Monday, June 13, 2016
Technische Universität Berlin
Institut für Mathematik
Straße des 17. Juni 136
10623 Berlin
room MA 041
Lecture - 14:15
Abstract:
What does it mean to program a category? Can one program localisations of categories including the derived category formalism and what is this good for? I will try to answer these questions by showing our applications, how far we got, and where we are heading.
Colloquium - 16:00
Abstract:
We study a particular graded ring structure on the set of all loopfree matroids on a
fixed labeled ground set, which occurs naturally in tropical geometry. The product is
given by matroid intersection and the additive structure is defined by assigning to each
matroid the indicator vector of its chains of flats. This ring has many striking properties,
which we will outline in this talk:
First and foremost, the Tutte polynomial, and thus many more matroid invariants,
induce Z-module homomorphisms from this ring.
Furthermore, the ring is graded by corank and a basis for each graded part is given
by the set of nested matroids of the corresponding corank. We will see that the number
of these matroids is a Eulerian number, thus establishing that, as a free Z-module, the
intersection ring has rank n!. It is generated in corank one, i.e. every matroid is a
linear combination of products of corank one matroids. Finally, we will see that the
multiplication induces an interesting duality (which has nothing to do with dualizing
matroids!), similar to Poincaré duality.
The talk will also include a very brief excursion into tropical geometry, demonstrating
the geometric intuition behind these ideas.