Colloquium

On rational and nonrational polytopes

by Kalle Karu, Assistant Professor, Harvard University, Cambridge, Massachusetts

Abstract

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