The ordinary braid group ${\mathrm Br}_n$ is a well-known algebraic structure which encodes configurations of $n$ non-touching strands (“braids”) up to continious transformations (“isotopies”). A classical result of Khovanov and Thomas states that there is a natural categorical action of ${\mathrm Br}_n$ on the derived category of the cotangent bundle of the variety of complete flags in ${\mathbb C}^n$. 

In this talk, I will introduce a new structure: the category ${\mathrm GBr}_n$ of generalised braids. These are the braids whose strands are allowed to touch in a certain way. They have multiple endpoint configurations and can be non-invertible, thus forming a category rather than a group. In the context of triangulated categories, it is natural to impose certain relations which result in the notion of a skein-triangulated representation of ${\mathrm GBr}_n$. A decade-old conjecture states that there is a skein-triangulated action of ${\mathrm GBr}_n$ on the cotangent bundles of the varieties of full and partial flags in ${\mathbb C}^n$. We prove this conjecture for $n = 3$. We also show that, in fact, any categorical action of ${\mathrm Br}_n$ can be lifted to a categorical action of ${\mathrm GBr}_n$, generalising a result of Ed Segal. This is a joint work with Rina Anno and Lorenzo De Biase.

Spatial navigation in preclinical and clinical Alzheimer’s disease - Relevance for topological data analysis?

Spatial navigation changes are one of the first symptoms of Alzheimer’s disease and also lead to significant safeguarding issues in patients after diagnosis. Despite their significant implications, spatial navigation changes in preclinical and clinical Alzheimer’s disease are still poorly understood. In the current talk, I will explain the spatial navigation processes in the brain and their relevance to Alzheimer’s disease. I will then introduce our Sea Hero Quest project, which created the first global benchmark data for spatial navigation in ~4.5 million people worldwide via a VR-based game. I will present data from the game, which has allowed to create personalised benchmark data for at-risk-of-Alzheimer’s people. The final part of my talk will explore how real-world environment & entropy impacts on dementia patients getting lost and how this has relevance for GPS technology based safeguarding and car driving in Alzheimer’s disease.

Positively folded galleries arise as images of retractions of buildings onto a fixed apartment and play a role in many areas of maths (such as in the study of affine Hecke algebras, Macdonald polynomials, MV-polytopes, and affine Deligne-Lusztig varieties). In this talk, we will define positively folded galleries, and then look at how these can be used to study affine flag varieties. We will also look at a new recursive description of the set of end alcoves of folded galleries with respect to alcove-induced orientations, which gives us a combinatorial description of certain double coset intersections in these affine flag varieties. This talk is based on joint work with Elizabeth Milićević, Petra Schwer and Anne Thomas.

For a family $A$ in $\{0,...,k\}^n$, its deletion shadow is the set obtained from $A$ by deleting from any of its vectors one coordinate. Given the size of $A$, how should we choose $A$ to minimise its deletion shadow? And what happens if instead we may delete only a coordinate that is zero? We discuss these problems, and give an exact solution to the second problem.