We will discuss TQFTs (at a basic level), then higher categorical extensions, and see how these lead naturally to the notion of Segal spaces.
We will finish presenting Nyikos' counterexample to
Bozyakova's conjecture: If e(Y) = L(Y) for every subspace Y of X, must X
be hereditarily D?
Last week in the Kinderseminar I talked about a rough estimate on volumes of certain hyperbolic 3-manifolds. This time I will describe a different approach for similar estimates (you will not need to remember that talk, don't worry!), which is, in some sense, complementary to that one, as it regards mapping tori. A theorem of Jeffrey Brock provides bounds for their volume in terms of how the monodromy map acts on the pants graph (a relative of the better known curve complex) of the base surface. I will describe the setting and the relevance of this result (in particular the one it has for me); hopefully, I will also tell you part of its proof.
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.
Raushan Buzyakova asked if a space is hereditarily D provided
that the extent and Lindelöf numbers coincide for every subspace. We
will introduce interval topologies on trees and present Nyikos'
counterexample to this conjecture.
This is the first of a series of talks based on Gary
Gruenhage's 'A survey of D-spaces' [1]. A space is D if for every
neighbourhood assignment we can choose a closed discrete set of points
whose assigned neighbourhoods cover the space. The mention of
neighbourhood assignments and a topological notion of smallness (that
is, of being closed and discrete) is peculiar among covering properties.
Despite being introduced in the 70's, we still don't know whether a
Lindelöf or a paracompact space must be D. In this talk, we will examine
some elementary properties of this class via extent and Lindelöf numbers.
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.