Forthcoming events in this series

Fri, 25 Nov 2022

15:00 - 16:00
L5

### Signal processing on cell complexes using discrete Morse theory

Celia Hacker
(EPFL)
Further Information

Celia is a PhD student under the supervision of Kathryn Hess since 2018.

Abstract

At the intersection of Topological Data Analysis and machine learning, the field of cellular signal processing has advanced rapidly in recent years. In this context, each signal on the cells of a complex is processed using the combinatorial Laplacian and the resulting Hodge decomposition. Meanwhile, discrete Morse theory has been widely used to speed up computations by reducing the size of complexes while preserving their global topological properties. In this talk, we introduce an approach to signal compression and reconstruction on complexes that leverages the tools of discrete Morse theory. The main goal is to reduce and reconstruct a cell complex together with a set of signals on its cells while preserving their global topological structure as much as possible. This is joint work with Stefania Ebli and Kelly Maggs.

Fri, 18 Nov 2022

15:00 - 16:00
L5

### Tensor-based frameworks for cancer genomics

Neriman Tokcan
(MIT & Harvard)
Further Information

(taken from https://nerimantokcan.com/)

Neriman Tokcan's research focuses on formulating novel, mathematically sound theoretical frameworks to perform analysis of multi-modal, multi-dimensional data while preserving the integrity of their structure. Her work on the generalization of matrix-based compression, noise elimination, and dimension reduction methods to higher dimensions. Her background is at the intersection of algebraic geometry, multi-linear algebra, combinatorics, and representation theory. I explore applications in bioinformatics and cancer genomics.

Currently, Neriman is working on the formulation of the novel, mathematically sound tensor-based frameworks, and the development of computational tools to model tumor microenvironments.

Neriman will join the University of Massachusetts Boston as a Tenure-Track Assistant Professor of Applied Mathematics in January 2023.

Abstract

The tumor microenvironment (TME) is a complex milieu around the tumor, whereby cancer cells interact with stromal, immune, vascular, and extracellular components. The TME is being increasingly recognized as a key determinant of tumor growth, disease progression, and response to therapies. We build a generalizable and robust tensor-based framework capable of integrating dissociated single-cell and spatially resolved RNA-seq data for a comprehensive analysis of the TME. Tensors are a generalization of matrices to higher dimensions. Tensor methods are known to be able to successfully incorporate data from multiple sources and perform a joint analysis of heterogeneous high-dimensional data sets. The methodologies developed as part of this effort will advance our understanding of the TME in multiple directions. These include cellular heterogeneity within the TME, crosstalks between cells, and tumor-intrinsic pathways stimulating tumor growth and immune evasion.

Fri, 11 Nov 2022

12:00 - 15:45
L2

### Centre for Topological Data Analysis Centre Meeting

Adam Brown, Heather Harrington, Živa Urbančič, David Beers.
(University of Oxford, Mathematical Institute)
Further Information

Details of speakers and schedule will be posted here nearer the time.

Abstract

Fri, 04 Nov 2022

15:00 - 16:00
L5

### Dynamics of neural circuits at different scales

Jānis Lazovskis
Further Information

Jānis Lazovskis is an Assistant Professor at RTU Riga Business School in Riga, Latvia, working in algebraic topology and topological data analysis, in particular dynamic data. His research focuses on the intersection of topology and neuroscience, simplifying and classifying in silico activity with graph theoretic and topological tools. Previously Jānis worked as a postdoc in Ran Levi's group at Aberdeen, and completed his PhD under Ben Antieau at the University of Illinois at Chicago. As an instructor and administrator of undergraduate mathematics courses, Jānis pushes for more inclusion and equity through better teaching methods and modified assessments.

