Ricerc@Sapienza

Many physical principles are an expression of an underlying variational principle; Euler made the expansive statement that
``...nothing at all takes place in the universe in which some rule of the maximum or minimum does not appear.''
When the maximum or minimum is attained -- for example, when potential energy is minimized -- the physical law is also described by a partial differential equation, the Euler-Lagrange equation.

At the same time, physical laws often have a purely geometric description. Einstein's field equations of General Relativity prescribe the Ricci curvature tensor of space-time. Soap bubbles minimize area and therefore, in physical terms, are the least energy configuration.
This property is geometrically expressed by the fact that soap bubbles have constant mean curvature. However, the roundness of soap bubbles was proved by Alexandrov
using PDE techniques, i.e., by analyzing the Euler-Lagrange equation. Therefore, one is often compelled to view physical phenomena while wearing three hats: that of the physicist, the geometer, and the analyst.
The examples of the Einstein field equations and minimal surfaces are particularly instructive, because their study drove many of the developments in the existence
and regularity theory for nonlinear PDEs in the last century. The goal of our proposal will be to bring together researchers who are working PDEs with connections to differential geometry and mathematical physics.