Armand Borel Knihy

Focusing on the analytic theory of automorphic forms, the book delves into the structure and properties of these forms on G=SL2(R) and the upper-half plane, particularly regarding discrete subgroups of finite covolume. It explores various topics, including the construction of fundamental domains, Poincaré series, cusp forms, and the spectral decomposition of associated spaces. The text also connects these concepts to infinite dimensional unitary representations, requiring a foundational understanding of functional analysis and basic Lie group theory.

Focusing on the compactification of noncompact symmetric and locally symmetric spaces, this book explores their significance across various mathematical fields such as analysis, number theory, algebraic geometry, and algebraic topology. It provides uniform constructions of known compactifications, emphasizing their geometric and topological structures. The work serves as a comprehensive guide to understanding these complex spaces and their applications, addressing the extensive literature on compactifications in a clear and structured manner.

This revised edition of "Linear Algebraic Groups" covers foundational topics in algebraic groups, Lie algebras, and transformation spaces. It explores solvable groups, linear algebraic group properties, and Chevally's structure theory. Expanded content includes central isogenies and rational points of isotropic reductive groups, requiring familiarity with algebraic geometry.

This book collects the papers published by A. Borel from 1983 to 1999. About half of them are research papers, written on his own or in collaboration, on various topics pertaining mainly to algebraic or Lie groups, homogeneous spaces, arithmetic groups (L2-spectrum, automorphic forms, cohomology and covolumes), L2-cohomology of symmetric or locally symmetric spaces, and to the Oppenheim conjecture. Other publications include surveys and personal recollections (of D. Montgomery, Harish-Chandra, and A. Weil), considerations on mathematics in general and several articles of a historical nature: on the School of Mathematics at the Institute for Advanced Study, on N. Bourbaki and on selected aspects of the works of H. Weyl, C. Chevalley, E. Kolchin, J. Leray, and A. Weil. The book concludes with an essay on H. Poincaré and special relativity. Some comments on, and corrections to, a number of papers have also been added.

Ce livre traite des notions algébriques, y compris les complexes, le théorème fondamental et les faisceaux. Il explore l'algèbre spectrale, tant pour les applications continues que pour les espaces fibres, tout en offrant des applications pratiques.

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.