Graduiertenkolleg: Methods for Discrete Structures

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Monday Lecture and Colloquium

Monday, May 27, 2013

Freie Universität Berlin
Institut für Informatik
Takustr. 9
14195 Berlin
room 005

Lecture - 14:15

Gil Kalai - Hebrew University of Jerusalem

Advances and Challenges in the combinatorial study of convex polytopes and related structures

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

Karim Adiprasito - Freie Universität Berlin

Equifacets and Equifaces of convex polytopes

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.

Letzte Aktualisierung: 24.05.2013