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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.

Geometry of Teichmüller space and mapping class groups
11 April 2018

Abstract: I will discuss work in progress with Dowdall, Leininger, and Sisto, in which we aim to develop a notion of geometrical finiteness for subgroups of mapping class groups. Motivated by the theory of convex cocompact subgroups, which are precisely those which determine hyperbolic surface group extensions, I will describe some hyperbolic properties of the surface group extensions coming from lattice Veech subgroups.

Geometry of Teichmüller space and mapping class groups
10 April 2018

Abstract: In this talk, we will consider cusp excursions of random geodesics for Weil--Petersson type metrics on orientable surfaces of finite type. In particular, we will explain how to derive bounds for the size of the "maximal" excursions. Furthermore, assuming polynomial mixing for the Weil--Petersson flow on non-exceptional moduli spaces, we will give similar bounds in the general setting. Our methods rely on deriving effective ergodic average for mixing flows. This is joint work with Carlos Matheus.

Graphs, surfaces, and cube complexes
9 July 2018

Abstract: The talk is based on a joint work with Jingyin Huang (MPIM Bonn). We show that two-dimensional Artin groups act geometrically on metrically systolic complexes. The latter are simply connected flag simplicial complexes with links satisfying some non-positive-curvature-like condition. As corollaries we obtain new results on two-dimensional Artin groups and all their finitely presented subgroups: the Dehn function is quadratic, the Conjugacy Problem is solvable. Furthermore, metric systolicity is used by us for studying quasi-isometric rigidity of such groups. In the particular case of large-type Artin groups the results are strengthened - such groups are systolic. In that case we conclude many other new properties, e.g. biautomaticity, existence of an EZ-boundary.

Computation in geometric topology
13 December 2017

Abstract: Demonstration of the software Regina.

Geometry of Teichmüller space and mapping class groups
13 April 2018

Abstract: A recent breakthrough of Ershov-He shows that the Johnson kernel subgroup of the mapping class group is finitely generated for g at least 12. In joint work with Ershov and Church, I have extended this to show that every term of the lower central series of the Torelli group is finitely generated once the genus is sufficiently large. A byproduct of our work is a proof that the Johnson kernel is finitely generated for g at least 4 which is remarkably simple (so simple, in fact, that I will be able to give in nearly complete detail in this talk).

Computation in geometric topology
15 December 2017

Abstract: Instead of the usual focus on time complexity, here we discuss the space complexity of topological algorithms. We explain how to test 2-manifolds for homeomorphism in logspace (joint with Elder, Kalka and Tillmann), and we prove a variety of positive and negative results for 3-dimensional problems on finite state tree automata (joint with Fellows).

Geometry of Teichmüller space and mapping class groups
12 April 2018

Abstract: We study pseudo-Anosov mapping classes with bounded normalized Weil-Petersson translation distance (and unbounded genus). In analogy with a result of Farb-Leininger-Margalit for Teichmüller translation distances, we show all such mapping classes fit together into a finite collection of cusped hyperbolic 3-manifolds, where the cusps are filled to become either vertical (transverse to fibers) or horizontal (parallel to fibers). After a reduction using work of Schlenker, Kojima-McShane and Brock-Bromberg, the argument uses the theory of branched surfaces in 3-manifolds. This is joint work with Chris Leininger, Juan Souto and Sam Taylor.

Computation in geometric topology
14 December 2017

Abstract: While the search for efficient algorithms to solve three-dimensional topological problems has been very active in the past decades, results on computational hardness have been comparatively very scarce. In this talk, we will present two NP-hardness proofs for problems on embeddings into three-manifolds. The first one is about finding an embedded non-orientable surface of Euler genus g in a triangulated three-manifold, and the second one is about deciding whether a two- or three-dimensional complex embeds into R^3. This hardness stands in stark contrast with the lower dimensional cases which can be solved in linear time, and a variety of computational problems in S^3 like unknot or three-sphere recognition which are in NP and co-NP (assuming the generalized Riemann hypothesis). This is based on joint work with Benjamin Burton, Yo'av Rieck, Eric Sedgwick, Martin Tancer and Uli Wagner.