On rational and nonrational polytopes

Let P be a polytope and f_i the number of i-dimensional faces of P. An interesting problem in combinatorics is to decide what conditions the numbers f_i must satisfy. This problem has a beautiful connection with algebraic geometry (due to R. Stanley). To a polytope P with rational vertices one can associate an algebraic variety. Then familiar conditions on the cohomology of the variety define conditions on the face numbers f_i. In this talk I discuss Stanley's proof of the rational case and its extension to the case of a nonrational polytope.

Refreshments will be served at 3:45 p.m. in the Faculty Lounge, Math Annex (Room 1115).