The International Research Fellowship Program enables U.S. scientists and engineers to conduct nine to twenty-four months of research abroad. The program's awards provide opportunities for joint research, and the use of unique or complementary facilities, expertise and experimental conditions abroad.
This award will support a twenty-four month research fellowship by Dr. Michael R. Munn to work with Dr. Peter M. Topping at Mathematics Institute at the University of Warwick in the UK.
The aim of this project is to work closely with Dr. Peter Topping and the Geometric Analysis group of the Mathematics Institute at the University of Warwick. First, the PI aims to continue his research studying the influence of volume growth on the topology of Riemannian manifolds with nonnegative Ricci curvature. Second, the PI aims to study the consequences of a lower bound on Ricci curvature in the more general setting of metric measure spaces. This concept is on the forefront of research in geometric analysis and has only recently been developed. As such there are many fundamental open questions. Lastly, the PI aims to study the behavior of Ricci flow in this more general setting and determine any topological or geometric consequences for the underlying metric space. Recently, Fields Medalist Perelman has used Hamilton?s Ricci flow to solve the century old Poincare conjecture. These fields are being actively pursued by researchers within the Mathematics Institute at Warwick, most notably by Dr. Topping, thus making Warwick an ideal place for this fellowship research. Dr. Topping is a leading expert in geometric analysis and has employed the techniques of geometric flows, such as Ricci flow and mean curvature flow, with great success. Dr. Topping is one of only a few mathematicians studying e nature of Ricci flow on metric measure spaces.
An understanding of Ricci curvature and Ricci flow in metric spaces is of great importance in geometric analysis and metric geometry. These original concepts have only recently been introduced and are on the forefront of research in geometric analysis. In Perelman?s work, he did not address the Ricci flow when singularities develop and this has been an important question of Hamilton, Lott, and others. In a very recent paper, Topping and McCann introduce a notion of Ricci flow for metric spaces. These advances are of great importance as they can be used to understand the development of these singularities. The PI?s current research involves prior work of Perelman?s involving Ricci curvature. The PI also extends these results to metric measure spaces. The Mathematics Institute at Warwick is a thriving environment and is undergoing rapid expansion. The large number of resources available within the Institute make this an ideal host site. Currently there is great interest in the development of these new ideas for Ricci flow in metric spaces and these techniques have applications across the physical sciences. The ideas related to Ricci flow are closely related to those of mean curvature flow as well as crystal flow. Understanding the development of singularities in this context is essential. In particular, singularities describe the development of flaws in crystals. A deeper understanding of the implications of this research will enable the PI to be a better professor at a university with science majors and engineers.
