Explore the intricacies of Serrin's overdetermined problem in rough domains through this illuminating lecture by Alessio Figalli from the Institute for Advanced Study. Delve into the classical Serrin's overdetermined theorem and its extensions to non-smooth domains. Discover the recent breakthrough in applying this theorem to Lipschitz domains, a long-standing open question in the field. Examine the innovative approach that proves the theorem's validity for sets of finite perimeter with a uniform upper bound on density, including domains with slit discontinuities. Gain insights into the collaborative work with Yi Zhang that led to this significant advancement in Analysis and Mathematical Physics.