# Research

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.

Elliptic Curves and Thompson's Sporadic Simple group; arXiv version; published version.

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;

Vertex operators for imaginary 𝔤𝔩2-subalgebras in the Monster Lie Algebra; arXiv link; Joint with Darlayne Addabbo, Lisa Carbone, Elizabeth Jurisich, and Scott H. Murray.

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.

Spectral theory of automorphic forms applied to string scattering amplitudes; joint with Holley Friedlander, Kim Klinger-Logan, Manish Pandey, and Runqiu Xu.

In Preparation.

Abstract: In string theory, elementary particles are represented by vibrational modes of a string. Strings interact by 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 orderVertex operators for imaginary 𝔤𝔩2-subalgebras in the Monster Lie algebra; Rocky Mountain Representation Theory Seminar.

Online Seminar; 16 March 2023.

Of Moonshine, Monsters, and Curves; Mathematical Sciences Seminar; IBA School of Mathematics and Computer Sciences.

Karachi, Pakistan; 13 Jan 2023.

Elliptic Curves and Moonshine; Number Theory Seminar; Kansas State University.

Online Seminar; October 5th, 2022.

Modular forms and Moonshine; Joint Mathematics Meetings; AMS Special Session “Modular Forms and Combinatorics.”

Virtual Conference; April 8th, 2022.

Rademacher sums and moonshine, Wednesday Zoom Seminar, Stockholm University.

Online Seminar, February 9th, 2022.

Moonshine and Elliptic Curves; Rutgers Lie Group/Quantum Mathematics Seminar.

Online Seminar; November 12th, 2021.

Moonshine and Arithmetic; John Conway Spirited Seminar Series, Lahore University of Management Sciences.

Online Seminar; November 1st, 2021.

What is... Moonshine? What is...? Seminar at the University of Queensland.

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

Weight 3/2 Moonshine for the Thompson group; Specialty Seminar in Partition Theory, q-Series, and Related Topics.

Online Seminar; September 30th, 2021.

What is... Moonshine? UW Algebra and Algebraic Geometry Seminar

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

Elliptic Curves and Moonshine; UW Algebra and Algebraic Geometry Seminar

Online Seminar; March 9th, 2021.

Elliptic Curves and Moonshine; University of Virginia Number Theory Seminar.

Online Seminar; February 19th, 2021.

Elliptic Curves and Moonshine; Joint Mathematics Meetings; AMS Special Session on Algebraic and Arithmetic Geometry.

Virtual Conference; January 8th, 2021.

Elliptic Curves and Moonshine; Number Theory Seminar, University of Georgia.

Online Seminar; September 16th, 2020.

Elliptic Curves and Moonshine; Vanderbilt University Number Theory Seminar.

Online Seminar; September 1st, 2020.

### (Contributed)

*in reverse chronological order"What is... moonshine?"; Women in Noncommutative Algebra and Representation Theory 3 (22w5033).

BIRS, Alberta; April 3-8, 2022.

Elliptic Curves and Moonshine; The Autumn Meeting of the Swedish Math Society.

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

Moonshine and Elliptic Curves; Women in automorphic forms at Universität Bielefeld.

Online Conference; September 20th, 2021.

Elliptic Curves and Thompson's Sporadic Group; New Developments in Number Theory.

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

Elliptic Curves and Thompson's group; Women at the Intersection of Mathematics and Theoretical Physics.

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

Elliptic Curves and Thompson's group; Modularity in Quantum Systems.

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

Elliptic Curves and Thompson's group; PAlmetto Joint Arithmetic, Modularity, and Analysis Series (PAJAMAS).

Online Conference; September 19-20th, 2020.

Elliptic Curves and Moonshine; Madison Moduli Weekend.

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

Measuring the Role of Curiosity in Student Engagement and Performance; The Advanced Graduate Teaching Fellowship Symposium.

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

# Theses.

On Moonshine and Elliptic Curves. (Ph.D. Dissertation; 2021)

Quivers: Representations, Mutations, and Applications. (Undergrad Thesis – Expository; 2015)