This book is about algebro-geometric solutions of completely integrable nonlinear partial differential equations in (1+1)-dimensions; also known as soliton equations. Explicitly treated integrable models include the KdV, AKNS, sine-Gordon, and Camassa-Holm hierarchies as well as the classical massive Thirring system. An extensive treatment of the class of algebro-geometric solutions in the stationary and time-dependent contexts is provided. The formalism presented includes trace formulas, Dubrovin-type initial value problems, Baker-Akhiezer functions, and theta function representations of all...

This book provides a comprehensive account of a key, perhaps the most important, theory that forms the basis of Taylor-Wiles proof of Fermat's last theorem. Hida begins with an overview of the theory of automorphic forms on linear algebraic groups and then covers the basic theory and recent results on elliptic modular forms, including a substantial simplification of the Taylor-Wiles proof by Fujiwara and Diamond. He offers a detailed exposition of the representation theory of profinite groups (including deformation theory), as well as the Euler characteristic formulas of Galois cohomology...

This second edition develops the foundations of finite group theory. For students already exposed to a first course in algebra, it serves as a text for a course on finite groups. For the reader with some mathematical sophistication but limited knowledge of finite group theory, the book supplies the basic background necessary to begin to read journal articles in the field. It also provides the specialist in finite group theory with a reference on the foundations of the subject. Unifying themes include the Classification Theorem and the classical linear groups. Lie theory appears in chapters on...

This is a modern introduction to Kaehlerian geometry and Hodge structure. Coverage begins with variables, complex manifolds, holomorphic vector bundles, sheaves and cohomology theory (with the latter being treated in a more theoretical way than is usual in geometry). The book culminates with the Hodge decomposition theorem. In between, the author proves the Kaehler identities, which leads to the hard Lefschetz theorem and the Hodge index theorem. The second part of the book investigates the meaning of these results in several directions.

The second volume of this modern account of Kaehlerian geometry and Hodge theory starts with the topology of families of algebraic varieties. The main results are the generalized Noether-Lefschetz theorems, the generic triviality of the Abel-Jacobi maps, and most importantly, Nori's connectivity theorem, which generalizes the above. The last part deals with the relationships between Hodge theory and algebraic cycles. The text is complemented by exercises offering useful results in complex algebraic geometry. Also available: Volume I 0-521-80260-1 Hardback $60.00 C

Starting from the classical finite-dimensional Galois theory of fields, this book develops Galois theory in a much more general context. The authors first formalize the categorical context in which a general Galois theorem holds, and then give applications to Galois theory for commutative rings, central extensions of groups, the topological theory of covering maps and a Galois theorem for toposes. The book is designed to be accessible to a wide audience, the prerequisites are first courses in algebra and general topology, together with some familiarity with the categorical notions of limit...

Incorporated in this volume are the first two books in Mukai's series on Moduli Theory. The notion of a moduli space is central to geometry. However, its influence is not confined there; for example, the theory of moduli spaces is a crucial ingredient in the proof of Fermat's last theorem. Researchers and graduate students working in areas ranging from Donaldson or Seiberg-Witten invariants to more concrete problems such as vector bundles on curves will find this to be a valuable resource. Among other things this volume includes an improved presentation of the classical foundations of...

This is a new edition of the now classic text. The already extensive treatment given in the first edition has been heavily revised by the author. The addition of two new sections, numerous new results and 150 references means that this represents an up-to-date and comprehensive account of random graph theory. The theory estimates the number of graphs of a given degree that exhibit certain properties. It not only has numerous combinatorial applications, but also serves as a model for the probabilistic treatment of more complicated random structures. This book, written by an acknowledged expert...

This classic textbook, now reissued, offers a clear exposition of modern probability theory and of the interplay between the properties of metric spaces and probability measures. The new edition has been made even more self-contained than before; it now includes a foundation of the real number system and the Stone-Weierstrass theorem on uniform approximation in algebras of functions. Several other sections have been revised and improved, and the comprehensive historical notes have been further amplified. A number of new exercises have been added, together with hints for solution.

R.A. Bailey covers in this study the mathematics of association schemes--an area lying between pure mathematics and statistics that relates to the optimal design of scientific experiments. The book is accessible to mathematicians as well as statisticians. Arising from a graduate course taught by the author, it appeals to students as well as researchers as a valuable reference work from which to learn about the statistical/combinatorial aspects of their work.