Georgia Institute of Technology College of Sciences

We describe minimal free resolutions of a lattice ideal associated with a graph and its initial ideal. These ideals are closely related to chip firing games and the Riemann-Roch theorem on graphs. Our motivations are twofold: describing information related to the Riemann-Roch theorem in terms of Betti numbers of the lattice ideal and the problem of explicit description of minimal free resolutions. This talk is based on joint work with Frank-Olaf Schreyer and John Wilmes. Analogous results were simultaneously and independently obtained by Fatemeh Mohammadi and Farbod Shokrieh.

A metrized complex of algebraic curves over a field K is, roughly speaking, a finite edge-weighted graph G together with a collection of marked complete nonsingular algebraic curves C_v over K, one for each vertex; the marked points on C_v correspond to edges of G incident to v. We will present a Riemann-Roch theorem for metrized complexes of curves which generalizes both the classical and tropical Riemann-Roch theorems, together with a semicontinuity theorem for the behavior of the rank function under specialization of divisors from smooth curves to metrized complexes. The statement and proof of the latter result make use of Berkovich's theory of non-archimedean analytic spaces. As an application of the above considerations, we formulate a partial generalization of the Eisenbud-Harris theory of limit linear series to semistable curves which are not necessarily of compact type. This is joint work with Omid Amini.

Let a,b,c >= 2 be integers satisfying 1/a + 1/b + 1/c > 1. Darmon and Granville proved that the generalized Fermat equation x^a + y^b = z^c has only finitely many coprime integer solutions; conjecturally something stronger is true: for a,b,c \geq 3 there are no non-trivial solutions and for (a,b,c) = (2,3,n) with n >= 10 the only solutions are the trivial solutions and (+- 3,-2,1) (or (+- 3,-2,+- 1) when n is even). I'll explain how the modular method used to prove Fermat's last theorem adapts to solve generalized Fermat equations and use it to solve the equation x^2 + y^3 = z^10.

Hyperbolic polynomials are real polynomials that can be thought of as generalized determinants. Each such polynomial determines a convex cone, the hyperbolicity cone. It is an open problem whether every hyperbolicity cone can be realized as a linear slice of the cone of psd matrices. We discuss the state of the art on this problem and describe an inner approximation for a hyperbolicity cone via a sums of squares relaxation that becomes exact if the hyperbolic polynomial possesses a symmetric determinantal representation. (Based on work in progress with Cynthia Vinzant)