CAREER: Ricci Flow and Ricci Solitons

Geometric flows describe how curved spaces evolve over time, much as heat flow smooths out temperature differences. These flows have become central tools in geometry and topology, most famously through Ricci flow and its role in the resolution of the Poincaré and Geometrization Conjectures. Under this project, the investigator will study fundamental questions concerning Ricci flow, singularity formation, and the topology of three-dimensional spaces with positive scalar curvature. A major goal is to understand the geometric models that arise near singularities, especially those that look cylindrical at large scales, and to develop tools to extend these techniques through singularities in higher dimensions. The project will also investigate how positive scalar curvature constrains the global shape of open three-dimensional spaces. Additionally, the project will promote the training and development of students and early-career researchers through graduate courses, undergraduate research mentoring, workshops, and outreach activities. This project includes two primary research directions. In the first direction, the investigator will study asymptotically cylindrical ancient solutions and singularity models in Ricci flow. The investigator will develop sharp asymptotic estimates near cylindrical ends, aiming to obtain a full Taylor-type expansion of the flow near the cylinder to arbitrarily high order. These estimates are expected to contribute to the classification of asymptotically cylindrical Ricci flows, including higher-dimensional models related to the investigator’s flying wing solitons, and to provide new foundations for performing surgery through cylindrical singularities in four-dimensional Ricci flow. In the second direction, the investigator will study open three-dimensional manifolds with positive scalar curvature. The investigator will use generalized singular Ricci flows to study Yau’s conjectural classification of such manifolds, with the goal of understanding how positive scalar curvature restricts their topology without imposing bounded geometry assumptions. 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.