Forthcoming events in this series


Fri, 24 May 2024

15:00 - 16:00
L5

Applying stratified homotopy theory in TDA

Lukas Waas
(Univeristy of Heidelberg)
Abstract

 

The natural occurrence of singular spaces in applications has led to recent investigations on performing topological data analysis (TDA) on singular data sets. However, unlike in the non-singular scenario, the homotopy type (and consequently homology) are rather course invariants of singular spaces, even in low dimension. This suggests the use of finer invariants of singular spaces for TDA, making use of stratified homotopy theory instead of classical homotopy theory.
After an introduction to stratified homotopy theory, I will describe the construction of a persistent stratified homotopy type obtained from a sample with two strata. This construction behaves much like its non-stratified counterpart (the Cech complex) and exhibits many properties (such as stability, and inference results) necessary for an application in TDA.
Since the persistent stratified homotopy type relies on an already stratified point-cloud, I will also discuss the question of stratification learning and present a convergence result which allows one to approximately recover the stratifications of a larger class of two-strata stratified spaces from sufficiently close non-stratified samples. In total, these results combine to a sampling theorem guaranteeing the (approximate) inference of (persistent) stratified homotopy types from non-stratified samples for many examples of stratified spaces arising from geometrical scenarios.

Fri, 17 May 2024

15:00 - 16:00
L5

Persistent Minimal Models in Rational Homotopy Theory

Kelly Spry Maggs
(École Polytechnique Fédérale de Lausanne (EPFL))
Abstract
One-parameter persistence and rational homotopy theory are two different ‘torsion-free’ algebraic models of space. Each enhances the cochain complex with additional algebraic structure— persistence equips cochain complexes with an action of a polynomial coefficient ring; rational homotopy theory equips cochains complexes with a graded-commutative product.
 
The persistent minimal model we introduce in this talk reconciles these two types of algebraic structures. Generalizing the classical case, we will describe how persistent minimal models are built by successively attaching the persistent rational homotopy groups into the persistent CDGA model. The attaching maps dualize to a new invariant called the persistent rational k-invariant.
 
This is joint work with Samuel Lavenir and Kathryn Hess: https://arxiv.org/abs/2312.08326


 

Fri, 03 May 2024

15:00 - 16:00
L5

Local systems for periodic data

Adam Onus
(Queen Mary University of London)
Abstract

 

Periodic point clouds naturally arise when modelling large homogenous structures like crystals. They are naturally attributed with a map to a d-dimensional torus given by the quotient of translational symmetries, however there are many surprisingly subtle problems one encounters when studying their (persistent) homology. It turns out that bisheaves are a useful tool to study periodic data sets, as they unify several different approaches to study such spaces. The theory of bisheaves and persistent local systems was recently introduced by MacPherson and Patel as a method to study data with an attributed map to a manifold through the fibres of this map. The theory allows one to study the data locally, while also naturally being able to appeal to local systems of (co)sheaves to study the global behaviour of this data. It is particularly useful, as it permits a persistence theory which generalises the notion of persistent homology. In this talk I will present recent work on the theory and implementation of bisheaves and local systems to study 1-periodic simplicial complexes. Finally, I will outline current work on generalising this theory to study more general periodic systems for d-periodic simplicial complexes for d>1. 

Fri, 26 Apr 2024

15:00 - 16:00
L5

Lagrangian Hofer metric and barcodes

Patricia Dietzsch
(ETH Zurich)
Further Information

Patricia is a Postdoc in Mathematics at ETH Zürich, having recently graduated under the supervision of Prof. Paul Biran.

Patricia is working in the field of symplectic topology. Some key words in her current research project are: Dehn twist, Seidel triangle, real Lefschetz fibrations and Fukaya categories. Besides this, she is a big fan of Hofer's metric, expecially of the Lagrangian Hofer metric and the many interesting open questions related to it. 

Abstract

 

This talk discusses an application of Persistence Homology in the field of Symplectic Topology. A major tool in Symplectic Topology are Floer homology groups. These are algebraic invariants that can be associated to pairs of Lagrangian submanifolds. A richer algebraic invariant can be obtained using 
filtered Lagrangian Floer theory. This gives rise to a persistence module and a barcode. Its bar lengths are invariants for the pair of Lagrangians. 
 
