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Geometry of Teichmüller space and mapping class groups
12 April 2018

Abstract: Veering triangulations were introduced by Agol as canonical ideal triangulations of pseudo-Anosov mapping tori which are punctured along singular orbits. Since these manifolds are hyperbolic, Agol asked whether veering triangulations are always geometric. That is, can the triangulation be realized in the hyperbolic metric with each tetrahedron positively oriented? Unfortunately, the answer is no since Hodgson, Issa, and Segerman subsequently produced examples of nongeometric veering triangulations. In this talk, I’ll discuss joint work with Futer and Worden that explains why this is generally the case.

Teichmuller Dynamics
21 March 2018

Abstract: In this talk, we discuss an extension of the results of Marmi-Moussa-Yoccoz on local conjugacy classes of interval exchange transformations of restricted Roh type. In particular, we answer a question of Krikorian about the codimension of the local conjugacy class of self-similar interval exchange maps associated to the Eierlegende Wollmilchsau and the Ornithorynque.
This is a joint work with G. Forni and S. Marmi.

Geometry of Teichmüller space and mapping class groups
10 April 2018

Abstract: Hierarchically hyperbolic spaces provide a uniform framework for working with many important examples, including mapping class groups, right angled Artin groups, Teichmüller space, and others. In this talk I'll provide an introduction to studying groups and spaces from this point of view. This discussion will center around recent work in which we classify quasiflats in these spaces, thereby resolving a number of well-known questions and conjectures. This is joint work with Mark Hagen and Alessandro Sisto.

Yair Minsky Short Course
14 March 2018

Abstract: We will discuss a number of aspects of the geometry of the Teichmüller space of Riemann surfaces: its asymptotic geometry, as seen through Thurston's compactification by measured laminations; its coarse geometry, as captured by the relative hyperbolicity properties of the complex of curves; and the dynamical geometry of its geodesic flows. The interactions between these points of view are important for understanding the different roles Teichmüller space plays, for example in the structure of hyperbolic 3-manifolds. After an overview of these topics we will spend some time talking about the Weil-Petersson geodesic flow, which remains a delicate and difficult to understand part of the theory, and where we are still left with more questions than answers.

Graphs, surfaces, and cube complexes
9 July 2018

Abstract: In the 90s, Sageev showed how to build an action on a cube complexfrom a collection of codimension one subgroups of a group G. We consider the case that the group G is hyperbolic, and show that vertex stabilizers are quasiconvex if and only if the codimension one subgroups are. We also give conditions under which the action can be promoted to a proper cocompact action.
This is joint work with Daniel Groves.

Geometry of Teichmüller space and mapping class groups
9 April 2018

Abstract: We show that the axis of a pseudo-Anosov homeomorphism in the mapping class group may not have the strong contractibility property. Specifically, we show that, after choosing a generating set carefully, one can find a pseudo-Anosov homeomorphism f, a sequence of points x_k and a sequence of radii R_k so that the ball B(x_k, R_k) is disjoint from the axis of f, but the closest point projection of B(x_k, R_k) to the axis of f has a diameter at least c log(R_k). Along the way we show how, in certain situations, the distance between two points in the mapping class group can be computed precisely, not just up to a multiplicative error. This is a joint work in Yvon Verberne.

Teichmuller Dynamics
21 March 2018

Full title: A Gap Theorem (Steinhaus Conjecture) for General Interval Exchange Transformations via Zippered Rectangles

Abstract: The Three Gap Theorem, also known as Steinhaus Conjecture, is a historic result asserting that for any α∈(0,1) and any integer N≥1, the fractional parts of the sequence 0,α,2α,⋯,(N−1)α partition the unit interval into N subintervals having at most three distinct lengths. We are presenting:

* a geometric interpretition in terms of sections to flows on surfaces that provides yet another proof of TGT

* a similar (and independent) approach by Alan Haynes, Jens Marklof, and Andreas Strombergsson to TGT that proves TGT and higher dimensional analogues (linear forms)

* a gap theorem for orbits of general interval exchanges maps with optimal bound

If there is time, we will touch on some applications of TGT that are of contemporary interest.