Past Topology Seminar

For mapping class groups of surfaces it is well-understood that their homology stability is closely related to the fact that they give rise to an infinite loop space. Indeed, they define an operad whose algebras group complete to infinite loop spaces.

In recent work with Basterra, Bobkova, Ponto and Yaekel we define operads with homology stability (OHS) more generally and prove that they are infinite loop space operads in the above sense. The strong homology stability results of Galatius and Randal-Williams for moduli spaces of manifolds can be used to construct examples of OHSs. As a consequence the map to K-theory defined by the action of the diffeomorphisms on the middle dimensional homology can be shown to be a map of infinite loop spaces.

Elliptic cohomology is a family of generalised cohomology theories
$Ell_E^*$ parametrised by an elliptic curve $E$ (over some ring $R$).
Just like many other cohomology theories, elliptic cohomology admits
equivariant versions. In this talk, I will recall an old conjectural
description of elliptic cohomology, due to G. Segal, S. Stolz and P.
Teichner. I will explain how that conjectural description led me to
suspect that there should exist a generalisation of equivariant
elliptic cohomology, where the group of equivariance gets replaced by
a fusion category. Finally, I will construct $C$-equivariant elliptic
cohomology when $C$ is a fusion category, and $R$ is a ring of
characteristc zero.

The cobordism hypothesis gives a correspondence between the
framed local topological field theories with values in C and a fully
dualizable objects in C.  Changing framing gives an O(n) action on the
space of local TFTs, and hence by the cobordism hypothesis it gives a
(homotopy coherent) action of O(n) on the space of fully dualizable
objects in C.  One example of this phenomenon is that O(3) acts on the
space of fusion categories.  In fact, O(3) acts on the larger space of
finite tensor categories.  I'll describe this action explicitly and
discuss its relationship to the double dual, Radford's theorem,
pivotal structures, and spherical structures.  This is part of work in
progress joint with Chris Douglas and Chris Schommer-Pries.
 

Given two actions of a group $G$ on trees $T_1,T_2$, Guirardel introduced the "core", a $G$--cocompact CAT(0) subspace of $T_1\times
T_2$.  The covolume of the core is a natural notion of "intersection number" for the two tree actions (for example, if $G$ is a surface group
and $T_1,T_2$ are Bass-Serre trees associated to splittings along some curves, this "intersection number" is the one you'd expect).  We
generalise this construction by considering a fixed finitely-presented group $G$ equipped with finitely many essential, cocompact actions on
CAT(0) cube complexes $X_1,...,X_d$.  Inside $X=X_1\times ... \times X_d$, we find a $G$--invariant subcomplex $C$ which, although not convex
or necessarily CAT(0), has each component isometrically embedded with respect to the $\ell_1$ metric on $X$ (the key point is this change from
the CAT(0) to the $\ell_1$ viewpoint).  In the case where $d=2$ and $X_1,X_2$ are simplicial trees, $C$ is the Guirardel core.  Many
features of the Guirardel core generalise, and I will summarise these. For example, if the cubulations $G\to Aut(X_i)$ are "essentially
different", then $C$ is connected and $G$--cocompact.  Time permitting, I will discuss an application, namely a new proof of Nielsen realisation
for finite subgroups of $Out(F_n)$.  This talk is based on ongoing joint work with Henry Wilton.

Large-scale homology computations are often rendered tractable by discrete Morse theory. Every discrete Morse function on a given cell complex X produces a Morse chain complex whose chain groups are spanned by critical cells and whose homology is isomorphic to that of X. However, the space-level information is typically lost because very little is known about how critical cells are attached to each other. In this talk, we discretize a beautiful construction of Cohen, Jones and Segal in order to completely recover the homotopy type of X from an overlaid discrete Morse function.

Coarse embeddings occur completely naturally in geometric group theory: every finitely generated subgroup of a finitely generated group is coarsely embedded. Since even very nice classes of groups - hyperbolic groups or right-angled Artin groups for example - are known to have 'wild' collections of subgroups, there are precious few invariants that one may use to prove a statement of the form '$H$ does not coarsely embed into $G$' for two finitely generated groups $G,H$.
The growth function and the asymptotic dimension are two coarse invariants which which have been extensively studied, and a more recent invariant is the separation profile of Benjamini-Schramm-Timar.

