Thursday October 8, 15:00 (UK time).

Zoom meeting Slack channel Slides Video (TBA)

Mehdi Yazdi (Oxford)

The complexity of determining knot genus in a fixed three-manifold

Abstract: The genus of a knot in a three-manifold is defined to be the minimum genus of a compact, orientable surface bounding that knot, if such a surface exists. In particular a knot can be untangled if and only if it has genus zero. We consider the computational complexity of determining knot genus. Such problems have been studied by several mathematicians; among them are the works of Hass-Lagarias-Pippenger, Agol-Hass-Thurston, Agol and Lackenby. For a fixed three-manifold the knot genus problem asks, given a knot \(K\) and an integer \(g\), whether the genus of \(K\) is equal to \(g\). Marc Lackenby proved that the knot genus problem for the three-sphere lies in NP.

We prove that this can be generalised to any fixed, compact, orientable three-manifold, answering a question of Agol-Hass-Thurston from 2002.

This is joint work with Marc Lackenby.

Thursday October 8, 15:30 (UK time).

Zoom meeting Slack channel Slides Video (TBA)

David Gabai (Princeton)

The fully marked surface theorem

Abstract: In his seminal 1976 paper Bill Thurston observed that a closed leaf \(S\) of a foliation has Euler characteristic equal, up to sign, to the Euler class of the foliation evaluated on \([S]\), the homology class represented by \(S\). We give a converse for taut foliations: if the underlying manifold is hyperbolic and if the Euler class of a taut foliation \(F\) evaluated on \([S]\) equals, up to sign, the Euler characteristic of \(S\), then there exists another taut foliation \(F'\) such that \(S\) is homologous to a union of leaves and such that the plane field of \(F'\) is homotopic to that of \(F\). In particular, \(F\) and \(F'\) have the same Euler class.

In the same paper Thurston proved that taut foliations on closed hyperbolic three-manifolds have Euler class of norm at most one, and conjectured that, conversely, any integral cohomology class with norm equal to one is the Euler class of a taut foliation. Work of Yazdi, together with our main result, give a negative answer to Thurston's conjecture.

This is joint work with Mehdi Yazdi.

Thursday October 15, 15:00 (UK time).

Zoom meeting Slack channel Slides Video (TBA)

Esmee te Winkel (Warwick)

Knots in the curve graph

Abstract: By a famous theorem of Thurston the space \(\PML\) of projective (measured) laminations on a five-times punctured sphere is a three-sphere. An elementary example of a projective lamination is a simple closed geodesic with the counting measure. This defines a map from the set of curves to \(\PML\), which extends to an injective map from the curve graph to \(\PML\). The topology of the image of the curve graph in \(\PML\) and its complement were previously studied by Gabai.

In this talk we will introduce certain finite subgraphs of the curve graph of the five-times punctured sphere and determine whether their image in \(\PML\) is knotted.

Thursday October 15, 15:30 (UK time).

Zoom meeting Slack channel Slides Video (TBA)

Rudradip Biswas (Manchester)

Generation of unbounded derived categories of modules over groups in Kropholler's hierarchy

Abstract: For a group \(G\) in Kropholler's hierarchy and a commutative ring \(R\), we will go through some recently discovered generation properties of \(D(\Mod-RG)\) in terms of localising and colocalising subcategories. If time permits, we will try to include a few comments on how these generation properties shed some light on some deep properties of \(D(\Mod-RG)\) as a triangulated category.

Thursday October 22, 15:00 (UK time).

Zoom meeting Slack channel Slides Video (TBA)

Yair Minsky (Yale)

Veering triangulations and their polynomials

Abstract: McMullen introduced certain polynomials associated to fibered three-manifolds, which package together the dynamical data associated to all the fibrations in a given fibered face of Thurston's norm ball. Agol's veering triangulations provide a good setting in which similar invariants can be defined.

In this introduction to Sam's talk, I will review this background, explain the definition of the "veering Polynomial" and the "taut Polynomial", the relationship between them, and how they recover McMullen's polynomial in the fibered face.

This is joint work with Michael Landry and Sam Taylor.

Thursday October 22, 15:30 (UK time).

Zoom meeting Slack channel Slides Video (TBA)

Sam Taylor (Temple)

The veering polynomial, the flow graph, and the Thurston norm

Abstract: This is a continuation of Yair’s talk on the veering polynomial. Here we show how the veering polynomial can be constructed as the Perron polynomial of a certain combinatorially defined directed graph, which we call the flow graph. This perspective will allows us to relate our polynomial to a face \(F\) of the Thurston norm ball and to see that the cone over \(F\) is spanned by surfaces that are "carried" by the veering triangulation. We’ll also discuss criteria for when the face \(F\) is fibered.

