About this Title
Ezra Miller, University of Minnesota, Minneapolis, MN, Victor Reiner, University of Minnesota, Minneapolis, MN and Bernd Sturmfels, University of California, Berkeley, Berkeley, CA, Editors
Publication: IAS/Park City Mathematics Series
Publication Year: 2007; Volume 13
ISBNs: 978-0-8218-3736-8 (print); 978-1-4704-3912-5 (online)
MathSciNet review: MR2383123
MSC: Primary 05-01; Secondary 05-06, 52-02
Geometric combinatorics describes a wide area of mathematics that is primarily the study of geometric objects and their combinatorial structure. Perhaps the most familiar examples are polytopes and simplicial complexes, but the subject is much broader. This volume is a compilation of expository articles at the interface between combinatorics and geometry, based on a three-week program of lectures at the Institute for Advanced Study/Park City Math Institute (IAS/PCMI) summer program on Geometric Combinatorics. The topics covered include posets, graphs, hyperplane arrangements, discrete Morse theory, and more. These objects are considered from multiple perspectives, such as in enumerative or topological contexts, or in the presence of discrete or continuous group actions.
Most of the exposition is aimed at graduate students or researchers learning the material for the first time. Many of the articles include substantial numbers of exercises, and all include numerous examples. The reader is led quickly to the state of the art and current active research by worldwide authorities on their respective subjects.
Graduate students and research mathematicians interested in combinatorics; discrete methods in geometry and topology.
Table of Contents
- What is geometric combinatorics?—An overview of the graduate summer school
- Lattice points, polyhedra, and complexity
- Root systems and generalized associahedra
- Topics in combinatorial differential topology and geometry
- Geometry of $q$ and $q,t$-analogs in combinatorial enumeration
- Chromatic numbers, morphism complexes, and Stiefel-Whitney characteristic classes
- Equivariant invariants and linear geometry
- An introduction to hyperplane arrangements
- Poset topology: Tools and applications
- Convex polytopes: Extremal constructions and $f$-vector shapes