The 33rd Annual Workshop in Geometric Topology will take place at Colorado College in Colorado Springs, CO. All talks will take place in the Kresge Lecture Hall in Tutt Science Center Room 122. (Click here for a campus map). The lecture hall is equipped with chalkboards and whiteboards, an overhead projector, and projector.
A tentative schedule for the conference can be found here: ConferenceSchedule
Principal Lectures by Nathan Dunfield
(Click on the Lecture titles below to download and watch the full versions of the lectures.)
Lecture 1: Fun with finite covers of 3-manifolds: connections between topology, geometry, and arithmetic.
Abstract: Following 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 recent 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.
Lecture 2: A tale of two norms.
Abstract: The first cohomology of a hyperbolic 3-manifold has two natural norms:
the Thurston norm, which measure topological complexity of surfaces representing the dual homology class, and the harmonic norm, which is just the L^2 norm on the corresponding space of harmonic 1-forms.
Bergeron-Sengun-Venkatesh recently showed that these two norms are closely related, at least when the injectivity radius is bounded below.
Their work was motivated by the connection of the harmonic norm to the Ray-Singer analytic torsion and issues of torsion growth discussed in the first talk. After carefully introducing both norms and the connection to torsion growth, I will discuss new results that refine and clarify the precise relationship between them; one tool here will be a third norm based on least-area surfaces. This is joint work with Jeff Brock.
Lecture 3: Floer homology, group orders, and taut foliations of hyperbolic 3-manifolds
Abstract: A bold conjecture of Boyer-Gorden-Watson and others posit that for any irreducible rational homology 3-sphere M the following three conditions are equivalent: (1) the fundamental group of M is left-orderable, (2) M has non-minimal Heegaard Floer homology, and (3) M admits a co-orientable taut foliation. Very recently, this conjecture was established for all graph manifolds by the combined work of Boyer-Clay and Hanselman-Rasmussen-Rasmussen-Watson. I will discuss a computational survey of these properties involving several hundred thousand hyperbolic 3-manifolds.
Twenty Minute Contributed Talks
Please click the following link for the collection of abstracts for the contributed talks: Abstracts