Abstract: In this talk I'll discuss recent results for the Monopolist problem which I've obtained in collaboration with Robert McCann and Kelvin Shuangjian Zhang. The Monopolist's problem is one of optimal pricing under incomplete information, but gives rise to an optimization problem in the calculus of variations/elliptic PDE. This model's novel mathematical behavior and free boundary structure has led to renewed interest within the mathematical community. I will discuss the configuration of the free boundaries in this problem and our description of prototypical solutions, including previously unanticipated behaviors of this model. This colloquium talk will be accessible to nonspecialists.