Graphs and Matroids Seminar - Erik Panzer

Title: The Hepp bound of a matroid: flags, volumes and integrals

Abstract:

Invariants of combinatorial structures can be very useful tools that capture some specific characteristics, and repackage them in a meaningful way. For example, the famous Tutte polynomial of a matroid or graph tracks the rank statistics of its submatroids, which has many applications, and relations like contraction-deletion establish a very close connection between the algebraic structure of the invariant (e.g. Tutte polynomials) and the actual matroid itself.

I will present an invariant, called the Hepp bound, that associates to a matroid a rational function in many variables (one variable for each element of the matroid). This invariant behaves nicely with respect to duality and 2-sums, and the residues at its poles factorize into the Hepp bounds of sub- and quotient matroids. It can be specialized to Crapo's beta invariant and it is also related to Derksen's invariant. The construction is motivated by the tropicalization of Feynman integrals from the quantum field theory of elementary particles physics. In the case of graphs, the Hepp bound therefore obeys further interesting relations that are known for Feynman integrals.

Due to this rich structure, the Hepp bound can be viewed from several distinct perspectives, each making certain properties emerge more directly than others. I will sketch 3 definitions:
1) enumerative - as a certain sum over flags of submatroids,
2) analytic - as an integral,
3) geometric - as a volume of a polytope.