Georgia Institute of Technology College of Sciences

Originally introduced independently by Hassler Whitney and Takeo Nakasawa, matroids are a combinatorial way of axiomatizing the notion of linear independence in vector spaces. If $K$ is a field and $n$ is a positive integer, any linear subspace of $K^n$ gives rise to a matroid; such matroid are called representable over $K$. Given a matroid $M$, one can ask over which fields $M$ is representable. More generally, one can ask about representability over partial fields in the sense of Semple and Whittle. Pendavingh and van Zwam introduced the universal partial field of a matroid $M$, which governs the representations of $M$ over all partial fields. Unfortunately, most matroids (asymptotically 100%, in fact) are not representable over any partial field, and in this case, the universal partial field gives no information.

Oliver Lorscheid and I have introduced a generalization of the universal partial field which we call the foundation of a matroid. The foundation of $M$ is a type of algebraic object which we call a pasture; pastures include both hyperfields and partial fields. Pastures form a natural class of field-like objects within Lorscheid's theory of ordered blueprints, and they have desirable categorical properties (e.g., existence of products and coproducts) that make them a natural context in which to study algebraic invariants of matroids. The foundation of a matroid $M$ represents the functor taking a pasture $F$ to the set of rescaling equivalence classes of $F$-representations of $M$; in particular, $M$ is representable over a pasture $F$ if and only if there is a homomorphism from the foundation of $M$ to $F$. (In layman's terms, what we're trying to do is recast as much as possible of the theory of matroids and their representations in functorial ``Grothendieck-style'' algebraic geometry, with the goal of gaining new conceptual insights into various phenomena which were previously understood only through lengthy case-by-case analyses and ad hoc computations.)

As a particular application of this point of view, I will explain the classification which Lorscheid and I have recently obtained of all possible foundations for ternary matroids (matroids representable over the field of three elements). The proof of this classification theorem relies crucially on Tutte's celebrated Homotopy Theorem. 

Function classes are collections of Boolean functions on a finite set. Recently, a method of studying function classes via commutative algebra, by associating a squarefree monomial ideal to a function class, was introduced by Yang. I will describe this connection, as well as some free resolutions and Betti numbers for these ideals for an interesting collection of function classes, corresponding to intersection-closed posets. This is joint work with Chris Eur, Greg Yang, and Mengyuan Zhang.

We introduce the notion of tropical curves of hyperelliptic type. These are tropical curves whose Jacobian is isomorphic to that of a hyperelliptic tropical curve, as polarized tropical abelian varieties. Using the tropical Torelli theorem (due to Caporaso and Viviani), this characterization may be phrased in terms of 3-edge connectiviations. We show that being of hyperelliptic type is independent of the edge lengths and is preserved when passing to genus ≥2 connected minors. The main result is an excluded minors characterization of tropical curves of hyperelliptic type.

Mori Dream Spaces are generalizations of toric varieties and, as the name suggests, Mori's minimal model program can be run for every divisor. It is known that for n≥5, the blow-up of Pn at r very general points is a Mori Dream Space iff r≤n+3. In this talk we proceed to blow up points as well as lines, by considering the blow-up X of P3 at 6 points in very general position and all the 15 lines through the 6 points. We find that the unique anticanonical section of X is a Jacobian K3 Kummer surface S of Picard number 17. We prove that there exists an infinite-order pseudo-automorphism of X, whose restriction to S is one of the 192 infinite-order automorphisms constructed by Keum.  A consequence is that there are infinitely many extremal effective divisors on X; in particular, X is not a Mori Dream Space. We show an application to the blow-up of Pn (n≥3) at (n+3) points and certain lines.  We relate this pseudo-automorphism to the structure of the birational automorphism group of P3. This is a joint work with Lei Yang.