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Using the Borsuk-Ulam Theorem: Lectures on Topological Methods in Combinatorics and Geometry (Universitext) [Paperback]

by Jiri Matousek, A. Björner, G.M. Ziegler
4.0 out of 5 stars  See all reviews (1 customer review)

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Book Description

October 10, 2008 3540003622 978-3540003625 1st ed. 2003

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.


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Using the Borsuk-Ulam Theorem: Lectures on Topological Methods in Combinatorics and Geometry (Universitext) + Thirty-three Miniatures: Mathematical and Algorithmic Applications of Linear Algebra (Student Mathematical Library)
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Editorial Reviews

Review

From the reviews:

[...]Matousek's lively little textbook now shows that Lovász' insight as well as beautiful work of many others (such as Vreci´ca and Zivaljevi´c, and Sarkaria) have opened up an exciting area of matheamtics that connects combinatorics, graph theory, algebraic topology and discrete geometry. What seemed like an ingenious trick in 1978 now presents itself as an instance of the "test set paradigm": to construct configuration spaces for combinatorial problems such that coloring, incidence or transversal problems may be translated into the (non-)existence of suitable equivariant maps.

The vivid account of this area and its ramifications by Matousek is an exciting, a coherent account of this area of topological combinatorics. It features a collection of mathematical gems written with a broad view of the subject and still with loving care for details. Recommended reading! [..]

Günter M. Ziegler (Berlin)

Zbl. MATH Volume 1060 Productions-no.: 05001

...."The book contains more than 100 exercises, many of them being compressed outlines of interesting results. As such, most of them are both challenging and interesting.

I think this book will be of interest mostly for mathematicians and graduate students. It can certainly be used in a graduate course or seminar. It has an extensive list of references and it covers many interesting and not-so-easy results, with proofs that are sometimes easier than in the original papers. I wouldn't say that this book is an "easy read", but it is very well written, very interesting, and very informative".

Mihaela Poplicher, MAA Online

"This book is intended to make elementary topological methods more accessible to those who work in other areas of mathematics. … This excellent book is useful for specialists in discrete geometry, combinatorics and computer science who want to learn how algebraic topology can be used in their discipline. It is also very suitable as a textbook for such a course." (Ferenc Fodor, Acta Scientiarum Mathematicarum, Vol. 71, 2005)

"The book is addressed to mathematicians, scientists, and engineers working on signal and image processing and medical imaging. It is written by experts and for experts, but each chapter has an introductory part written for non-specialists, giving them the possibility to find what the chapter is dealing with. The book contains important contributions to the areas mentioned in the title: sampling, wavelets and tomography." (S. Cobzas, Studia universitatis Babes-Bolyai Mathematica, Vol. XLIX (2), 2004)

"This book … is based on a couple of graduate courses in ‘topological combinatorics’ taught by the author … . The book contains more than 100 exercises, many of them being compressed outlines of interesting results. As such, most of them are both challenging and interesting. … It has an extensive list of references and it covers many interesting and no-so-easy results, with proofs that are sometimes easier than in the original papers. … it is very well written, very interesting, and very informative." (Mihaela Poplicher, MAA Online, January, 2005)

"This excellent textbook is based on lecture notes by the author on applications of some topological methods in combinatorics and geometry. … All chapters include exercises and historical and bibliographical notes. Many hints for further reading are given. … topologists will find here some nice applications for their lectures." (Zdzislaw Dzedzej, Mathematical Reviews, 2004 i)

"Matoušek’s lively little textbook now shows that Lovász’ insight as well as beautiful work of many others … have opened up an exciting area of mathematics that connects combinatorics, graph theory, algebraic topology and discrete geometry. … The vivid account of this area and its ramifications by Matoušek is an exciting, a coherent account of this area of topological combinatorics. It features a collection of mathematical gems written with a broad view of the subject and still with loving care for details. Recommended reading!" (Günter M. Ziegler, Zentralblatt MATH, Vol. 1016, 2003)

"This textbook explains elementary but powerful topological methods based on the Borsuk-Ulam theorem and its generalizations. It covers many substantial results, sometimes with proofs simpler than those in the original papers." (L'ENSEIGNEMENT MATHEMATIQUE, Vol. 49 (1-2), 2003)

From the Back Cover

"The "Kneser conjecture" -- posed by Martin Kneser in 1955 in the Jahresbericht der DMV -- is an innocent-looking problem about partitioning the k-subsets of an n-set into intersecting subfamilies. Its striking solution by L. Lovász featured an unexpected use of the Borsuk-Ulam theorem, that is, of a genuinely topological result about continuous antipodal maps of spheres.

Matousek's lively little textbook now shows that Lovász' insight as well as beautiful work of many others (such as Vrecica and Zivaljevic, and Sarkaria) have opened up an exciting area of mathematics that connects combinatorics, graph theory, algebraic topology and discrete geometry. What seemed like an ingenious trick in 1978 now presents itself as an instance of the "test set paradigm": to construct configuration spaces for combinatorial problems such that coloring, incidence or transversal problems may be translated into the (non-)existence of suitable equivariant maps.

The vivid account of this area and its ramifications by Matousek is an exciting, a coherent account of this area of topological combinatorics. It features a collection of mathematical gems written with a broad view of the subject and still with loving care for details. Recommended reading! […]"

Günter M.Ziegler (Berlin)
Zbl. MATH Volume 1060 Productions-no.: 05001


Product Details

  • Series: Universitext
  • Paperback: 214 pages
  • Publisher: Springer; 1st ed. 2003 edition (October 10, 2008)
  • Language: English
  • ISBN-10: 3540003622
  • ISBN-13: 978-3540003625
  • Product Dimensions: 9.1 x 6.1 x 0.6 inches
  • Shipping Weight: 11.4 ounces (View shipping rates and policies)
  • Average Customer Review: 4.0 out of 5 stars  See all reviews (1 customer review)
  • Amazon Best Sellers Rank: #658,880 in Books (See Top 100 in Books)

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4 of 5 people found the following review helpful
4.0 out of 5 stars Insight Pedagogy and Honesty October 16, 2007
Format:Paperback
While allowing you to learn new concepts and definitions in a classical sense , this book shows what mathematics is about. It makes connections and gives you insight.
If you think that mathematics is made out of connections and is an "Art of variations" you will be rewarded. Those interested in the "frontier" between discrete and continuous will be interested too.
Definitely pedagogical and moreover intellectually honest.
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