Abstract:

Let \Delta be a (d-1)-dimensional simplicial manifold with n vertices. One of the problems in combinatorics is: what is the maximum possible number of i -dimensional faces of \Delta ? What is the maximum possible Euler characteristic of \Delta ?

The Upper Bound Conjecture (UBC) asserts that if \Delta is a Eulerian simplicial manifold then for any i the number of i -dimensional faces of \Delta is not greater than the number of i -faces of the cyclic d -polytope with n vertices.

In this talk we will outline the proof of the UBC for all Eulerian manifolds. We also will sketch the proof of the analog of the UBC for arbitrary (non-Eulerian) simplicial manifolds and the (partial) proof of a conjecture by K\"{u}hnel concerning the maximum Euler characteristic of even-dimensional simplicial manifolds.

Coffee, Cookies, Company at the faculty lounge at 3:30.