Orient BlackSwan

About the BookAbout the Author(s)Table of Contents

This book presents a course in the geometry of convex polytopes in arbitrary dimension, suitable for an advanced undergraduate or beginning graduate student. The book starts with the basics of polytope theory. Schlegel and Gale diagrams are introduced as geometric tools to visualize polytopes in high dimension and to unearth bizarre phenomena in polytopes. The heart of the book is a treatment of the secondary polytope of a point configuration and its connections to the state polytope of the toric ideal defined by the configuration. These polytopes are relatively recent constructs with numerous connections to discrete geometry, classical algebraic geometry, symplectic geometry, and combinatorics. The connections rely on Gröbner bases of toric ideals and other methods from commutative algebra.
The book is self-contained and does not require any background beyond basic linear algebra. With numerous figures and exercises, it can be used as a textbook for courses on geometric, combinatorial, and computational aspects of the theory of polytopes.

Rekha R Thomas is Professor of Mathematics at the University of Washington, Seattle, WA.

Chapter 1. Abstract algebra: Groups, rings and fields
Chapter 2. Convex polytopes: Definitions and examples
Chapter 3. Faces of polytopes
Chapter 4. Schlegel diagrams
Chapter 5. Gale diagrams
Chapter 6. Bizarre polytopes
Chapter 7. Triangulations of point configurations
Chapter 8. The secondary polytope
Chapter 9. The permutahedron
Chapter 10. Abstract algebra: Polynomial rings
Chapter 11. Gröbner bases I
Chapter 12. Gröbner bases II
Chapter 13. Initial complexes of toric ideals
Chapter 14. State polytopes of toric ideals