Georgia Institute of Technology College of Sciences

This talk is about an application of combinatorial algebraic geometry to complex/arithmetic dynamics. The n-th Gleason polynomial G_n is a polynomial in one variable with Z-coefficients, whose roots correspond to degree-2 polynomials with an n-periodic ramification point. Per_n is an affine algebraic curve, defined over Q, parametrizing degree-2 rational maps with an n-periodic ramification point. Two long-standing open questions in complex dynamics are: (1) Is G_n is irreducible over Q? (2) Is Per_n connected? We show that if G_n is irreducible over Q, then Per_n is irreducible over C, and is therefore connected. In order to do this, we find a Q-rational smooth point on a projective completion of Per_n — this Q-rational smooth point represents a special degeneration of degree-2 self-maps.

Traditionally, chip-firing is a discrete dynamical system where poker chips move around the vertices of a graph. One fascinating result is that number of configurations of a fixed number of chips, modulo a firing equivalence relation, is the number of spanning trees of the graph. This relationship gives the set of spanning trees group-like properties.

In this talk, I will discuss how chip-firing ideas can be generalized from graphs to regular matroids, where bases play the role of spanning trees. This will lead to an overview of joint work with Ding, Tóthmérész, and Yuen on the consistency of the Backman-Baker-Yuen Sandpile Torsor. 

============(Below is the information on the pre-talk.)============

Title (pre-talk): Transforming Spanning Trees Using Mathematicians and Coffee Cups

Abstract (pre-talk): There is a fascinating structure to the set of spanning trees of a plane graph, which allows this set to behave much like a group. Perhaps most incredibly, there is a sense in which this structure is canonical.
In this talk, I will show you how spanning trees can be transformed after introducing mathematicians and coffee cups on some of the vertices. This is a variant of the rotor-routing process which takes advantage of a special property of plane graphs.

A Springer fiber is the set of complete flags in Cn which are fixed by a given nilpotent matrix. It is a fundamental object of study in geometric representation theory and algebraic combinatorics. The irreducible components of a Springer fiber are indexed by combinatorial objects called standard Young tableaux. It is an open problem to describe geometric properties of these components (such as their singular loci and cohomology classes) in terms of the combinatorics of tableaux. We initiate a new approach to this problem by characterizing which irreducible components are equal to Richardson varieties, which are comparatively much better understood. Another motivation comes from Lusztig's recent study of the cell decomposition of the totally nonnegative part of a Springer fiber into totally positive Richardson cells. This is joint work in progress with Martha Precup.

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.