Past Topology Seminar

16 November 2015

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.

9 November 2015
Yanki Lekili

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.

2 November 2015
Bruce Bartlett

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.

26 October 2015

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.

19 October 2015
David Cimasoni

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.

12 October 2015
Cornelia Drutu

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.

29 June 2015
Ramses Fernandez-Valencia

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.

15 June 2015
Brian Bowditch
We describe some results regarding the quasi-isometric rigidity of
Teichm\"uller space in either the Teichm\"uller metric or the Weil-Petersson
metric; as well as some other spaces canonically associated to a surface.
A key feature which these spaces have in common is that they admit
a ternary operation, which in an appropriate sense, satisfies the
axioms of a median algebra, up to bounded distance.  This allows
us to set many of the arguments in a general context.
We note that quasi-isometric rigidity of the Teichm\"uller metric has recently
been obtained independently by Eskin, Masur and Rafi by different methods.
8 June 2015

p, li { white-space: pre-wrap; 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.