Abstract

Models of animal brains are increasingly common and mapped in increasing detail. To simplify analysis of their function, we consider subregions and show that they perform well as classifiers of overall activity, with only a fraction of the neurons. The uniqueness of such ''reliable'' regions seems to be related to the types of connections that pairs of neurons form in them. By focusing on topologically significant structures and reciprocally connected neurons we find even stronger classification results. This is ongoing work across several institutions, including EPFL, the Blue Brain Project, and the University of Aberdeen.

Fri, 28 Oct 2022

15:00 - 16:00
L5

### Topological Data Analytic Frameworks for Discovering Biophysical Signatures in 3D Shapes and Images

Lorin Crawford
(Brown University)
Further Information

Lorin Crawford is the RGSS Assistant Professor of Biostatistics at Brown University. He is affiliated with the Center for Statistical Sciences, Center for Computational Molecular Biology, and the Robert J. and Nancy D. Carney Institute for Brain Science.

Abstract
Fri, 21 Oct 2022

15:00 - 16:00
L5

### Kan Extensions and Kan Ensembles in Machine Learning

Dan Shiebler
(Abnormal Security)
Further Information

Right now Dan works as the Head of Machine Learning at Abnormal Security. Previously. He led the Web Ads Machine Learning team at Twitter. Before that he worked as a Staff ML Engineer at Twitter Cortex and a Senior Data Scientist at TrueMotion.

His PhD research at the University of Oxford focused on applications of Category Theory to Machine Learning (advised by Jeremy Gibbons and Cezar Ionescu). Before that he worked as a Computer Vision Researcher at the Serre Lab.

You can find out more about Dan here: https://danshiebler.com/

Abstract

A common problem in data science is "use this function defined over this small set to generate predictions over that larger set." Extrapolation, interpolation, statistical inference and forecasting all reduce to this problem. The Kan extension is a powerful tool in category theory that generalizes this notion. In this work we explore several applications of Kan extensions to data science. We begin by deriving simple classification and clustering algorithms as Kan extensions and experimenting with these algorithms on real data. Next, we build more complex and resilient algorithms from these simple parts.

Fri, 14 Oct 2022

15:00 - 16:00
L5

### Applied Topology for Discrete Structures

Emilie Purvine
(Pacific Northwest National Laboratory)
Further Information

(From PNNL website)

Emilie's academic background is in pure mathematics, with a BS from University of Wisconsin - Madison and a PhD from Rutgers University, her research since joining PNNL in 2011 has focused on applications of combinatorics and computational topology together with theoretical advances needed to support the applications. Over her time at PNNL, Purvine has served as both a primary investigator and technical staff member on several projects in applications ranging from computational chemistry and biology to cybersecurity and power grid modeling. She has authored over 40 technical publications and is currently an associate editor for the Notices of the American Mathematical Society. Purvine also coordinates PNNL’s Postgraduate Organization which plans career development seminars, an annual research symposium, and promotes networking and mentorship for PNNL’s post bachelors, post masters, and post doctorate research associates.

Abstract

Discrete structures have a long history of use in applied mathematics. Graphs and hypergraphs provide models of social networks, biological systems, academic collaborations, and much more. Network science, and more recently hypernetwork science, have been used to great effect in analyzing these types of discrete structures. Separately, the field of applied topology has gathered many successes through the development of persistent homology, mapper, sheaves, and other concepts. Recent work by our group has focused on the convergence of these two areas, developing and applying topological concepts to study discrete structures that model real data.

This talk will survey our body of work in this area showing our work in both the theoretical and applied spaces. Theory topics will include an introduction to hypernetwork science and its relation to traditional network science, topological interpretations of graphs and hypergraphs, and dynamics of topology and network structures. I will show examples of how we are applying each of these concepts to real data sets.

Fri, 10 Jun 2022
15:00
L3

### Directed networks through simplicial paths and Hochschild homology

Henri Riihimäki
(KTH Royal Institute of Technology)
Abstract

Directed graphs are a model for various phenomena in the
sciences. In topological data analysis particularly the advent of
applying topological tools to networks of brain neurons has spawned
interest in constructing topological spaces out of digraphs, developing
computational tools for obtaining topological information, and using
these to understand networks. At the end of the day, (homological)
computations of the spaces reveal something about the geometric
realisation, thereby losing the directionality information.

