\( \newcommand{\Sp}{\operatorname{Sp}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\GL}{\operatorname{GL}} \newcommand{\Sol}{\operatorname{Sol}} \newcommand{\PGL}{\operatorname{PGL}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\SU}{\operatorname{SU}} \newcommand{\PU}{\operatorname{PU}} \newcommand{\Pin}{\operatorname{Pin}} \newcommand{\Vol}{\operatorname{Vol}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\CAT}{\operatorname{CAT}} \newcommand{\Out}{\operatorname{Out}} \newcommand{\vcd}{\operatorname{vcd}} \newcommand{\Id}{\operatorname{Id}} \newcommand{\Flat}{\operatorname{Flat}} \newcommand{\CP}{\mathbb{CP}} \newcommand{\CC}{\mathbb{C}} \newcommand{\EE}{\mathbb{E}} \newcommand{\HH}{\mathbb{H}} \newcommand{\RR}{\mathbb{R}} \newcommand{\QQ}{\mathbb{Q}} \newcommand{\ZZ}{\mathbb{Z}} \newcommand{\calG}{\mathcal{G}} \newcommand{\calO}{\mathcal{O}} \newcommand{\from}{\colon} \newcommand{\cross}{\times} \newcommand{\group}[2]{\langle #1 \mid #2 \rangle} \)

Geometry and Topology Online

Warwick Mathematics Institute

International Centre for Mathematical Sciences

Summer, 2020

Please contact Saul Schleimer if you would like to speak or to suggest a speaker. Please contact Gillian Kerr if you have questions about using Zoom. The ICMS page for the seminar has a time zone calculator, a link to the slack channel, links to the videos, and other invariant information.

While this page is the main page for the seminar, I will attempt to also maintain an up-to-date listing at researchseminars.org.

The seminar will run weekly, with two 25 minute talks separated by a five minute break. The first talk will start on the hour, and the second on the half hour. We'll open the Zoom session 15 minutes before we start; we'll close the Zoom session about 30 minutes after we finish. Note that no password is required; links to the zoom session for each talk are below.

Tuesday July 21, 16:00 (UK time).

Zoom meeting Slack channel Slides Video (TBA)

Michael Landry (WUSTL)

Faces of the Thurston norm ball up to isotopy

Abstract: Let \(M\) be a three-manifold with nondegenerate Thurston norm \(x\) on its second homology. There is a partial dictionary between the combinatorics of the polyhedral unit ball of \(x\) and the topological features of \(M\). This dictionary is quite incomplete, but its existing entries are tantalizing.

Currently, most of the entries of this dictionary concern fibered faces of the unit ball. Thurston proved that these organize all fibrations of \(M\) over the circle. Fried and Mosher tell us more: for each fibered face \(F\) there is a (canonical) pseudo-Anosov flow whose Euler class computes the norm \(x\) in the cone over \(F\). Furthermore, the flow "sees" certain least-complexity surfaces. Further work of Mosher shows that, under certain conditions, pseudo-Anosov flows can naturally specify nonfibered faces of the unit ball.

After giving some of this background we will discuss results from my recent preprint (https://arxiv.org/abs/2006.16328). We show that Agol's veering triangulations can be used to determine faces of Thurston norm balls, to compute the Thurston norm over those faces, and to collate all isotopy classes of least-complexity surfaces over those faces. This analysis includes nonfibered faces.

Tuesday July 28, 16:00 (UK time).

Zoom meeting Slack channel Slides Video (TBA) Paper

Rich Schwartz (Brown)

The spheres of \(\Sol\)

Abstract: We give a complete characterization of the cut locus of the identity in \(\Sol\), one of the strangest of the eight Thurston geometries. As a corollary we prove that the metric spheres in \(\Sol\) are in fact topological spheres.

This is joint work with Matei Coiculescu.

Tuesday July 28, 16:30 (UK time).

Zoom meeting Slack channel Slides Video (TBA) Paper

Scott Taylor (Colby)

Equivariant Heegaard genus of reducible three-manifolds

Abstract: Suppose that \(M\) is a closed, connected, oriented three-manifold which comes with a group action by a finite group of (orientation preserving) diffeomorphisms. The equivariant Heegaard genus of \(M\) is then the minimal genus of an equivariant Heegaard surface. The equivariant sphere theorem, together with recent work of Scharlemann, suggests that equivariant Heegaard genus might be additive under equivariant connected sum, while analogies with tunnel number suggest it should not be.

