Explore the advanced mathematical concepts of resurgence and quantum modularity in this hour-long lecture by Campbell Wheeler at the Institut des Hautes Etudes Scientifiques (IHES). Delve into the improved analytic properties of solutions to specific q-difference equations and their impact on computing conjectural formulae for Stokes constants of associated asymptotic series. Examine examples from the perspective of quantum modular forms, drawing from the work of Garoufalidis, Gu, Mariño, Kashaev, and Zagier. Gain insights into the application of these concepts to invariants associated with simple knots, and understand the broader implications for mathematical research in this field.