Women in Math Colloquium Series
The classification of compact Riemannian manifolds with positive or non-negative sectional curvature is a long-standing problem in Riemannian geometry. In particular, restricting our attention to closed, simply-connected manifolds, there are no topological obstructions that allow us to distinguish between positive and non-negative curvature, that is, we have no examples of manifolds that admit a metric of non-negative curvature that do not admit a metric of positive curvature. However, with the introduction of symmetries, we are able to distinguish between these two classes. A natural first step in this program is to assume the existence of a large isometric group action, with the definition of large being purposefully ambiguous. Obviously, large dimension with respect to that of the manifold or large rank are two possibilities. Another way to view large is in terms of the size of the largest isotropy subgroup of the action. In particular, for a rank k torus action on an n-dimensional manifold, the largest possible isotropy subgroup is of dimension n — k. For an action with an orbit whose isotropy subgroup is of dimension n — k (respectively, n — k — 1), we say it is isotropy-maximal (respectively, almost isotropy-maximal). In this talk, I'll discuss some recent joint work with Zheting Dong and Christine Escher concerning isotropy-maximal and almost isotropy-maximal manifolds of non-negative curvature.
Funded by NSF DMS-1844267
Colloquium Tea at 3pm in Math Lounge (3 MLH)