TTT115: Online
Hosted by Sheffield

Talks were online and we used Gather.town for discussions during the break and after the talks.

Date

17 December 2021

Speakers

Rachael Boyd (Cambridge)
Homological stability for Temperley-Lieb algebras

Abstract: Many sequences of groups and spaces satisfy a phenomenon called 'homological stability'. I will present joint work with Hepworth, in which we abstract this notion to sequences of algebras, and prove homological stability for the sequence of Temperley-Lieb algebras. The proof uses a new technique of 'inductive resolutions', to overcome the lack of flatness of the Temperley-Lieb algebras.

I will also introduce the 'complex of planar injective words' which plays a key role in our work. Time permitting, I will explore some connections to representation theory and combinatorics that arose from our work.

Álvaro Torras Casas (Cardiff)
The persistence Mayer-Vietoris spectral sequence

Abstract: Persistent homology has been used to develop models and extract insights from data in a variety of applications, such as in neuroscience, image processing or analysis of signals. Part of this wide use comes from its simplicity, as complex data related to filtered complexes is summarised into a barcode.

An inconvenient of persistent homology is that it becomes computationally expensive when working with large datasets; for example, when analysing 3D images. This is why there is a need to develop algorithms that can handle these computations by using many computational cores working at the same time. Essentially, one would wish to divide the underlying data, compute the local barcode for each division and proceed to merge these barcodes to obtain the global persistent homology.

In this talk I will explain how this can be done by using the persistence Mayer-Vietoris spectral sequence. I will also explain a variation of this spectral sequence which is closely related to the homotopy colimit spectral sequence; it turns out that this object is the best choice for dividing computations on simplicial complexes. I will briefly show some experiments using PerMaViss, a Python module implementing some of these ideas.

Andy Tonks (Leicester)
Comparing localisations across adjunctions

Abstract: In the literature there are several examples of constructions that preserve localisations, and many more for which there is at least a canonical map comparing the localisation of some construction with the corresponding construction on the localised object.

In this talk I discuss recent work together with Carles Casacuberta and Oriol Raventós providing a general framework for such comparisons. This works very nicely not only for ordinary categories but also for model categories. We observe, for example, that a localisation L preserves algebras over a monad T if and only if T preserves the L-equivalences, recovering several classical examples – and the general framework leads us to 'dual' results concerning localisations that  preserve coalgebras, or colocalisations that preserve algebras or coalgebras.

We investigate similar questions for homotopy localisations in the context of model categories.