The P.I. will study the geometric properties of the Ricci flow in dimension bigger than three from three aspects. The first one is about the Ricci flow on manifolds with positive isotropic curvature. We want to work out a classification of closed 4-manifolds with positive isotropic curvature. The second one is about the Ricci flow on cohomogeneity one manifolds. One question is to explore when nonnegativity of sectional curvature is preserved and when the sectional curvature becomes positive. Another question is to explore when Hamilton and Ivey's pinching theorem will hold. The third one is about the relation between the Ricci flow and quantum field theory; especially we will focus on the duality property of the Ricci flow with symmetry. We will look for mathematical applications of new ideas in physics which are related to the Ricci flow.
The Ricci flow is a very important mathematical field and has the potential to solve very challenging problems relating to the geometric and topological properties of spaces of dimensions three and four. Recently the fundamental work of Perelman has drawn a lot of attention to the field. The Ricci flow has a close connection with string theory; more precisely, Ricci flow is the first order approximation of the so-called renormalization group flow (RG flow). The proposed study will improve our understanding of the Ricci flow and demonstrate that Riemannian Ricci flow is a powerful tool in studying geometric-topological properties of manifolds of dimension greater than three. Furthermore we hope the proposed study will improve our understanding of the RG flow in string theory.
