q-series, Resurgence and Modularity

q-series, Resurgence and Modularity

Explore the resurgent structure of formal power series associated with q-series in this advanced mathematics lecture. Delve into Zagier's "quantum modular forms" paper, focusing on the q-series σ(q) from Ramanujan's "Lost" Notebook. Examine how this simple resurgent structure potentially encodes modularity properties. Investigate similar resurgent structures in other q-series, including the Kontsevich-Zagier q-series for trefoil and those derived from fermionic spectral traces of quantum-mechanical operators. Discover the connections between resurgent structures and modularity properties across various mathematical contexts, providing insights into advanced topics in number theory and quantum mathematics.