Abstract

From the revolutionary work of Thurston and Perelman, we know 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.  After a passing wave at the breakthrough results of Agol, I will focus on exciting new questions about the geometric and arithmetic meaning of torsion in the homology of finite covers of hyperbolic 3-manifolds, motivated by the recent work of Bergeron, Venkatesh, Le, and others. I will include some of my own results in this area that are joint work with F. Calegari and J. Brock.

About the speaker

Dunfield is a professor of mathematics at the University of Illinois at Urbana–Champaign. He obtained his Ph.D. from the University of Chicago in 1999.  He then was a Benjamin Peirce Assistant Professor at Harvard University (1999–2003) and an associate professor at the California Institute of Technology (2003–2007). After that, he moved to U. Illinois, where he was promoted to professor in 2018.