Past Topology Seminar

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. 

Bordered Floer homology is a variant of Heegaard Floer homology adapted to manifolds with boundary. I will describe a class of three-manifolds with torus boundary for which these invariants may be recast in terms of immersed curves in a punctured torus. This makes it possible to recast the paring theorem in bordered Floer homology in terms of intersection between curves leading, in turn, to some new observations about Heegaard Floer homology. This is joint work with Jonathan Hanselman and Jake Rasmussen. 

For the programme see 

https://people.maths.ox.ac.uk/tillmann/OAC-manifolds.html

  
The Cartan model computes the equivariant cohomology of a smooth manifold X with 
differentiable action of a compact Lie group G, from the invariant functions on 
the Lie algebra with values in differential forms and a deformation of the de Rham 
differential. Before extracting invariants, the Cartan differential does not square 
to zero. Unrecognised was the fact that the full complex is a curved algebra, 
computing the quotient by G of the algebra of differential forms on X. This 
generates, for example, a gauged version of string topology. Another instance of 
the construction, applied to deformation quantisation of symplectic manifolds, 
gives the BRST construction of the symplectic quotient. Finally, the theory for a 
X point with an additional quadratic curving computes the representation category 
of the compact group G.

I will explain some recent work using minimal surfaces to address problems in 3-manifold topology.  Given a Heegaard splitting, one can sweep out a three-manifold by surfaces isotopic to the splitting, and run the min-max procedure of Almgren-Pitts and Simon-Smith to construct a smooth embedded minimal surface.   If the original splitting were strongly irreducible (as introduced by Casson-Gordon), H. Rubinstein sketched an argument in the 80s showing that the limiting minimal surface should be isotopic to the original splitting.  I will explain some results in this direction and how jointly with T. Colding and D. Gabai we can use such min-max minimal surfaces to complete the classification problem for Heegaard splittings of non-Haken hyperbolic 3-manifolds.

In dimension three, convex surface theory implies that every tight contact structure on a connected sum M # N can be constructed as a connected sum of tight contact structures on M and N. I will explain some examples showing that this is not true in any dimension greater than three.  The proof is based on a recent higher-dimensional version of a classic result of Eliashberg about the symplectic fillings of contact manifolds obtained by subcritical surgery. This is joint work with Paolo Ghiggini and Klaus Niederkrüger.

We will discuss the concept of $\ell^2$-stability of a group and show some of its rigidity consequences.  We provide moreover some very concrete examples of lattices in product of trees that have many interesting properties, $\ell^2$-stability being only one of them.

It is well-known that a complete Riemannian manifold M which is locally isometric to a symmetric space is covered by a symmetric space. We will prove that a discrete version of this property (called local to global rigidity) holds for a large class of vertex-transitive graphs, including Cayley graphs of torsion-free lattices in simple Lie groups, and Cayley graph of torsion-free virtually nilpotent groups. By contrast, we will exhibit various examples of Cayley graphs of finitely presented groups (e.g. PGL(5, Z)) which fail to have this property, answering a question of Benjamini, Ellis, and Georgakopoulos. This is a joint work with Mikael de la Salle.

We study symplectic invariants of the open symplectic manifolds X
obtained by plumbing cotangent bundles of spheres according to a
plumbing tree. We prove that certain models for the Fukaya category F(X)
of closed exact Lagrangians in X and the wrapped Fukaya category W(X)
are related by Koszul duality. As an application, we give explicit
computations of symplectic cohomology essentially for all trees. This is
joint work with Tolga Etg\"u.

Recent developments in 3-dimensional topological quantum field theory allow us to understand the vector spaces assigned to surfaces as spaces of string diagrams. In the Reshetikhin-Turaev model, these string diagrams live inside a handlebody bounding the surface, while in the Turaev-Viro model, they live on the surface itself. There is a "lifting map" from the former to the latter, which sheds new light on a number of constructions. Joint with Gerrit Goosen.

I will describe a “cubical flat torus theorem” for a group G acting properly and cocompactly on a CAT(0) cube complex.
This states that every “highest” free abelian subgroup of G acts properly and cocompactly on a convex subcomplex that is quasi-isometric to a Euclidean space.
I will describe some simple consequences, as well as the original motivation which was to prove the “bounded packing property” for cyclic subgroups of G.
This is joint work with Daniel Woodhouse.

In the first part of this talk, we will give a very gentle introduction to the Ising model. Then , we will explain a very simple proof of a combinatorial formula for the 2D Ising model partition function using the language of Kac-Ward matrices. This approach can be used for general weighted graphs embedded in surfaces, and extends to the study of several other observables. This is a joint work with Dima Chelkak and Adrien Kassel.
 

One way of understanding groups is by investigating their actions on special spaces, such as Hilbert and Banach spaces, non-positively curved spaces etc. Classical properties like Kazhdan property (T) and the Haagerup property are formulated in terms of such actions and turn out to be relevant in a wide range of areas, from the conjectures of Baum-Connes and Novikov to constructions of expanders. In this talk I shall overview various generalisations of property (T) and Haagerup to Banach spaces, especially in connection with classes of groups acting on non-positively curved spaces.

If you do not know quasicircles, you will understand what they are.
If you hate quasicircles, you will change your mind.
If you already love quasicircles, they will astonish you once more.

