The projects in this proposal aim to deepen the understanding of certain natural geometric evolution equations generalizing the Ricci flow, with a focus on understanding aspects of four dimensional geometry. One example is the gradient flow of the square norm of the curvature tensor, a natural fourth-order parabolic equation whose critical points unify various important classes of metrics on four-manifolds. Another important example is a generalization of Kahler Ricci flow to non-Kahler complex manifolds introduced in prior work of the PI and G. Tian. By utilizing methods from the well-developed theory of Ricci flow as well as techniques from complex geometry, the PI proposes to refine his existing work to understand the singularity formation of these flows. Aside from the intrinsic value of understanding physically natural equations, one possible direct application is to understand the topology of complex surfaces.
The method of geometric flows is a relatively new technique for understanding the structure of geometric objects. Ultimately, by understanding these equations one can gain a deep understanding of topological structures. Furthermore, these equations typically have a physical motivation, and hence by understanding them we gain insight into natural physical processes. Even beyond these theoretical uses geometric flows have recently seen industrial application. The proposed research will thus add to our overall understanding in these important areas.
A general goal in the field of geometry is the construction and classification of "special" or "distinguished" objects of some kind. For instance the round sphere is distinguished amongst all possible stretchings of the sphere as the one enclosing the given interior volume with the least possible surface area. A modern approach to discovering and classifying such special objects is to follow the action of some physical process which takes a generic object and constructs a special one. As an example consider the changing surface of a soap film which is not spherical, but as time progresses approaches a spherical shape. The primary focus of this project was to understand processes of this kind based on higher dimensional geometric and physical principles. In particular the relevant geometric structures either arose or have played a significant role in modern physical theories such as quantum mechanics, general relativity, and string theory. Our academic activity has resulted in a significant strengthening of our knowledge of the geometric content of some of these natural physical processes, even providing new insight and directions for further research from a purely physical point of view. The training of graduate students played a significant role in this project as well. Three graduate students were directly funded by this project, all of whom have received a broad training in advanced methods in geometry and partial differential equations which have broad applicability in science and mathematics. They either have completed or are in the process of completing research projects related to the general themes discussed above. This training has prepared them to bring advanced mathematical expertise to bear on either an academic position or a role in industry. We also delivered many lectures of a more public nature designed to increase awareness of and excitement about the forefront of mathematical research.
