Publications

Inverse Eigenvalue Problems, Floquet Isospectrality and the Hilbert-Chow Morphism

Let $A$ be an $n\times n$ matrix over an algebraically closed field of characteristic $0$ or $>n$. Using deformation theory, we determine exactly when there exists a nonzero diagonal matrix $D$ such that $A$ and $A+D$ have the same spectrum: this fails if and only if, for each size $k$, all $k\times k$ principal minors of $A$ are equal. This yields a new result about the spectral theory of discrete periodic Schrödinger operators.

Elimination Without Eliminating: Computing Complements of Real Hypersurfaces Using Pseudo-Witness Sets

Complements of real hypersurfaces arise naturally in applications – for example, the complement of the real discriminant of a polynomial system can tell you how many real solutions the system has. We develop a new numerical method for computing this complement even without a symbolic representation for the hypersurface.

Semigroup Graded Stillman's Conjecture

In 2000, Mike Stillman conjectured that the projective dimension of a homogeneous ideal in a standard graded polynomial ring can be bounded just in terms of the number and degrees of the generators. We resolve Stillman’s conjecture for families of polynomial rings that are graded by any abelian group under mild conditions. Conversely, we show that these conditions are necessary for the existence of a Stillman bound.

LikelihoodGeometry: Macaulay2 package

A Macaulay2 package for computing the likelihood correspondence of statistical models.

Feature Propagation on Knowledge Graphs Using Cellular Sheaves

We utilize a sheaf-theoretic technique to optimally extend vector representations of old entities in knowledge graphs to previously unseen entities. This technique compares well to other more complicated state-of-the-art inductive reasoning methods.

Likelihood Correspondence of Toric Statistical Models

Characteristics of the maximum likelihood estimation problems are reflected in the geometry of a variety called the likelihood correspondence. We construct the ideal of this variety for some popular examples of statistical models.

Syzygies of Curves in Products of Projective Spaces

Motivated by toric geometry, we lift machinery for understanding syzygies of curves in projective space to the setting of products of projective spaces. Using this machinery, we show an analogue of an influential result of Gruson, Peskine, and Lazarsfeld that gives a bound on the regularity of a possibly singular curve given its degree and the dimension of the ambient projective space.

Virtual Criterion for Generalized Eagon-Northcott Complexes

Any matrix gives rise to an associated family of canonical complexes associated to it, including the Eagon-Northcott and Buchsbaum-Rim. Famously, there is a simple criterion for checking the exactness of these complexes. We give generalization of this criterion for these complexes to be virtual resolutions, which are “exact enough” to be relevant to the geometry of toric varieties.

Quarternion-Valued Breather Soliton, Rational, and Periodic QKdV Solutions

The KdV equation is a prototypical example of an exactly solvable PDE, due in part to its admittance of soliton solutions. This paper examines and classifies the dynamics of quaternion-valued solutions of the noncommutative KdV equation.

Aggregate Dispersions to Enhance the Intrachain Order in Surfactant-Stabilized Aqueous Colloids of Poly(3-hexylthiophene)

Quaternion-Valued KdV Solutions

My bachelor thesis.