A Course in Topological Combinatorics

by Mark de Longueville

Overview

A Course in Topological Combinatorics is the first undergraduate textbook on the field of topological combinatorics, a subject that has become an active and innovative research area in mathematics over the last thirty years with growing applications in math, computer science, and other applied areas. Topological combinatorics is concerned with solutions to combinatorial problems by applying topological tools. In most cases these solutions are very elegant and the connection between combinatorics and topology often arises as an unexpected surprise.

The textbook covers topics such as fair division, graph coloring problems, evasiveness of graph properties, and embedding problems from discrete geometry. The text contains a large number of figures that support the understanding of concepts and proofs. In many cases several alternative proofs for the same result are given, and each chapter ends with a series of exercises. The extensive appendix makes the book completely self-contained.

The textbook is well suited for advanced undergraduate or beginning graduate mathematics students. Previous knowledge in topology or graph theory is helpful but not necessary. The text may be used as a basis for a one- or two-semester course as well as a supplementary text for a topology or combinatorics class.

Product Details

ISBN-13:	9781441979094
Publisher:	Springer New York
Publication date:	09/18/2012
Series:	Universitext
Edition description:	2012
Pages:	240
Product dimensions:	6.00(w) x 9.20(h) x 0.80(d)

About the Author

Mark de Longueville is professor of Mathematics at Hochschule für Technik und Wirtschaft Berlin and faculty member of the Berlin branch of New York University.

Table of Contents

Preface.- List of Symbols and Typical Notation.- 1 Fair-Division Problems.- 2 Graph-Coloring Problems.- 3 Evasiveness of Graph Properties.- 4 Embedding and Mapping Problems.- A Basic Concepts from Graph Theory.- B Crash Course in Topology.- C Partially Ordered Sets, Order Complexes, and Their Topology.- D Groups and Group Actions.- E Some Results and Applications from Smith Theory.- References.- Index.

Editorial Reviews

From the reviews:

“The present book … presents a sequence of combinatorial themes which have shown an affinity for topological methods … . This book is filled with extremely attractive mathematics … and bringing topology into the play of combinatorics and graph theory is a wonderfully elegant manoeuvre. Here it is carried out coherently, and on a pretty grand scale, and we are thus afforded the opportunity to encounter (algebraic) topology in a very seductive uniform context. What a marvelous thing!” (Michael Berg, MAA Reviews, July, 2013)

“In the book’s four main chapters, Longueville (Univ. of Applied Sciences, Germany) addresses fair-division problems; graph coloring; graph property evasiveness; and embeddings and mappings. … Basic results of algebraic topology already have powerful consequences for analysis, but the subject’s arcana can look like art for art’s sake. The author’s charting of a novel application domain for a core subject makes this book an essential acquisition. Summing Up: Essential. Upper-division undergraduates and above.” (D. V. Feldman, Choice, Vol. 50 (8), April, 2013)

“Topological combinatorics is concerned with the applications of the many powerful techniques of algebraic topology to problems in combinatorics. … The present book aims to give a clear and vivid presentation of some of the most beautiful and accessible results from the area. The text, based upon some courses by the author at Freie Universität Berlin, is designed for an advanced undergraduate student.” (Hirokazu Nishimura, zbMATH, Vol. 1273, 2013)

From the Publisher