Requests for Financial Support: April 15, 2016
Requests for On-Campus Housing: May 18, 2016
The featured speaker this year is Nathan Dunfield, Associate Professor at the University of Illinois at Urbana-Champaign. He will give a series of three one hour lectures.
Titles and Abstracts for Principal Lectures
Lecture 1: Fun with finite covers of 3-manifolds: connections between topology, geometry, and arithmetic.
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.
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
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.
Participants are invited to contribute a twenty to thirty minute talk. Abstracts may be provided on the registration form or e-mailed directly to Fred Tinsley (firstname.lastname@example.org).
Financial support is available to help defer the travel and living expenses of participants who do not have other funding for their research. Such support can be requested on the registration form. Requests for support must be received by April 8, 2016. Graduate students and recent PhDs in geometric topology are especially encouraged to apply.
The workshop will be supported by a grant from the National Science Foundation (DMS-1461385), the Office of the Dean at Colorado College, and the Department of Mathematics and Computer Science at Colorado College.
To defray local costs, a $25 registration fee will be collected on site.
Fredric Ancel, University of Wisconsin-Milwaukee
Greg Friedman, Texas Christian University
Craig Guilbault, University of Wisconsin-Milwaukee
Eric Swenson, Brigham Young University
Frederick Tinsley, Colorado College
Gerard Venema, Calvin College
Contact Fred Tinsley (email@example.com) if you have questions about the workshop or comments on this web site.