We explain how these numbers can be used to estimate the Lagrangian Hofer distance between the two Lagrangians: It is a well-known stability result  that the bar lengths are lower bounds of the distance. We show how to get an upper bound of the distance in terms of the bar lengths in the special case of equators in a cylinder.
Fri, 08 Mar 2024

15:00 - 16:00
L6

Topological Perspectives to Characterizing Generalization in Deep Neural Networks

Tolga Birdal
((Imperial College)
Further Information

 

Dr. Tolga Birdal is an Assistant Professor in the Department of Computing at Imperial College London, with prior experience as a Senior Postdoctoral Research Fellow at Stanford University in Prof. Leonidas Guibas's Geometric Computing Group. Tolga has defended his master's and Ph.D. theses at the Computer Vision Group under Chair for Computer Aided Medical Procedures, Technical University of Munich led by Prof. Nassir Navab. He was also a Doktorand at Siemens AG under supervision of Dr. Slobodan Ilic working on “Geometric Methods for 3D Reconstruction from Large Point Clouds”. His research interests center on geometric machine learning and 3D computer vision, with a theoretical focus on exploring the boundaries of geometric computing, non-Euclidean inference, and the foundations of deep learning. Dr. Birdal has published extensively in leading academic journals and conference proceedings, including NeurIPS, CVPR, ICLR, ICCV, ECCV, T-PAMI, and IJCV. Aside from his academic life, Tolga has co-founded multiple companies including Befunky, a widely used web-based image editing platform.

Abstract

 

Training deep learning models involves searching for a good model over the space of possible architectures and their parameters. Discovering models that exhibit robust generalization to unseen data and tasks is of paramount for accurate and reliable machine learning. Generalization, a hallmark of model efficacy, is conventionally gauged by a model's performance on data beyond its training set. Yet, the reliance on vast training datasets raises a pivotal question: how can deep learning models transcend the notorious hurdle of 'memorization' to generalize effectively? Is it feasible to assess and guarantee the generalization prowess of deep neural networks in advance of empirical testing, and notably, without any recourse to test data? This inquiry is not merely theoretical; it underpins the practical utility of deep learning across myriad applications. In this talk, I will show that scrutinizing the training dynamics of neural networks through the lens of topology, specifically using 'persistent-homology dimension', leads to novel bounds on the generalization gap and can help demystifying the inner workings of neural networks. Our work bridges deep learning with the abstract realms of topology and learning theory, while relating to information theory through compression.

 

Fri, 01 Mar 2024

15:00 - 16:00
L6

Applied Topology TBC

Zoe Cooperband
(University of Pennsylvania)
Further Information

Dr  Zoe Copperband is a member of the Penn Engineering GRASP Laboratory. Her recent preprint, Towards Homological Methods in Graphic Statics, can be found here.

Fri, 16 Feb 2024

15:00 - 16:00
L5

Morse Theory for Tubular Neighborhoods

Antoine Commaret
(INRIA Sophia-Antipolis)
Abstract
Given a set $X$ inside a Riemaniann manifold $M$ and a smooth function $f : X -> \mathbb{R}$, Morse Theory studies the evolution of the topology of the closed sublevel sets filtration $X_c = X \cap f^{-1}(-\infty, c]$ when $c \in \mathbb{R}$ varies using properties on $f$ and $X$ when the function is sufficiently generic. Such functions are called Morse Functions . In that case, the sets $X_c$ have the homotopy type of a CW-complex with cells added at every critical point. In particular, the persistent homology diagram associated to the sublevel sets filtration of a Morse Function is easily understood. 
 
In this talk, we will give a broad overview of the classical Morse Theory, i.e when $X$ is itself a manifold, before discussing how this regularity assumption can be relaxed. When $M$ is a Euclidean space, we will describe how to define a notion of Morse Functions, first on sets with positive reach (a result from Joseph Fu, 1988), and then for any tubular neighborhood of a set at a regular value of its distance function, i.e when $X = \{ x \in M, d_Y(x) \leq \varepsilon \}$ where $Y \subset M$ is a compact set and $\varepsilon > 0$ is a regular value of $d_Y$ the distance to $Y$ function.
 
 
If needed, here are three references :
 
Morse Theory , John Milnor, 1963
 
Curvature Measures and Generalized Morse Theory, Joseph Fu, 1988
Morse Theory for Tubular Neighborhoods, Antoine Commaret, 2024, Arxiv preprint https://arxiv.org/abs/2401.04034
Fri, 02 Feb 2024

15:00 - 16:00
L5

Algebraic and Geometric Models for Space Communications

Prof. Justin Curry
(University at Albany)
Further Information

Justin Curry is a tenured Associate Professor in the Department of Mathematics and Statistics at the University at Albany SUNY.

His research is primarily in the development of theoretical foundations for Topological Data Analysis via sheaf theory and category theory.

Abstract

In this talk I will describe a new model for time-varying graphs (TVGs) based on persistent topology and cosheaves. In its simplest form, this model presents TVGs as matrices with entries in the semi-ring of subsets of time; applying the classic Kleene star construction yields novel summary statistics for space networks (such as STARLINK) called "lifetime curves." In its more complex form, this model leads to a natural featurization and discrimination of certain Earth-Moon-Mars communication scenarios using zig-zag persistent homology. Finally, and if time allows, I will describe recent work with David Spivak and NASA, which provides a complete description of delay tolerant networking (DTN) in terms of an enriched double category.

