Curvature, Symmetry, and Topology

The plane geometry that one learns in high school gives one the first introduction to Euclidean geometry, one of the three classical geometries. Euclidean geometry has applications to architecture, engineering, computer science and crystallography, as well as various branches of modern mathematics. The other two geometries are Spherical and Hyperbolic. Spherical geometry is central to the study of geophysics and astronomy, and crucial for navigation. Hyperbolic geometry has modern applications to the theory of special relativity in Physics as well as Art and Design. Global Riemannian Geometry generalizes these three geometries, all of which have constant curvature, that is, the way in which they "bend" does not change. In particular, this area of mathematics provides a framework to study spaces of varying curvature. One of the major challenges in this area of study is to understand how local geometric invariants such as curvature relate to global topological invariants such as fundamental group, which indicates whether or not the space has 1-dimensional "holes". Manifolds with curvature bounds have been studied intensively since the conception of Global Riemannian geometry. Over the last 30 years, one approach to the study of manifolds with lower curvature bounds has been the introduction of symmetries and is the main focus of this project. The PI will also continue her outreach work with middle and high school students, as well as graduate training, and the organization of workshops and conferences. The principal investigator will pursue a program in which she carefully studies and analyzes symmetries of Riemannian manifolds with lower curvature bounds, considering sectional, Ricci, scalar, and the intermediate curvatures that interpolate between sectional and Ricci and between sectional and scalar, and some of their corresponding generalizations to Alexandrov spaces, with an eye to gaining a deeper understanding of this largely unknown class of spaces. She will study not only how continuous and discrete symmetries relate to the topology of such spaces but also try to find new examples of Riemannian manifolds of positive Ricci curvature and almost non-negative sectional curvature using symmetries and topology as tools to do so. 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.