However, digraphs can also be associated with path algebras. We can now
consider applying Hochschild homology to extract information, hopefully
obtaining something more refined in terms of the combinatorics of the
directed edges and paths in the digraph. Unfortunately, Hochschild
homology tends to vanish beyond degree 1. We can overcome this by
considering different higher paths of simplices, and thus introduce
Hochschild homology of digraphs in higher degrees. Moreover, this
procedure gives an implementable persistence pipeline for network
analysis. This is a joint work with Luigi Caputi.

Fri, 03 Jun 2022
15:00
L3

### Projected barcodes : a new class of invariants and distances for multi-parameter persistence modules

Nicolas Berkouk
(École Polytechnique Fédérale de Lausanne (EPFL))
Abstract

In this talk, we will present a new class of invariants of multi-parameter persistence modules : \emph{projected barcodes}. Relying on Grothendieck's six operations for sheaves, projected barcodes are defined as derived pushforwards of persistence modules onto $\R$ (which can be seen as sheaves on a vector space in a precise sense). We will prove that the well-known fibered barcode is a particular instance of projected barcodes. Moreover, our construction is able to distinguish persistence modules that have the same fibered barcodes but are not isomorphic. We will present a systematic study of the stability of projected barcodes. Given F a subset of the 1-Lipschitz functions, this leads us to define a new class of well-behaved distances between persistence modules, the  F-Integral Sheaf Metrics (F-ISM), as the supremum over p in F of the bottleneck distance of the projected barcodes by p of two persistence modules.

In the case where M is the collection in all degrees of the sublevel-sets persistence modules of a function f : X -> R^n, we prove that the projected barcode of M by a linear map p : R^n \to R is nothing but the collection of sublevel-sets barcodes of the post-composition of f by p. In particular, it can be computed using already existing softwares, without having to compute entirely M. We also provide an explicit formula for the gradient with respect to p of the bottleneck distance between projected barcodes, allowing to use a gradient ascent scheme of approximation for the linear ISM. This is joint work with François Petit.

Fri, 27 May 2022
15:00
L3

### Applied Topology TBC

Luis Scoccola
(Northeastern University)
Fri, 20 May 2022

15:00 - 16:00
L3

### Approximating Persistent Homology for Large Datasets

Anthea Monod
(Imperial College London)
Abstract

Persistent homology is an important methodology from topological data analysis which adapts theory from algebraic topology to data settings and has been successfully implemented in many applications. It produces a statistical summary in the form of a persistence diagram, which captures the shape and size of the data. Despite its widespread use, persistent homology is simply impossible to implement when a dataset is very large. In this talk, I will address the problem of finding a representative persistence diagram for prohibitively large datasets. We adapt the classical statistical method of bootstrapping, namely, drawing and studying smaller multiple subsamples from the large dataset. We show that the mean of the persistence diagrams of subsamples—taken as a mean persistence measure computed from the subsamples—is a valid approximation of the true persistent homology of the larger dataset. We give the rate of convergence of the mean persistence diagram to the true persistence diagram in terms of the number of subsamples and size of each subsample. Given the complex algebraic and geometric nature of persistent homology, we adapt the convexity and stability properties in the space of persistence diagrams together with random set theory to achieve our theoretical results for the general setting of point cloud data. We demonstrate our approach on simulated and real data, including an application of shape clustering on complex large-scale point cloud data.

This is joint work with Yueqi Cao (Imperial College London).

