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Geometric Combinatorics
Edited by: Ezra Miller and Victor Reiner, University of Minnesota, Minneapolis, MN, and Bernd Sturmfels, University of California, Berkeley, CA
A co-publication of the AMS and IAS/Park City Mathematics Institute.
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IAS/Park City Mathematics Series
2007; 691 pp; hardcover
Volume: 13
ISBN-10: 0-8218-3736-2
ISBN-13: 978-0-8218-3736-8
List Price: US$104
Member Price: US$83.20
Order Code: PCMS/13
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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.

Titles in this series are co-published 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 multi-faceted 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 Charney-Davis 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 Stiefel-Whitney characteristic classes
  • Preamble
  • Introduction
  • The functor Hom\((-,-)\)
  • Stiefel-Whitney 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 3-dimensional polytopes
  • Shapes of \(f\)-vectors
  • 2-simple 2-simplicial 4-polytopes
  • \(f\)-vectors of 4-polytopes
  • Projected products of polygons
  • A short introduction to polymake
  • Bibliography
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