Geometric Analysis under lower Ricci Curvature bounds

This project explores a fundamental question in modern mathematics: how the shape of space is influenced when certain natural curvature conditions are imposed. Curvature is a way to measure how a surface or a space bends. In everyday life, we encounter positively curved spaces such as spheres, flat ones like a sheet of paper, or negatively curved ones like a saddle. But when we move to higher dimensions, the picture becomes far more intricate. The focus of this project is on a specific notion of curvature, called Ricci curvature. It plays a central role not only in mathematics but also in physics, where it appears in Einsteins equations of general relativity. A lower bound on Ricci curvature provides a kind of minimum bending condition that strongly influences the overall shape and structure of a space. Over the past decades, researchers have discovered deep links between Ricci curvature, analysis, geometry, and probability. Despite this progress, many central questions remain open, especially in four or more dimensions. For example: How does Ricci curvature shape the possible topologies of spaces? Can it rule out certain exotic structures? What happens when spaces with controlled curvature collapse? This project aims to tackle several of these long-standing challenges. It combines classical tools of geometry with modern techniques from analysis and metric geometry, and it brings together an international network of collaborators. The results are expected to advance our understanding of geometric structures under curvature constraints and shed light on questions that connect mathematics, physics, and our broader understanding of space itself.