Research

"It doesn't matter how long it takes, if the end result is a good theorem."

-John Tate

Research

I study the intersections of Number Theory, Commutative and Non-commutative Algebra, and Representation Theory. My current work focuses on the classification of higher-dimensional tori and toric varieties over arbitrary fields and on the Weak Lefschetz Property (WLP) for graded algebras, especially through their connections to the geometry and coordinate rings of toric varieties. Using tools from algebraic geometry, Galois cohomology, and representation theory, I emphasize constructive methods that uncover the symmetries and invariants of algebraic and arithmetic structures.

Publications/Preprints

On the Weak Lefschetz Property for Ideals Generated by Powers of General Linear Forms
(with Matthew Booth and Adela Vraciu)
To appear in Journal of Commutative Algebra
[arXiv link]

We describe the initial ideals of almost complete intersections generated by powers of general linear forms and prove that the Weak Lefschetz Property (WLP) in a fixed degree d holds when the number of variables n is sufficiently large relative to d. In particular, for ideals generated by squares, we determine precisely the range for which the WLP holds. Additionally, we provide bounds for the degree 3 case.

Classifying Torsors of Tori via Brauer Groups
(with Alexander Duncan)
Submitted
[arXiv link]

Using Mackey functors, we develop a general framework for classifying torsors of algebraic tori in terms of Brauer groups of finite field extensions of the base field. This generalizes Blunk’s description of tori associated with degree 6 del Pezzo surfaces to all retract rational tori—essentially the largest class for which such a classification is possible.

Current Research and Future Goals

Torsors of del Pezzo Varieties
I am studying the forms of tori associated with del Pezzo varieties. Under specified conditions, these tori are retract rational, and we provide a classification in terms of elements of the Brauer group. This work lays the foundation for a broader program aimed at classifying forms of tori and their associated toric varieties over arbitrary fields via separable algebras. The project is ongoing and will appear on the arXiv in due course.

Proposed Research Directions

Advancing the Methodology for the Classification of Arbitrary Tori: I plan to expand the current classification framework to arbitrary tori using elements of the Brauer group, providing a classification up to Brauer equivalence. Our ongoing work presents a framework that classifies all retract rational tori.
Determining the Sharpness of the Bound for WLP in Arbitrary Degree: I aim to investigate the sharpness of the bound for the Weak Lefschetz Property when the ideal of an almost complete intersection is generated by powers of general linear forms of arbitrary degree.

Long-Term Vision
My long-term goal is to bridge algebraic geometry, non-commutative algebra, and Mackey functors to achieve a comprehensive classification of toric varieties over arbitrary fields. I seek to develop a unified theoretical framework that integrates these disciplines, deepening our understanding of their connections and advancing applications toward this classification.

Master's Thesis

Aug 2019 – June 2020
Master’s Thesis: Connectivity of the Tropical Double Ramification Cycle
Supervised by Dr. Dmitry Zakharov
Department of Mathematics, Central Michigan University, Mount Pleasant, MI, USA

My research investigated the connectivity properties of the tropical double ramification (DR) cycle, a polyhedral object in tropical geometry, within the moduli space of tropical curves of genus ggg with nnn marked points. Using graph-theoretic and combinatorial methods, I showed that the tropical DR cycle maintains connectivity in codimension one for specific parameter choices. This work integrates techniques from algebraic geometry, tropical geometry, and combinatorics, focusing on the construction and enumeration of cones in polyhedral spaces.

Expository Articles On Mackey Functors

These notes were created to facilitate my understanding of Mackey functors at the onset of my project. They serve as an introductory resource for those interested in the topic.

Theory_Of_Cohomological_Mackey_Functors.pdf

These are the notes I prepared during class project of representation theory.

Representation_theory_project_report.pdf