A. Borel – författare

Contains sections on Automorphic representations and L-functions as well as Arithmetical algebraic geometry and L-functions.

The book by Borel and Wallach is a classic treatment of the use of cohomology in representation theory, particularly in the setting of automorphic forms and discrete subgroups. The authors begin with general material, covering Lie algebra cohomology, as well as continuous and differentiable cohomology. Much of the machinery is designed for the study of the cohomology of locally symmetric spaces, realized as double coset spaces, where the quotient is by a maximal compact subgroup and by a discrete subgroup. Such spaces are central to applications to number theory and the study of automorphic forms. The authors give a careful presentation of relative Lie algebra cohomology of admissible and of unitary $G$-modules. As part of the general development, the Langlands classification of irreducible admissible representations is given. Computations of important examples are another valuable part of the book.In the twenty years between the first and second editions of this work, there was immense progress in the use of homological algebra to construct admissible representations and in the study of arithmetic groups. The second edition is a corrected and expanded version of the original, which was an important catalyst in the growth 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 two intervening decades.

Treatise on Analysis, Volume 10–VIII provides information pertinent to the study of the most common boundary problems for partial differential equations. This book presents the study of Cauchy''s problem in its most elementary form.Comprised of one chapter, this volume begins with an overview of Hilbert-von Neumann spectral theory and explores all possible boundary conditions related to spectral theory. This text then examines the link of Cauchy''s problem with the behavior of the equation''s characteristics. This book discusses as well the case of linear elliptic operators. The reader is also introduced to Sobolev spaces and some of their generalizations that provide an essential tool in the study of these elliptic problems, and their manipulation requires delicate upper bounds to obtain the best possible results.This book is a valuable resource for mathematicians.

Armand Borel’s mathematical work centered on the theory of Lie groups. Because of the increasingly important place of this theory in the whole of mathematics, Borel’s work influenced some of the most important developments of contemporary mathematics. His first great achievement was to apply to Lie groups and homogenous spaces the powerful techniques of algebraic topology developed by Leray, Cartan, and Steenrod. In 1992, Borel was awarded the International Balzan Prize for Mathematics "for his fundamental contributions to the theory of Lie groups, algebraic groups and arithmetic groups, and for his indefatigable action in favor of high quality in mathematical research and of the propagation of new ideas."He wrote more than 145 articles before 1982, which were collected in three volumes published in 1983. A fourth volume of subsequent articles was published in 2001. Volume I collects the papers written from 1948 to 1958.