Sam Ballas
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MATHEMATICS COLLOQUIUM
Speaker: Sam Ballas 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. |