Spaces with Ricci curvature bounded below

Differential geometry is a branch of mathematics that studies the properties of smooth spaces, known as manifolds, and it has profound applications throughout mathematics, physics, and other scientific disciplines. In general, those quantities that measure the shape of the manifold are called curvature. The study of one particular kind of curvature -- Ricci curvature -- is one of central topics in differential geometry and has significant connections to general relativity. This project seeks to study the geometry and topology of spaces whose Ricci curvature can't be too low, i.e., where this curvature has a lower bound. The notion of Ricci curvature lower bounds can also be extended to singular spaces (e.g., spaces with cone tips) which may not have manifold structures. Investigating these singular spaces can in turn advance the understanding of Ricci curvature on smooth manifolds. The broader impacts of this project include raising awareness of the importance of singular spaces through seminar and conference presentations, bringing modern concepts and ideas in geometry to students in an accessible way, as well as mentoring undergraduates in these areas. In one direction, the project will study the fundamental groups of complete and non-compact manifolds with nonnegative Ricci curvature. Specifically, this will investigate the relation between the structure of fundamental groups (for example, finite generation and virtual nilpotency) and the equivariant asymptotic geometry. The project will also aim to understand the geometry of Ricci limit spaces, especially that of singular sets with large Hausdorff dimension. Lastly, the project will explore the fundamental groups of closed manifolds with Ricci curvature bounded below, including their stability under the Gromov-Hausdorff topology and uniform control on the group structure. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.