Mathematics Colloquium, Spring 2004
Special Lecture presented by

Face numbers of simplicial manifolds

The f-vector of a simplicial complex K is a vector whose i-th coordinate is the number of faces of dimension i in K (for i = 0, 1, ...). One of the central problems in geometric combinatorics is to characterize the collection of f-vectors for certain families of simplicial complexes or, at least, to find substantial necessary conditions. This problem was addressed in works of Klee, McMullen, Stanley, and others. In the talk I will present the state-of-the-art results on the subject for several classes of simplicial complexes, among them simplicial manifolds and pseudomanifolds with isolated singularities. In particular I will discuss the topics of

(1) the Upper Bound Conjecture originally proposed by Motzkin for polytopes and later extended by Klee to all Eulerian complexes, asserting that, in the class of all Eulerian complexes of dimension d - 1 on n vertices, the boundary of the cyclic polytope has the largest number of i-dimensional faces for all i;

(2) related conjectures by Kuhnel and Sparla on the Euler characteristic of simplicial manifolds.

Monday, January 26, 2004, 245 Altgeld Hall, 4:00 p.m.