Afshin Goodarzi: On the Face Numbers of Simplicial Complexes

Simplicial complexes provide a link between combinatorics, algebra and topology. On one hand they are useful in homology-computation of a big variety of topological spaces. On the other hand, they are corresponded to square-free monomial ideals via Stanley-Reisner theory.  New theoretical/computational developments such as Gröbner bases and discrete Morse theory increase the interest in simplicial complexes. 

A classical problem in combinatorics, with many advantages in algebra and topology, is to give a numerical characterization of the face numbers of various classes of simplicial complexes. For instance one may consider the classes of (sequentially) Cohen-Macaulay complexes (from algebra), complexes with r-colorable underlying graph (from combinatorics), or complexes that triangulate a specific topological space (such as spheres, homology spheres, balls, ...) and study their face numbers. 

In this talk, I shall give a survey on the subject and mention some recent results and progresses.