The complex Langevin (CL) method is a numerical approach to alleviate the numerical sign problem in the computation of path integrals in lattice field theories. Mathematically, it is a simple numerical tool to compute a wide class of high-dimensional and oscillatory integrals with the form of an ensemble average. The method was developed in 1980s. However, after it was proposed, it had very few applications due to its subtle nature. It is often observed that the CL method converges but the limiting result is incorrect. Less than one decade ago, the CL method was improved by gauge cooling method and dynamical stabilization, after which the CL method acquired much more attention and was later successfully applied to a number of fields including finite density quantum chromodynamics, superstring theory, and the spin-orbit coupling. In this talk, I will take the mathematical perspective to explain the basic idea of the CL method and the reason of its failure. The current limitation of the method and the possible remedies will also be discussed.

Dr. Zhenning Cai is currently an assistant professor at Department of Mathematics, National University of Singapore (NUS). Before moving to NUS, he worked as a postdoc at RWTH Aachen University, University of Maryland and Duke University. Dr. Cai received his Bachelor and PhD degrees from Peking University in 2013. Dr. Cai's research interest includes numerical methods for the gas kinetic theory; real-time simulation of open quantum systems; complex Langevin method in the lattice field theory.