Research

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

-John Tate

Current Research Focus

My doctoral research, supervised by Dr. Alexander Duncan at the University of South Carolina, delves into the intersections of Number theory, Non-commutative and commutative algebras, and Group representation theory. Specifically, I am investigating:

Classification of Higher Dimensional Tori and Toric Varieties: Developing a framework for classifying higher-dimensional tori and toric varieties over arbitrary fields, with implications for the study of algebraic groups and their representations.

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.

Torsors of del Pezzo Varieties

In preparation

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

In this thesis, we studied the connectivity properties of a polyhedral complex known as the tropical double ramification (DR) cycle. We proved that, for a specific choice of parameters, the tropical DR cycle exhibits the same connectedness properties as its classical counterpart.

Proposed Research Directions

Advancing the Methodology for the Classification of Arbitrary Tori : I plan to expand our classification framework to classify arbitrary tori in terms of elements of the Brauer group. Specifically, I aim to provide a classification of these tori via certain algebras up to Brauer equivalence. In our current work on the arXiv, we present a framework that offers a classification of all retract rational tori.

Determining the sharpness of the bound for WLP in arbitrary degree: I also plan to investigate the sharpness of the bound for WLP when the ideal of the almost complete intersection is generated by by powers of general linear forms of arbitrary degree.

Long-Term Vision

My long-term research goal is to bridge the fields of algebraic geometry, non-commutative algebra, and the theory of Mackey functors to achieve a comprehensive classification of toric varieties over arbitrary fields. I aim to develop a unified theoretical framework that integrates these disciplines, deepening our understanding of their connections and advancing applications toward this classification. For further context and inspiration, see Twisted froms of toric varieties by Alexander Duncan.

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