Forbidden Configurations in Discrete Geometry
Cambridge University Press, 5/17/2018
EAN 9781108439138, ISBN10: 1108439136
Paperback, 238 pages, 22.8 x 15.2 x 1 cm
Language: English
This book surveys the mathematical and computational properties of finite sets of points in the plane, covering recent breakthroughs on important problems in discrete geometry, and listing many open problems. It unifies these mathematical and computational views using forbidden configurations, which are patterns that cannot appear in sets with a given property, and explores the implications of this unified view. Written with minimal prerequisites and featuring plenty of figures, this engaging book will be of interest to undergraduate students and researchers in mathematics and computer science. Most topics are introduced with a related puzzle or brain-teaser. The topics range from abstract issues of collinearity, convexity, and general position to more applied areas including robust statistical estimation and network visualization, with connections to related areas of mathematics including number theory, graph theory, and the theory of permutation patterns. Pseudocode is included for many algorithms that compute properties of point sets.
1. A happy ending
2. Overview
3. Configurations
4. Subconfigurations
5. Properties, parameters, and obstacles
6. Computing with configurations
7. Complexity theory
8. Collinearity
9. General position
10. General-position partitions
11. Convexity
12. More on convexity
13. Integer realizations
14. Stretched permutations
15. Configurations from graphs
16. Universality
17. Stabbing
18. The big picture.