The PATCH seminar, a joint venture with Bryn Mawr College, Haverford College, and the University of Pennsylvania, has its first meeting of Spring 2012 on Tuesday, February 28 at Temple. The two speakers are Morwen Thistlethwaite of the University of Tennessee and Nancy Hingston of The College of New Jersey and IAS.
Thistlethwaite will speak at 4:15 PM, on
Title: Finding and deforming representations of 3-manifold groups
Abstract: Some assorted methods are described for finding exact specifications of representations of 3-manifold groups into classical matrix groups. These include (i) a method for finding hyperbolic structures on links that does not involve an ideal triangulation of the link complement, and (ii) deformations away from the hyperbolic structure of certain closed hyperbolic 3-manifolds.
Hingston will speak at 5:30, on
Title: Loop products and dynamics
Abstract: A metric on a compact manifold M gives rise to a length function on the free loop space LM whose critical points are the closed geodesics on M in the given metric. Morse theory gives a link between Hamiltonian dynamics and the topology of loop spaces, between iteration of closed geodesics and the algebraic structure given by the Chas-Sullivan product on the homology of LM. Geometry reveals the existence of a related product on the cohomology of LM. A number of known results on the existence of closed geodesics are naturally expressed in terms of nilpotence of products. We use products to prove a resonance result for the loop homology of spheres. I will not assume any prior knowledge of loop products. Mark Goresky, Hans-Bert Rademacher, and (work in progress) Ralph Cohen and Nathalie Wahl are collaborators.