Beijing Institute of Mathematical Sciences and Applications

Resurgence theory provides a powerful framework for understanding the hidden structures behind divergent asymptotic expansions and their analytic continuation. This course will introduce the fundamental ideas of resurgence theory and explore its applications in differential equations, geometry, and mathematical physics.

We begin with the basic theory of Borel–Laplace summation, Gevrey asymptotics, and the analytic structure of Borel transforms. The course will then introduce Écalle’s alien calculus, including alien derivatives and their role in describing the relations between different asymptotic sectors and Stokes phenomena. Classical examples arising from ordinary differential equations will be studied to illustrate how resurgence reveals exponentially small effects invisible to perturbative expansions.

The second part of the course focuses on finite-dimensional oscillatory and exponential integrals. We will discuss the interplay between Picard–Lefschetz theory, complex saddle points, and resurgence. In particular, we will examine how the geometry of Lefschetz thimbles and the topology of critical points are related to Stokes structures and alien calculus.

The course will further explore infinite-dimensional settings, especially path integrals and heat kernel asymptotics on manifolds. Topics include complexification of Riemannian manifolds, complex geodesics, semiclassical expansions, and resurgence structures arising from heat kernels and quantum field theoretic models.

Finally, we will discuss connections between resurgence and probability theory, including applications to stochastic processes, asymptotic analysis of probabilistic models, and the role of resurgence methods in understanding non-perturbative phenomena in random systems.