ACO Seminar - Luis Ferroni

— 4:00pm

Location:
In Person - Wean Hall 8220

Speaker:
LUIS FERRONI, Postdoctoral Researcher, Institute for Advanced Study, Princeton
https://sites.google.com/view/ferroniluis/home/

In this talk we introduce a polytope that encodes all matroids of a fixed size and rank. The polytope is constructed using the representation of each matroid as a combination of Schubert matroids within the valuative group. Linear maps in the polytope's ambient space correspond to valuative invariants on matroids. Each matroid is represented as a lattice point, but only certain special matroids correspond to the vertices of the polytope, giving rise to a new notion of "extremality." 

We argue that in cases where a conjecture in matroid theory posits the positivity of a specific invariant, extremal matroids should be examined first. For instance, we will explain why the counterexamples to the Ehrhart positivity conjecture by De Loera, Haws, and Köppe, as well as the counterexamples to the Merino-Welsh conjecture on Tutte polynomials, both correspond to vertices of these polytopes. In general, if the conjecture involves a valuative invariant, extremal matroids are the correct candidates to be counterexamples.

Furthermore, the framework of these polytopes allows us to address a well-known folklore question in matroid theory. We show the existence of a valuative invariant that serves as a test for representability. In other words, we show that there are valuative invariants that are non-negative for all realizable matroids but fail to remain non-negative in general. 

This is joint work with Alex Fink. 

4:00 pm - Jane Street-sponsored tea and cookies in the math lounge (please bring your own mug if you have one).

Event Website:
https://aco.math.cmu.edu/abs-24-25/mar13.html

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