Past Topology Seminar

1 June 2020
15:45
Benson Farb
Abstract

In some ways the theory of mapping class groups of 4-manifolds is in 2020 at the same place where the theory of mapping class groups of 2-manifolds was in 1973, before Thurston changed everything.  In this talk I will describe some first steps in an ongoing joint project with Eduard Looijenga where we are trying to understand mapping class groups of certain algebraic surfaces (e.g. rational elliptic surfaces, and also K3 surfaces) from the Thurstonian point of view.

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18 May 2020
15:45
Abstract

Just as differential equations often boundary conditions of various types, so too do quantum field theories often admit boundary theories. I will explain these notions and then discuss a theorem proved with Constantin Teleman, essentially characterizing certain 3-dimensional topological field theories which admit nonzero boundary theories. One application is to gapped systems in condensed matter physics.

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11 May 2020
15:45
Irena Matkovic
Abstract

Tight contact structures on knot complements arise both from Legendrian realizations of the knot in the standard tight contact structure and from the non-loose Legendrian realizations in the overtwisted structures on the sphere. In this talk, we will deal with negative torus knots. We wish to concentrate on the relations between these various Legendrian realizations of a knot and the contact structures on the surgeries along the knot. In particular, we will build every contact structure by a single Legendrian surgery, and relate the knot properties to the properties of surgeries; namely, tightness, fillability and non-vanishing Heegaard Floer invariant.

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4 May 2020
15:45
Ian Agol
Abstract

Addressing a question of Baker and Reid,

we give a criterion to show that an arithmetic group 

has a congruence subgroup that is algebraically

fibered. Some examples to which the criterion applies

include a hyperbolic 4-manifold group containing infinitely

many Bianchi groups, and a complex hyperbolic surface group.

This is joint work with Matthew Stover.

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27 April 2020
15:45
Martin Palmer
Abstract

For a bundle E over a manifold M, the associated "configuration-section spaces" are spaces of configurations of points in M together with a section of E over the complement of the configuration. One often considers subspaces where the behaviour of the section near a configuration point -- a kind of "monodromy" -- is restricted or prescribed. These are examples of "non-local configuration spaces", and may be interpreted physically as moduli spaces of "fields with prescribed singularities" in an ambient manifold.

An important class of examples is given by Hurwitz spaces, which are moduli spaces of branched G-coverings of the 2-disc, and which are homotopy equivalent to certain configuration-section spaces on the 2-disc. Ellenberg, Venkatesh and Westerland proved that, under certain conditions, Hurwitz spaces are (rationally) homologically stable; from this they then deduced an asymptotic version of the Cohen-Lenstra conjecture for function fields, a purely number-theoretical result.

We will discuss another homological stability result for configuration-section spaces, which holds (with integral coefficients) whenever the base manifold M is connected and open. We will also show that the "stabilisation maps" are split-injective (in all degrees) whenever dim(M) is at least 3 and M is either simply-connected or its handle dimension is at most dim(M) - 2.

This represents joint work with Ulrike Tillmann.

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9 March 2020
15:45
Radhika Gupta
Abstract

The mapping class group of a surface is associated to its Teichmüller space. In turn, its boundary consists of projective measured laminations. Similarly, the group of outer automorphisms of a free group is associated to its Outer space. Now the boundary contains equivalence classes of arationaltrees as a subset. There exist distinct projective measured laminations that have the same underlying geodesic lamination, which is also minimal and filling. Such geodesic laminations are called `non-uniquely ergodic'. I will talk briefly about laminations on surfaces and then present a construction of non-uniquely ergodic phenomenon for arational trees. This is joint work with Mladen Bestvina and Jing Tao.

2 March 2020
15:45
Hannah Schwartz
Abstract

We will first construct pairs of homotopic 2-spheres smoothly embedded in a 4-manifold that are smoothly equivalent (via an ambient diffeomorphism preserving homology) but not even topologically isotopic. Indeed, these examples show that Gabai's recent "4D Lightbulb Theorem" does not hold without the 2-torsion hypothesis. We will proceed to discuss two distinct ways of obstructing such an isotopy, as well as related invariants which can be used to obstruct an isotopy between pairs of properly embedded disks (rather than spheres) in a 4-manifold.

17 February 2020
15:45
David Hume
Abstract


Given two metric spaces $X$ and $Y$, it is natural to ask how faithfully, from the point of view of the metric, one can embed $X$ into $Y$. One way of making this precise is asking whether there exists a coarse embedding of $X$ into $Y$. Positive results are plentiful and diverse, from Assouad's embedding theorem for doubling metric spaces to the elementary fact that any finitely generated subgroup of a finitely generated group is coarsely embedded with respect to word metrics. Moreover, the consequences of admitting a coarse embedding into a sufficiently nice space can be very strong. By contrast, there are few invariants which provide obstructions to coarse embeddings, leaving many seemingly elementary geometric questions open.
I will present new families of invariants which resolve some of these questions. Highlights of the talk include a new algebraic dichotomy for connected unimodular Lie groups, and a method of calculating a lower bound on the conformal dimension of a compact Ahlfors-regular metric space.
 

10 February 2020
15:45
Abstract

Witten-Reshetikhin-Turaev quantum invariants of links and 3 dimensional manifolds are obtained from quantum sl(2). There exist different versions of quantum sl(2) leading to other families of invariants. We will briefly overview the original construction and then discuss two variants. First one, so called unrolled quantum sl(2), allows construction of invariants of 3-manifolds involving C* flat connections. In simplest case it recovers Reidemeister torsion. The second one is the non restricted version at a root of unity. It enables construction of invariants of links equipped with a gauge class of SL(2,C) flat connection. This is based respectively on joined work with Costantino, Geer, Patureau and Geer, Patureau, Reshetikhin.

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