CAREER: Aspects of Riemannian Geometry and Manifolds with Density

Syracuse University, Syracuse NY

Investigators

Abstract

Riemannian geometry investigates the generalization of familiar geometric notions such as length, angle, and volume to more abstract, often high-dimensional, spaces called manifolds. Manifolds and their geometric properties are not only central to various branches of mathematics but also are ubiquitous as mathematical models in the sciences. Perhaps the greatest example of this is in general relativity, where one of Einstein's great breakthroughs was the discovery that gravity is modeled by the mathematical notion of curvature. This research project focuses on curvature of Riemannian manifolds. Specific emphasis will be placed on manifolds with density, which can be envisioned as Riemannian manifolds composed of a material with variable, as opposed to uniform, density. Despite the appearance of manifolds with density in various areas of mathematics and applications, including proof of the Poincare conjecture, they are not yet well understood. This project addresses this gap by helping to develop a geometric theory of manifolds with density. The project also uses geometry as a theme for a coherent program of educational activities that include: (1) professional development workshops in teaching and learning for in-service secondary education teachers, which will help support teachers in transition to recently developed K-12 standards; (2) innovation in introductory differential geometry curricula at the undergraduate and graduate level that, by emphasizing applications, will also foster new interaction between mathematics and science researchers; and (3) training of mathematics graduate students in best strategies for engaging students in introductory calculus courses. Ricci curvature for manifolds with density has been an active area of recent research. This includes Ricci solitons, which are both fixed points of the Ricci flow and examples of spaces with constant weighted Ricci curvature, and the study of weighted Ricci curvature bounds. The investigator will continue study of the classification of shrinking Ricci solitons in dimension four and higher. In addition to playing a key role in development of monotonic functionals for the Ricci flow, weighted Ricci curvature bounds also appear in the theory of optimal transport, isoperimetric inequalities, and general relativity and cosmology. Despite the vast amount of research in Ricci curvature of manifolds with density, there was no corresponding theory of weighted sectional curvature until recently; another goal of this project is to further develop the theory of weighted sectional curvature bounds, including investigating applications and connections to other areas of mathematics. Recent collaborative work of the investigator also introduces a new geometric approach to manifolds with density that places a certain torsion free affine connection as the fundamental object of study. This approach not only provides new insight into the weighted Ricci and sectional curvatures but also suggests new natural structures that promise novel results, such as a weighted holonomy group, which will be investigated.