This book discusses the mathematical interests of Joachim Schwermer, who throughout his career has focused on the cohomology of arithmetic groups, automorphic forms and the geometry of arithmetic manifolds. To mark his 66th birthday, the editors brought together mathematical experts to offer an overview of the current state of research in these and related areas. The result is this book, with contributions ranging from topology to arithmetic. It probes the relation between cohomology of arithmetic groups and automorphic forms and their L-functions, and spans the range from classical Bianchi groups to the theory of Shimura varieties. It is a valuable reference for both experts in the fields and for graduate students and postdocs wanting to discover where the current frontiers lie.

Cohomology of arithmetic groups serves as a tool in studying possible relations between the theory of automorphic forms and the arithmetic of algebraic varieties resp. the geometry of locally symmetric spaces. These proceedings will serve as a guide to this still rapidly developing area of mathematics. Besides two survey articles, the contributions are original research papers.

Aimed at second year graduate students, this text introduces them to cohomology theory (involving a rich interplay between algebra and topology) with a minimum of prerequisites. No homological algebra is assumed beyond what is normally learned in a first course in algebraic topology, and the basics of the subject, as well as exercises, are given prior to discussion of more specialized topics.

This collection of papers is based on lectures delivered at the Tata Institute of Fundamental Research (TIFR) as part of a special year on arithmetic groups, $L$-functions and automorphic forms. The volume opens with an article by Cogdell and Piatetski-Shapiro on Converse Theorems for $GL_n$ and applications to liftings. It ends with some remarks on the Riemann Hypothesis by Ram Murty. Other talks cover topics such as Hecke theory for Jacobi forms, restriction maps and $L$-values, congruences for Hilbert modular forms, Whittaker models for $p$-adic $GL(4)$, the Seigel formula, newforms for the Maass Spezialchar, an algebraic Chebotarev density theorem, a converse theorem for Dirichlet series with poles, Kirillov theory for $GL_2(\mathcal{D})$, and the $L^2$ Euler characteristic of arithmetic quotients. The present volume is the latest in the Tata Institute's tradition of recognized contributions to number theory.

It has been nearly twenty years since the first edition of this work. In the intervening years, there has been immense progress in the use of homological algebra to construct admissible representations and in the study of arithmetic groups. This second edition is a corrected and expanded version of the original, which was an important catalyst in the expansion of the field. Besides the fundamental material on cohomology and discrete subgroups present in the first edition, this edition also contains expositions of some of the most important developments of the last two decades.

Arithmetic groups are generalisations, to the setting of algebraic groups over a global field, of the subgroups of finite index in the general linear group with entries in the ring of integers of an algebraic number field. They are rich, diverse structures and they arise in many areas of study. This text enables you to build a solid, rigorous foundation in the subject. It first develops essential geometric and number theoretical components to the investigations of arithmetic groups, and then examines a number of different themes, including reduction theory, (semi)-stable lattices, arithmetic groups in forms of the special linear group, unipotent groups and tori, and reduction theory for adelic coset spaces. Also included is a thorough treatment of the construction of geometric cycles in arithmetically defined locally symmetric spaces, and some associated cohomological questions. Written by a renowned expert, this book is a valuable reference for researchers and graduate students.

The International Conference on Fundamental Sciences: Mathematics and Theoretical Physics provided a forum for reviewing some of the significant developments in mathematics and theoretical physics in the 20th century; for the leading theorists in these fields to expound and discuss their views on new ideas and trends in the basic sciences as the new millennium approached; for increasing public awareness of the importance of basic research in mathematics and theoretical physics; and for promoting a high level of interest in mathematics and theoretical physics among school students and teachers. This was a major conference, with invited lectures by some of the leading experts in various fields of mathematics and theoretical physics.

This is the third supplementary volume to Kluwer's highly acclaimed twelve-volume Encyclopaedia of Mathematics. This additional volume contains nearly 500 new entries written by experts and covers developments and topics not included in the previous volumes. These entries are arranged alphabetically throughout and a detailed index is included. This supplementary volume enhances the existing twelve volumes, and together, these thirteen volumes represent the most authoritative, comprehensive and up-to-date Encyclopaedia of Mathematics available.

This book constitutes the refereed proceedings of the 4th International Algorithmic Number Theory Symposium, ANTS-IV, held in Leiden, The Netherlands, in July 2000. The book presents 36 contributed papers which have gone through a thorough round of reviewing, selection and revision. Also included are 4 invited survey papers. Among the topics addressed are gcd algorithms, primality, factoring, sieve methods, cryptography, linear algebra, lattices, algebraic number fields, class groups and fields, elliptic curves, polynomials, function fields, and power sums.