Dating back to work of Berthelot, rigid cohomology appeared as a common generalization of Monsky-Washnitzer cohomology and crystalline cohomology. It is a p-adic Weil cohomology suitable for computing Zeta and L-functions for algebraic varieties on finite fields. Moreover, it is effective, in the sense that it gives algorithms to compute the number of rational points of such varieties. This is the first book to give a complete treatment of the theory, from full discussion of all the basics to descriptions of the very latest developments. Results and proofs are included that are not available elsewhere, local computations are explained, and many worked examples are given. This accessible tract will be of interest to researchers working in arithmetic geometry, p-adic cohomology theory, and related cryptographic areas.

Edward Witten once said that Elliptic Cohomology was a piece of 21st Century Mathematics that happened to fall into the 20th Century. He also likened our understanding of it to what we know of the topography of an archipelago; the peaks are beautiful and clearly connected to each other, but the exact connections are buried, as yet invisible. This very active subject has connections to algebraic topology, theoretical physics, number theory and algebraic geometry, and all these connections are represented in the sixteen papers in this volume. A variety of distinct perspectives are offered, with topics including equivariant complex elliptic cohomology, the physics of M-theory, the modular characteristics of vertex operator algebras, and higher chromatic analogues of elliptic cohomology. This is the first collection of papers on elliptic cohomology in almost twenty years and gives a broad picture of the state of the art in this important field of mathematics.

The book provides an introduction to the theory of cluster sets, a branch of topological analysis which has made great strides in recent years. The cluster set of a function at a particular point is the set of limit values of the function at that point which may be either a boundary point or (in the case of a non-analytic function) an interior point of the function's domain. In topological analysis, its main application is to problems arising in the theory of functions of a complex variable, with particular reference to boundary behaviour such as the theory of prime ends under conformal mapping. An important and novel feature of the book is the discussion of more general applications to non-analytic functions, including arbitrary functions. The authors assume a general familiarity with classical function theory but include the more specialised material required for the development of the theory of cluster sets, so making the treatment accessible to graduate students.

Concerned with two fundamental problems in low-dimensional topology, the D(2)-problem and the realization problem, F.E.A. Johnson develops general methods and provides complete solutions in some instances. His book is suitable for graduate students wanting to learn low-dimensional homotopy theory as well as established researchers in the field.

This text provides an introduction to basic concepts in differential topology, differential geometry, and differential equations, and some of the main basic theorems in all three areas: for instance, the existence, uniqueness, and smoothness theorems for differential equations and the flow of a vector field; the basic theory of vector bundles including the existence of tubular neighborhoods for a submanifold; the calculus of differential forms; basic notions of symplectic manifolds, including the canonical 2-form; sprays and covariant derivatives for Riemannian and pseudo-Riemannian manifolds; applications to the exponential map, including the Cartan-Hadamard theorem and the first basic theorem of calculus of variations. Although the book grew out of the author's earlier book "Differential and Riemannian Manifolds", the focus has now changed from the general theory of manifolds to general differential geometry, and includes new chapters on Jacobi lifts, tensorial splitting of the double tangent bundle, curvature and the variation formula, a generalization of the Cartan-Hadamard theorem, the semiparallelogram law of Bruhat-Tits and its equivalence with seminegative curvature and the exponential map distance increasing property, a major example of seminegative curvature (the space of positive definite symmetric real matrices), automorphisms and symmetries, and immersions and submersions. These are all covered for infinite-dimensional manifolds, modeled on Banach and Hilbert Spaces, at no cost in complications, and some gain in the elegance of the proofs. In the finite-dimensional case, differential forms of top degree are discussed, leading to Stokes' theorem (even for manifolds with singular boundary), and several of its applications to the differential or Riemannian case. Basic formulas concerning the Laplacian are given, exhibiting several of its features in immersions and submersions.

