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Geometric Combinatorics

Ezra Miller, Victor Reiner, Bernd Sturmfels, editors
American Mathematical Society
Publication Date: 
Number of Pages: 
IAS/Park City Mathematics Series 13
[Reviewed by
Gizem Karaali
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This book is an outcome of a recent (2004) summer program in geometric combinatorics at the IAS/Park City Mathematics Institute. 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. 

In a nutshell, geometric combinatorics deals with geometric objects and their combinatorial properties. However this vague description does not do justice to the beauty and diversity of the field. The book under review gives the reader a much more developed and balanced perspective.

The volume begins with a brief piece by two of the editors (Miller and Reiner) which can be viewed simultaneously as an introduction to the text and to the field. This is followed by lectures by Alexander Barvinok on lattice points and polyhedra. The next set of lecture notes is by Sergei Fomin and Nathan Reading and introduces root systems, generalized associahedra and cluster algebras. Robin Forman’s lectures on combinatorial differential topology and geometry are then followed up by Mike Haiman and Alexander Woo’s discussion of combinatorial enumeration and q and q-t analogues. Other sections include Dimitry Kozlov’s lectures on chromatic numbers, morphism complexes and Stiefel-Whitney characteristic classes, and Robert MacPherson’s lectures on equivariant invariants and linear geometry, Michelle Wachs’ lectures on poset topology. Richard Stanley contributes a six-lecture introduction to hyperplane arrangements, and Gunter Ziegler’s piece on convex polytopes wraps up the volume.

As can be recognized by anyone mildly familiar with the field, the above comprises a formidable list of experts; furthermore, each and every one of them writes quite eloquently. As Miller and Reiner emphasize in their introduction, the volume does not manage to cover everything under the sun that will fit under the umbrella term Geometric Combinatorics. However it makes for an excellent introduction for anyone interested in any of the topics that are indeed covered. Highly recommended as a starting point for all students of mathematics, professional and otherwise.

Gizem Karaali is assistant professor of Mathematics at Pomona College.
  • 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