# Using the Borsuk Ulam Theorem

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.

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.

# A Journey Through Discrete Mathematics

We refer to the detailed, elementary exposition in Matoušek's book “Using the
BorsukUlam Theorem” [52]. Current research continues this line of work, using
more advanced methods from Equivariant Algebraic Combinatorics; see for ...

This collection of high-quality articles in the field of combinatorics, geometry, algebraic topology and theoretical computer science is a tribute to Jiří Matoušek, who passed away prematurely in March 2015. It is a collaborative effort by his colleagues and friends, who have paid particular attention to clarity of exposition – something Jirka would have approved of. The original research articles, surveys and expository articles, written by leading experts in their respective fields, map Jiří Matoušek’s numerous areas of mathematical interest.

# Poincar s Legacies

There are many proofs of this theorem, but I will focus on the proof that is based
on the Borsuk-Ulam theorem: Theorem 1.7.3 (Borsuk-Ulam theorem). Let f : S” ...
Proof of the Ham Sandwich theorem using the Borsuk-Ulam theorem. We can ...

There are many bits and pieces of folklore in mathematics that are passed down from advisor to student, or from collaborator to collaborator, but which are too fuzzy and non-rigorous to be discussed in the formal literature. Traditionally, it was a matter of luck and location as to who learned such folklore mathematics. But today, such bits and pieces can be communicated effectively and efficiently via the semiformal medium of research blogging. This book grew from such a blog. In 2007, Terry Tao began a mathematical blog to cover a variety of topics, ranging from his own research and other recent developments in mathematics, to lecture notes for his classes, to non-technical puzzles and expository articles. The articles from the first year of that blog have already been published by the AMS. The posts from 2008 are being published in two volumes. This book is Part II of the second-year posts, focusing on geometry, topology, and partial differential equations. The major part of the book consists of lecture notes from Tao's course on the Poincare conjecture and its recent spectacular solution by Perelman. The course incorporates a review of many of the basic concepts and results needed from Riemannian geometry and, to a lesser extent, from parabolic PDE. The aim is to cover in detail the high-level features of the argument, along with selected specific components of that argument, while sketching the remaining elements, with ample references to more complete treatments. The lectures are as self-contained as possible, focusing more on the ``big picture'' than on technical details. In addition to these lectures, a variety of other topics are discussed, including expository articles on topics such as gauge theory, the Kakeya needle problem, and the Black-Scholes equation. Some selected comments and feedback from blog readers have also been incorporated into the articles. The book is suitable for graduate students and research mathematicians interested in broad exposure to mathematical topics.

# Fixed Point Theory

In fact, the book could even serve as an introduction to algebraic topology among others. It is certain that the book will be a standard work on Fixed Point Theory for many years to come.

The theory of Fixed Points is one of the most powerful tools of modern mathematics. This book contains a clear, detailed and well-organized presentation of the major results, together with an entertaining set of historical notes and an extensive bibliography describing further developments and applications. From the reviews: "I recommend this excellent volume on fixed point theory to anyone interested in this core subject of nonlinear analysis." --MATHEMATICAL REVIEWS

# Lectures on Hyperbolic Geometry

Marcus, D. A.: Number Fields Martinez, A.: An Introduction to Semiclas- sical and
Microfocal Analysis MatouSek, J.: Using the Borsuk-Ulam Theorem Matsuki, K.:
Introduction to the Mori Program Mc Carthy, P. J.: Introduction to Arithmetical ...

The core of the book is the study of the space of the hyperbolic manifolds endowed with the Chabauty and the geometric topology, and in particular the proof of the hypberbolic surgery theorem in dimension three, based on the representation of three-mainfolds as glued ideal tetrahedra. The development of this main theme requires setting a wide background forming the body of the book: the classical geometry of the hyperbolic space, the Fenchel-Nielsen parametrization of the Teichmüller space, Mostow's rigidity theorem, Margulis' lemma. As a conclusion some features of bounded cohomology, flat fiber bundles and amenable groups are mentioned.

# Computational Complexity

Lovàsz gives a topological proof (using the famous Borsuk-Ulam FixedPoint
Theorem) that determines the chromatic number of the Kneser graph exactly.
From his proof one can indeed obtain an algorithm for solving chromatic number
on all ...

This beginning graduate textbook describes both recent achievements and classical results of computational complexity theory. Requiring essentially no background apart from mathematical maturity, the book can be used as a reference for self-study for anyone interested in complexity, including physicists, mathematicians, and other scientists, as well as a textbook for a variety of courses and seminars. More than 300 exercises are included with a selected hint set. The book starts with a broad introduction to the field and progresses to advanced results. Contents include: definition of Turing machines and basic time and space complexity classes, probabilistic algorithms, interactive proofs, cryptography, quantum computation, lower bounds for concrete computational models (decision trees, communication complexity, constant depth, algebraic and monotone circuits, proof complexity), average-case complexity and hardness amplification, derandomization and pseudorandom constructions, and the PCP theorem.

# Fundamenta Mathematicae

ing F: So-Ro defined by F(a) = a – f(a) is in some sense “small”) then the Borsuk-
Ulam theorem holds forf. Granas [1] uses the term “compact fields” for such
mappings and the precise definition will be given later. Granas theorem can now
be ...

# Journal of the London Mathematical Society

The Borsuk-Ulam theorem has recently been used by Gleason [4] to show that
given an integer q, every element of a ... this more general result by Gleason's
method, using a generalisation of the Borsuk-Ulam theorem, in which the
antipodal ...

# Equipartition of Mass Distributions by Hyperplanes

For d = 3, the answer is again affirmative [Had66, YDEP89); the proof in [YDEP89
. uses the Borsuk-Ulam theorem on the 2-dimensional sphere. For d > 5, the
answer is negative [Avi84]. The open question for d = 4 motivated the work ...