Geometric Combinatorics

Author(s): 
E. Miller, V. Reiner, B. Sturmfels
Publisher: 
American Mathematical Society
Year: 
2007
ISBN: 
978-0-8218-3736-8
Price (tentative): 
USD 99
MSC main category: 
00 General
Review: 

The notion of geometric combinatorics is quickly getting a much broader meaning. At present it covers not only a structure of polytopes and simplicial complexes but many further topics and interesting connections to other fields of mathematics. It is worth looking at the contents of this book, which contains written versions of the lecture series presented at a three-week program organised at the IAS/Park City Mathematics Institute in 2004. Counting of lattice points in polyhedra and connections to computational complexity is discussed in lectures by A. Barvinok. Root systems, generalised associahedra and combinatorics of clusters form the topic of the lectures by S. Fomin and N. Reading. Combinatorial problems inspired by topics from differential topology (Morse theory) and differential geometry (the Hopf conjecture) are studied by R. Forman. M. Haiman and A. Woo treat topics around Catalan numbers and Macdonald polynomials (the positivity conjecture). D. N. Kozlov discusses in his lectures chromatic numbers, morphism complexes and Stiefel-Whitney characteristic classes. Lectures by R. MacPherson cover topics such as equivariant homology, intersection homology, moment graphs and linear graphs and their cohomology. R. P. Stanley discusses topics connected with hyperplane arrangements and M. L. Wachs treats poset topology. The book ends with a contribution by G. M. Ziegler on convex polytopes. The book contains an enormous amount of interesting material (including a substantial numbers of exercises).

Reviewer: 
vs