This is the softcover reprint of the English translation of 1971 (available from Springer since 1989) of the first 4 chapters of Bourbaki's Topologie générale. It gives all the basics of the subject, starting from definitions. Important classes of topological spaces are studied, uniform structures are introduced and applied to topological groups. Real numbers are constructed and their properties established. Part II, comprising the later chapters, Ch. 5-10, is also available in English in softcover.

An algebraic structure consists of a set of elements, with some rule of combining them, or some special property of selected subsets of the entire set. Many algebraic structures, such as vector space and group, come to everyday use of a modern physicist. Catering to the needs of graduate students and researchers in the field of mathematical physics and theoretical physics, this comprehensive and valuable text discusses the essential concepts of algebraic structures such as metric space, group, modular numbers, algebraic integers, field, vector space, Boolean algebra, measure space and Lebesgue integral. Important topics including finite and infinite dimensional vector spaces, finite groups and their representations, unitary groups and their representations and representations of the Lorentz group, homotopy and homology of topological spaces are covered extensively. Rich pedagogy includes various problems interspersed throughout the book for better understanding of concepts.

Unit-I 0. Historical Background .... 1-4 1. Groups and Their Basic Properties .... 1-65 2. Subgroups .... 66-80 3. Cyclic Groups .... 81-93 4. Coset Decomposition, Lagrange’s and Fermat’s Theorem .... 94-113 5. Normal Subgroups .... 114-125 6. Quotient Groups .... 126-131 Unit-II 7. Homomorphism and Isomorphism of Groups, Fundamental Theorem of Homomorphism .... 132-151 8. Transformation and Permutation Group Sn (n < 5), Cayley’s Theorem .... 152-186 9. Group Automorphism, Inner Automorphism, Group of Automorphisms .... 187-206 Unit-III 10. Definition and Basic Properties of Rings, Subrings .... 207-232 11. Ring Homomorphism, Ideals, Quotient Ring .... 233-259 12. Polynomial Ringh .... 260-296 13. Integral Domain .... 297-310 14. Field .... 311-330 Unit-IV 15. Definition and Examples of Vector Space, Subspaces, Sum and Direct sum of Subspaces, Linear Span, Linear Dependence, Linear Independence and Their basic Properties .... 331-360 16. Basis, Finite Dimensional Vector Space and Dimension (Existence Theorem, Extension Theorem, Inoriance of the number of Elements), DImension of sum of Subspaces, Quonient Space and It Dimension .... 361-380 Unit-V 17. Linear Transformation and Its Representation as a Matrix .... 381-403 18. Algebra of Linear transformations, Rank-Nullity Theorem, Change of basis, Dual Space, Bi-dual Space and Natural Isomorphism Adjoint of a Linear Transformation .... 404-438 19. Eigen-Values and Eigen-Vectors of a Linear Transformation, Diagonalization .... 439-472

Mathematical Physics is an introduction to such basic mathematical structures as groups, vector spaces, topological spaces, measure spaces, and Hilbert space. Geroch uses category theory to emphasize both the interrelationships among different structures and the unity of mathematics. Perhaps the most valuable feature of the book is the illuminating intuitive discussion of the "whys" of proofs and of axioms and definitions. This book, based on Geroch's University of Chicago course, will be especially helpful to those working in theoretical physics, including such areas as relativity, particle physics, and astrophysics.

Architecture of Mathematics describes the logical structure of Mathematics from its foundations to its real-world applications. It describes the many interweaving relationships between different areas of mathematics and its practical applications, and as such provides unique reading for professional mathematicians and nonmathematicians alike. This book can be a very important resource both for the teaching of mathematics and as a means to outline the research links between different subjects within and beyond the subject. Features All notions and properties are introduced logically and sequentially, to help the reader gradually build understanding. Focusses on illustrative examples that explain the meaning of mathematical objects and their properties. Suitable as a supplementary resource for teaching undergraduate mathematics, and as an aid to interdisciplinary research. Forming the reader's understanding of Mathematics as a unified science, the book helps to increase his general mathematical culture.

A polynomial identity for an algebra (or a ring) A A is a polynomial in noncommutative variables that vanishes under any evaluation in A A. An algebra satisfying a nontrivial polynomial identity is called a PI algebra, and this is the main object of study in this book, which can be used by graduate students and researchers alike. The book is divided into four parts. Part 1 contains foundational material on representation theory and noncommutative algebra. In addition to setting the stage for the rest of the book, this part can be used for an introductory course in noncommutative algebra. An expert reader may use Part 1 as reference and start with the main topics in the remaining parts. Part 2 discusses the combinatorial aspects of the theory, the growth theorem, and Shirshov's bases. Here methods of representation theory of the symmetric group play a major role. Part 3 contains the main body of structure theorems for PI algebras, theorems of Kaplansky and Posner, the theory of central polynomials, M. Artin's theorem on Azumaya algebras, and the geometric part on the variety of semisimple representations, including the foundations of the theory of Cayley–Hamilton algebras. Part 4 is devoted first to the proof of the theorem of Razmyslov, Kemer, and Braun on the nilpotency of the nil radical for finitely generated PI algebras over Noetherian rings, then to the theory of Kemer and the Specht problem. Finally, the authors discuss PI exponent and codimension growth. This part uses some nontrivial analytic tools coming from probability theory. The appendix presents the counterexamples of Golod and Shafarevich to the Burnside problem.

This book is intended for graduate students and research mathematicians interested in set theory.

This book constitutes the refereed proceedings of the 26th International Symposium on Mathematical Foundations of Computer Science, MFCS 2001, held in Marianske Lazne, Czech Republic in August 2001. The 51 revised full papers presented together with 10 invited contributions were carefully reviewed and selected from a total of 118 submissions. All current aspects of theoretical computer science are addressed ranging from mathematical logic and programming theory to algorithms, discrete mathematics, and complexity theory. Besides classical issues, modern topics like quantum computing are discussed as well.

This book is concerned with recent trends in the representation theory of algebras and its exciting interaction with geometry, topology, commutative algebra, Lie algebras, quantum groups, homological algebra, invariant theory, combinatorics, model theory and theoretical physics. The collection of articles, written by leading researchers in the field, is conceived as a sort of handbook providing easy access to the present state of knowledge and stimulating further development. The topics under discussion include diagram algebras, Brauer algebras, cellular algebras, quasi-hereditary algebras, Hall algebras, Hecke algebras, symplectic reflection algebras, Cherednik algebras, Kashiwara crystals, Fock spaces, preprojective algebras, cluster algebras, rank varieties, varieties of algebras and modules, moduli of representations of quivers, semi-invariants of quivers, Cohen-Macaulay modules, singularities, coherent sheaves, derived categories, spectral representation theory, Coxeter polynomials, Auslander-Reiten theory, Calabi-Yau triangulated categories, Poincare duality spaces, selfinjective algebras, periodic algebras, stable module categories, Hochschild cohomologies, deformations of algebras, Galois coverings of algebras, tilting theory, algebras of small homological dimensions, representation types of algebras, and model theory. This book consists of fifteen self-contained expository survey articles and is addressed to researchers and graduate students in algebra as well as a broader mathematical community. They contain a large number of open problems and give new perspectives for research in the field.