach which more accurately mirrors real-world social problems.

These lecture notes from the 1985 AMS Short Course examine a variety of topics from the contemporary theory of actuarial mathematics. Recent clarification in the concepts of probability and statistics has laid a much richer foundation for this theory. Other factors that have shaped the theory include the continuing advances in computer science, the flourishing mathematical theory of risk, developments in stochastic processes, and recent growth in the theory of finance. In turn, actuarial concepts have been applied to other areas such as biostatistics, demography, economic, and reliability engineering.

In the 100 years since the founding of the AMS, the American mathematical community has grown from a small group heavily dependent on European mathematicians to a large and influential group that in many areas sets the standard for the rest of the world. By the 1930s, there was a flourishing mathematical community to welcome the influx of mathematicians fleeing Europe. This volume, the first in the History of Mathematics series, brings together a variety of perspectives on the political, social, and mathematical forces that have shaped the American mathematical community in the past century.

The first section of the book deals with some of the influential mathematics departments in the United States. Functioning as centers of research and training, these departments played a major role in shaping the mathematical life in this country. The second section deals with an extraordinary conference held at Princeton in 1946 to commemorate the university's bicentennial. The influence of women in American mathematics, the burgeoning of differential geometry in the last 50 years, and discussions of the work of von Karman and Weiner are among other topics covered.

Computational complexity theory is the study of the quantitative laws that govern computing. During the last 25 years, this field has grown into a rich mathematical theory. Currently one of the most active research areas in computer science, complexity theory is of considerable interest to mathematicians as well, since some of the key open problems in this field raise basic questions about the nature of mathematics. Many experts in complexity theory believe that, in coming decades, the strongest influence on the development of mathematics will come from the extended use of computing and from concepts and problems arising in computer science.This volume contains the proceedings of the AMS Short Course on Computational Complexity Theory, held at the Joint Mathematics Meetings in Atlanta in January 1988. The purpose of the short course was to provide an overview of complexity theory and to describe some of the current developments in the field. The papers presented here represent contributions by some of the top experts in this burgeoning area of research.

The present book is the third and final volume in the ""A Century of Mathematics in America"" collection. The theme of the second volume is continued here in articles describing the mathematics and the mathematical personalities in some of the nations' prominent departments: Johns Hopkins, Clark, Columbia, MIT, Michigan, Texas, and the Institute for Advanced Study.

This volume contains the proceedings of a highly successful AMS Short Course on Chaos and Fractals, held during the AMS Centennial Celebration in Providence, Rhode Island in August 1988. Chaos and fractals have been the subject of great interest in recent years and have proven to be useful in a variety of areas of mathematics and the sciences. The purpose of the short course was to provide a solid introduction to the mathematics underlying the notions of chaos and fractals. The papers in this book range over such topics as dynamical systems theory, Julia sets, the Mandelbrot set, attractors, the Smale horseshoe, calculus on fractals, and applications to data compression. The authors represented here are some of the top experts in this field. Aimed at beginning graduate students, college and university mathematics instructors, and non-mathematics researchers, this book provides readable expositions of several exciting topics of contemporary research.

This volume, a joint publication with the American Institute of Physics, contains the proceedings of a symposium honoring the memory of Josiah Willard Gibbs, one of the giants of theoretical physics. Three articles provide perspectives on Gibbs, the man, and on the place his work occupies in the history of science. There are also contributions from leading scientists on statistical mechanics, thermodynamics, geophysics, number theory, general relativity, and economics.