Lectures on Sphere Arrangements – the Discrete Geometric Side (Fields Institute Monographs, 32) - Softcover

Bezdek, Károly

Softcover

ISBN 10: 1493900323 ISBN 13: 9781493900329

Publisher: Springer, 2015

Synopsis

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.

About this title

From the Back Cover

This monograph gives a short introduction to parts of modern 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 30 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 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 basic topics. The first two deal with 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 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 as well as to some other important research areas such as that on coverings by planks (with close ties to geometric analysis). The fourth basic topic is discussed under covering balls by cylinders.

Edition details

• Publisher: Springer
• Publication date: 2015
• Language: English
• ISBN 10: 1493900323
• ISBN 13: 9781493900329
• Binding: Paperback
• Number of pages: 187