This workshop, sponsored by AIM and the NSF, will be devoted to emerging interactions between modular forms and quantum topology along with their connections to related areas such as representation theory, combinatorics, and mathematical physics. The workshop's primary motivation stems from foundational work of Lawrence and Zagier, Witten, Habiro, and others, as well as recent advances at the intersection of modular forms and knot theory, e.g., the Volume and Modularity Conjectures. Further motivating developments include Nahm’s conjecture on the modularity of certain $q$-hypergeometric series, quantum and holomorphic quantum modular forms, properties of the Habiro ring, and resurgence. An important goal of the workshop is to bring together researchers at various career stages to collaboratively work on related problems from complementary mathematical perspectives.
The main topics for the workshop include
- Volume and Modularity Conjectures
- Habiro ring and colored Jones polynomials, identities, resurgence, and (holomorphic) quantum modularity
- The interplay among q-hypergeometric series, modular forms, representation theory, and combinatorics
This event will be run as an AIM-style workshop. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.