Monday, May 18, 2009
Technische Universität Berlin
Fakultät II, Institut für Mathematik
Str. des 17. Juni 136,
room MA 041
Lecture - 14:15
The talk deals with the study of convex polytopes from a combinatorial point of view. The face vector of a polytope counts the faces of each fixed dimension. The flag vector captures more combinatorial information--it counts sequences of faces, of specified dimensions, ordered by inclusion. The problems of characterizing the face vectors and flag vectors of polytopes are open for dimensions four and up... wide open, in the sense that we lack even reasonable conjectures. In this talk I will highlight results, examples, techniques, and connections with other areas of mathematics.
Colloquium - 16:00
A coloring of a fine mixed subdivision of a simplex gives rise to an acyclic system of permutations on the edges of the simplex. In particular, we prove that a system on the edges of an equilateral triangle is achievable through a coloring if and only if it is acyclic, and provide evidence to conjecture that the same result is true for simplices in any dimension. Our work is related to the results on triangulation of products of simplices, Schubert calculus, tropical hyperplane arrangements and tropical oriented matroids, obtained by Santos, Ardila-Billey, Develin-Sturmfels and Ardila-Develin, among others. It also settles an special case of a conjecture of Ardila-Billey about the positions of the simplices in a fine mixed subdivision and provides precise explanation about the behavior of a generic pseudo tropical hyperplane at infinity.