IAS/Park City Mathematics Series 2007; 691 pp; hardcover Volume: 13 ISBN10: 0821837362 ISBN13: 9780821837368 List Price: US$104 Member Price: US$83.20 Order Code: PCMS/13
 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 threeweek 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. Titles in this series are copublished with the Institute for Advanced Study/Park City Mathematics Institute. Members of the Mathematical Association of America (MAA) and the National Council of Teachers of Mathematics (NCTM) receive a 20% discount from list price. Readership Graduate students and research mathematicians interested in combinatorics; discrete methods in geometry and topology. Reviews "The editors have done an excellent job in bringing together many leaders of the field and encouraging them to write expository lecture notes on various topics that expertly showcase the multifaceted world of this vast and rapidly growing field of mathematics."  MAA Reviews Table of Contents  What is geometric combinatorics?An overview of the graduate summer school
 Bibliography
A. Barvinok, Lattice points, polyhedra, and complexity  Introduction
 Inspirational examples. Valuations
 Identities in the algebra of polyhedra
 Generating functions and cones. Continued fractions
 Rational polyhedra and rational functions
 Computing generating functions fast
 Bibliography
S. Fomin and N. Reading, Root systems and generalized associahedra  Introduction
 Reflections and roots
 Dynkin diagrams and Coxeter groups
 Associahedra and mutations
 Cluster algebras
 Enumerative problems
 Bibliography
R. Forman, Topics in combinatorial differential topology and geometry  Introduction
 Discrete Morse theory
 Discrete Morse theory, continued
 Discrete Morse theory and evasiveness
 The CharneyDavis conjectures
 From analysis to combinatorics
 Bibliography
M. Haiman and A. Woo, Geometry of \(q\) and \(q,t\)analogs in combinatorial enumeration  Introduction
 Kostka numbers and \(q\)analogs
 Catalan numbers, trees, Lagrange inversion, and their \(q\)analogs
 Macdonald polynomials
 Connecting Macdonald polynomials and \(q\)Lagrange inversion; \((q,t)\)analogs
 Positivity and combinatorics?
 Bibliography
D. N. Kozlov, Chromatic numbers, morphism complexes, and StiefelWhitney characteristic classes  Preamble
 Introduction
 The functor Hom\((,)\)
 StiefelWhitney classes and first applications
 The spectral sequence approach
 The proof of the Lovász conjecture
 Summary and outlook
 Bibliography
R. MacPherson, Equivariant invariants and linear geometry  Introduction
 Equivariant homology and intersection homology (Geometry of pseudomanifolds)
 Moment graphs (Geometry of orbits)
 The cohomology of a linear graph (Polynomial and linear geometry)
 Computing intersection homology (Polynomial and linear geometry II)
 Cohomology as functions on a variety (Geometry of subspace arrangements)
 Bibliography
R. P. Stanley, An introduction to hyperplane arrangements  Basic definitions, the intersection poset and the characteristic polynomial
 Properties of the intersection poset and graphical arrangements
 Matroids and geometric lattices
 Broken circuits, modular elements, and supersolvability
 Finite fields
 Separating hyperplanes
 Bibliography
M. L. Wachs, Poset topology: Tools and applications  Introduction
 Basic definitions, results, and examples
 Group actions on posets
 Shellability and edge labelings
 Recursive techniques
 Poset operations and maps
 Bibliography
G. M. Ziegler, Convex polytopes: Extremal constructions and \(f\)vector shapes  Introduction
 Constructing 3dimensional polytopes
 Shapes of \(f\)vectors
 2simple 2simplicial 4polytopes
 \(f\)vectors of 4polytopes
 Projected products of polygons
 A short introduction to polymake
 Bibliography
