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Automorphic q-series and their application

Final Report Summary - AQSER (Automorphic q-series and their application)

In OBJECTIVE 1, I investigated many q-series coming from combinatorics and additionally studied their asymptotic and modularity properties. In particular, I solved the following problems:
- asymptotics for stacks (with Mahlburg), unimodal sequences (with Folsom and Rhoades), and Fishburn matrices (with Li and Rhoades); my PhD student Ciolan investigated the asymptotic behavior of partitions into powers.
- I proved (with Dousse,) Dyson's conjecture pertaining to the profile of Dyson's crank. This question has been open for many years and required new techniques using Jacobi forms that have sub-
sequently been used by other researchers, for example it was used by Hamel and Manschot.
- I investigated modularity results pertaining to unimodal sequences (with Lovejoy) and Capparelli type identities (with Mahlburg).
- I showed combinatorial identities including companions to Capparelli's identities, Schur double series representation, and universal mock theta functions (with Andrews, Lovejoy, and Mahlburg).
- I proved (mock) modularity of many combinatorial generating functions including functions introduced by Andrews, Dixit, Schultz, and Yee (with Jennings-Shaffer and Mahlburg). Thus, I now have a clear understanding about the interplay of combinatorial q-series and modular forms.
In OBJECTIVE 2, I investigated (with Dousse, Lovejoy, and Mahlburg) further percolation models and studied the role of overpartitions. We showed there is a relation between mock theta functions and sequence conditions. I (jointly with Kane, Parry, and Rhoades) further established asymptotics of radial limits for certain functions introduced by Wright. These functions appear in bootstrap per-
colation models and are related to partitions without sequences of k consecutive part sizes.
In OBJECTIVE 3, I investigated the role of the harmonic Maass and modular forms in Lie theory.
- I obtained a better understanding of the relationship between functions motivated by Lie theory and automorphic objects. I have considered various “traces” associated to the Lie theory of vertex
operator algebras (VOAs) and have in some cases been able to demonstrate their modularity. This work is helping to explain the interplay between modular forms and Lie theory. As these results pertain to VOAs they can apply to other algebraic structures besides Lie algebras, for instance to VOAs associated to lattices.
- I studied how false theta functions play a role in W-algebras and found candidates for higher-
dimension false theta functions (with Kaszian and Milas), having wide-reaching applications to
physics.
- I investigated the role of Jacobi Poincaré series in Moonshine (with Duncan and Rolen). A main
difficulty here was to show convergence of the Poincaré series.
- I solved the motivating question from the grant pertaining to the modularity of Kac-Wakimoto
characters. We now fully understand these. These results have a major impact on Objective 4.
In OBJECTIVE 4, I studied the following: - Asymptotic behavior of certain inverse theta functions (with Manschot).
- Indefinite theta functions arising from Gromov-Witten theory (with Rolen and Zwegers). These give us natural candidates for higher dimensional indefinite theta functions, which we investigated further. We came across certain false theta functions that contribute a better understanding of
multiple centered black holes.
- I also studied quantum modular properties (modular symmetries on the rationals) of quantum
invariants of plumbed 3-manifolds introduced recently by Gukov, Pei, Putrov, and Vafa. In
particular, I (jointly with Mahlburg and Milas) solved a quantum modularity conjecture of Gukov.