New approaches to nonperturbative physics Resurgent instantons and their stokes data

Perturbation theory is one of the most widely used approaches for obtaining approximate solutions to problems in physics. However, to obtain exact solutions, we also need to include nonperturbative phenomena, which are notoriously hard to compute. Quite recently, a relatively novel approach to this problem has gained more attention in the physics community. This approach is called resurgence, a mathematical framework that explains how nonperturbative phenomena can be extracted from perturbative physics. Resurgence tells us that perturbative and nonperturbative contributions should be unified in a more general structure called a transseries. In this thesis, we study these transseries and their so-called Stokes phenomenon in several distinct problems. The first of these is the Painlevé I equation, in which we show how its two-parameter transseries solutions are affected by the Stokes phenomenon. The second problem that we address is that of one-dimensional quantum mechanics, where we provide a new method for computing all nonperturbative contributions in a single sweep. Then we study the structures that underlie so-called large order relations, which connect perturbative and nonperturbative contributions in resurgence. Finally, we elucidate the relation between parametric resurgence and the wall-crossing phenomenon that string theorists study in supersymmetric quantum field theories.