Research

Publications

• Matching Fields in Macaulay2

• Subalgebra Bases in Macaulay2

• Restricted Chain-Order Polytopes via Combinatorial Mutations

• Combinatorial Mutations of Gelfand-Tsetlin Polytopes, Feigin-Fourier-Littelmann-Vinberg Polytopes, and Block Diagonal Matching Field Polytopes

• Toric Degenerations of Partial Flag Varieties and Combinatorial Mutations of Matching Field Polytopes

• The Equivariant Ehrhart Theory of Polytopes with Order-Two Symmetries

• A Study of Nonlinear Multiview Varieties

• Matroid Stratifications of Hypergraph Varieties, their Realization Spaces, and Discrete Conditional Independence Models

• Standard Monomial Theory and Toric Degenerations of Richardson Varieties in the Grassmannian

• Standard Monomial Theory and Toric Degenerations of Richardson Varieties in Flag Varieties

• Conditional Probabilities via Line Arrangements and Point Configurations

• Combinatorial Mutations and Block Diagonal Polytopes 

• Standard Monomial Theory and Toric Degenerations of Richardson Varieties inside Grassmannians and Flag Varieties

• Toric Degenerations of Flag Varieties from Matching Field Tableaux

• Toric Degenerations of Grassmannian and Schubert Varieties from Matching Field Tableaux 

• Standard Monomial Theory and Toric Degenerations of Schubert Varieties from Matching Field Tableaux 

• Conditional Independence Ideals with Hidden Variables 

Work in Progress

• Algorithms in Equivariant Ehrhart Theory

• Metric Properties of Markov Bases

• Subalgebra and Khovanskii Bases Equivalence

• On Matroid Varieties and Point-Line Configurations

• The Algebaic Matorid of the Grassmannian via Matching Fields

• The Full Equivariant Ehrhart Theory of the Hypersimplex

• Matroid Stratification of Toric Maximum Likelihood Degree

Introduction to Research Topics

Toric Degenerations of Grassmannian, Flag Varieties and Schubert Varieties via Matching Fields

In these works, we expand the collection of known examples of toric degenerations of Grassmannians and flag varieties. A toric degeneration of a variety X is flat family F over the affine line such that the special fiber (over zero) is a toric variety and all other fibers are isomorphic to X.  Toric degenerations are useful for studying varieties because:

1. flatness of the degeneration guarantees that many properties such as the Hilbert function are preserved in the toric fiber,

2. algebraic invariants of the toric variety are often encoded combinatorially in the moment polytope or fan.

Toric degenerations are often difficult to calculate so we begin by looking at the Grassmannian, whose combinatorial structure simplifies computations. 

Mohammadi and Shaw used matching fields, introduced by Sturmfels and Zelevinsky, that arise from top dimensional cones of the tropicalisation of the Grassmannian Gr(3,n) to find toric degenerations. They defined a class of matching fields called block-diagonal matching fields that have a simple combinatorial description but exhibit all but one of the possible Gröbner degenerations for Gr(3,6) arising from all matching fields, up to isomorphism.

In my work, I have shown that block diagonal matching fields give rise to toric degenerations for all Grassmannians Gr(k,n). With careful work, these ideas can be extended to find toric degenerations of Schubert varieties inside the Grassmannian and flag varieties of type A and their associated Schubert and Richardson varieties. 

From the perspective of computing toric degenerations, we can ask the following natural question: how can we produce a new toric degenerations from a given a one? Or, in other words: what relates different toric degenerations? And is there an algorithm? In work of Escobar and Harada, with an appendix by Itlen, it shown that the polytopes, or more generally Newton-Okounkov bodies, associated to adjacent prime cones of tropical varieties are related by combinatorial mutations. In my work, I have studied combinatorial mutations that pass between matching field polytopes. In doing so, we can prove that large families of matching fields give rise to toric degenerations. These proofs rely only upon making small, piecewise linear, modifications to polytopes.

Conditional Independence Ideals with Hidden Variables.

Let X, Y and Z be random variables. We say that X and Y are conditionally independent given Z if P(X = x, Y = y | Z = z) = P(X = x | Z = z) P(Y = y | Z = z) for all x, y and z. In other words, the joint distribution of X and Y given Z factors as product. We view conditional independence as a constraint on the random variables. Given a large set of random variables, a conditional independence (CI) model is a collection of CI constraints on the variables. It is common to ask, which probability distributions satisfy a given set of CI statements? In the case when all variables are discrete with finite range, a distribution satisfies a CI statement if and only if the probabilities satisfy certain polynomial equations. Collecting all such polynomials yields an ideal called the CI ideal. 

Once we have an ideal, we can use techniques from commutative algebra and geometry to study its vanishing locus. In my projects, the CI statements are imbued with additional information. We say that a variable is hidden if its marginal probabilities are not observed in the ideal. This naturally leads to the definition of so-called determinantal hypergraph ideals which generalize many familiar families of ideals including binomial edge ideals, determinantal facet ideals and some ideals of adjacent minors.

Understanding hypergraph ideals is, in general, very difficult so we often fix a class of conditional independence statements with hidden variables and write down their associated ideals. For small cases we can use computational methods to compute their irreducible decomposition. From a more theoretical point of view, we can use

• Gröbner bases,

• Perturbation arguments,

• Line arrangements,

• Matroid realisation spaces,

to study the decomposition of a hypergraph ideal. In our work we give frameworks for studying hypergraph ideals and for particularly interesting collections of CI statements with hidden variables, we give a description of prime decomposition of the associated CI ideal.