The heart of the book is a lengthy introduction to the representation theory of finite dimensional algebras, in which the techniques of quivers with relations and almost split sequences are discussed in some detail.

This is the first of two volumes providing an introduction to modern developments in the representation theory of finite groups and associative algebras, which have transformed the subject into a study of categories of modules. Thus, Dr. Benson's unique perspective in this book incorporates homological algebra and the theory of representations of finite-dimensional algebras. This volume is primarily concerned with the exposition of the necessary background material, and the heart of the discussion is a lengthy introduction to the (Auslander-Reiten) representation theory of finite dimensional algebras, in which the techniques of quivers with relations and almost-split sequences are discussed in some detail.

This book provides a detailed but concise account of the theory of structure of finite p-groups admitting p-automorphisms with few fixed points. The relevant preliminary material on Lie rings is introduced and the main theorems of the book on the solubility of finite p-groups are then presented. The proofs involve notions such as viewing automorphisms as linear transformations, associated Lie rings, powerful p-groups, and the correspondences of A. I. Mal'cev and M. Lazard given by the Baker-Hausdorff formula. Many exercises are included. This book is suitable for graduate students and researchers working in the fields of group theory and Lie rings.

This work consists of two sections on the moduli spaces of vector bundles. The first part tackles the classification of vector bundles on algebraic curves. The author also discusses the construction and elementary properties of the moduli spaces of stable bundles. In particular Le Potier constructs HilbertSHGrothendieck schemes of vector bundles, and treats Mumford's geometric invariant theory. The second part centers on the structure of the moduli space of semistable sheaves on the projective plane. The author sketches existence conditions for sheaves of given rank, and Chern class and construction ideas in the general context of projective algebraic surfaces. Professor Le Potier provides a treatment of vector bundles that will be welcomed by experienced algebraic geometers and novices alike.

Cohomology of Drinfeld Modular Varieties aims to provide an introduction to this subject and to the Langlands correspondence for function fields. These varieties are the analogues for function fields of the Shimura varieties over number fields. The Langlands correspondence is a conjectured link between automorphic forms and Galois representations over a global field. By analogy with the number-theoretic case, one expects to establish the conjecture for function fields by studying the cohomology of Drinfeld modular varieties, which has been done by Drinfeld himself for the rank two case. This second volume is concerned with the ArthurSHSelberg trace formula, and to the proof in some cases of the Ramanujan-Petersson conjecture and the global Langlands conjecture for function fields. The author uses techniques that are extensions of those used to study Shimura varieties. Though the author considers only the simpler case of function rather than number fields, many important features of the number field case can be illustrated. Several appendices on background material keep the work reasonably self-contained. This book will be of much interest to all researchers in algebraic number theory and representation theory.

The ends of a topological space are the directions in which it becomes noncompact by tending to infinity. The tame ends of manifolds are particularly interesting, both for their own sake, and for their use in the classification of high-dimensional compact manifolds. The book is devoted to the related theory and practice of ends, dealing with manifolds and CW complexes in topology and chain complexes in algebra. The first part develops a homotopy model of the behavior at infinity of a noncompact space. The second part studies tame ends in topology. The authors show tame ends to have a uniform structure, with a periodic shift map. They use approximate fibrations to prove that tame manifold ends are the infinite cyclic covers of compact manifolds. The third part translates these topological considerations into an appropriate algebraic context, relating tameness to homological properties and algebraic K- and L-theory. This book will appeal to researchers in topology and geometry.

The Novikov Conjecture is the single most important unsolved problem in the topology of high-dimensional non-simply connected manifolds. The two volumes Novikov Conjectures, Index Theorems, and Rigidity are the outgrowth of a conference held at the Mathematisches Forschungsinstitut Oberwolfach (Germany) in September, 1993, on the subject of the title. They are intended to give a snapshot of the status of work on the Novikov Conjecture and related topics from many points of view: geometric topology, homotopy theory, algebra, geometry, analysis. Volume 2 contains: fundamental long research papers on bounded K-theory and the assembly map in algebraic K-theory, and on Epsilon surgery theory; shorter research and survey papers on various topics related to the Novikov Conjecture.

