Optimal geometric structures and the evolution of shapes are governed by partial differential equations. These same types of equations come up over and over again across many diverse areas in science, engineering, and mathematics. The geometric invariance makes the equations canonical and means that they also describe phenomena seemingly unrelated to geometry. Often the geometry unlocks the structure of the equation and leads to fundamental tools in PDE. Conversely, analysis has played a central role in the development of geometry. Understanding the equations and their fundamental properties requires simultaneous insight into both analysis and geometry and the interplay between the two.

In this talk, we will discuss this principle for several fundamental equations. We start by seeing how a long-standing problem in geometry leads to optimal regularity for viscosity solutions of a degenerate elliptic PDE, then turn to using PDE to understand optimal shapes and geometric evolution.