Past Junior Topology and Group Theory Seminar

A bit more than ten years ago, Peter Oszváth and Zoltán Szabó defined Heegaard-Floer homology, a gauge theory inspired invariant of three-manifolds that is designed to be more computable than its cousins, the Donaldson and Seiberg-Witten invariants for four-manifolds. This invariant is defined in terms of a Heegaard splitting of the three-manifold. In this talk I will show how Heegaard-Floer homology is defined (modulo the analysis that goes into it) and explain some of the directions in which people have taken this theory, such as knot theory and fitting Heegaard-Floer homology into the scheme of topological field theories.

Working together with the Blue Brain Project at the EPFL, I'm trying to develop new topological methods for neural modelling. As a mathematician, however, I'm really motivated by how these questions in neuroscience can inspire new mathematics. I will introduce new work that I am doing, together with Kathryn Hess and Ran Levi, on brain plasticity and learning processes, and discuss some of the topological and geometric features that are appearing in our investigations.

A group is said to be quasirandom if all its unitary representations have “large” dimension. After introducing quasirandom groups and their basic properties, I shall turn to recent applications in two directions: constructions of expanders and non-existence of large product-free sets.

It is an open question whether a group with a finite classifying space is hyperbolic or contains a Baumslag Solitar Subgroup. An idea of Gromov was to use aperiodic tilings of the plane to try and disprove this conjecture. I will be looking at some of the attempts to find a counterexample.

I will talk about random walks on groups and define the Poisson boundary of such. Studying it gives criteria for amenability or growth. I will outline how this can be used and describe recent related results and still open questions.

This talk consists of three parts. As a motivation, we are first going to introduce von Neumann algebras associated with discrete groups and briefly describe their interplay with measurable group theory. Next, we are going to consider group von Neumann algebras of general locally compact groups and highlight crucial differences between the discrete and the non-discrete case. Finally, we present some recent results on group von Neumann algebras associated with certain locally compact HNN-extensions.

In this talk I will describe an attempt to construct a conformal field theory with target space a symmetric product of $R^D$ (referred to by physicists as orbifold sigma model). The construction uses branched covers of $S^2$ to lift the well studied formulation of a sigma model on $S^2$, in terms of vertex operator algebras, to higher genus surfaces. I will motivate and explain this construction.

Group actions play an important role in both topological problems and coarse geometric conjectures. I will introduce the idea of a partial action of a group on a metric space and explain, in the case of certain classes of coarsely disconnected spaces, how partial actions can be used to give a geometric proof of a result of Willett and Yu concerning the coarse Baum-Connes conjecture.

The integers (while wonderful in many others respects) do not make for fascinating Geometric Group Theory. They are, however, essentially the only infinite finitely generated group which is both hyperbolic and amenable. In the class of locally compact topological groups, the intersection of these two notions is richer, and the major aim of this talk will be to give the structure of a classification of such groups due to Caprace-de Cornulier-Monod-Tessera, beginning with Milnor's proof that any connected Lie group admitting a left-invariant negatively curved Riemannian metric is necessarily soluble.

To continue the day's questions of how complex groups can be I will be looking about some decision problems. I will prove that certain properties of finitely presented groups are undecidable. These properties are called Markov properties and include many nice properties one may want a group to have. I will also hopefully go into an algorithm of Whitehead on deciding if a set of n words generates F_n.

I will outline Bergeron-Wise’s proof that the Virtual Haken Conjecture follows from Wise’s Conjecture on virtual specialness of non-positively curved cube complexes. If time permits, I will sketch some highlights from the proof of Wise’s Conjecture due to Agol and based on the Weak Separation Theorem of Agol-Groves-Manning.

Self-similarity is a fundamental idea in many areas of mathematics. In this talk I will explain how it has entered group theory and the links between self-similar groups and other areas of research. There will also be pretty pictures.

First of all, I will give an overview of what the phenomenon of homological stability is and why it's useful, with plenty of examples. I will then introduce configuration spaces -- of various different kinds -- and give an overview of what is known about their homological stability properties. A "configuration" here can be more than just a finite collection of points in a background space: in particular, the points may be equipped with a certain non-local structure (an "orientation"), or one can consider unlinked embedded copies of a fixed manifold instead of just points. If by some miracle time permits, I may also say something about homological stability with local coefficients, in general and in particular for configuration spaces.

The theory of equations over groups goes back to the very beginning of group theory and is linked to many deep problems in mathematics, such as the Diophantine problem over rationals. In this talk, we shall survey some of the key results on equations over groups, give an outline of the Makanin-Razborov process (an algorithm for solving equations over free groups) and its connections to other results in group theory and low-dimensional topology.

We will discuss (very) recent work by Hensel; Przytycki and Webb, who describe unicorn paths in the arc graph and show that they form 1-slim triangles and are invariant under taking subpaths. We deduce that all arc graphs are 7-hyperbolic. Considering the same paths in the arc and curve graph, this also shows that all curve graphs are 17-hyperbolic, including closed surfaces.

The eponymous result is due to Bridson and Vogtmann, and was proven in their paper "Automorphisms of Automorphism Groups of Free Groups" (Journal of Algebra 229). While I'll remind you all the basic definitions, it would be very helpful to be already somewhat familiar with the outer space.

I'll discuss work of Wise and Ollivier-Wise that gives cubulations of certain small cancellation and random groups, which in turn shows that they do not have property (T).

I will be looking at some conjectures and theorems closely related to the h-cobordism theorem and will try to show some connections between them and some group theoretic conjectures.

We will start with the square torus, move on to all regular polygons, and then look at a large family of flat surfaces called Bouw-Möller surfaces, made by gluing together many polygons. On each surface, we will consider the action of a certain shearing action on geodesic paths on the surface, and a certain corresponding sequence.

A subgroup $H$ of a group $G$ is said to be engulfed if there is a
finite-index subgroup $K$ other than $G$ itself such that $H<K$, or
equivalently if $H$ is not dense in the profinite topology on $G$.  In
this talk I will present a variety of methods for showing that a
subgroup of a discrete group is engulfed, and demonstrate how these
methods can be used to study finite-sheeted covering spaces of
topological spaces.

Let F be a free group, and N a normal subgroup of F with derived subgroup N'. The Magnus embedding gives a way of seeing F/N' as a subgroup of a wreath product of a free abelian group over over F/N. The aim is to show that the Magnus embedding is a quasi-isometric embedding (hence "Q.I." in the title). For this I will use an alternative geometric definition of the embedding (hence "picture"), which I will show is equivalent to the definition which uses Fox calculus. Please note that we will assume no prior knowledge of calculus.

In this part, I will redefine the quantum representations for $G = SU(2)$ making no mention of flat connections at all, instead appealing to a purely combinatorial construction using the knot theory of the Jones polynomial.

Using these, I will discuss some of the properties of the representations, their strengths and their shortcomings. One of their main properties, conjectured by Vladimir Turaev and proved by Jørgen Ellegaard Andersen, is that the collection of the representations forms an infinite-dimensional faithful representation. As it is still an open question whether or not mapping class groups admit faithful finite-dimensional representations, it becomes natural to consider the kernels of the individual representations. Furthermore, I will hopefully discuss Andersen's proof that mapping class groups of closed surfaces do not have Kazhdan's Property (T), which makes essential use of quantum representations.