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.
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-Safarevich 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; Joint with Darlayne Addabbo, Lisa Carbone, Elizabeth Jurisich, and Scott H. Murray.; 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}. 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.
Parking Function Games; joint with Alexander Clifton and Pamela E. Harris.
In Preparation.
Abstract: We introduce two new parking schemes inspired by the classical games of Monopoly and backgammon. Consider a one-way street with n parking spots and a queue of n cars waiting to park, where each car has a preferred parking spot on the street and enters in sequence to park in that spot if it is available. A preference list is considered a parking function if all cars can park under a certain parking scheme. We prove that all preference lists are parking functions for both the Monopoly and backgammon parking schemes. However, the final order in which the cars park, known as the outcome, is a permutation of {1,2,...,n} and differs significantly between the two schemes.
For a fixed outcome π, we provide a product formula for the number of Monopoly parking functions that result in π. Furthermore, we show that the number of backgammon parking functions that result in the order π is bounded below by 2^(n-1) and bounded above by the n-th Catalan number. We also explore specific outcomes where the number of backgammon parking functions involves (sums of) Motzkin numbers.
If a car does not park in its preferred parking spot, we define its displacement as the difference between the spot where it ultimately parks and its preferred spot. We study displacement multisets that encode the displacements for each unlucky car. We provide formulas for the number of backgammon parking functions with displacement multisets containing at most two values, as well as for Monopoly parking functions when the displacement multiset is a singleton.
Spectral theory of automorphic forms applied to string scattering amplitudes; joint with Holley Friedlander, Kim Klinger-Logan, and Manish Pandey.
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.
John Baez's advice on giving good talks:
"People should leave your talks feeling happier and wiser than when they came in. So often it's the opposite. Be an exception. Your talks should be clear, concise, fun, exciting, and never ever run over time."
Monster Lie Algebra: Friend or Foe; Recent Developments in Logarithmic Conformal Field Theory.
Banff International Research Station; Banff, Alberta; July 2025
Automorphic forms and string scattering amplitudes; Joint Mathematics Meetings; AMS Special Session “Rethinking Number Theory.”
Seattle, Washington; January 10th, 2025.
Reflections and perspectives: a look back at my teaching journey; Emory University Teaching Seminar.
Emory University; October 2024.
Of Moonshine and Denominator Identities. "Unveiling Infinite Symmetries: Mini-Course on Algebraic Structures of Generalized Kac-Moody Type";
Canadian Mathematical Society Summer Meeting; July 2024
Monster Lie Algebra: Friend or Foe? Mathematical Physics Seminar; Perimeter Institute for Theoretical Physics.
Waterloo, Ontario; November 30th, 2023.
Spectral theory of automorphic forms applied to string scattering amplitudes; AWM Research Symposium; Special Session, "Rethinking Number Theory".
Clark Atlanta University; September 30, 2023.
Vertex 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.
Monster Lie Algebra: Friend or Foe; Clifford Lectures - The Web of Modularity.
Tulane University, New Orleans; Feb 2024.
"What is... moonshine?"; Women in Noncommutative Algebra & Representation Theory 3 (22w5033).
Banff International Research Station, 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.)
On Moonshine and Elliptic Curves. (Ph.D. Dissertation; 2021)
Quivers: Representations, Mutations, and Applications. (Undergrad Thesis – Expository; 2015)