Fri, 13 May 2022

15:00 - 16:00
L2

### Non-Euclidean Data Analysis (and a lot of questions)

John Aston
(University of Cambridge)
Abstract

The statistical analysis of data which lies in a non-Euclidean space has become increasingly common over the last decade, starting from the point of view of shape analysis, but also being driven by a number of novel application areas. However, while there are a number of interesting avenues this analysis has taken, particularly around positive definite matrix data and data which lies in function spaces, it has increasingly raised more questions than answers. In this talk, I'll introduce some non-Euclidean data from applications in brain imaging and in linguistics, but spend considerable time asking questions, where I hope the interaction of statistics and topological data analysis (understood broadly) could potentially start to bring understanding into the applications themselves.

Fri, 06 May 2022

15:00 - 16:00
L4

### Applied Topology TBC

(University of Oxford, Mathematical Institute)
Fri, 29 Apr 2022

15:00 - 16:00
L4

### Signed barcodes for multiparameter persistence

Magnus Botnan
(Free University of Amsterdam)
Abstract

Moving from persistent homology in one parameter to multiparameter persistence comes at a significant increase in complexity. In particular, the notion of a barcode does not generalize straightforwardly. However, in this talk, I will show how it is possible to assign a unique barcode to a multiparameter persistence module if one is willing to take Z-linear combinations of intervals. The theoretical discussion will be complemented by numerical experiments. This is joint work with Steffen Oppermann and Steve Oudot.

Fri, 11 Mar 2022

15:00 - 16:00
L6

### Applied Topology TBC

Anthea Monod
(Imperial College London)
Fri, 04 Mar 2022

15:00 - 16:00
L6

### Open questions on protein topology in its natural environment.

Christopher Prior
(Durham University)
Abstract

Small angle x-ray scattering is one of the most flexible and readily available experimental methods for obtaining information on the structure of proteins in solution. In the advent of powerful predictive methods such as the alphaFold and rossettaFold algorithms, this information has become increasingly in demand, owing to the need to characterise the more flexible and varying components of proteins which resist characterisation by these and more standard experimental techniques. To deal with structures about little of which is known a parsimonious method of representing the tertiary fold of a protein backbone as a discrete curve has been developed. It represents the fundamental local Ramachandran constraints through a pair of parameters and is able to generate millions of potentially realistic protein geometries in a short space of time. The data obtained from these methods provides a treasure trove of information on the potential range of topological structures available to proteins, which is much more constrained that that available to self-avoiding walks, but still far more complex than currently understood from existing data. I will introduce this method and its considerations then attempt to pose some questions I think topological data analysis might help answer. Along the way I will ask why roadies might also help give us some insight….

Fri, 25 Feb 2022

15:00 - 16:00
L6

### Homotopy, Homology, and Persistent Homology using Cech’s Closure Spaces

Peter Bubenik
(University of Florida)
Abstract

We use Cech closure spaces, also known as pretopological spaces, to develop a uniform framework that encompasses the discrete homology of metric spaces, the singular homology of topological spaces, and the homology of (directed) clique complexes, along with their respective homotopy theories. We obtain nine homology and six homotopy theories of closure spaces. We show how metric spaces and more general structures such as weighted directed graphs produce filtered closure spaces. For filtered closure spaces, our homology theories produce persistence modules. We extend the definition of Gromov-Hausdorff distance to filtered closure spaces and use it to prove that our persistence modules and their persistence diagrams are stable. We also extend the definitions Vietoris-Rips and Cech complexes to closure spaces and prove that their persistent homology is stable.

This is joint work with Nikola Milicevic.

Fri, 18 Feb 2022

14:00 - 17:00
L2

### Centre for Topological Data Analysis Group Meeting

Various Speakers
(Mathematical Institute)
Fri, 11 Feb 2022

15:00 - 16:00
L2

### Topology-Based Graph Learning

Bastian Rieck
(Helmholtz Zentrum München)
Abstract

Topological data analysis is starting to establish itself as a powerful and effective framework in machine learning , supporting the analysis of neural networks, but also driving the development of novel algorithms that incorporate topological characteristics. As a problem class, graph representation learning is of particular interest here, since graphs are inherently amenable to a topological description in terms of their connected components and cycles. This talk will provide
an overview of how to address graph learning tasks using machine learning techniques, with a specific focus on how to make such techniques 'topology-aware.' We will discuss how to learn filtrations for graphs and how to incorporate topological information into modern graph neural networks, resulting in provably more expressive algorithms. This talk aims to be accessible to an audience of TDA enthusiasts; prior knowledge of machine learning is helpful but not required.

