# Research

Photo taken in Atlanta, GA

My research involves moonshine and related aspects of number theory. I'm particularly interested in the question: "Can moonshine help solve problems in number theory?"

Slides for a 5-min lightning talk on my work can be found here. For a quick introduction to moonshine, you can watch this 30-minute talk I gave on the topic.

# Publications.

1. 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;

1. Vertex operators for imaginary gl 2's in the Monster Lie Algebra; joint with Darlayne Addabbo, Lisa Carbone, Elizabeth Jurisich, and Scott H. Murray.

In Preparation.

Abstract: The Monster Lie algebra m is a quotient of the physical space of the vertex operator algebra V:=V^\natural \otimes V_{I,I}. We construct families of elements that generate gl 2-subalgebras of V corresponding to imaginary simple roots of m. We prove the existence of primary vectors in V^\natural of each homogeneous weight and for 0<n<100, we show that these can be used to construct the corresponding gl 2-subalgebras. Furthermore, we show that the Monster acts non-trivially on each of these subalgebras and that this induces an action on their image in m.

# Talks.

### (Invited)

*in reverse chronological order
1. Modular forms and Moonshine; Joint Mathematics Meetings; AMS Special Session “Modular Forms and Combinatorics.”

Virtual Conference; April 8th, 2022.

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

Online Seminar, February 9th, 2022.

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

Online Seminar; November 12th, 2021.

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

Online Seminar; November 1st, 2021.

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

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

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

Online Seminar; September 30th, 2021.

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

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

1. Elliptic Curves and Moonshine; UW Algebra and Algebraic Geometry Seminar

Online Seminar; March 9th, 2021.

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

Online Seminar; February 19th, 2021.

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

Virtual Conference; January 8th, 2021.

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

Online Seminar; September 16th, 2020.

1. Elliptic Curves and Moonshine; Vanderbilt University Number Theory Seminar.

Online Seminar; September 1st, 2020.

### (Contributed)

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

BIRS, Alberta; April 3-8, 2022.

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

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

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

Online Conference; September 20th, 2021.

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

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

1. 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.)

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

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

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

Online Conference; September 19-20th, 2020.

1. Elliptic Curves and Moonshine; Madison Moduli Weekend.

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

1. 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.

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

2. Quivers: Representations, Mutations, and Applications.