LMS - IMA Joint Meeting: 1-2 October 2020
The lectures are aimed at a general mathematical audience. All interested, whether LMS and/or IMA members or not, are most welcome to attend this event.
13:30 Introduction and housekeeping
13:45 - 14:30 Kathryn Hess (EPFL)
14:40 - 15:25 Vidit Nanda (Oxford)
15:45 - 16:30 Ran Levi (Aberdeen)
9:30 - 10:15 Gueorgui Mihaylov (GSK & King's College London)
10:30 - 11:15 Ulrike Tilmann (Oxford)
11:15 Thanks and Farewells
The London Mathematical Society and the Institute of Mathematics and its Applications will hold their next Joint Meeting online on 1st and 2nd October 2020. This year’s topic is “Topological methods in Data Science". The meeting will take place over two days; the afternoon of 1st October the morning of 2nd October, and will comprise five talks of 45 minutes each including questions.
Kathryn Hess (EPFL), Gueorgui Mihaylov (King's College London); Vidit Nanda (University of Oxford); Ran Levi (University of Aberdeen) Ulrike Tillmann (University of Oxford).
Kathryn Hess (EPFL):
Title: Trees, barcodes, and symmetric groups
Abstract: Motivated by the desire to automate classification of neuron morphologies, we designed a topological signature, the Topological Morphology Descriptor (TMD), that assigns a barcode to any geometric tree (i.e, any finite binary tree embedded in R^3). We showed that the TMD effectively determines the reliability of clusterings of random and neuronal trees. Moreover, using the TMD we performed an objective, stable classification of pyramidal cells in the rat neocortex, based only on the shape of their dendrites.
We have also reverse-engineered the TMD, in order to digitally synthesize dendrites, to compensate for the dearth of available biological reconstructions. The algorithm we developed, called Topological Neuron Synthesis (TNS), stochastically generates a geometric tree from a barcode, in a biologically grounded manner. The synthesized cortical dendrites are statistically indistinguishable from the corresponding reconstructed dendrites in terms of morpho-electrical properties and the networks they form. We synthesized cortical networks of structurally altered dendrites, revealing principles linking branching properties to the structure of large-scale networks.
In this talk I will provide an overview of the TMD and the TNS and then present recent theoretical and computational results concerning their behavior and properties, in which symmetric groups play a key role.
This is joint work with Adélie Garin and Lida Kanari, building on earlier collaborations led by Lida Kanari.
Vidit Nanda (University of Oxford)
Title: Geometric anomaly detection in data.
Abstract: Fitting low-dimensional manifolds to high-dimensional point clouds is among the most fundamental practices in contemporary data analysis. The underlying assumption behind the success of all such techniques is that the points are well-approximated by a single low-dimensional manifold. This talk will describe a topological technique for automatically detecting data points which fail to satisfy this manifold hypothesis. As a consequence, it becomes possible to decompose the data points into different clusters, each of which can be well-approximated by a single manifold. This is joint work with B Stolz, J Tanner and H Harrington.
Ran Levi (University of Aberdeen):
Title: Combinatorial Structures in Neural Networks
Abstract: Understanding structure and function in a natural neural network is a formidable task that has been occupying the attention of scientists and mathematicians for many decades. Algebraic topology entered that scene relatively recently, equipped with its own tools as well as those of adjacent subjects such as graph theory and combinatorics. In this talk I will review some of the work I've done with collaborators where we employ various combinatorial constructions to networks emerging from digital reconstructions of sections of brain tissue developed by the Blue Brain Project. I will demonstrate how these combinatorial objects, analysed by methods of algebraic topology, reveal interesting structural features of a neural network, as well as hold the potential to inform on its functionality.
Gueorgui Mihaylov (GSK & King's College, London):
Title: A gauge theory of complex systems
Abstract: Emergent phenomena in complex systems recently became a very relevant, fascinating and strongly multidisciplinary field of research. Understanding, modelling and quantifying the complexity of biological, economical and industrial systems, estimating the risks or exploiting potential, often unexpected benefits of their emergent large-scale behaviour (safety and reliability vs efficiency and productivity) proved to be a non-trivial and highly impactful area of scientific interest.
In this talk we introduce a construction of a gauge field theory of complex adaptive systems based on a suitable simplicial formulation of discrete differential geometry. The main idea is adapting and applying standard concepts of the theory of geometric structures over differentiable manifolds (principal and associated bundles) to model complexity. Suitably defined discrete analogues of local geometric obstructions and global topological obstructions (characteristic classes) that impede fibre bundles to be trivial, play crucial role in our construction.
Two industrial examples in which this geometric viewpoint on complex systems proved to be very efficient will be briefly introduced.
Ulrike Tilmann (Oxford)
Title: Homology of random geometric complexes
Abstract: Topological data analysis studies the shape of data, like a point cloud. But to know whether the detected topological features are significant one needs to understand what is to be expected, the null-hypothesis. This leads naturally to the study of random graphs constrained by some underlying geometry and their higher dimensional analogue, random simplicial complexes.
Given a random configuration of n points on a manifold one might then ask how large one needs to choose r-balls around these points so that the manifold is covered, or the associated Cech complex has homology isomorphic to that of the manifold. We will be interested in the asymptotic behaviour of the radius r as the number of points n goes to infinity. Surprisingly, the threshold formulae one gets depend on the homological degree and whether the manifold has boundary or not, but not on its geometry or topology.
This talk is based on joint work with Henry-Louis de Kergorlay and Oliver Vipond building on previous results by Bobrowski, Oliveira, Weinberger, and others.
The meeting will be jointly chaired by the Presidents of the IMA and LMS.
Joint LMS-IMA Organising Committee: Alina Vdovina (Newcastle), Brita Nucinkis (RHUL), Helen Wilson (UCL) and Richard Pinch (IMA),
To attend, please register here: https://www.smartsurvey.co.uk/s/LMSIMADATASCIENCE/