The main objective of this project is the study of nonlinear parabolic equations which come from differential geometry problems, such as the evolution of a hypersurface of Euclidean space by functions of its principal curvatures, the Ricci flow and the Yamabe flow. More precisely, the PI focuses on singularity analysis of possible finite time singularities occurring in those evolution equations. Such equations appear in quantum field theory, plasma physics, thin liquid film dynamics. More precisely, one of the things the PI wants to study is the regularity of nonlinear geometric flows, such as finding the minimal geometric conditions that will guarantee the smooth existence of a solution to the Ricci flow and the mean curvature flow. The other thing the PI would like to understand are the ancient solutions to nonlinear geometric flows and their classification. It is well known that ancient solutions arise as singularity models (blown up limits) at finite time singularities. Their classification is crucial for better understanding the singularities that may 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 ultra-violet regime. One special class of ancient solutions are Ricci solitons. The PI would like to study those, having an ultimate goal of classifying generic singularities of a generic Ricci flow.
The project the PI proposes links many different active fields of mathematics, such as nonlinear analysis, differential geometry and topology. The proposed research activity on singularity analysis and regularity of nonlinear parabolic geometric evolution equations may result in interesting applications in geometry and topology. There may be potential application in physics as well. It is well known that the Ricci flow theory has lead to a solution of the Poincare conjecture in topology. The hope is that geometric flows may help solving other important topological question such as the classification of manifolds in higher dimensions. One of the main obstacles in order to even approach such a difficult question like that by using the flow theory is understanding the singularity formation and the classification of singularities, since one can not hope the flow will exist forever. Most likely it will develop singularities in finite time. The PI proposes to understand the formation of singularities in the flows such as the Ricci flow, mean curvature flow, the Yamabe flow and therefore contribute to finding a way to approach the big mentioned problem above.
