Current Research

I study moduli problems in algebraic geometry and representation theory using methods from computational algebra. I'm interested in combinatorial algebraic geometry, mathematical biology, and tropical geometry. CV

Papers and Preprints

The toric geometry of triangulated polygons in Euclidean space: In this paper we give a geometric description of the toric degenerations moduli of points on the projective line, equivalently polygons in euclidean 3-space. We describe the stratefied symplectic structure of these spaces, and the action of a compact torus in terms of the geometry of euclidean polygons. Joint work with Benjamin Howard and John Millson.

Presentations of semigroup algebras of weighted trees: We find presentations for subalgebras of invariants of the coordinate algebras of binary symmetric models of phylogenetic trees studied by Buczynska and Wiesniewski. These algebras arise as toric degenerations of projective coordinate rings of weight varieties of the Grassmanian of two planes, and as toric degenerations of rings of invariants of Cox-Nagata rings of the moduli of semistable quasiparabolic rank 2 bundles on the projective line.

Gorenstein semigroup algebras of weighted trees: We classify exactly when the toric degenerations of projective coordinate rings of the moduli of points on the line are Gorenstein. These algebras arise as toric deformations of algebras of invariants of the Cox-Nagata ring of the blow-up of n-1 points on n-3 projective space. As a corollary, we find exactly when these families of rings are Gorenstein as well.

The algebra of conformal blocks: We study and generalize the connection between Hilbert functions from phylogenetic algebraic geometry and the Verlinde formula from mathematical physics, as discovered by Sturmfels and Xu. In order to accomplish this we introduce deformations of algebras of non-abelian theta functions for a general simple complex Lie algebra, structured on the moduli stack of stable punctured curves. We also study the relationship between these algebras and branching algebras of the associated simply connected reductive group.

The m-dissimilarity map and representation theory of SL_m:We give another proof that m-dissimilarity vectors of weighted trees are points on the tropical Grassmanian, as conjectured by Cools, and proved by Giraldo in response to a question of Sturmfels and Pachter. We accomplish this by relating m-dissimilarity vectors to the representation theory of SL_m by means of specially constructed valuations.

Toric Deformations and tropical geometry of branching algebras: We construct polyhedral families of valuations on the branching algebra of a morphism of reductive groups. This establishes a connection between the combinatorial rules for studying a branching problem and the tropical geometry of the branching algebra. In the special case when the branching problem comes from the inclusion of a Levi subgroup or a diagonal subgroup, we use the dual canonical basis of Lusztig and Kashiwara to build toric deformations of the branching algebra.

Coordinate rings for the moduli of SL_2(\C) quasi-parabolic principal bundles on a curve and toric fiber products: We continue the program to understand the commutative algebra of the projective coordinate rings of line bundles on the moduli of quasi-parabolic principal bundles on a marked projective curve. We prove a general theorem about presentations of these rings, which implies that for generic marked curves the square of any effective line bundle has projective coordinate ring generated in degree 1 with a presenting ideal generated in degree 3. When the genus of the curve is less than or equal to 2, we find that the square of any such line bundle gives a Koszul projective coordinate ring. Both theorems are obtained by studying toric degenerations of the projective coordinate ring. This leads us to formalize the properties of the polytopes used in proving our results by constructing a category of polytopes with term-orders, and studying its closure properties under fiber products.

Dissimilarity maps on trees and the representation theory of GL_n(\C): We revisit the representation theory in type A used previously to establish that the dissimilarity vectors of phylogenetic trees are points on the tropical Grassmannian variety. We use a different version of this construction to show that the space of phylogenetic trees maps to the tropical varieties of every flag variety of GL_n(\C). Using this map, we interpret the tropicalization of the semistandard tableaux basis of an irreducible representation of GL_n(\C) as combinatorial invariants of phylogenetic trees.