In this talk I will describe a new spectrum of coarse invariants, which include both the separation profile and the growth function, and can be used to tackle many interesting problems, for instance: Does there exist a coarse embedding of the Baumslag-Solitar group $BS(1,2)$ or the lamplighter group $\mathbb{Z}_2\wr\mathbb{Z}$ into a hyperbolic group?

This is part of an ongoing collaboration with John Mackay and Romain Tessera.
 

I will begin the talk by reviewing the definition of commutative K-theory, a generalized cohomology theory introduced by Adem and Gomez. It is a refinement of topological K-theory, where the transition functions of a vector bundle satisfy a commutativity condition. The theory is represented by an infinite loop space which is called a “classifying space for commutativity”.  I will describe the homotopy type of this infinite loop space. Then I will discuss the graded ring structure on its homotopy groups, which corresponds to the tensor product of vector bundles.
 

Configuration spaces of points in a manifold are well studied. Giving the points thickness has obvious physical meaning: the configuration space of non-overlapping particles is equivalent to the phase space, or energy landscape, of a hard spheres gas. But despite their intrinsic appeal, relatively little is known so far about the topology of such spaces. I will overview some recent work in this area, including a theorem with Yuliy Baryshnikov and Peter Bubenik that related the topology of these spaces to mechanically balanced, or jammed, configurations. I will also discuss work in progress with Robert MacPherson on hard disks in an infinite strip, where we understand the asymptotics of the Betti numbers as the number of disks tends to infinity. In the end, we see a kind of topological analogue of a liquid-gas phase transition.

(Joint work with Marco Golla and József Bodnár)
We will give a general overview of the plethora of concordance invariants which can be extracted from Ozsváth-Szabó-Rasmussen's knot Floer homology. 
We will then focus on the $\nu^+$ invariant and prove some of its useful properties. 
Furthermore we will show how it can be used to obstruct the existence of cobordisms between algebraic knots.

I will report on joint work with Michael Weiss (https://arxiv.org/pdf/1503.00498.pdf):

Let K be a subset of a smooth manifold M. In some cases, functor calculus methods lead to a homotopical formula for M \ K in terms of the spaces M \ S,  where S runs through the finite subsets of K. This is for example the case when K is a smooth compact sub manifold of co-dimension greater or equal to three.

A cornerstone of the study of mapping class groups is the
Nielsen--Thurston classification theorem. I will outline a
polynomial-time algorithm that determines the Nielsen--Thurston type and
the canonical curve system of a mapping class. Time permitting, I shall
describe a polynomial-time algorithm to compute the quotient orbifold of
a periodic mapping class, and I shall discuss the conjugacy problem for
the mapping class group. This is joint work with Mark Bell.

Decorate knot cobordisms functorially induce maps on knot Floer homology.
We compute these maps for elementary cobordisms, and hence give a formula for 
the Alexander and Maslov grading shifts. We also show a non-vanishing result in the case of
concordances and present some applications to invertible concordances. 
This is joint work with Marco Marengon.
 

We characterize cutting arcs on ber surfaces that produce new ber surfaces,
and the changes in monodromy resulting from such cuts. As a corollary, we
characterize band surgeries between bered links and introduce an operation called
generalized Hopf banding. We further characterize generalized crossing changes between
bered links, and the resulting changes in monodromy.

This is joint work with Matt Rathbun, Kai Ishihara and Koya Shimokawa

 In this introduction for topologists, we explain the role that extensions of L-infinity algebras by taking homotopy fibers plays in physics. This first appeared with the work of physicists D'Auria and Fre in 1982, but is beautifully captured by the "brane bouquet" of Fiorenza, Sati and Schreiber which shows how physical objects such as "strings", "D-branes" and "M-branes" can be classified by taking successive homotopy fibers of an especially simple L-infinity algebra called the "supertranslation algebra". We then conclude by describing our joint work with Schreiber where we build the brane bouquet out of the homotopy theory of an even simpler L-infinity algebra called the superpoint.

