Georgia Institute of Technology College of Sciences

Previous work of Chan-Church-Grochow and Baker-Wang shows that the set of spanning trees in a plane graph $G$ is naturally a torsor for the Jacobian group of $G$. Informally, this means that the set of spanning trees of $G$ naturally forms a group, except that there is no distinguished identity element. We generalize this fact to graphs embedded on orientable surfaces of arbitrary genus, which can be identified with ribbon graphs. In this generalization, the set of spanning trees of $G$ is replaced by the set of spanning quasi-trees of the ribbon graph, and the Jacobian group of $G$ is replaced by the Jacobian group of the associated regular orthogonal matroid $M$.

Our proof shows, more generally, that the family of "BBY torsors'' constructed by Backman-Baker-Yuen and later generalized by Ding admit natural generalizations to regular orthogonal matroids. In addition to shedding light on the role of planarity in the earlier work mentioned above, our results represent one of the first substantial applications of orthogonal matroids to a natural combinatorial problem about graphs. 

 Joint work with Matt Baker and Donggyu Kim. 

For a central simple algebra $A$ over a field $K$, there are two major invariants, viz., period and index. For a field $K$, the Brauer-$l$-dimension of $K$ for a prime number $l$, is the smallest natural number $d$ such that for every finite field extension $L/K$ and every central simple $L$-algebra $A$ (of period a power of $l$), we have that index($A$) divides period$(A)^d$.

If $K$ is a number field or a local field, then classical results from class field theory tell us that the Brauer-$l$-dimension of $K$ is 1. This invariant is expected to grow under a field extension, bounded by the transcendence degree. Some recent works in this area include that of Harbater-Hartmann-Krashen for $K$ a complete discretely valued field, in the good characteristic case. In the bad characteristic case, for such fields $K$, Parimala-Suresh have given some bounds.

Also, the u-invariant of $K$ is the maximal dimension of anisotropic quadratic forms over $K$. For example, the u-invariant of $\mathbb{C}$ is 1, for $F$ a non-real global or local field the u-invariant of $F$ is 1, 2, 4, or 8, etc.

In this talk, I will present similar bounds for the Brauer-l-dimension and the strong u-invariant of a complete non-Archimedean valued field $K$ with residue field $\kappa$.

The flag variety G/B plays an outsized role in representation theory, combinatorics, geometry and algebra. Hessenberg varieties form a special class of subvarietes of the flag variety, arising in diverse contexts. The cohomology ring of a semisimple Hessenberg variety is recognized to be a representation of an associated finite group, and is related to the expansion of some special polynomials in terms of other well-known polynomial bases. These varieties may have pathological behavior, and their basic properties have been characterized only in restricted cases. Matrix Hessenberg schemes in type A consist of a lift of these varieties to G = Gl(n, C), where we can use the coordinate ring of matrices to study them.

In this talk, we present a full characterization of matrix Hessenberg schemes over the minimal sheet of Lie(G) in type A. We show that each semisimple matrix Hessenberg scheme lies in a flat family with a nilpotent matrix Hessenberg scheme, which in turn allows us to study their geometric properties. We describe the schemes fully in terms of Schubert varieties and opposite Schubert varieties, both well-known subvarieties of G/B. More subtly we characterize combinatorially which matrix Hessenberg schemes are reduced. These results are joint with Martha Precup at Washington University, St. Louis.