NSF Award
2204324
The plane geometry we learn in high school gives us an introduction to Euclidean geometry, one of the three classical geometries. Euclidean geometry has applications to 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 vital for navigation. Hyperbolic geometry has modern applications to the theory of special relativity in Physics. Global Riemannian Geometry generalizes these three geometries. One of the major challenges in this area of study is to understand how local geometric invariants such as curvature, that is, how the space under consideration "bends", 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. One relatively recent 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 project will also continue the PI's outreach work with middle and high school students, as well as graduate training, and the organization of workshops and conferences with an emphasis on the inclusion of women and under-represented groups.<br/><br/>The project will pursue a program in which she carefully studies and analyzes symmetries of Riemannian manifolds with lower curvature bounds, considering sectional, Ricci, scalar, and intermediate scalar curvature lower bounds and some of their corresponding generalizations to Alexandrov spaces, with an eye to gaining a deeper understanding of this largely unknown class of spaces. The project will study not only how continuous and discrete symmetries relate to the topology of such spaces, but also aim 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. The project also includes training and mentoring of students as well as conference and workshop organization with an emphasis on inclusivity.<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.
