Featured presentations from notable and upcoming researchers in the field.
The Borel conjecture predicts that closed, aspherical manifolds (i.e., those with contractible universal cover) are topologically rigid: they are determined up to homeomorphism by their fundamental group. I will discuss the smooth version of this conjecture (concerning manifolds up to diffeomorphism), which is true in dimensions ≤ 3 but long known to be false in all dimensions ≥ 5. I will explain joint work with Davis, Huang, Ruberman, and Sunukjian that resolves the remaining 4-dimensional case by detecting exotic smooth structures on certain closed aspherical 4-manifolds.
View Submission
The most basic Riemannian manifolds are those admitting a complete metric of constant curvature. The classification of closed manifolds with metrics of positive and zero curvature is has been relatively well-understood for quite a long time. Constant curvature -1 manifolds, hyperbolic manifolds, remain quite a bit more mysterious, particularly in high dimensions. I will give a (biased) narrative regarding what we know, including a number of exciting recent results with connections to dynamics and geometric group theory, and look forward to some problems I hope to see solved in the coming years.
View Submission
The Zimmer program seeks to classify smooth actions of arithmetic groups, like SL(n,Z), on compact manifolds. Separately, the Nielsen realization problem asks when a subgroup of a mapping class group Mod(M) can be realized by a group of diffeomorphisms of M. In many natural situations, the mapping class group is closely related to an arithmetic group, and the realization problem is tied to the Zimmer program. I will discuss examples of this connection and describe some recent results and open questions.
View Submission