N3.12

An important property of Gromov hyperbolic spaces is the fact that every path for which all sufficiently long subpaths are quasi-geodesics is itself a quasi-geodesic. Gromov showed that this property is actually a characterization of hyperbolic spaces. In this talk, we will consider a weakened version of this local-to-global behaviour, called the Morse local-to-global property. The class of spaces that satisfy the Morse local-to-global property include several examples of interest, such as CAT(0) spaces, Mapping Class Groups, fundamental groups of closed 3-manifolds and more. The leverage offered by knowing that a space satisfies this property allows us to import several results and techniques from the theory of hyperbolic groups. In particular, we obtain results relating to stable subgroups, normal subgroups and algorithmic properties.

In this talk, I will introduce the notion of a sheaf on a topological space. I will then explain why "topological spaces" are an artificial limitation on enjoying life (esp. cohomology) to the fullest and what to do about that (answer: sites). Sheaves also fail our needs, but they have a suitable natural upgrade (i.e. stacks).
This talk will be heavily peppered with examples that come from the world around you (music, torsors, etc.).
 

In 2010 Coecke, Sadrzadeh, and Clark formulated a new model of natural language which operates by combining the syntactics of grammar and the semantics of individual words to produce a unified ''meaning'' of sentences. This they did by using category theory to understand the component parts of language and to amalgamate the components together to form what they called a ''distributional compositional categorical model of meaning''. In this talk I shall introduce the model of Coecke et. al., and use it to compare the meaning of sentences in Irish and in English (and thus ascertain when a sentence is the translation of another sentence) using a cosine similarity score.

The Irish language is a member of the Gaelic family of languages, originating in Ireland and is the official language of the Republic of Ireland.

In my talk I will discuss the use of topological methods in the analysis of neural data. I will show how to obtain good state spaces for Head Direction Cells and Grid Cells. Topological decoding shows how neural firing patterns determine behaviour. This is a local to global situation which gives rise to some reflections.

Lines and planes can be fitted to data by minimising the sum of squared distances from the data to the geometric object.  But what about fitting objects from topology such as simplicial complexes?  I will present a method of fitting topological objects to data using a maximum likelihood approach, generalising the sum of squared distances.  A simplicial mixture model (SMM) is specified by a set of vertex positions and a weighted set of simplices between them.  The fitting process uses the expectation-maximisation (EM) algorithm to iteratively improve the parameters.

Remarkably, if we allow degenerate simplices then any distribution in Euclidean space can be approximated arbitrarily closely using a SMM with only a small number of vertices.  This theorem is proved using a form of kernel density estimation on the n-simplex.

We'll discuss the large charge expansion in CFTs with supersymmetry, focussing on 1908.10306 by Grassi, Komargodski and Tizzano.

Configuration spaces of points in Euclidean space or on a manifold are well studied in algebraic topology. But what if the points have some positive thickness? This is a natural setting from the point of view of physics, since this the energy landscape of a hard-spheres system. Such systems are observed experimentally to go through phase transitions, but little is known mathematically.

In this talk, I will focus on two special cases where we have started to learn some things about the homology: (1) hard disks in an infinite strip, and (2) hard squares in a square or rectangle. We will discuss some theorems and conjectures, and also some computational results. We suggest definitions for "homological solid, liquid, and gas" regimes based on what we have learned so far.

This is joint work with Hannah Alpert, Ulrich Bauer, Robert MacPherson, and Kelly Spendlove.

The goal of topological data analysis is to apply tools form algebraic topology to reveal geometric structures hidden within high dimensional data. Mapper is among its most widely and successfully applied tools providing, a framework for the geometric analysis of point cloud data. Given a number of input parameters, the Mapper algorithm constructs a graph, giving rise to a visual representation of the structure of the data.  The Mapper graph is a topological representation, where the placement of individual vertices and edges is not important, while geometric features such as loops and flares are revealed.

However, Mappers method is rather ad hoc, and would therefore benefit from a formal approach governing how to make the necessary choices. In this talk I will present joint work with Francisco Belchì, Jacek Brodzki, and Mahesan Niranjan. We study how sensitive to perturbations of the data the graph returned by the Mapper algorithm is given a particular tuning of parameters and how this depend on the choice of those parameters. Treating Mapper as a clustering generalisation, we develop a notion of instability of Mapper and study how it is affected by the choices. In particular, we obtain concrete reasons for high values of Mapper instability and experimentally demonstrate how Mapper instability can be used to determine good Mapper outputs.

Our approach tackles directly the inherent instability of the choice of clustering procedure and requires very few assumption on the specifics of the data or chosen Mapper construction, making it applicable to any Mapper-type algorithm.