Ancient Ricci Flow and Ricci Solitons

This project explores a powerful mathematical process called Ricci flow, which smooths out geometric shapes over time--like heat evening out the temperature on a surface. Ricci flow has led to major breakthroughs, including Perelman's resolution of the century-old Poincaré Conjecture, and continues to transform our understanding of the geometry and topology of space. A major challenge lies in understanding the "singularities" that inevitably form during the flow, where the shape becomes infinitely curved, and hold vital clues about the hidden structure of space. A central idea in Perelman's proof is the use of surgeries to continue Ricci flows through singularities. The PI aims to extend this surgery construction in higher dimensions, with the goal of uncovering new geometric and topological applications. The research will be complemented by mentoring graduate students and organizing workshops and conferences. The research project is split into two parts: The first project focuses on classifying ancient Ricci flows that are asymptotic to cylinders, which serve as potential singularity models. These asymptotically cylindrical flows include the classical rotationally symmetric examples such as the Bryant soliton and Perelman's ovals, as well as new examples recently constructed by the PI, known as flying wings. The flying wings are asymptotic to cylinders with more than one R-factor and break the rotational symmetry. The PI aims to develop a general method to estimate the asymptotic behavior of such flows without relying on rotational symmetry or positive curvature assumptions. These techniques may ultimately enable an extension of Perelman's 3D Ricci flow with surgeries to 4D, providing a broader and more robust topological toolkit. The second project builds on the PI's ongoing work with collaborators to understand 3D open manifolds with positive scalar curvature. The PI has developed a theory of generalized singular Ricci flows that extends the flow to arbitrary 3D manifolds, including those with unbounded geometry. Building on this, the PI will investigate the removal of the bounded geometry assumption and work toward a complete classification of all such manifolds. Please report errors in award information by writing awardsearch@nsf.gov. 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.