 
The dimension of a finite-dimensional vector space V can be computed as the trace of the identity endomorphism id_V.  This dimension is also the value F_V(S^1) of the circle in the 1-dimensional field theory F_V associated to the vector space.  The trace of any endomorphism f:V-->V can be interpreted as the value of that field theory on a circle with a defect point labeled by the endomorphism f.  This last invariant makes sense even when the vector space is infinite-dimensional, and gives the trace of a trace-class operator on Hilbert space.  We introduce a 2-dimensional analog of this invariant, the `2-trace'.  The 2-dimension of a finite-dimensional separable k-algebra A is the dimension of the center of the algebra.  This 2-dimension is also the value F_A(S^1 x S^1) of the torus in the 2-dimensional field theory F_A associated to the algebra. Given a 2-endomorphism p of the algebra (that is an element of the center), the 2-trace of p is the value of the field theory on a torus with a defect point labeled by p.  Generalizations of this invariant to other defect configurations make sense even when the algebra is not finite-dimensional or separable, and this leads to a general notion of 2-trace class and 2-trace in any 2-category.  This is joint work with Andre Henriques.

Let X be a complex algebraic variety containing a point x. One of the central ideas of deformation theory is that the local structure of X near the point x can be encoded by a differential graded Lie algebra. In this talk, Jacob Lurie will explain this idea and discuss some generalizations to more exotic contexts.

I am going to discuss Rips' conjecture that all finitely presented groups with quadratic Dehn functions have decidable conjugacy problem.

This is a joint work with A.Yu. Olshanskii.
 

In this talk I'll give a general presentation of my recent work that a purely loxodromic Kleinian group of Hausdorff dimension<1 is a classical Schottky group. This gives a complete classification of all Kleinian groups of dimension<1. The proof uses my earlier result on the classification of Kleinian groups of sufficiently small Hausdorff dimension. This result in conjunction to another work (joint with Anderson) provides a resolution to Bers uniformization conjecture. No prior knowledge on the subject is assumed.

We want to discuss a collection of results around the following Question: Given a smooth compact manifold $M$ without boundary, does $M$ admit a Riemannian metric of positive scalar curvature?

We focus on the case of spin manifolds. The spin structure, together with a chosen Riemannian metric, allows to construct a specific geometric differential operator, called Dirac operator. If the metric has positive scalar curvature, then 0 is not in the spectrum of this operator; this in turn implies that a topological invariant, the index, vanishes.
 

We use a refined version, acting on sections of a bundle of modules over a $C^*$-algebra; and then the index takes values in the K-theory of this algebra. This index is the image under the Baum-Connes assembly map of a topological object, the K-theoretic fundamental class.

The talk will present results of the following type:
 
If $M$ has a submanifold $N$ of codimension $k$ whose Dirac operator has non-trivial index, what conditions imply that $M$ does not admit a metric of positive scalar curvature? How is this related to the Baum-Connes assembly map? 

We will present previous results of Zeidler ($k=1$), Hanke-Pape-S. ($k=2$), Engel and new generalizations. Moreover, we will show how these results fit in the context of the Baum-Connes assembly maps for the manifold and the submanifold. 
 

It is known that if the boundary of a 1-ended
hyperbolic group G has a local cut point then G splits over a 2-ended group. We prove a similar theorem for CAT(0)
groups, namely that if a finite set of points separates the boundary of a 1-ended CAT(0) group G
then G splits over a 2-ended group. Along the way we prove two results of independent interest: we show that continua separated
by finite sets of points admit a tree-like decomposition and we show a splitting theorem for nesting actions on R-trees.
This is joint work with Eric Swenson.

We will prove an upper bound for the volume of a minimal
hypersurface in a closed Riemannian manifold conformally equivalent to
a manifold with $Ric > -(n-1)$.  In the second part of the talk we will
construct a sweepout of a closed 3-manifold with positive Ricci
curvature by 1-cycles of controlled length and prove an upper bound
for the length of a stationary geodesic net. These are joint works
with Parker Glynn-Adey (Toronto) and Xin Zhou (MIT).

Motivated by work of Borel and Serre on arithmetic groups, Bestvina and Feighn defined a bordification of Outer space; this is an enlargement of Outer space which is highly-connected at infinity and on which the action of $Out(F_n)$ extends, with compact quotient. They conclude that $Out(F_n)$ satisfies a type of duality between homology and cohomology.  We show that Bestvina and Feighn’s  bordification can be realized as a deformation retract of Outer space instead of an extension, answering some questions left open by Bestvina and Feighn and considerably simplifying their proof that the bordification is highly connected at infinity.

Finiteness properties of groups come in many flavours, I will discuss topological finiteness properties. These relate to the finiteness of skelata in a classifying space. Groups with interesting finiteness properties have been found in many ways, however all such examples contains free abelian subgroups of high rank. I will discuss some constructions of groups discussing the various ways we can reduce the rank of a free abelian subgroup.