This is joint work with Michael Landry and Yair Minsky.

Thursday October 29, 15:00 (UK time).

Zoom meeting Slack channel Notebook Video (TBA)

Mark Bell (Independent)

Computations in big mapping class groups

Abstract: We will take a brief look at some of the computations that are possible in big mapping class groups. In particular we will discuss the implementation of Bigger - a Python package which allows you to study and manipulate laminations and mapping classes on infinite-type surfaces.

Thursday November 5, 15:00 (UK time).

Zoom meeting Slack channel Slides Video (TBA)

Chenxi Wu (Rutgers)

Bounds on asymptotic translation length on free factor and free splitting complexes

Abstract: The free factor and free splitting complexes are analogies for the curve complex on surfaces. We found some upper bound on the asymptotic translation length on these complexes when the train track maps have homotopic mapping tori, analogous to an upper bound we found earlier in the setting of curve complexes.

This is joint work with Hyungryul Baik and Dongryul Kim.

Thursday November 5, 15:30 (UK time).

Zoom meeting Slack channel Slides Video (TBA)

Ivan Dynnikov (Steklov)

An algorithm to compare Legendrian knots

Abstract: We have worked out a general method to decide whether two given Legendrian knots are Legendrian equivalent. The method yields a formal algorithmic solution to the problem (with very high algorithmic complexity) and, in certain circumstances, allows one to distinguish Legendrian knots practically, including some cases in which the computation of any known algebraic invariant except for the two classical ones (Thurston--Bennequin's and Maslov's) is infeasible. We use this, in particular, to provide an example of an annulus embedded in the three-sphere and tangent to the contact structure along the whole boundary, such that the two connected components of the boundary are not equivalent as Legendrian knots.

The talk is based on joint works with Maxim Prasolov and Vladimir Shastin.

Thursday December 3, 15:00 (UK time).

Zoom meeting Slack channel Slides Video (TBA)

Tara Brendle (Glasgow)

The mapping class group of connect sums of \(S^2 \times S^1\)

Abstract: Let \(M_n\) denote the connect sum of \(n\) copies of \(S^2 \times S^1\). Laudenbach showed that the mapping class group \(\Mod(M_n)\) is an extension of the group \(\Out(F_n)\) by \((\ZZ/2)^n\), where the latter group is the "sphere twist" subgroup of \(\Mod(M_n)\).

We prove that this extension splits. In this talk, I will describe the splitting and discuss some simplifications of Laudenbach's original proof that arise from our techniques.

This is joint work with N. Broaddus and A. Putman.

Thursday December 3, 15:30 (UK time).

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

Ying Hu (UNO)

Euler classes of taut foliations on \(\QQ\)-homology spheres and Dehn fillings

Abstract: The Euler class of an oriented plane field over a three-manifold is a second cohomology class, which determines the plane field up to isomorphism. In this talk, we will discuss the Euler class of taut foliations on a \(\QQ\)-homology sphere. We view \(\QQ\)-homology spheres as Dehn fillings on knot manifolds and give necessary and sufficient conditions for the Euler class of taut foliations on such manifolds to vanish. We will also apply these results to study the orderability of three-manifold groups.

Thursday December 10, 15:00 (UK time).

Zoom meeting Slack channel Slides Video (TBA)

Ruth Charney (Brandeis)

Outer space for right-angled Artin groups

Abstract: Right-angled Artin groups (RAAGs) span a range of groups from free groups to free abelian groups. Thus, their (outer) automorphism groups range from \(\Out(F_n)\) to \(\GL(n,\ZZ)\). Automorphism groups of RAAGs have been well-studied over the past decade from a purely algebraic viewpoint. To allow for a more geometric approach, one needs to construct a contractible space with a proper action of the group.

In this pair of talks we will construct such a space, namely an analogue of Culler-Vogtmann’s Outer Space for arbitrary RAAGs.

This is joint work with Corey Bregman and Karen Vogtmann.

Thursday December 10, 15:30 (UK time).

Zoom meeting Slack channel Slides Video (TBA)

Corey Bregman (Brandeis)

Outer space for right-angled Artin groups

Abstract: Right-angled Artin groups (RAAGs) span a range of groups from free groups to free abelian groups. Thus, their (outer) automorphism groups range from \(\Out(F_n)\) to \(\GL(n,\ZZ)\). Automorphism groups of RAAGs have been well-studied over the past decade from a purely algebraic viewpoint. To allow for a more geometric approach, one needs to construct a contractible space with a proper action of the group.

In this pair of talks we will construct such a space, namely an analogue of Culler-Vogtmann’s Outer Space for arbitrary RAAGs.

This is joint work with Ruth Charney and Karen Vogtmann.