Monday, October 19, 2015
Technische Universität Berlin
Institut für Mathematik
Straße des 17. Juni 136
10623 Berlin
room MA 041
Lecture - 14:15
Abstract:
The d-dimensional simplicial, terminal, and reflexive polytopes with
at least 3d-2 vertices are classified. In particular, it turns out
that all of them are smooth Fano polytopes. This improves on previous
results of (Casagrande, 2006) and (Øbro, 2008). Smooth Fano polytopes
play a role in algebraic geometry and mathematical physics. In
addition to this, I would like to argue that they provide interesting
testing ground for helping to sharpen the tools of geometric
combinatorics.
This talks is based on joint work with Benjamin Assarf, Andreas
Paffenholz and Julian Pfeifle.
Colloquium - 16:00
Abstract:
I will talk about the application of syzygy tool from commutative algebra in two concrete examples arisen in divisor theory of graphs and system reliability theory. Attached to any finite graph G, there is a natural lattice (the ``lattice of integral cuts'') whose Delaunay cell decomposition is related to graphic hyperplane arrangements. There is also a canonical ideal IG which encodes the linear equivalences of divisors on G. The ideas from potential theory on graphs and from the theory of Delaunay decompositions for lattices can be used to read the Betti numbers of IG in terms of the number of faces of various dimensions in the graphic hyperplane arrangement, or equivalently, the number of orbits of the Delaunay cells of various dimensions in the cut lattice. The ideal IG is also appearing in the theory of system reliability, and its Hilbert series encodes the reliability of the system which is the probability that G is connected (assuming that the vertices are reliable but each edge may fail with the probability pe). This develops new connections between the theory of oriented matroid, the theory of divisors on graphs, and the theory of system reliability.