Complex Polynomials explores the geometric theory of polynomials and rational functions in the plane. Early chapters build the foundations of complex variable theory, melding together ideas from algebra, topology, and analysis. Throughout the book, the author introduces a variety of ideas and constructs theories around them, incorporating much of the classical theory of polynomials as he proceeds. These ideas are used to study a number of unsolved problems. Several solutions to problems are given, including a comprehensive account of the geometric convolution theory.

Over the past two decades a general mathematical theory of infinite electrical networks has been developed. This is the first book to present the salient features of this theory in a coherent exposition. Using the basic tools of functional analysis and graph theory, the author presents the fundamental developments of the past two decades and discusses applications to other areas of mathematics. The first half of the book presents existence and uniqueness theorems for both infinite-power and finite-power voltage-current regimes, and the second half discusses methods for solving problems in...

The theory of modular forms and especially the so-called Ramanujan Conjectures have recently been applied to resolve problems in combinatorics, computer science, analysis, and number theory. Professor Sarnak begins by developing the necessary background material in modular forms. He then considers in detail the solution of three problems: the Rusiewisz problem concerning finitely additive rotationally invariant measures on the sphere; the explicit construction of highly connected but sparse graphs, e.g. expander graphs and Ramanujan graphs; and the Linnik problem concerning the distribution...

This book presents the definitive account of the applications of this algebra to the surgery classification of topological manifolds. The central result is the identification of a manifold structure in the homotopy type of a Poincare duality space with a local quadratic structure in the chain homotopy type of the universal cover. The difference between the homotopy types of manifolds and Poincare duality spaces is identified with the fibre of the algebraic L-theory assembly map, which passes from local to global quadratic duality structures on chain complexes. The algebraic L-theory assembly...

Sporadic Groups provides for the first time a self-contained treatment of the foundations of the theory of sporadic groups accessible to mathematicians with a basic background in finite groups, such as in the author's text Finite Group Theory. Introductory material useful for studying the sporadics, such as a discussion of large extraspecial 2-subgroups and Tits' coset geometries, opens the book. A construction of the Mathieu groups as the automorphism groups of Steiner systems follows. The Golay and Todd modules and the 2-local geometry for M24 are discussed. This is followed by the standard...

This volume explores Diophantine approximation on smooth manifolds embedded in Euclidean space, developing a coherent body of theory comparable to that of classical Diophantine approximation. In particular, the book deals with Khintchine-type theorems and with the Hausdorff dimension of the associated null sets. After setting out the necessary background material, the authors give a full discussion of Hausdorff dimension and its uses in Diophantine approximation. They employ a wide range of techniques from the number theory arsenal to obtain the upper and lower bounds required, highlighting...

Building on min-max methods, Professor Ghoussoub systematically develops a general theory that can be applied in a variety of situations. In so doing he also presents a whole new array of duality and perturbation methods. The prerequisites for following this book are relatively few; an appendix sketching certain methods in analysis makes the book self-contained.

Affine differential geometry has undergone a period of revival and rapid progress in the past decade. This book is a self-contained and systematic account of affine differential geometry from a contemporary view. It covers not only the classical theory, but also introduces the modern developments of the past decade. The authors have concentrated on the significant features of the subject and their relationship and application to such areas as Riemannian, Euclidean, Lorentzian and projective differential geometry. In so doing, they also provide a modern introduction to the latter. Some of the...

This ground-breaking work combines the classic (the zeta-function) with the modern (the spectral theory) to create a comprehensive but elementary treatment of spectral resolution. The story starts with a basic but unabridged treatment of the spectral resolution of the non-Euclidean Laplacian and the trace formulas. The author achieves this by the use of standard tools from analysis rather than any heavy machinery, forging a substantial aid for beginners in spectral theory. These ideas are then utilized to unveil a new image of the zeta-function, revealing it as the main gem of a necklace...

Representation theory has applications to number theory, combinatorics and many areas of algebra. The aim of this text is to present some of the key results in the representation theory of finite groups. Professor Alperin concentrates on local representation theory, emphasizing module theory throughout. In this way many deep results can be obtained rather quickly. After two introductory chapters, the basic results of Green are proved, which in turn lead in due course to Brauer's First Main Theorem. A proof of the module form of Brauer's Second Main Theorem is then presented, followed by a...