Fri, 04 Feb 2022

11:00 - 12:00
L6

### Computing the Extended Persistent Homology Transform of binary images

Katharine Turner
(Australian National University)
Further Information

Abstract

The Persistent Homology Transform, and the Euler Characteristic Transform are topological analogs of the Radon transform that can be used in statsistical shape analysis. In this talk I will consider an interesting variant called the Extended Persistent Homology Transform (XPHT) which replaces the normal persistent homology with extended persistent homology. We are particularly interested in the application of the XPHT to binary images. This paper outlines an algorithm for efficient calculation of the XPHT exploting relationships between the PHT of the boundary curves to the XPHT of the foreground.

Fri, 28 Jan 2022

15:00 - 16:00
L6

### Topological Tools for Signal Processing

Sarah Tymochko
(Michigan State University)
Abstract

Topological data analysis (TDA) is a field with tools to quantify the shape of data in a manner that is concise and robust using concepts from algebraic topology. Persistent homology, one of the most popular tools in TDA, has proven useful in applications to time series data, detecting shape that changes over time and quantifying features like periodicity. In this talk, I will present two applications using tools from TDA to study time series data: the first using zigzag persistence, a generalization of persistent homology, to study bifurcations in dynamical systems and the second, using the shape of weighted, directed networks to distinguish periodic and chaotic behavior in time series data.

Fri, 21 Jan 2022

15:00 - 16:00
L6

### A Multivariate CLT for Dissociated Sums with Applications to Random Complexes

(Mathematical Institute)
Abstract

Acyclic partial matchings on simplicial complexes play an important role in topological data analysis by facilitating efficient computation of (persistent) homology groups. Here we describe probabilistic properties of critical simplex counts for such matchings on clique complexes of Bernoulli random graphs. In order to accomplish this goal, we generalise the notion of a dissociated sum to a multivariate setting and prove an abstract multivariate central limit theorem using Stein's method. As a consequence of this general result, we are able to extract central limit theorems not only for critical simplex counts, but also for generalised U-statistics (and hence for clique counts in Bernoulli random graphs) as well as simplex counts in the link of a fixed simplex in a random clique complex.

Fri, 10 Dec 2021

15:00 - 16:00
Virtual

### A topological approach to signatures

Darrick Lee
(EPFL)
Abstract

The path signature is a characterization of paths that originated in Chen's iterated integral cochain model for path spaces and loop spaces. More recently, it has been used to form the foundations of rough paths in stochastic analysis, and provides an effective feature map for sequential data in machine learning. In this talk, we return to the topological foundations in Chen's construction to develop generalizations of the signature.

Fri, 03 Dec 2021

15:00 - 16:00
Virtual

Carina Curto
(Penn State)
Fri, 26 Nov 2021

15:00 - 16:00
Virtual

### Morse inequalities for the Koszul complex of multi-persistence

Claudia Landi
(University of Modena and Reggio Emilia)
Abstract

In this talk, I'll present inequalities bounding the number of critical cells in a filtered cell complex on the one hand, and the entries of the Betti tables of the multi-parameter persistence modules of such filtrations on the other hand. Using the Mayer-Vietoris spectral sequence we first obtain strong and weak Morse inequalities involving the above quantities, and then we improve the weak inequalities achieving a sharp lower bound for the number of critical cells. Furthermore, we prove a sharp upper bound for the minimal number of critical cells, expressed again in terms of the entries of Betti tables. This is joint work with Andrea Guidolin (KTH, Stockholm). The full paper is posted online as arxiv:2108.11427.