Research

  Research

My research lies at the intersection of algebraic geometry, arithmetic geometry, and commutative algebra, with a central focus on rationality problems and toric geometry. Broadly, I study how cohomological and combinatorial methods can classify algebraic structures and determine their rationality over arbitrary base fields. These questions trace back to some of the deepest ideas in algebraic geometry: understanding when varieties admit simple birational models, and how symmetries and group actions shape their classification.

A recurring theme in my work is the search for constructive frameworks that make these abstract relationships explicit—linking algebra, geometry, and arithmetic through concrete invariants.

 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 soon.

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.

Additional Problems: G-Varieties and Mackey/Tambara Functors

Here are some further open problems I am thinking about. If any of these directions catch your attention, I’d be glad to collaborate.

• How do Mackey and Tambara functors encode the birational geometry of G-varieties?

• Can one classify equivariant contractions of del Pezzo and toric surfaces purely in Mackey functor terms?

• What are the obstructions to G-rationality or stable G-rationality detectable from the functor data?

• How do subgroup restrictions change the Mori cone, and can this be organized categorically?

• Can we connect the representation theory of G with the structure of equivariant derived categories?