NSF Award
2316659
A fundamental theme in geometry is to study the relationship between the curvature and the shape of the space. This project will investigate in dimensions four and higher how “positivity curvature” determines, partially or in full, the global shape of the underlying space, aiming to extend the famous Gauss-Bonnet Theorem in dimension two and Hamilton and Perelman’s work on geometrization in dimension three to higher dimensions. In this project, the PI will expand the study of geometrization to manifolds of dimensions four and higher, making use of tools such as curvature operators and Ricci flows. In addition, the PI will organize a variety of activities for middle and high school students in the local Wichita area and engage in mentoring programs for undergraduate and graduate students at Wichita State University aimed at broadening participation, especially amongst underrepresented minority students.<br/> <br/>The research objective of this project is to investigate the classification of four and higher-dimensional spaces, including compact Riemannian manifolds, Einstein manifolds, and shrinking gradient Ricci solitons, whose curvature satisfies a positivity condition, such as positive isotropic curvature, positive sectional curvature, positive Ricci curvature, and positive curvature operator of the second kind. The outcome is a better understanding of the relationship between curvature and topology. Primary strategies include analyzing solutions to the Ricci flow, applying the maximum principle to partial differential equations satisfied by geometric quantities, and understanding the relationship between various notions of positive curvature via tensor algebra and Lie algebra.<br/><br/>This project is jointly funded by the Launching Early-Career Academic Pathways in the Mathematical and Physical Sciences (LEAPS) and the Established Program to Stimulate Competitive Research (EPSCoR).<br/><br/>This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
