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. 


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-Safarevich groups of quadratic twists of certain elliptic curves.

Citation: Journal of Number Theory, Volume 224, 2021, Pages 274-306, ISSN 0022-314X,;

Abstract: The Monster Lie algebra 𝔪 is a quotient of the physical space of the vertex algebra V=V♮⊗V_{1,1}. For each imaginary simple root (1,n) of 𝔪, we construct vertex algebra elements that project to bases for subalgebras of 𝔪 isomorphic to 𝔤𝔩2. Our method requires the existence of pairs of primary vectors in V♮ satisfying some natural conditions, which we prove. We show that the action of the Monster finite simple group 𝕄 on the subspace of primary vectors in V♮ induces an 𝕄-action on the set of 𝔤𝔩2 subalgebras corresponding to a fixed imaginary simple root. We use the generating function for dimensions of subspaces of primary vectors of V♮ to prove that this action is non-trivial for small values of n.

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, and Manish Pandey.



*in reverse chronological order


*in reverse chronological order