Veronica Fantini - q-series, Resurgence and Modularity

In Zagier's paper titled "quantum modular forms", one of the first examples of quantum modular form is related to the q-series\n$$\n \\sigma(q)=1+\\sum_{n=0}^\\infty (-1)^n q^{n+1} (q)_n\n$$\nfrom Ramanujan's "Lost" Notebook. In this talk, I will discuss the resurgent structure of the formal power series associated with the q-series $\\sigma(q)$: it is a simple resurgent structure which conjecturally encodes the modularity properties already studied by Zagier.\nFurthermore, the same resurgent structure appears when considering formal power series associated to other q-series, such as the Kontsevich--Zagier q-series for trefoil and the q-series coming from the fermionic spectral traces of quantum-mechanical operators related with the quantization of the mirror curve of toric CY 3-folds (recently studied by C. Rella arXiv:2212.10606). Hence we expect to find analougus modularity properties by studying their resurgent structures.