Past Topology Seminar

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.