Forthcoming events in this series


Fri, 06 Mar 2020

15:00 - 16:00
N3.12

Estimating the reach of a submanifold

John Harvey
(Swansea University)
Abstract

The reach is an important geometric invariant of submanifolds of Euclidean space. It is a real-valued global invariant incorporating information about the second fundamental form of the embedding and the location of the first critical point of the distance from the submanifold. In the subject of geometric inference, the reach plays a crucial role. I will give a new method of estimating the reach of a submanifold, developed jointly with Clément Berenfeld, Marc Hoffmann and Krishnan Shankar.

Fri, 21 Feb 2020

15:00 - 16:00
N3.12

Two Models of Random Simplicial Complexes

Lewis Mead
(Queen Mary University of London)
Abstract

The talk will introduce two general models of random simplicial complexes which extend the highly studied Erdös-Rényi model for random graphs. These models include the well known probabilistic models of random simplicial complexes from Costa-Farber, Kahle, and Linial-Meshulam as special cases. These models turn out to have a satisfying Alexander duality relation between them prompting the hope that information can be transferred for free between them. This turns out to not quite be the case with vanishing probability parameters, but when all parameters are uniformly bounded the duality relation works a treat. Time permitting I may talk about the Rado simplicial complex, the unique (with probability one) infinite random simplicial complex.
This talk is based on various bits of joint work with Michael Farber, Tahl Nowik, and Lewin Strauss.

Fri, 24 Jan 2020

15:00 - 16:00
N3.12

The topology and geometry of molecular conformational spaces and energy landscapes

Ingrid Membrillo-Solis
(University of Southampton)
Abstract

Molecules are dynamical systems that can adopt a variety of three dimensional conformations which, in general, differ in energy and physical properties. The identification of energetically favourable conformations is fundamental in molecular physics and computational chemistry, since it is closely related to important open problems such as the prediction of the folding of proteins and virtual screening for drug design.
In this talk I will present theoretical and data-driven approaches to the study of molecular conformational spaces and their associated energy landscapes. I will show that the topology of the internal molecular conformational space might change after taking its quotient by the group action of a discrete group of symmetries. I will also show that geometric and topological tools for data analysis such as procrustes analysis, local dimensionality reduction, persistent homology and discrete Morse theory provide with efficient methods to study the mathematical structures underlying the molecular conformational spaces and their energy landscapes.
 

Fri, 06 Dec 2019

15:00 - 16:00
N3.12

Measuring the stability of Mapper type algorithms

Matt Burfitt
(University of Southampton)
Abstract

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.

Fri, 22 Nov 2019

15:00 - 16:00
N3.12

Configuration spaces of particles and phase transitions

Matt Kahle
(Ohio State University)
Abstract

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.

Fri, 15 Nov 2019

15:00 - 16:00
N3.12

The Topology of Brain cells

Nils Baas
(NTNU)
Abstract

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.

Fri, 08 Nov 2019

15:00 - 16:00
N3.12

Simplicial Mixture Models - Fitting topology to data

James Griffin
(University of Coventry)
Abstract

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.

Fri, 01 Nov 2019

15:00 - 16:00
N3.12

The Persistence Mayer-Vietoris spectral sequence

Alvaro Torras Casas
(Cardiff University)
Abstract

In this talk, linear algebra for persistence modules will be introduced, together with a generalization of persistent homology. This theory permits us to handle the Mayer-Vietoris spectral sequence for persistence modules, and solve any extension problems that might arise. The result of this approach is a distributive algorithm for computing persistent homology. That is, one can break down the underlying data into different covering subsets, compute the persistent homology for each cover, and join everything together. This approach has the added advantage that one can recover extra geometrical information related to the barcodes. This addresses the common complaint that persistent homology barcodes are 'too blind' to the geometry of the data.

Fri, 21 Jun 2019

15:30 - 16:00
N3.12

Smoothness of Persistence

Jacob Leygonie
((Oxford University))
Abstract

We can see the simplest setting of persistence from a functional point of view: given a fixed finite simplicial complex, we have the barcode function which, given a filter function over this complex, returns the corresponding persistent diagram. The bottleneck distance induces a topology on the space of persistence diagrams, and makes the barcode function a continuous map: this is a consequence of the stability Theorem. In this presentation, I will present ongoing work that seeks to deepen our understanding of the analytic properties of the barcode function, in particular whether it can be said to be smooth. Namely, if we smoothly vary the filter function, do we get smooth changes in the resulting persistent diagram? I will introduce a notion of differentiability/smoothness for barcode valued maps, and then explain why the barcode function is smooth (but not everywhere) with respect to the choice of filter function. I will finally explain why these notions are of interest in practical optimisation/learning situations. 

