Positive curvature and torus symmetry

As a mathematician faced with a research problem or an educational task, the PI strives to abstract away inessential information, break down complicated structures into simple components, and identify the right tools for the job. In this project, the PI plans to apply methods from disparate fields such as discrete mathematics and homotopy theory to problems in geometry, especially problems involving the impact of symmetry and local curvature conditions on the global shape of high-dimensional objects called manifolds. Crucial to this progress are communication and collaboration with experts from across the country and around the globe whose areas of expertise both overlap and supplement the PI’s. Additional aspects of this project include growing and diversifying the body of students, researchers, and experts in STEM fields who will positively impact the advancement of the research goals of this project, the future of STEM education, and our society’s ability more broadly to tackle difficult scientific problems. The PI will analyze local-to-global principles in geometry. Goals involve analyzing the interaction of (local) positive curvature conditions in Riemannian geometry and (global) algebraic topological and symmetric structures. The PI will apply tools from homotopy theory, equivariant cohomology theory, matroid theory, and topological graph theory. These methods have applications in the Grove Symmetry Program but do not use curvature, so this work has the potential to apply in other areas of geometry. 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.