Publications

Ultraproducts of Graded Algebras

A common argument for establishing bounds within commutative algebra is the construction of a “limiting polynomial ring” that allows one to trade asymptotic behavior over many polynomial rings for exact behavior in the limit ring. We define a degree bounded ultraproduct that packages this argument up and establish several strong properties about it using its unique position within logic. We give numerous out-of-the-box algebraic tools to use on such rings.

Multigraded 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.

Likelihood Correspondence of 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.

Extending Transductive Knowledge Graph Embedding Models for Inductive Logical Relational Inference

We utilize a sheaf-theoretic technique to optimally extend vector representations of old entities in knowledge graphs to previously unseen entities. This technique also gives an efficient way of predicting facts about these new entities and compares well to other more complicated state-of-the-art inductive reasoning methods.

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.