Explore the mathematical foundations of supersolids through the stationary Gross-Pitaevskii equation in this 43-minute conference lecture. Delve into the analysis of extended functions that do not tend to zero at infinity, which model the condensate component of infinite Bose gases. Learn how to combine partial differential equation techniques with statistical mechanics methods to investigate the phase diagram of this complex system. Discover the proof of a unique freezing transition for interaction potentials that are stable but not positive definite, and examine the conjecture that solutions in the solid phase exhibit periodic behavior. Understand how these periodic solutions describe supersolids, a fascinating state of matter that has garnered significant attention in recent experimental research, through work developed in collaboration with Phan Thành Nam from Munich.