Publications

On virtual resolutions of points in a product of two projective spaces: arxiv: https://arxiv.org/abs/2402.12495; Appears in the Journal of Pure and Applied Algebra: https://authors.elsevier.com/sd/article/S0022-4049(25)00136-7
Ideals preserved by linear changes of coordinates in positive characteristic: arxiv: https://arxiv.org/abs/2404.10544; To appear in Communications in Algebra.

Expository notes

Strong multiplicity one for classical modular forms: my undergraduate senior thesis

Abstract

Classical modular forms are analytic functions of the upper half plane invariant under the action of congruence subgroups of SL₂(ℤ) up to a factor of automorphy. In this setting, the Hecke operators on the space of modular forms are normal operators on the subspace of cusp forms, the modular forms that vanish at cusps. Cusp forms satisfy the strong multiplicity one property, which roughly says that normalized eigenforms of the Hecke operators are uniquely determined by their eigenvalues. Strong multiplicity one, and other multiplicity one results, are often presented in the language of automorphic forms and representation theory. We introduce the theory of classical modular forms and provide a self-contained proof of the strong multiplicity one property for classical modular forms, following the proof in Modular Forms by Toshitsune Miyake. This approach avoids using the language of automorphic forms and representation theory by studying L-functions associated to modular forms, and their Euler products. An application of the strong multiplicity one property yields a basis for the space of cusp forms, which is nice since the space of modular forms decomposes into the direct sum of the space of cusp forms and the space of Eisenstein series.
Grothendieck spectral sequences: notes for a reading course
Representation Theory (in progress): notes for a course David Ben-Zvi taught

details

These are notes I took from M 392C Representation Theory via SL2 taught by David Ben-Zvi in Fall 2025. Here is a description of the course from the syllabus: I will present an unorthodox introduction to the representation theory of Lie groups and Lie algebras, focussing entirely on the group of two by two matrices with determinant one. Thus we hope to cover a breadth of topics in representation theory, that are usually sacrificed for the depth of treating general Lie groups. We will start with finite dimensional representations of SL_2(C) (or equivalently SU_2) and their relation to the geometry of the Riemann sphere. We will then move on to unitary (infinite dimensional) representations of SL_2(C) and especially SL_2(R), and their relation to the geometry of the upper half plane and modular forms. We will (in an ideal universe) conclude with the beautiful geometry of the tree, associated to SL_2(Q_p), which is a p-adic analog of the upper half plane. Besides its intrinsic beauty, this subject is ubiquitous in number theory, topology and physics. In final projects we will explore some of these connections as well as generalizations to other Lie groups.

Activities

July 2025: I participated in UT Austin's 2025 Summer Minicourses. I held a two-week long series of lectures (nine lectures) on homological algebra, spectral sequences, and the derived category. These were meant to be introductory lectures aimed towards incoming graduate students or advanced undergraduate students with a background in algebra and category theory (of which the latter was covered in the week prior to the start of my lecture series). The lectures are on YouTube.
February-April 2025: I participated in UT Austin's Spring 2025 Directed Reading Program as a mentor. I mentored Linh Dam on Fourier Analysis following Stein and Shakarchi's text, culminating in her giving a presentation on a Fourier-theoretic proof of the isoperimetric inequality.
April 2024: I have completed my undergraduate honors thesis: Strong multiplicity one for classical modular forms. It is effectively a primer for the theory of modular forms as it appears in Diamond and Shurman and provides a proof of the strong multiplicity one property for classical modular forms. I appreciate comments!

Abstract

Classical modular forms are analytic functions of the upper half plane invariant under the action of congruence subgroups of SL₂(ℤ) up to a factor of automorphy. In this setting, the Hecke operators on the space of modular forms are normal operators on the subspace of cusp forms, the modular forms that vanish at cusps. Cusp forms satisfy the strong multiplicity one property, which roughly says that normalized eigenforms of the Hecke operators are uniquely determined by their eigenvalues. Strong multiplicity one, and other multiplicity one results, are often presented in the language of automorphic forms and representation theory. We introduce the theory of classical modular forms and provide a self-contained proof of the strong multiplicity one property for classical modular forms, following the proof in Modular Forms by Toshitsune Miyake. This approach avoids using the language of automorphic forms and representation theory by studying L-functions associated to modular forms, and their Euler products. An application of the strong multiplicity one property yields a basis for the space of cusp forms, which is nice since the space of modular forms decomposes into the direct sum of the space of cusp forms and the space of Eisenstein series.; The second project I worked on at the 2023 University of Minnesota REU produced a preprint: Ideals preserved by linear changes of coordinates in positive characteristic. To appear in Communications in Algebra.
February 2024: The first project I worked on at the 2023 University of Minnesota REU produced a preprint: On virtual resolutions of points in a product of projective spaces. To appear in the Journal of Pure and Applied Algebra.
January 2024: I presented at JMM 2024 in San Francisco, in the AMS Contributed Paper Session on Commutative Algebra and Related Topics.
June-July 2023: I participated in the Summer 2023 University of Minnesota Combinatorics and Algebra REU; see this page for details about both projects I worked on.
June-July 2022: I participated in the Summer 2022 Georgia Institute of Technology mathematics REU; see this page for details about the REU as well as this page for our poster.

Music

I play clarinet and saxophone recreationally. See this page for more.

Games

I play lots of video games. See this page for more.