توپولوژی گروه‌های لی، جلد اول و دوم (ترجمه‌های مونوگراف‌های ریاضی، جلد ۹۱)

This book provides historical background and a complete overview of the qualitative theory of foliations and differential dynamical systems. Senior mathematics majors and graduate students with background in multivariate calculus, algebraic and differential topology, differential geometry, and linear algebra will find this book an accessible introduction. Upon finishing the book, readers will be prepared to take up research in this area. Readers will appreciate the book for its highly visual presentation of examples in low dimensions. The author focuses particularly on foliations with compact leaves, covering all the important basic results. Specific topics covered include: dynamical systems on the torus and the three-sphere, local and global stability theorems for foliations, the existence of compact leaves on three-spheres, and foliated cobordisms on three-spheres. Also included is a short introduction to the theory of differentiable manifolds. The theory of Abelian functions, which was at the center of nineteenth-century mathematics, is again attracting attention. However, today it is frequently seen not just as a chapter of the general theory of functions, but as an area of application of the ideas and methods of commutative algebra. This book presents an exposition of the fundamentals of the theory of Abelian functions based on the methods of the classical theory of functions. This theory includes the theory of elliptic functions as a special case. Among the topics covered are theta functions, Jacobians, and Picard varieties. The author has aimed the book primarily at intermediate and advanced graduate students, but it would also be accessible to the beginning graduate student or advanced undergraduate who has a solid background in functions of one complex variable. This book will prove especially useful to those who are not familiar with the analytic roots of the subject. In addition, the detailed historical introduction cultivates a deep understanding of the subject. Thorough and self-contained, the book will provide readers with an excellent complement to the usual algebraic approach. Lie groups are very general mathematical objects that appear in numerous areas such as topology, functional analysis, and algebra, as well as differential geometry and differential topology. The purpose of these two parts is to provide a guide to the topology of Lie groups and homogeneous spaces by bringing together a wide range of results relating to them. The first part thoroughly studies topological properties of the classical groups as typical examples of Lie groups. In the second part, the authors study general properties of compact Lie groups, particularly the exceptional groups. Lie groups are very general mathematical objects that appear in numerous areas such as topology, functional analysis, and algebra, as well as differential geometry and differential topology. This book provides a guide to the topology of Lie groups and homogeneous spaces by bringing together a wide range of results relating to them. Offers historical background and an overview of the qualitative theory of foliations and differential dynamical systems. This book focuses particularly on foliations with compact leaves. It contains topics that include: dynamical systems on the torus and the three-sphere, and local and global stability theorems for foliations. Content: Nonsingular dynamical systems on the torus $C^r$ manifolds and tangent spaces Dynamical systems and limit sets Foliations Stability theorems on foliations The existence of compact leaves Foliations and differential forms Cobordisms of foliations. Presents an exposition of the fundamentals of the theory of Abelian functions based on the methods of the classical theory of functions. This book covers such topics as theta functions, Jacobians, and Picard varieties. A.i. Markushevich ; [translated From The Russian By G. Bluher ; Translation Edited By Ralph P. Boas]. Translation Of: Vvedenie V Klassicheskui︠u︡ Teorii︠u︡ Abelevykh Funkt︠s︡iĭ.