|Oct. 24 and 31||Sam Lin
|Rank Rigidity in Dimension Three|
|Oct. 5, 4pm||Alexandre Girouard
|Discretization and spectrum of compact Riemannian manifolds with boundary|
|Sept. 26||Petra Bonfire-Taylor
(Dartmouth College, Thayer Engineering School)
|Quasiconformal Homogeneity, Active Learning and Programming|
|Sept. 20, 9am||Moon Duchin
|Discrete geometry, with applications to voting|
Fixing K=-1, 0, 1, a complete Riemannian manifold is said to have higher hyperbolic, Euclidean, or spherical rank if every geodesic admits a normal parallel field making curvature K with the geodesic. Examples for manifolds of higher rank includes locally symmetric spaces. The main goal of the study of rank rigidity is to show that with suitable assumptions, locally symmetric spaces are the only manifolds of higher rank.
Traditionally, rank rigidity has been studied in conjunction with the sectional curvature bounds. After giving a short survey of the historical results, I will talk about how rank rigidity holds in dimension three without a priori sectional curvature bounds.
October 5, 2017: Alexandre Girouard "Discretization and spectrum of compact Riemannian manifolds with boundary"
Abstract: In this talk I will introduce a notion of discretization which allows one to associate a graph to any compact Riemannian manifold with boundary. The eigenvalues of the Dirichlet-to-Neumann map on the manifold are then comparable to those of a natural spectral problem on its discretization. Applications to the construction of a sequence of surfaces with large eigenvalues will be discussed. This is a joint work with Bruno Colbois (Neuchâtel) and Binoy Raveendran.
September 26, 2017: Petra Bonfire-Taylor "Quasiconformal Homogeneity, Active Learning and Programming"
What do the seemingly distinct topics in the title have to do with one another? They are all tied together via the active learning piece. If you don’t know what active learning is, you’ll find out in this talk. Ditto for quasiconformal homogeneity. And while you probably know all about programming, I’ll demonstrate some novel tools we are developing at Thayer to facilitate active learning in a programming class.
Quasiconformal Homogeneity: A quasiconformal homeomorphism between domains is a mapping that behaves as much like a conformal mapping as possible in that, infinitesimally, it distorts spheres at worst into ellipsoids with bounded ratio between major and minor axes. A domain is quasiconformally homogeneous if any two of its points can be mapped onto one another via a quasiconformal homeomorphism of the domain to itself. I’ll speak about geometric and topological constraints that quasiconformal homogeneity imparts on domains (or, more generally, hyperbolic manifolds) and attempts to classify them. The audience will be involved in this presentation.
Abstract: Several specialties in mathematics and computer science are built around discrete or metric-space generalizations of classical geometry of manifolds. In math, comparison geometry (with roots in ideas of Alexandrov and others) has taken off in geometric group theory, proving very fruitful for the large-scale study of discrete groups. In CS, there's a burgeoning field of discrete differential geometry, which looks at geometry of meshes and builds up theory of curvature, Laplacians, and so on. I'll survey some ideas in these areas and will explain possible applications to electoral redistricting.