The Novikov Conjecture is the single most important unsolved problem in the topology of high-dimensional non-simply connected manifolds. These two volumes are the outgrowth of a conference held at the Mathematisches Forschungsinstitut Oberwolfach (Germany) in September 1993, on the subject of 'Novikov Conjectures, Index Theorems and Rigidity'. They are intended to give a snapshot of the status of work on the Novikov Conjecture and related topics from many points of view: geometric topology, homotopy theory, algebra, geometry and analysis. Volume 1 contains: * A detailed historical survey and bibliography of the Novikov Conjecture and of related subsequent developments, including an annotated reprint (both in the original Russian and in English translation) of Novikov's original 1970 statement of his conjecture * An annotated problem list * The texts of several important unpublished classic papers by Milnor, Browder, and Kasparov * Research/survey papers on the Novikov Conjecture by Ferry/Weinberger, Gromov, Mishchenko, Quinn, Ranicki, and Rosenberg.

Successive waves of migrant concepts, largely from mathematical physics, have stimulated the study of vector bundles over algebraic varieties in the past few years. But the subject has retained its roots in old questions concerning subvarieties of projective space. The 1993 Durham Symposium on vector bundles in algebraic geometry brought together some of the leading researchers in the field to further explore these interactions. This book is a collection of survey articles by the main speakers at the Symposium and presents to the mathematical world an overview of the key areas of research involving vector bundles. Topics include augmented bundles and coherent systems which link gauge theory and geometric invariant theory; Donaldson invariants of algebraic surfaces; Floer homology and quantum cohomology; conformal field theory and the moduli spaces of bundles on curves; the Horrocks-Mumford bundle and codimension 2 subvarieties in p4 and p5; and exceptional bundles and stable sheaves on projective space. This book will appeal greatly to mathematicians working in algebraic geometry and areas adjoining mathematical physics.

In geometric knot theory, a central issue is to study the various geometric properties of knots when the knots have certain thickness. This setting makes a knot more like one that is tied with a uniform physical rope. These problems are mostly motivated by the recent applications of knot theory in fields such as biology and polymer chemistry. In this book, the authors first give a brief review of the basic concepts and terminologies such as the thickness of a knot and the rope length of a knot. They then review the main results in this field. The topics include results on the global minimum rope length of knots, various lower and upper rope length bounds of knots in terms of their crossing numbers, and lower and upper bounds on the total curvatures of thick knots. Some special families of knots or under different settings, such as lattice knots and smooth knots are also considered. While some proofs are omitted or only outlined due to the page limitation of the book, many important ideas, methods, and theorems are explained in depth. At the end of the book, a list of some open problems in this field is given.

Etale cohomology is an important branch in arithmetic geometry. This book covers the main materials in SGA 1, SGA 4, SGA 4 1/2 and SGA 5 on etale cohomology theory, which includes decent theory, etale fundamental groups, Galois cohomology, etale cohomology, derived categories, base change theorems, duality, and -adic cohomology. The prerequisites for reading this book are basic algebraic geometry and advanced commutative algebra.

This book presents a definitive account of the applications of the algebraic L-theory 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 map is used to give a purely algebraic formulation of the Novikov conjectures on the homotopy invariance of the higher signatures; any other formulation necessarily factors through this one. The book is designed as an introduction to the subject, accessible to graduate students in topology; no previous acquaintance with surgery theory is assumed, and every algebraic concept is justified by its occurrence in topology.

Traditionally the Adams-Novikov spectral sequence has been a tool which has enabled the computation of generators and relations to describe homotopy groups. Here a natural geometric description of the sequence is given in terms of cobordism theory and manifolds with singularities. The author brings together many interesting results not widely known outside the USSR, including some recent work by Vershinin. This book will be of great interest to researchers into algebraic topology.

