## Fun with finite covers of 3-manifolds: connections between topology, geometry, and arithmetic

Nathan Dunfield

University of Illinois, Urbana-Champaign

November 23, 2015

Following the revolutionary work of Thurston and Perelman, the topology of 3-manifolds is deeply intertwined with their geometry. In particular, hyperbolic geometry, the non-Euclidean geometry of constant negative curvature, plays a central role. In turn, hyperbolic geometry opens the door to applying tools from number theory, specifically automorphic forms, to what might seem like purely topological questions.