The basic idea is to associate with a combinatorial object Xan algebraic structure A(X), and derive from algebraic properties of A(X)combinatorial consequences for X. For example, Stanley's proof of the UpperBound Theorem for simplicial spheres uses the Cohen-Macaulay property of theface ring associated with a simplicial complex.

We will review the basics of Stanley's theory, illustrate themon examples, and time permitting, discuss more recent advances of this theory.

(All needed terms and background will be given in thetalk.)