Research

Photo taken in Atlanta, GA

My research involves moonshine and related aspects of number theory and representation theory. For a quick introduction to moonshine, you can watch this 30-minute talk I gave on the topic. 

Papers.

Abstract: We characterize all infinite-dimensional graded virtual modules for Thompson’s sporadic simple group, whose graded traces are certain weight 3/2 weakly holomorphic modular forms satisfying special properties. We then use these modules to detect the non-triviality of Mordell--Weil, Selmer, and Tate--Shafarevich groups of quadratic twists of certain elliptic curves.

Citation: Journal of Number Theory, Volume 224, 2021, Pages 274-306, ISSN 0022-314X, https://doi.org/10.1016/j.jnt.2021.01.015;

Accepted for publication in Journal of Pure and Applied Algebra; arXiv link

Abstract: The Monster Lie algebra 𝔪 is a quotient of the physical space of the vertex algebra V:=V♮⊗V_{1,1}. It is known to be generated by 𝔤𝔩2-subalgebras corresponding to its real and imaginary root vectors. 

For each imaginary simple root (1,n) of 𝔪, we describe elements in V that project under the quotient map to sl_2-triples in 𝔪. We extend each of these triples to a set of generators of subalgebras of 𝔪 isomorphic to gl_2 corresponding to each (1,n).

This requires the existence of primary vectors in V satisfying some natural conditions, which we prove in a number of cases.

We show that the action of the Monster M on V♮induces an M-orbit of each gl_2 subalgebra corresponding to a fixed imaginary simple root. We conjecture that this underlying M-action is non-trivial and prove this in many cases.

In Preparation. 

Abstract: In string theory, elementary particles are represented by vibrational modes of a string. Strings interact through various joining and splitting processes, and the probabilities that certain scattering processes occur are given by string scattering amplitudes. When computing a low-energy expansion of these string scattering amplitudes, coefficient functions arise that are automorphic functions appearing as solutions to various differential equations and whose expressions involve combinations of Eisenstein series on an arithmetic quotient of the exceptional group E_8. The first few solutions to these differential equations are known on SL_2(R). We describe work toward a spectral solution in the SL_3(R) case. This project was initiated at the Rethinking Number Theory 3 workshop and is in collaboration with Holley Friedlander, Kim Klinger-Logan,  Manish Pandey, and Runqiu Xu.

Talks. 

(Invited)

*in reverse chronological order

Waterloo, Ontario; November 30th, 2023.

Clark Atlanta University; September 30, 2023.

Online Seminar; 16 March 2023.

Karachi, Pakistan; 13 Jan 2023.

Online Seminar; October 5th, 2022.

Virtual Conference; April 8th, 2022.

Online Seminar, February 9th, 2022. 

Online Seminar; November 12th, 2021.

Online Seminar; November 1st, 2021.

120-min; Online Seminar; October 11th, 2021.

Online Seminar; September 30th, 2021.

30-min; Virtual "Pre-Seminar" talk aimed at grad students; March 9th, 2021. 

Online Seminar; March 9th, 2021. 

Online Seminar; February 19th, 2021. 

Virtual Conference; January 8th, 2021. 

Online Seminar; September 16th, 2020. 

Online Seminar; September 1st, 2020.

(Contributed)

*in reverse chronological order

BIRS, Alberta; April 3-8, 2022.

Online Meeting/Uppsala, Sweden; Nov 19th, 2021

Online Conference; September 20th, 2021. 

Virtual Contributed talk series; March 2nd, 2021. (Organized by POINT.)

Online Conference; February 22nd, 2021 (``Gong show'' style mini-presentation.)

Online Conference; December 15th, 2020 (``Gong show'' style mini-presentation.)

Online Conference; September 19-20th, 2020.

Online Conference; September 26-27th, 2020.  (5-minute "Lightning talk." Slides)

Virtual Symposium;  May 5th, 2020. (A Teaching-as-Research Project; Slides.)

Theses.