Dr. Robin Neumayer，Northwestern University

Functional and geometric inequalities play an important role in many problems arising in the calculus of variations, partial differential equations, and geometry. The isoperimetric inequality is a classical example of a geometric inequality: among allsubsets of Euclidean space with a fixed volume, balls have the smallest perimeter. A key point in the study of functional and geometric inequalities is to understand their optimal form, which typically involves studying minimizers of an energy functional andtheir stability properties. These lectures will focus on two main themes: applications of functional and geometric inequalities in various contexts, and the interplay between the geometry of Riemannian manifolds and their optimal functional inequalities. The lecture will be a broad overview of the field.