Monday, October 31, 2016
Technische Universität Berlin
Institut für Mathematik
Straße des 17. Juni 136
10623 Berlin
room MA 041
Lecture - 14:15
Abstract:
Polycrystalline materials, such as metals, are composed of crystal grains of varying size and shape. Typically, the occurring grain cells have the combinatorial types of 3-dimensional simple polytopes, and together they tile 3-dimensional space.
We will see that some of the occurring grain types are substantially more frequent than others - where the frequent types turn out to be
"combinatorially round". Here, the classification of grain types gives us, as an application of combinatorial low-dimensional topology, a new
starting point for a topological microstructure analysis of steel.
Colloquium - 16:00
Abstract:
We define the excess degree e(P) of a d-polytope P as 2f_1-df_0, where f_0 and f_1 denote the number of vertices and edges of the polytope, respectively.
We first prove that the excess degree of a d-polytope does not take every natural number: the smallest values are 0 and d-2, with the value d-1 only occurring when d=3 or 5. On the other hand, if d is even, the excess degree takes every even natural number from d\sqrt{d} onwards; while, if d is odd, the excess degree takes every natural number from d\sqrt{2d} onwards.
Our study of the excess degree is then applied in four different settings. We show that polytopes with small excess (i.e., e(P)=d-2,d-1) behave in a similar manner to simple polytopes in terms of (Minkowski) decomposability: each such polytope is either decomposable or a pyramid, and their duals are always indecomposable. This is best possible since there are d-polytopes with excess d which are decomposable and d-polytopes which are not. Secondly, we characterise the decomposable d-polytopes with at most 2d+1 vertices. Here we are only concerned with combinatorial decomposability: a polytope P is (combinatorially) decomposable if whenever Q is combinatorially equivalent to P (equivalent face lattices), Q is decomposable.Thirdly, we characterise all pairs (f_0,f_1) for which there exists a 5-polytope with f_0 vertices and f_1 edges. And, fourthly, we characterise the d-polytopes with at most 2d+2 vertices and the minimum number of edges, answering in this way a 1969 conjecture of Grunbaum.