Georgia Institute of Technology College of Sciences

The theory of $p$-adic differential equations first rose to prominence after Dwork used them to prove the rationality of zeta functions of a positive characteristic variety in 1960. Since then, there has been growing interest in the category of convergent $F$-isocrystals and the subcategory of overconvergent $F$-isocrystals due to this subcategory having good cohomology theory with finiteness properties. Recent work by Grubb, Kedlaya and Upton examines when a convergent $F$-isocrystal is overconvergent by restricting to smooth curves on the scheme under a mild tameness assumption (measured by the Swan conductor). In my talk, I will introduce the above categories and talk about work in progress about bounding the Swan conductor of an overconvergent $F$-isocrystal in terms of data associated with the corresponding convergent $F$-isocrystal.

Shellability is a fundamental concept in combinatorial topology and algebraic combinatorics. Two foundational results are Bruggesser–Mani’s line shellings of polytopes and Björner’s theorem that the order complex of a geometric lattice is shellable. 

Inspired by Bruggesser–Mani’s line shellings of polytopes, we introduce line shellings for the lattice of flats of a matroid:  given a normal complex for a Bergman fan of a matroid induced by a building set, we show that the lexicographic order of the coordinates of its vertices is a shelling order.  This yields a new geometric proof of Björner’s classical result and establishes shellability for all nested set complexes for matroids.

This is joint work with Galen Dorpalen-Barry, Anastasia Nathanson, Ethan Partida, and Noah Prime.

The Plücker embedding exhibits the Grassmannian Gr(r, n) as a closed subvariety of projective space. A theorem of Hodge shows that its homogeneous ideal has as a quadratic Gröbner basis the so-called multiple-exchange relations between Plücker coordinates. Since the set of these polynomials is quite large and unwieldy, it is often preferable to work with a smaller set of single-exchange Plücker relations. An even smaller set of polynomials is the collection of local (or 3-term) exchange relations. We will recall and clarify the relationships between these three. We go on to examine the situation over hyperfields. In their pioneering work, Baker and Bowler showed that the theories of matroids, oriented matroids, valuated matroids etc. can be collectively understood under a common banner as the theory of Grassmanians over hyperfields. Their work gives a good accounting of the relationship between single- and local-exchange relations in this generalized setting. We will discuss what can be said about the multiple-exchange relations. This leads to considerations of elementary linear-algebraic facts in the hyperfield setting. All results may be suitably extended to the flag setting---which we will discuss, time permitting. The talk is based on joint work with Nathan Bowler and Changxin Ding.

The chromatic polynomial of a graph, which counts colorings of the graph, has a habit of showing up in unexpected places in geometry, e.g. in the theory of hyperplane arrangements. This sometimes has interesting purely combinatorial consequences, such as Huh's proof of Hoggar/Read's conjecture on coefficients of chromatic polynomials. 

I'll discuss a new incarnation of chromatic polynomials. To a graph G, we can naturally associate a sequence of intersection numbers on moduli spaces of stable curves. Surprisingly, we prove that these recover values of the chromatic polynomial of G at negative integers. 

I'll also discuss how this leads to new algebraic invariants of directed graphs.

(Joint with Bernhard Reinke)