Lorenzo Vecchi (Università di Bologna)

Matroids encode in a combinatorial way the notion of linear independence and can be seen as a generalization of matrices, graphs and hyperplane arrangements.
The main protagonist of this talk is an invariant called the Chow ring of a matroid, whose definition is given in analogy with the one arising from Algebraic Geometry. Long-standing combinatorial conjectures were solved by the introduction of this and other related geometric tools, which in turn have remarkable combinatorial features; for example, their Hilbert series seem to be real-rooted.
After a friendly introduction to Matroid Theory, the plan of the talk is to answer the following questions.
1) How can we study the Hilbert series without actually building the whole graded vector space?While trying to answer this question, different algebraic and combinatorial objects will arise along the way, like the Kazhdan-Lusztig-Stanley polynomials. Help will come both from Poset Theory and Polytope Theory.
2) After obtaining these combinatorial answers, which tools can be lifted back to the higher categorical level we started from?In particular, we are concerned with questions regarding properties of some functors in a new category of matroids. Time permitting, we will also transform all these invariant into graded representations of the group of symmetries of the matroid.
This is based on a joint work with Luis Ferroni, Jacob Matherne, and Matthew Stevens and an ongoing project with Ben Elias, Dane Miyata, and Nicholas Proudfoot.