Abstract: The talk will feature several results by Grove-Petersen that address the structure of spaces with curvature ≥ 1 and radius ≥ pi/2.  The story starts with the Grove-Shiohama diameter sphere theorem, continues with Perel’man’s development of Alexandrov space geometry, and ends with some rigidity results when the space in question has boundary. A few conjectures will also be mentioned. 

Abstract: TBA

Abstract: I will discuss some recent and not so recent rigidity results in both geometry and dynamics, discuss their relationships and pose some open problems.  As one example, consider a symmetric space with sectional curvatures between 0 and 1. Then any another metric with the same extremal curvatures which agrees with the given one outside a proper convex subset is isometric to the symmetric one (joint work with Connell, Islam and Nguyen).  The arguments here are entirely geometric.  As a second example, consider a compact quotient X of quaternionic hyperbolic space and a perturbation g of its metric.  If g has positive hyperbolic rank, then g is isometric to X  (joint work with Connell and Nguyen).  Here dynamics and Carnot metrics enter in an essential way, via the geodesic flow. 

Abstract:  We will discuss the classification of compact simply connected five dimensional manifolds with non-negative sectional curvature, which admit an isometric action by a 2-torus. The classification is up to equivariant diffeomorphisms of the T^2 actions. This is joint work with Karsten Grove.