Monday, May 27, 2013
Freie Universität Berlin
Institut für Informatik
Lecture - 14:15
The lecture will discuss some of the main problems and some of the most disturbing problems in the combinatorics of convex polytopes and more general structures. Among the issues i will discuss are: f-vectors, flag vectors, neighborliness, low-dimensional skeleta, the g-conjecture for spheres, some problems around the Hirsch conjecture, problems on special classes of polytopes.
Colloquium - 16:00
A classic observation in the theory of convex polytopes is that for every 3-dimensional polytope, we have that
3p_3+2p_4+p_5 > 11 (1)
where p_3, p_4 and p_5 denote the number of triangle, quadrilateral resp. pentagon faces of the polytope. In particular, not every facet of a 3-polytope can be a hexagon. In a delightful 1967 paper, Perles and Shephard proved generalizations to equation (1), and demonstrated, for example, that not every facet of a 7-polytope can be a 6-dimensional crosspolytope. Since then, their results have been refined in several ways, but mesmerizing problems remain open.
I will discuss the classic approaches to the problem, and relations to results of Brooks, Gao-Yau and others on the existence of negatively curved metrics on manifolds of dimension 3 and higher.