The field of differential geometry is the study of curved objects, and this project focuses on the use of geometric evolution equations or `flows' to understand these objects. More specifically, these equations govern mathematically-defined processes that smoothly modify objects in a way that is driven by geometrically meaningful quantities such as length, area, volume or curvature. The equations that the PI studies have very nice regularity properties, in the sense that bumpy objects often become smoother as they evolve; in fact, it is reasonable to expect that the object will gain more symmetries as the flow proceeds, and this will help us better understand the class of geometric objects we started with. Unfortunately, these equations also have the potential to develop singularities in finite time, meaning that the solution may acquire sharp points and corners, and one cannot expect in general to have a smooth solution to the equation for a long time. In order for us to achieve our ultimate goal, and better understand the underlying geometric object using the flow techniques, we need to understand how to deal with singularities that may arise. These and similar topics have been the subject of several workshops in geometric analysis at Rutgers University organized by the PI (together with colleagues), attracting many graduate students and featuring both research talks and advanced graduate-course level lectures.One of the aims of this project is the classification of ancient solutions to nonlinear geometric flows, such as, the Ricci flow and the mean curvature flow. The PI proposes to combine the PDE techniques and geometric estimates to study the ancient solutions of such flows. Their classification is crucial for better understanding the singularities that occur in finite time. Ancient solutions to the two-dimensional Ricci flow describe trajectories of the renormalization group equations of certain asymptotically free local quantum field theories in the ultraviolet regime. The PI will classify ancient closed non-collapsed solutions to the three-dimensional Ricci flow, which would settle down a conjecture stated by Perelman in one of his papers. The PI shall also find minimal conditions that guarantee smooth existence of a solution to the Ricci flow and the mean curvature flow. The PI also wants to understand singularity formation in the following sense: what the stable singularity models in four dimensional Ricci flow are. This is related to understanding the singularity formation of a four-dimensional Ricci flow starting at generic initial data.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.
|Effective start/end date||9/1/18 → 8/31/21|
- National Science Foundation (National Science Foundation (NSF))
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