**Author**: Ji?í Matoušek

**Publisher:** Springer Science & Business Media

**ISBN:**

**Category:** Mathematics

**Page:** 486

**View:** 922

The main topics in this introductory text to discrete geometry include basics on convex sets, convex polytopes and hyperplane arrangements, combinatorial complexity of geometric configurations, intersection patterns and transversals of convex sets, geometric Ramsey-type results, and embeddings of finite metric spaces into normed spaces. In each area, the text explains several key results and methods.

This book, based on the authors' lecture series at a 2006 satellite meeting of the International Congress of Mathematicians, offers a comprehensive survey of core areas of combinatorial geometry. These lecture notes aptly describe both the history and the state of the art of these topics. These combinatorial techniques have found applications in areas of computer science ranging from graph drawing to frequency allocation in cellular networks.

Cohomology and homology modulo 2 helps the reader grasp more readily the basics of a major tool in algebraic topology. Compared to a more general approach to (co)homology this refreshing approach has many pedagogical advantages: 1. It leads more quickly to the essentials of the subject, 2. An absence of signs and orientation considerations simplifies the theory, 3. Computations and advanced applications can be presented at an earlier stage, 4. Simple geometrical interpretations of (co)chains. Mod 2 (co)homology was developed in the first quarter of the twentieth century as an alternative to integral homology, before both became particular cases of (co)homology with arbitrary coefficients. The first chapters of this book may serve as a basis for a graduate-level introductory course to (co)homology. Simplicial and singular mod 2 (co)homology are introduced, with their products and Steenrod squares, as well as equivariant cohomology. Classical applications include Brouwer's fixed point theorem, Poincaré duality, Borsuk-Ulam theorem, Hopf invariant, Smith theory, Kervaire invariant, etc. The cohomology of flag manifolds is treated in detail (without spectral sequences), including the relationship between Stiefel-Whitney classes and Schubert calculus. More recent developments are also covered, including topological complexity, face spaces, equivariant Morse theory, conjugation spaces, polygon spaces, amongst others. Each chapter ends with exercises, with some hints and answers at the end of the book.

This monograph gives a short introduction to the relevant modern parts of discrete geometry, in addition to leading the reader to the frontiers of geometric research on sphere arrangements. The readership is aimed at advanced undergraduate and early graduate students, as well as interested researchers. It contains more than 40 open research problems ideal for graduate students and researchers in mathematics and computer science. Additionally, this book may be considered ideal for a one-semester advanced undergraduate or graduate level course. The core part of this book is based on three lectures given by the author at the Fields Institute during the thematic program on “Discrete Geometry and Applications” and contains four core topics. The first two topics surround active areas that have been outstanding from the birth of discrete geometry, namely dense sphere packings and tilings. Sphere packings and tilings have a very strong connection to number theory, coding, groups, and mathematical programming. Extending the tradition of studying packings of spheres, is the investigation of the monotonicity of volume under contractions of arbitrary arrangements of spheres. The third major topic of this book can be found under the sections on ball-polyhedra that study the possibility of extending the theory of convex polytopes to the family of intersections of congruent balls. This section of the text is connected in many ways to the above-mentioned major topics and it is also connected to some other important research areas as the one on coverings by planks (with close ties to geometric analysis). This fourth core topic is discussed under covering balls by cylinders.

The proceedings covers computational complexity, cryptography, parallel and distributed computing, machine learning, logic, coding theory, theoretical databases, information technology, networks, quantum computing, and much more.