Thursday, August 1, 2019
08/01/2019 - 10:00am
In this talk I will define the notion of Generalized Tetrahedral Property which extends Sormani’s Tetrahedral Property. This definition retains all the results of the original TP proven by Portegies-Sormani: it provides a lower bound on the sliced filling volume and a lower bound on the volumes of balls. Hence, sequences with uniform GTP have subsequences which converge in both Gromov-Hausdorff and Sormani-Wenger Intrinsic Flat sense to the same non-collapsed metric space. Through some examples we will see that the main motivation to extend the TP is to include Euclidean cones over metric spaces with small diameter. (Joint work with Jesús Nuñez-Zimbrón)
08/01/2019 - 1:00pm
Intuition drawn from quantum mechanics and geometric optics raises the following long-standing problem: is the length spectrum of a closed Riemannian manifold encoded in its Laplace spectrum? That is, can you hear the length spectrum of a manifold? The answer to this question is known to be positive for a generic (i.e. sufficiently “bumpy”) manifold; however, little seems to be known in general. This is especially true for spaces with large symmetry groups. In this talk we’ll report on the progress we’ve made answering this question for symmetric spaces of the compact type and other homogeneous spaces.
08/01/2019 - 4:00pm
In a 1978 paper, Gromov explained how the Lipschitz constant of a map S^n -> Y, where Y is a simply connected compact Riemannian manifold, restricts its homotopy class. He later conjectured that the bounds obtained there are asymptotically sharp. For example, this would imply that if Y is a closed oriented n-manifold with a disk cut out, then one could always find a C(Y)L-Lipschitz map which is homotopic to L^n times the puncture, even though you can only get