Monday, May 27, 2013
Freie Universität Berlin
Institut für Informatik
Takustr. 9
14195 Berlin
room 005
Lecture - 14:15
Abstract:
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
Abstract:
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.