The liquid drop (LD) model, an old problem of Gamow for the shape of atomic nuclei, has recently resurfaced within the framework of the modern calculus of variations. The problem takes the form of a nonlocal isoperimetric problem on all 3-space with nonlocal interactions of Coulombic type.

In this talk, we first state and motivate the LD problem, and then summarize the state of the art for global minimizers.

We then address certain recent anisotropic variants of the LD problem in the small mass regime, with a particular focus on the minimality of the Wulff shape.

In the second half of the talk, we address related nonlocal problems involving competing interaction potentials of algebraic type. These problems are directly related to a wide class of self-assembly/aggregation models for interacting particle systems (eg. swarming). In particular, we will discuss the strong attraction limit.

This talk includes joint work with Almut Burchard (Toronto), Elias Hess-Childs (McGill), Robin Neumayer (IAS and Northwestern), and Ihsan Topaloglu (Virginia Commonwealth).