This updated and expanded second edition of an established text presents a detailed exposition of the modern theory of supermanifolds, including a rigorous account of the superanalogs of all the basic structures of ordinary manifold theory.

This is the first unified treatment in book form of the lower K-groups of Bass and the lower L-groups of the author. These groups arise as the Grothendieck groups of modules and quadratic forms which are components of the K- and L-groups of polynomial extensions. They are important in the topology of non-compact manifolds such as Euclidean spaces, being the value groups for Whitehead torsion, the Siebemann end obstruction and the Wall finiteness and surgery obstructions. Some of the applications to topology are included, such as the obstruction theories for splitting homotopy equivalences and for fibering compact manifolds over the circle. Only elementary algebraic constructions are used, which are always motivated by topology. The material is accessible to a wide mathematical audience, especially graduate students and research workers in topology and algebra.

J. Frank Adams had a profound influence on algebraic topology, and his works continue to shape its development. The International Symposium on Algebraic Topology held in Manchester during July 1990 was dedicated to his memory, and virtually all of the world's leading experts took part. This two volume work constitutes the proceedings of the symposium; the articles contained here range from overviews to reports of work still in progress, as well as a survey and complete bibliography of Adams' own work. These proceedings form an important compendium of current research in algebraic topology, and one that demonstrates the depth of Adams' many contributions to the subject. This second volume is oriented towards stable homotopy theory, the Steenrod algebra and the Adams spectral sequence. In the first volume the theme is mainly unstable homotopy theory, homological and categorical algebra.

J. Frank Adams had a profound influence on algebraic topology, and his works continue to shape its development. The International Symposium on Algebraic Topology held in Manchester during July 1990 was dedicated to his memory, and virtually all of the world's leading experts took part. This two volume work constitutes the proceedings of the symposium; the articles contained here range from overviews to reports of work still in progress, as well as a survey and complete bibliography of Adams' own work. These proceedings form an important compendium of current research in algebraic topology, and one that demonstrates the depth of Adams' many contributions to the subject. Here in the first volume the theme is mainly unstable homotopy theory, homological and categorical algebra. The second volume is oriented towards stable homotopy theory, the Steenrod algebra and the Adams spectral sequence.

Aimed at graduate students, this textbook provides an accessible and comprehensive introduction to operator theory. Rather than discuss the subject in the abstract, this textbook covers the subject through twenty examples of a wide variety of operators, discussing the norm, spectrum, commutant, invariant subspaces, and interesting properties of each operator. The text is supplemented by over 600 end-of-chapter exercises, designed to help the reader master the topics covered in the chapter, as well as providing an opportunity to further explore the vast operator theory literature. Each chapter also contains well-researched historical facts which place each chapter within the broader context of the development of the field as a whole.

This is the first of two volumes which will provide an introduction to modern developments in the representation theory of finite groups and associative algebras. The subject is viewed from the perspective of homological algebra and the theory of representations of finite dimensional algebras; the author emphasises modular representations and the homological algebra associated with their categories. This volume is self-contained and independent of its successor, being primarily concerned with the exposition of the necessary background material. The heart of the book is a lengthy introduction to the (Auslander-Reiten) representation theory of finite dimensional algebras, in which the techniques of quivers with relations and almost split sequences are discussed in detail. Much of the material presented here has never appeared in book form. Consequently students and research workers studying group theory and indeed algebra in general will be grateful to Dr Benson for supplying an exposition of a good deal of the essential results of modern representation theory.

Written by Arthur Ogus on the basis of notes from Pierre Berthelot's seminar on crystalline cohomology at Princeton University in the spring of 1974, this book constitutes an informal introduction to a significant branch of algebraic geometry. Specifically, it provides the basic tools used in the study of crystalline cohomology of algebraic varieties in positive characteristic. Originally published in 1978. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.