Extreme phenomena in Banach space geometry: small and big slices and their applications to Lipschitz-free spaces and their duals

This project aims to study small and big slice phenomena in the unit ball of Banach spaces and to improve our understanding of their influence on the structure of Lipschitz-free spaces and spaces of Lipschitz functions. These phenomena are closely related to two of the most studied properties of Banach spaces, the Radon-Nikodym property and the Daugavet property, which provide bridges between some significant areas of Functional Analysis such as operator and measure theory, differentiability, and geometry. Their celebrated characterizations in Lipschitz-free spaces have recently highlighted their importance in non-linear geometry and shown their high potential for applications to resolve several deep problems in metric geometry and optimization. The project is in line with the latest breakthroughs by several research teams on the topic, which have uncovered stunning interactions between the two seemingly opposite worlds and have opened many new perspectives of research.