About

I am a first year graduate student at the University of Texas at Austin. I graduated from the University of Florida in 2024, with a bachelors degree in mathematics and minors in physics and computer science. Here is my cv (as of 12/27/23).

I am interested in algebraic geometry, homological algebra, commutative algebra, representation theory, modular forms, harmonic analysis, functional analysis, and mathematical physics. I will eventually specialize somewhere, but I believe that a broad background in mathematics is nice to have.

My pronouns are he/him.

Email: sivakumars [at] ufl.edu
GitHub: github.com/sivakumarsai

Activities

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/corrections (there are certainly mistakes I did not catch).

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 in the 2023 University of Minnesota REU produced a preprint: Ideals preserved by linear changes of coordinates in positive characteristic. It has been submitted to Communications in Algebra.

February 2024

The first project I worked on in the 2023 University of Minnesota REU produced a preprint: On virtual resolutions of points in a product of projective spaces. It has been submitted to 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.

TBD