Lectures in Geometric Combinatorics (Student Mathematical Library, Vol. 33)

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. 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. 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.--book Jacket. Ias/park City Mathematics Institute -- Preface -- Ch. 1. Abstract Algebra : Groups, Rings And Fields -- Ch. 2. Convex Polytopes : Definitions And Examples -- Ch. 3. Faces Of Polytopes -- Ch. 4. Schlegel Diagrams -- Ch. 5. Gale Diagrams -- Ch. 6. Bizarre Polytopes -- Ch. 7. Triangulations Of Point Configurations -- Ch. 8. The Secondary Polytope -- Ch. 9. The Permutahedron -- Ch. 10. Abstract Algebra : Polynomial Rings -- Ch. 11. Gröbner Bases 1 -- Ch. 12. Gröbner Bases 2 -- Ch. 13. Initial Complexes Of Toric Ideals -- Ch. 14. State Polytopes Of Toric Ideals -- Bibliography -- Index. Rekha R. Thomas. Includes Bibliographical References (p. 139-140) And Index. Preface 9 Chapter 1. Abstract Algebra: Groups, Rings and Fields 11 Chapter 2. Convex Polytopes: Definitions and Examples 17 Chapter 3. Faces of Polytopes 25 Chapter 4. Schlegel Diagrams 35 Chapter 5. Gale Diagrams 47 Chapter 6. Bizarre Polytopes 57 Chapter 7. Triangulations of Point Configurations 67 Chapter 8. The Secondary Polytope 83 Chapter 9. The Permutahedron 95 Chapter 10. Abstract Algebra: Polynomial Rings 103 Chapter 11. Gröbner bases I 111 Chapter 12. Gröbner bases II 121 Chapter 13. Initial Complexes of Toric Ideals 127 Chapter 14. State Polytopes of Toric Ideals 137 Bibliography 149 Index 151