Fri, 21 Jun 2019

15:00 - 15:30
N3.12

Outlier Robust Subsampling Techniques for Persistent Homology

Bernadette Stolz-Pretzer
((Oxford University))
Abstract

The amount and complexity of biological data has increased rapidly in recent years with the availability of improved biological tools. When applying persistent homology to large data sets, many of the currently available algorithms however fail due to computational complexity preventing many interesting biological applications. De Silva and Carlsson (2004) introduced the so called Witness Complex that reduces computational complexity by building simplicial complexes on a small subset of landmark points selected from the original data set. The landmark points are chosen from the data either at random or using the so called maxmin algorithm. These approaches are not ideal as the random selection tends to favour dense areas of the point cloud while the maxmin algorithm often selects outliers as landmarks. Both of these problems need to be addressed in order to make the method more applicable to biological data. We study new ways of selecting landmarks from a large data set that are robust to outliers. We further examine the effects of the different subselection methods on the persistent homology of the data.

Fri, 14 Jun 2019

15:00 - 16:00
N3.12

Multiparameter persistence vs parametrised persistence

Jeffrey Giansiracusa
(Swansea University)
Abstract

One of the key properties of 1-parameter persistent homology is that its output can entirely encoded in a purely combinatorial way via persistence diagrams or barcodes.  However, many applications of topological data analysis naturally present themselves with more than 1 parameter. Multiparameter persistence suggests itself as the natural invariant to use, but the problem here is that the moduli space of multiparameter persistence diagrams has a much more complicated structure and we lack a combinatorial diagrammatic description.  An alternative approach was suggested by work of Giansiracusa-Moon-Lazar, where they investigated calculating a series of 1-parameter persistence diagrams as the other parameter is varied. In this talk I will discuss work in progress to produce a refinement of their perspective, making use the Algebraic Stability Theorem for persistent homology and work of Bauer-Lesnick on induced matchings.

Fri, 07 Jun 2019

15:00 - 15:30
N3.12

Persistence Paths and Signature Features in Topological Data Analysis

Ilya Chevyrev
((Oxford University))
Abstract

In this talk I will introduce the concept of the path signature and motivate its recent use in analysis of time-ordered data. I will then describe a new feature map for barcodes in persistent homology by first realizing each barcode as a path in a vector space, and then computing its signature which takes values in the tensor algebra over that vector space. The composition of these two operations — barcode to path, path to tensor series — results in a feature map that has several desirable properties for statistical learning, such as universality and characteristicness.

Fri, 24 May 2019

15:30 - 16:00
N3.12

Random Geometric Complexes

Oliver Vipond
((Oxford University))
Abstract

I will give an introduction to the asymptotic behaviour of random geometric complexes. In the specific case of a simplicial complex realised as the Cech complex of a point process sampled from a closed Riemannian manifold, we will explore conditions which guarantee the homology of the Cech complex coincides with the homology of the underlying manifold. We will see techniques which were originally developed to study random geometric graphs, which together with ideas from Morse Theory establish homological connectivity thresholds.

Fri, 24 May 2019
15:00
N3.12

Spectrograms and Persistent Homology

Wojciech Reise
(EPFL)
Abstract

I will give an overview of audio identification methods on spectral representations of songs. I will outline the persistent homology-based approaches that I propose and their shortcomings. I hope that the review of previous work will help spark a discussion on new possible representations and filtrations.

Fri, 10 May 2019

15:00 - 16:00
N3.12

Sheaf Laplacians as sums of semidefinite matrices

Jakob Hansen
(University of Pennsylvania)
Abstract

The class of sheaf Laplacians can be characterized as the convex closure of a certain set of sparse semidefinite matrices. From this viewpoint, the study of sheaf Laplacians becomes a question of linear algebra on sparse matrices. I will discuss the applications of this perspective to the problems of approximating, sparsifying, and learning sheaves.

Fri, 03 May 2019

15:00 - 16:00
N3.12

Persistence of Random Structures

Primoz Skraba
(Queen Mary University London)
Abstract

This talk will cover the connections of persistence with the topology of random structures. This includes an overview of various results from stochastic topology as well as the role persistence ideas  play in the analysis. This will include results on the maximally persistent classes and minimum spanning acycles/generalised trees.