We will describe some examples showing that equivariant Heegaard genus can be sub-additive, additive, or super-additive. Building on recent work with Tomova, We’ll sketch machinery that gives rise both to sharp bounds on the addivity of equivariant Heegaard genus and to a closely related invariant that is in fact additive.

Tuesday August 4, 16:00 (UK time).

Zoom meeting Slack channel Slides Video (TBA) Paper

Marissa Loving (Georgia Tech)

Covers and curves

Abstract: It is a celebrated result of Scott that every closed curve on a hyperbolic surface \(S\) lifts to a simple closed curve on some finite cover. In the spirit of this work we pose the following question: "What information about two covers \(X\) and \(Y\) of \(S\) can be derived by understanding how curves on \(S\) lift simply to \(X\) and \(Y\)?" In this talk, we will explore the answer to this question for regular finite covers of a closed hyperbolic surface.

This is joint work with Tarik Aougab, Max Lahn, and Yang (Sunny) Xiao.

Tuesday August 4, 16:30 (UK time).

Zoom meeting Slack channel Slides Video (TBA)

Sarah Dean Rasmussen (Cambridge)

Taut foliations from left orders, in Heegaard genus two

Abstract: Suppose that \(M\) is a closed, connected, oriented three-manifold which is not graph. All previously known constructions of taut foliations on such \(M\) used branched surfaces. These branched surfaces come from sutured manifold hierarchies, following Gabai, come from spanning surfaces of knot exteriors, following Roberts, or come from one-vertex triangulations with foliar orientations, following Dunfield.

In this talk, we give a new construction that does not use branched surfaces. Instead, we build a taut foliation from the data of a Heegaard diagram for \(M\) and a left order on the fundamental group \(\pi_1(M)\). We glue an \(\RR\)-transverse foliation (over a thickened Heegaard surface) to a pair of handlebody foliations; we then suitably cancel any singularities. For Heegaard diagrams satisfying mild conditions, this can be done reliably in Heegaard genus two. In some cases this construction can be extended to higher Heegaard genus. This helps explain numerical results of Dunfield: (i) tens of thousands of Heegaard-genus two hyperbolic L-spaces certifiably fail to admit fundamental group left orders and (ii) no hyperbolic L-space is known to admit a fundamental group left order.

Tuesday August 11, 16:00 (UK time).

Zoom meeting Slack channel Slides Video (TBA) Paper Paper

Andras Stipsicz (Renyi)

Connected Floer homology of covering involutions

Abstract: We use the covering involution of double branched covers of knots to define a knot invariant inspired by connected Heegaard Floer homology. Using this, we obtain novel concordance results.

This is joint work with Antonio Alfieri and Sungkyung Kang.

Tuesday August 18, 16:00 (UK time).

Zoom meeting Slack channel Slides Video (TBA)

Kasia Jankiewicz (Chicago)

Generalized Tits conjecture for Artin groups

Abstract: The Tits conjecture, proved by Crisp and Paris, states that the subgroup of an Artin group generated by powers of the standard generators is the "obvious" right-angled Artin group (RAAG). We aim to generalize this: the subgroup generated by a collection of naturally distinguished elements, specifically powers of the Garside elements, is a RAAG. I will discuss our partial results, for certain families of Artin groups.

This is joint work with Kevin Schreve.

Tuesday August 18, 16:30 (UK time).

Zoom meeting Slack channel Slides Video (TBA)

Jing Tao (Oklahoma)

The Nielsen-Thurston classification, revisited

Abstract: I will explain a new proof of the Nielsen-Thurston classification of mapping classes, using the Thurston metric on Teichmuller space.

This is joint work with Camille Horbez.

Tuesday August 25, 16:30 (UK time).

Zoom meeting Slack channel Slides (TBA) Video (TBA)

Dawid Kielak (Oxford)

Poincaré duality groups

Abstract: It is a classical fact that a Poincaré duality group, in dimension two, is a surface group. In this talk I will discuss a relatively short new proof of this.

This is joint work with Peter Kropholler.

Information on past talks. This line was last edited on Fri 14 Aug 2020 08:50:54 BST