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Sam Ballas


Speaker: Sam Ballas
Title: Geometric Structures on Manifolds
Affiliation: University of California Santa Barbara
Date: Friday, January 15, 2016
Place and Time: Room 101, Love Building, 3:35-4:30 pm
Refreshments: Room 204, Love Building, 3:00 pm

Abstract. A classical problem in the interplay between geometry and topology is to determine with what types of geometry a fixed manifold can be endowed. The case of surfaces goes back to the late 19th and early 20th centuries through the work of Riemann, Klein, Poincare, and others. One of the seminal results in this area is that every closed surface admits exactly one of three types of homogeneous Riemannian structures that depends only on the sign of its Euler characteristic. There is an analogous result for 3-manifolds, conjectured by Thurston in the 70's and proven by Perelman in 2003. However, the statement is not as simple as in dimension 2, as it requires cutting the manifold into pieces, each of which admits a homogenous Riemannian structure. In this talk we will survey the development from the field as well as describe recent results allowing one to "geometrize" certain 3-manifolds without the need to cut them into pieces.