Research overview

I work in arithmetic geometry, a field that leverages the tools developed in algebraic geometry to gain deep insights into number theory. Specifically, I am interested in the geometry of characteristic-p and p-adic Shimura varieties, and how number theorists apply them in the Langlands correspondence. Shimura varieties are algebro-geometric objects living in multiple mathematical worlds; namely, number theory (via automorphic forms), algebraic geometry (as varieties), and representation theory (via Galois representations). This makes them a Rosetta stone of modern mathematics, enabling us to provide answers in one field by translating the questions into the language of another. Special fibres of Shimura varieties often have direct moduli interpretations in many cases, making them relevant to the study of elliptic curves and abelian varieties.

Here is a more detailed Research Statement of my ongoing projects.

I worked in commutative algebra and finite geometry during my M.S. at the University of Michigan. Before that, my undergraduate research at U.C. Santa Cruz was in theoretical physics.

Publications and Preprints

Arithmetic geometry:

Traverso's Isogeny Conjecture for Some Unitary p-Divisible Groups (2025)

The isogeny cutoff of a p-divisible group X (defined over an algebraically closed field of characteristic p) measures the amount of p-torsion necessary to determine its isogeny class. The minimal height of X measures its distance to the closest minimal p-divisible group (in the sense of Oort). In this paper, we study these invariants for supersingular unitary p-divisible groups of signature (a, b). We provide a complete description of the possible minimal heights. As an application, we establish bounds on the isogeny cutoffs for these p-divisible groups. Finally, we rephrase our results in the language of the BTm stratifications of unitary Shimura varieties of signature (a, b).

This is joint work with Emerald Andrews, Deewang Bhamidipati, Maria (Mia) Fox, Steven Groen, and Heidi Goodson. This is a continuation of our collaboration from RNT - 3. We benefitted greatly from an AIM SQuaRe to start this project.

Ekedahl-Oort strata and the supersingular locus in the GU(q-2,2) Shimura variety (2024)

This paper concerns the characteristic-p fibers of GU(q-2,2) Shimura varieties, which classify abelian varieties with additional structure. These Shimura varieties admit two stratifications of interest: the Ekedahl-Oort stratification, based on the isomorphism class of the p-torsion subgroup scheme, and the Newton stratification, based on the isogeny class of the p-divisible group. It is natural to ask which Ekedahl-Oort strata intersect the unique closed Newton stratum, called the supersingular locus. In this paper, we present several novel techniques that give information about the interaction between the two stratifications for general signature (q-2,2), and as an application, we completely answer this question for the signature (3,2).

This is joint work with Emerald Andrews, Deewang Bhamidipati, Maria (Mia) Fox, Steven Groen, and Heidi Goodson. This preprint was a result of RNT - 3, where Dr Fox was our project leader.

Past research:

Variants of normality and steadfastness deform (22 pages in Michigan Math Journal, March 2025. DOI: 10.1307/mmj/20226231)

The cancellation problem asks whether A[X_1, X_2, . . . , X_n] ≅ B[Y_1, Y_2, . . . , Y_n] =>A ≅ B. Hamann introduced the class of steadfast rings as the rings for which a version of the cancellation problem considered by Abhyankar, Eakin, and Heinzer holds. By the work of Asanuma, Hamann, and Swan, steadfastness can be characterized in terms of p-seminormality, which is a variant of normality introduced by Swan. We prove that p-seminormality and steadfastness deform for reduced Noetherian local rings. We also prove that p-seminormality and steadfastness are stable under adjoining formal power series variables for reduced (not necessarily Noetherian) rings. Our methods also give new proofs of the facts that normality and weak normality deform, which are of independent interest.

This is joint work with Alexander Bauman, Havi Ellers, Gary Hu, Takumi Murayama, and Ying Wang.

This work was a result of MREG 2021.

Maximal skew sets of lines on a Hermitian surface and a modified Bron-Kerbosch algorithm (Preprint 2022)

In this paper, we study maximal sets of skew lines on Hermitian surfaces. We give a new algorithm to compute these sets and give some computational results for Hermitian surfaces of degrees 3, 4, and 5. In more generality, this algorithm solves a new variant of the clique listing problem, which may be more approachable than the classical problem. Finally, we explicitly construct a large skew set of lines on Hermitian varieties of any degree and use it to give a lower bound on the largest size of maximal skew sets and a lower bound on the possible number of maximal skew sets.

This is joint work with Anna Brosowsky, Haoyu Du, Madhav Krishna, Janet Page, and Tim Ryan.

This work was a result of LoG(M) 2022.

Other scholarly works

Research Prelude: The geometry of the basic locus in certain unitary Shimura Varieties

This is a semi-expository survey article finished towards achieving candidacy, partially fulfilling the Research Prelude requirement of the math Ph.D. program at CSU. Prof. Rachel Pries was my pimay advisor. The article focuses on how the Ekedahl-Oort and the Newton stratification can be used to better inform the geometry of the basic locus of unitary Shimura Varieties of PEL type for different signatures. The following different cases are considered:

a. General treatment of Ekedahl-Oort and Newton stratification for PEL type unitary Shimura Varieties for unramified primes..

b. PEL type unitary Shimura Variety of signature (2,2) when the prime p is assumed to be unramified.

c. PEL type GU(n-1,1) Shimura Variety when the prime p is assumed to be unramified.

d. PEL type GU(n-1,1) Shimura Variety when the prime p is assumed to be ramified.

Part B: The different incarnations of Ekedahl-Oort stratification

This is a semi-expository, semi-computational write-up done towards achieving candidacy, in partial fulfillment of the Part B requirement of the math Ph.D. program at CSU. I was primarily advised by Prof. Rachel Pries, while Prof. Jeff Achter served as my co-advisor. The article focuses on different presentations of the EO type when we are working on Siegel Modular Varieties.

Undergraduate Thesis: Arithmetic Attractor Mechanism of BPS Black Holes

We solve the attractor equations for the case of BPS black holes in the N = 2 SUGRA theory with moduli space given by SO(2,2)/U(1)×U(1). We analyze the associated mathematical structure given by pairs of binary quadratic forms by relating them to the attractor points. We finally explore the connection between Hurwitz class numbers of these pairs of binary quadratic forms counted by attractors with a certain Siegel 2-form, inspired by Kudla-Millson theory. In the process, we make use of relevant works by J. Morales, M. Bhargava, D. Zagier, G. Moore, et. al. in further understanding the arithmetic nature of the attractor mechanism.