To the uninitiated, algebraic topology might seem fiendishly complex, but its utility is beyond doubt. This brilliant exposition goes back to basics to explain how the subject has been used to further our understanding in some key areas. A number of important results in combinatorics, discrete geometry, and theoretical computer science have been proved using algebraic topology. While the results are quite famous, their proofs are not so widely understood. This book is the first textbook treatment of a significant part of these results. It focuses on so-called equivariant methods, based on the Borsuk-Ulam theorem and its generalizations. The topological tools are intentionally kept on a very elementary level. No prior knowledge of algebraic topology is assumed, only a background in undergraduate mathematics, and the required topological notions and results are gradually explained.

This revised and enlarged fifth edition features four new chapters, which contain highly original and delightful proofs for classics such as the spectral theorem from linear algebra, some more recent jewels like the non-existence of the Borromean rings and other surprises. From the Reviews "... Inside PFTB (Proofs from The Book) is indeed a glimpse of mathematical heaven, where clever insights and beautiful ideas combine in astonishing and glorious ways. There is vast wealth within its pages, one gem after another. ... Aigner and Ziegler... write: "... all we offer is the examples that we have selected, hoping that our readers will share our enthusiasm about brilliant ideas, clever insights and wonderful observations." I do. ... " Notices of the AMS, August 1999 "... This book is a pleasure to hold and to look at: ample margins, nice photos, instructive pictures and beautiful drawings ... It is a pleasure to read as well: the style is clear and entertaining, the level is close to elementary, the necessary background is given separately and the proofs are brilliant. ..." LMS Newsletter, January 1999 "Martin Aigner and Günter Ziegler succeeded admirably in putting together a broad collection of theorems and their proofs that would undoubtedly be in the Book of Erdös. The theorems are so fundamental, their proofs so elegant and the remaining open questio ns so intriguing that every mathematician, regardless of speciality, can benefit from reading this book. ... " SIGACT News, December 2011.

Symmetry has a strong impact on the number and shape of solutions to variational problems. This has been observed, for instance, in the search for periodic solutions of Hamiltonian systems or of the nonlinear wave equation; when one is interested in elliptic equations on symmetric domains or in the corresponding semiflows; and when one is looking for "special" solutions of these problems. This book is concerned with Lusternik-Schnirelmann theory and Morse-Conley theory for group invariant functionals. These topological methods are developed in detail with new calculations of the equivariant Lusternik-Schnirelmann category and versions of the Borsuk-Ulam theorem for very general classes of symmetry groups. The Morse-Conley theory is applied to bifurcation problems, in particular to the bifurcation of steady states and hetero-clinic orbits of O(3)-symmetric flows; and to the existence of periodic solutions nearequilibria of symmetric Hamiltonian systems. Some familiarity with the usualminimax theory and basic algebraic topology is assumed.

This is the most readable and thorough graduate textbook and reference for combinatorics, covering enumeration, graphs, sets, and methods.

This book provides an accessible introduction to algebraic topology, a field at the intersection of topology, geometry and algebra, together with its applications. Moreover, it covers several related topics that are in fact important in the overall scheme of algebraic topology. Comprising eighteen chapters and two appendices, the book integrates various concepts of algebraic topology, supported by examples, exercises, applications and historical notes. Primarily intended as a textbook, the book offers a valuable resource for undergraduate, postgraduate and advanced mathematics students alike. Focusing more on the geometric than on algebraic aspects of the subject, as well as its natural development, the book conveys the basic language of modern algebraic topology by exploring homotopy, homology and cohomology theories, and examines a variety of spaces: spheres, projective spaces, classical groups and their quotient spaces, function spaces, polyhedra, topological groups, Lie groups and cell complexes, etc. The book studies a variety of maps, which are continuous functions between spaces. It also reveals the importance of algebraic topology in contemporary mathematics, theoretical physics, computer science, chemistry, economics, and the biological and medical sciences, and encourages students to engage in further study.

Handbook of Combinatorics

These notes are based on lectures given at the University of Minnesota and the Courant Institute of Mathematical Sciences, New York University, in 1966 and 1967. Our aim has been to present some qualitative aspects of nonlinear analysis, which we think are important, in as simple and direct a manner as possible. Thus we have neither striven for results of the utmost generality nor complicated the text by introducing an excess number of new concepts. In this way we hope to make the ideas presented accessible to persons who enjoy mathematics and its applications but are not specialists in nonlinear analysis. To accomplish this goal in a small book we have had to sketch the ideas of a few proofs and to specialize the general theory of nonlinear analysis on finite and infinite dimensional differentiable manifolds. The interested reader will find this theory discussed in the monographs, Lectures on Nonlinear Functional Analysis by J. T. Schwartz, and Foundations of Global Nonlinear Analysis by R. S. Palais. Furthermore, our choice of material was necessarily selective, for example, iterative results such as Newton's method and Nash's implicit function theorem have been omitted. Nonetheless we believe that the material discussed here has sufficient beauty to induce the reader to further excursions into nonlinear analysis. Our text is divided into four chapters and two appendices. Chapter 1 is intended to be a partial answer to the question: What are some of the problems of nonlinear analysis and how have they been studied in previous generations? Chapter 2 introduces the concepts of the degree of a continuous mapping, and the theory of critical points of real-valued functions in finite dimensional Euclidean spaces Rn . In Chapter 3 we show how the ideas of Chapter 2 can be carried over to infinite dimensional spaces. Appendices 1 and 2 at the end of the book include some preliminary material necessary to the understanding of Chapters 2 and 1. In Chapter 4 we select a few specific nonlinear problems and indicate just how the methods of the previous chapters can be used to study these problems. The first-mentioned problem in Chapter 4, global univalence, is of great interest outside of mathematics (for example, to mathematical economists in the study of international trade and to applied mathematicians studying elastic deformations). Similarly the topics of differential equations (ordinary and partial) discussed in Chapter 4 are basic to the understanding of physical processes of nature.