Overview

The concept of shape is one of the big themes permeating the whole history of Mathematics. Among the very many ways it has been formalized, since the ancient times and through the centuries,the notion of curvature is one of the most successful and significant ones. Starting with pioneering work by Gauss and Riemann in the first half of the 19th century, this concept has been proposed and developed at increasing levels of abstraction, and has then emerged one hundred years ago in the context of Einstein's description of gravitational forces within his general theory of relativity. The language of differential geometry has proven to be a fundamental ingredient in the most diverse physical theories, when trying to describe the mysteries of the world we live in.

Along this journey lots of questions have been raised and novel techniques have been developed to answer them. In particular, over the last fifty years we have witnessed the massive use of analytic techniques, based on the analysis of partial differential equations, to attack fundamental problems in Geometry, and the interconnections between these two worlds, their mutual contaminations, have dramatically come out as one of the main trends in our discipline. Among the very many milestones that exemplify this phenomenon, we mentionthe proof of the positive mass theorem by Schoen and Yau,the proof of the Poincaré conjecture by Perelman and, more recently, the proof of the Willmore conjecture by Marques and Neves. The whole field which we now call Geometric Analysis is more vital than ever, new intriguing questions keep arising and breakthroughs happen more often that has ever been the case in the past.

In this project, we focus on two broad themes that have on the hand strongconnections with some of the most exciting recent developments in the field, and on the other displaythe potential of opening new avenues in the decades to come. First,we aim at discovering patterns in the mysterious landscape that is emerging from the solution of Yau’s conjecture about minimal hypersurfaces in Riemannian manifolds. Second,we wish to shed some light on the world of positive scalar curvature manifolds, which sits somewhere inbetween the rigid world of convex objects and the flexible world of geometric topology. The methodology we employ relies on a combination of elliptic and parabolic techniques.

Our goals are to provide significant contributions to these themes, and to attract highly promising young researchers soto favor the development of Geometric Analysis in the European context.