## Mid-Term Report Summary - AQSER (Automorphic q-series and their application)

In Objective 1, I investigated many q-series coming from combinatorics and in particular studied their asymptotic and modularity properties. In particular I solved the following problems:

- asymptotics for stacks (with K. Mahlburg , Louisiana State Univ.), unimodal sequences (with A. Folsom, Amherst College and R. Rhoades, Shasta College), and Fishburn matrices (with Y. Li, TU Darmstadt and R. Rhoades);

- I proved (with J. Dousse, Universität Zürich) 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 which subsequently have been used by other researchers - e.g. T. Hamel (physics), J. Manschot (physics) - for example for generating functions arising in physics;

- I investigated modularity results pertaining to unimodal sequences (with J. Lovejoy, Univ. Paris VII) and Capparelli type identities (with K. Mahlburg);

- I showed combinatorial indentities including companions to Capparelli's identities, Schur double series representation, and universal mock theta functions (with G. Andrews, Pennsytlvania State Univ., K. Mahlburg, and J. Lovejoy).

Thus, I now have a clear understanding about the interplay of combinatorial q-series and modular forms.

In Objective 2, I investigated (with J. Dousse, K. Mahlburg, and J. Lovejoy) further percolation models and in particular studied the role of overpartitions. We showed there is a relation between mock theta functions and sequence conditions.

This settles the outset goal of Objective 1.

In Objective 3, I investigated the role of the harmonic Maass and modular forms in Lie theory. In particular:

- I studied how false theta functions play a role in W-algebras and found candidates for higher-dimension false theta functions (with A. Milas, Univ. Albany);

- I investigated the role of Jacobi Poincaré series in Moonshine (with J. Duncan, Emory Univ. and L. Rolen, Pennsytlvania State Univ.). A main difficulty here was to show convergence of the Poincaré series.

- and I solved the motivating question from the grant pertaining to the modularity of Kac-Wakimoto characters. We now fully understand these. These results have big impact on Objective 4.

In Objective 4, I studied the

- asymptotic behavior of certain inverse theta functions (with J. Manschot, Trinity College Dublin);

- indefinite theta functions arising from Gromov-Witten theory (with L. Rolen and S. Zwegers, Univ. Köln). These give us natural candidates for higher dimensional indefinite theta functions which we investigate further. In Objective 3, we came across certain false theta functions which contribute a better understanding of multiple centered black holes.

I am standing well in most of the proposed time frame. The progress in Objective 3 was skewed down due to the birth of my thrid child. I have natural candidates for q-series from Lie theory whose quasimodularity properties are still to be understood. In Objective 4, I am now changing direction by investigating indefinite theta functions which naturally occured in our case.

- asymptotics for stacks (with K. Mahlburg , Louisiana State Univ.), unimodal sequences (with A. Folsom, Amherst College and R. Rhoades, Shasta College), and Fishburn matrices (with Y. Li, TU Darmstadt and R. Rhoades);

- I proved (with J. Dousse, Universität Zürich) 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 which subsequently have been used by other researchers - e.g. T. Hamel (physics), J. Manschot (physics) - for example for generating functions arising in physics;

- I investigated modularity results pertaining to unimodal sequences (with J. Lovejoy, Univ. Paris VII) and Capparelli type identities (with K. Mahlburg);

- I showed combinatorial indentities including companions to Capparelli's identities, Schur double series representation, and universal mock theta functions (with G. Andrews, Pennsytlvania State Univ., K. Mahlburg, and J. Lovejoy).

Thus, I now have a clear understanding about the interplay of combinatorial q-series and modular forms.

In Objective 2, I investigated (with J. Dousse, K. Mahlburg, and J. Lovejoy) further percolation models and in particular studied the role of overpartitions. We showed there is a relation between mock theta functions and sequence conditions.

This settles the outset goal of Objective 1.

In Objective 3, I investigated the role of the harmonic Maass and modular forms in Lie theory. In particular:

- I studied how false theta functions play a role in W-algebras and found candidates for higher-dimension false theta functions (with A. Milas, Univ. Albany);

- I investigated the role of Jacobi Poincaré series in Moonshine (with J. Duncan, Emory Univ. and L. Rolen, Pennsytlvania State Univ.). A main difficulty here was to show convergence of the Poincaré series.

- and I solved the motivating question from the grant pertaining to the modularity of Kac-Wakimoto characters. We now fully understand these. These results have big impact on Objective 4.

In Objective 4, I studied the

- asymptotic behavior of certain inverse theta functions (with J. Manschot, Trinity College Dublin);

- indefinite theta functions arising from Gromov-Witten theory (with L. Rolen and S. Zwegers, Univ. Köln). These give us natural candidates for higher dimensional indefinite theta functions which we investigate further. In Objective 3, we came across certain false theta functions which contribute a better understanding of multiple centered black holes.

I am standing well in most of the proposed time frame. The progress in Objective 3 was skewed down due to the birth of my thrid child. I have natural candidates for q-series from Lie theory whose quasimodularity properties are still to be understood. In Objective 4, I am now changing direction by investigating indefinite theta functions which naturally occured in our case.