I will describe analytic properties of a certain class of series in a parameter q. Physically, these series arise as quantum invariants of 3-manifolds via perturbative Chern-Simons theory, and mathematically, they come from q-hypergeometric functions, which come from the geometry of varieties via their periods. For all prime numbers, these analytic properties are conjectured (proved in some case) to give rise to refinements of q-de Rham cohomology classes (Habiro cohomology classes). Over the complex numbers, these series are resummable with completely explicit Stokes phenomenon. A nice application of these ideas is the construction of a global regulator from the third algebraic K-theory of a number field, which stores all reasonable known regulators into one collection of elements in the field. This is based on joint work with Garoufalidis-Scholze-Zagier and Andersen-Fantini-Kontsevich

https://indico.math.cnrs.fr/event/16742/