We give a classification of open Klein topological conformal field theories in terms of Calabi-Yau $A_\infty$-categories endowed with an involution. Given an open Klein topological conformal field theory, there is a universal open-closed extension whose closed part is the involutive version of the Hochschild chains associated to the open part.

*/ /*-->*/ Let G be a locally compact Hausdorff topological group. Examples are Lie groups, p-adic groups, adelic groups, and discrete groups. The BC (Baum-Connes) conjecture proposes an answer to the problem of calculating the K-theory of the convolution C* algebra of G. Validity of the conjecture has implications in several different areas of mathematics --- e.g. Novikov conjecture, Gromov-Lawson-Rosenberg conjecture, Dirac exhaustion of the discrete series, Kadison-Kaplansky conjecture. An expander is a sequence  of finite graphs which is efficiently connected. Any discrete group which contains an expander as a sub-graph of its Cayley graph is a counter-example to  the BC conjecture with coefficients. Such discrete groups have been constructed by Gromov-Arjantseva-Delzant and by Damian Osajda. This talk will indicate how to make a correction in BC with coefficients. There are no known counter-examples to the corrected conjecture, and all previously known confirming examples remain confirming examples.

Representations of free loop groups possess an operation, akin to
tensor product, under which they form a braided tensor category. I
will discuss a similar operation, which is present on the category of
representations of the based loop groups, and which equips it with the
structure of a monoidal cateogory. Finally, I will present a recent
result, according to which the Drinfel'd centre of the category of
representations of a based loop group is equivalent to the category of
representations of the corresponding free loop group.

Erdos and Renyi introduced a model for studying random graphs of a given "density" and proved that there is a sharp threshold at which lower density random graphs are disconnected and higher density ones are connected.  Motivated by ideas in geometric group theory we will explain some new threshold theorems we have discovered for random graphs.  We will then, explain applications of these results to the geometry of Coxeter groups.  Some of this talk will be on joint work with Hagen and Sisto; other parts are joint work with Hagen, Susse, and Falgas-Ravry.

The triangulation conjecture stated that any n-dimensional topological manifold is homeomorphic to a simplicial complex. It is true in dimensions at most 3, but false in dimension 4 by the work of Casson and Freedman. In this talk I will explain the proof that the conjecture is also false in higher dimensions. This result is based on previous work of Galewski-Stern and Matumoto, who reduced the problem to a question in low dimensions (the existence of elements of order 2 and Rokhlin invariant one in the 3-dimensional homology cobordism group). The low-dimensional question can be answered in the negative using a variant of Floer homology, Pin(2)-equivariant Seiberg-Witten Floer homology. At the end I will also discuss a related version of Heegaard Floer homology, which is more computable.

In this talk I will show how given a finitely generated relatively hyperbolic group G, one can construct a finite generating set X of G for which (G,X) has a number of metric properties, provided that the parabolic subgroups have these properties. I will discuss the applications of these properties to the growth series, language of geodesics, biautomatic structures and conjugacy problem. This is joint work with Yago Antolin.

Unordered configuration spaces on (connected) manifolds are basic objects
that appear in connection with many different areas of topology. When the
manifold M is non-compact, a theorem of McDuff and Segal states that these
spaces satisfy a phenomenon known as homological stability: fixing q, the
homology groups H_q(C_k(M)) are eventually independent of k. Here, C_k(M)
denotes the space of k-point configurations and homology is taken with
coefficients in Z. However, this statement is in general false for closed
manifolds M, although some conditional results in this direction are known.

I will explain some recent joint work with Federico Cantero, in which we
extend all the previously known results in this situation. One key idea is
to introduce so-called "replication maps" between configuration spaces,
which in a sense replace the "stabilisation maps" that exist only in the
case of non-compact manifolds. One corollary of our results is to recover a
"homological periodicity" theorem of Nagpal -- taking homology with field
coefficients and fixing q, the sequence of homology groups H_q(C_k(M)) is
eventually periodic in k -- and we obtain a much simpler estimate for the
period. Another result is that homological stability holds with Z[1/2]
coefficients whenever M is odd-dimensional, and in fact we improve this to
stability with Z coefficients for 3- and 7-dimensional manifolds.

If G is a semi-simple Lie group, it is known that all lattices
are arithmetic unless (up to finite index) G=SO(n,1) or SU(n,1).
Non-arithmetic lattices have been constructed in SO(n,1) for
all n and there are infinitely many non-arithmetic lattices in
SU(1,1). Mostow and Deligne-Mostow constructed 9 commensurability
classes of non-arithmetic lattices in SU(2,1) and a single
example in SU(3,1). The problem is open for n at least 4.
I will survey the history of this problem, and then describe
recent joint work with Martin Deraux and Julien Paupert, where
we construct 10 new commensurability classes of non-arithmetic
lattices in SU(2,1). These are the first examples to be constructed
since the work of Deligne and Mostow in 1986.

A permutation group is called sharply n-transitive if it acts 

freely and transitively on the set of ordered n-tuples of distinct 

points. The investigation of such permutation groups is a classical 

branch of group theory; it led Emile Mathieu to the discovery of the 

smallest finite simple sporadic groups in the 1860's. In this talk I 

will discuss the case where the permutation group is assumed to be a 

locally compact transformation group, and explain how this set-up is 

related to Gromov hyperbolicity and to arithmetic lattices in products 

of trees.