Workshop

I will introduce the audience to pseudo-Anosov flows and related structures on 3-manifolds, and present a simple proof of a theorem of Barbot that says such flows can be completely reconstructed from a group acting on the plane.  This is the setting for much of my recent joint work (with Barthelmé, Bonatti, Fenley and others); time permitting I will give some idea of this theory.  

Bieberbach's theorem states that a compact Euclidean manifold is virtually a torus. What happens if we relax the setting and ask which kinds of manifolds can arise as quotients of affine space by affine group actions? Margulis first found free groups acting affinely with manifold quotient on R^3, and the picture in three dimensions is now well-understood (a satisfying picture is given by Danciger-Guéritaud-Kassel), but remains more elusive in higher dimensions. We will discuss a small piece of the puzzle: given a positive representation of a free group in SO(2n,2n-1), we construct a large set of cocycles twisted by the representation that determine proper actions of the free group by affine transformations on R^(4n-1). We also describe fundamental domains for these actions, bounded by higher-dimensional versions of Drumm's crooked planes. This is joint work with Jean-Philippe Burelle.

The L-space conjecture makes a prediction about which rational homology spheres can admit a taut foliation. But where could the predicted taut foliations "come from"? Must they be compatible with “natural” geometric structures on the 3-manifold? In this talk, I'll discuss forthcoming work with John Baldwin and Matt Hedden, where we address a type of geography problem for taut foliations. In particular, we show that when K is a fibered strongly quasipositive knot, large surgeries along K can never admit a taut foliation which is ‘’compatible’’ with the natural flow on the Dehn surgered manifold. I'll explain why this is surprising, and if time permits, sketch the proof. No background will be assumed — all are welcome!

The so-called Problem of Compact Quotients asks for which homogeneous spaces G/H there exists a discrete subgroup Gamma of G such that Gamma\G/H is a compact manifold. We will discuss two methods for addressing this problem: a geometric one making links with Anosov representations, and a topological one based on weak triviality properties of certain sphere bundles. This is joint work with Nicolas Tholozan and Yosuke Morita.

A conjecture of Ghys asserts that any two transitive Anosov flows with orientable invariant foliations are almost orbit equivalent. In this talk, I will outline a connection between pseudo-Anosov flows and veering triangulations, and use it to reformulate Ghys’ conjecture in terms of properties of the veering ancestry graph. I will then discuss an algorithm whose implementation can be used to construct large subgraphs of this graph, and conclude by presenting the experimental data obtained.
This is joint work with Henry Segerman.

In dynamics, the speed of mixing depends on the dynamical features of the map and the regularity of the observables. Notably, two classical linear models—the Bernoulli doubling map and the CAT map—exhibit double exponential mixing for analytic observables. Are ergodic linear maps the only ones with this property? In dimension one, we provide a full classification for maps from the space of volume-preserving finite Blaschke products acting on the circle (as well as for free semigroup actions generated by a finite collection of such maps). In higher dimensions, we identify a necessary condition for double exponential mixing and present several families of examples and non-examples. Key ideas of the proof involve the Koopman precomposition operator on spaces of hyperfunctions (elements of the dual space of analytic functions), which turns out to be non-self-adjoint, compact, and quasi-nilpotent, with spectrum reduced to zero.

It is natural to ask how many isotopy classes of embedded essential surfaces lie in a given 3-manifold. The first bounds on the number of such surfaces were exponential, using normal surfaces. More recently, by restricting to alternating link complements in 3-sphere, Hass, Thompson and Tsvietkova obtained polynomial bounds, but for a limited class of surfaces: closed and spanning ones. In this talk, we discuss how to complete the picture for classical alternating links, and how to extend these results to other classes of cusped 3-manifolds. We give explicit polynomial bounds on all embedded essential surfaces, closed or any boundary slope, orientable or non-orientable. Our 3-manifolds are complements of links with alternating diagrams on wide classes of surfaces in broad families of 3-manifolds. This includes all alternating links in 3-sphere as well as many non-alternating ones, alternating virtual knots, many toroidally alternating knots, and most Dehn fillings of such manifolds.This is joint work with A. Tsvietkova.