Fri, 26 Jan 2024

15:00 - 16:00
L5

Expanding statistics in phylogenetic tree space

Gillian Grindstaff
(Mathematical Institute)
Abstract
For a fixed set of n leaves, the moduli space of weighted phylogenetic trees is a fan in the n-pointed metric cone. As introduced in 2001 by Billera, Holmes, and Vogtmann, the BHV space of phylogenetic trees endows this moduli space with a piecewise Euclidean, CAT(0), geodesic metric. This has be used to define a growing number of statistics on point clouds of phylogenetic trees, including those obtained from different data sets, different gene sequence alignments, or different inference methods. However, the combinatorial complexity of BHV space, which can be most easily represented as a highly singular cube complex, impedes traditional optimization and Euclidean statistics: the number of cubes grows exponentially in the number of leaves. Accordingly, many important geometric objects in this space are also difficult to compute, as they are similarly large and combinatorially complex. In this talk, I’ll discuss specialized regions of tree space and their subspace embeddings, including affine hyperplanes, partial leaf sets, and balls of fixed radius in BHV tree space. Characterizing and computing these spaces can allow us to extend geometric statistics to areas such as supertree contruction, compatibility testing, and phylosymbiosis.


 

Fri, 19 Jan 2024

15:00 - 16:00
L4

The Function-Rips Multifiltration as an Estimator

Steve Oudot
(INRIA - Ecole Normale Supérieure)
Abstract

Say we want to view the function-Rips multifiltration as an estimator. Then, what is the target? And what kind of consistency, bias, or convergence rate, should we expect? In this talk I will present on-going joint work with Ethan André (Ecole Normale Supérieure) that aims at laying the algebro-topological ground to start answering these questions.

Fri, 01 Dec 2023

15:00 - 16:00
L5

Computing algebraic distances and associated invariants for persistence

Martina Scolamiero
(KTH Stockholm)
Further Information

Martina Scolamiero is an Assistant Professor in Mathametics with specialization in Geometry and Mathematical Statistics in Artificial Intelligence.

Her research is in Applied and Computational Topology, mainly working on defining topological invariants which are suitable for data analysis, understanding their statistical properties and their applicability in Machine Learning. Martina is also interested in applications of topological methods to Neuroscience and Psychiatry.

Abstract

Pseudo metrics between persistence modules can be defined starting from Noise Systems [1].  Such metrics are used to compare the modules directly or to extract stable vectorisations. While the stability property directly follows from the axioms of Noise Systems, finding algorithms or closed formulas to compute the distances or associated vectorizations  is often a difficult problem, especially in the multi-parameter setting. In this seminar I will show how extra properties of Noise Systems can be used to define algorithms. In particular I will describe how to compute stable vectorisations with respect to Wasserstein distances [2]. Lastly I will discuss ongoing work (with D. Lundin and R. Corbet) for the computation of a geometric distance (the Volume Noise distance) and associated invariants on interval modules.

[1] M. Scolamiero, W. Chachólski, A. Lundman, R. Ramanujam, S. Oberg. Multidimensional Persistence and Noise, (2016) Foundations of Computational Mathematics, Vol 17, Issue 6, pages 1367-1406. doi:10.1007/s10208-016-9323-y.

[2] J. Agerberg, A. Guidolin, I. Ren and M. Scolamiero. Algebraic Wasserstein distances and stable homological invariants of data. (2023) arXiv: 2301.06484.

Fri, 24 Nov 2023

15:00 - 16:00
L5

Indecomposables in multiparameter persistence

Ulrich Bauer
(TU Munich)
Further Information

Ulrich Bauer is an associate professor (W3) in the department of mathematics at the Technical University of Munich (TUM), leading the Applied & Computational Topology group. His research revolves around application-motivated concepts and computational methods in topology and geometry, popularized by application areas such as topological data analysis. Some of his key research areas are persistent homology, discrete Morse theory, and geometric complexes.

Abstract

I will discuss various aspects of multi-parameter persistence related to representation theory and decompositions into indecomposable summands, based on joint work with Magnus Botnan, Steffen Oppermann, Johan Steen, Luis Scoccola, and Benedikt Fluhr.

A classification of indecomposables is infeasible; the category of two-parameter persistence modules has wild representation type. We show [1] that this is still the case if the structure maps in one parameter direction are epimorphisms, a property that is commonly satisfied by degree 0 persistent homology and related to filtered hierarchical clustering. Furthermore, we show [2] that indecomposable persistence modules are dense in the interleaving distance, and that being nearly-indecomposable is a generic property of persistence modules. On the other hand, the two-parameter persistence modules arising from interleaved sets (relative interleaved set cohomology) have a very well-behaved structure [3] that is encoded as a complete invariant in the extended persistence diagram. This perspective reveals some important but largely overlooked insights about persistent homology; in particular, it highlights a strong reason for working at the level of chain complexes, in a derived category [4].

 

[1] Ulrich Bauer, Magnus B. Botnan, Steffen Oppermann, and Johan Steen, Cotorsion torsion triples and the representation theory of filtered hierarchical clustering, Adv. Math. 369 (2020), 107171, 51. MR4091895

[2] Ulrich Bauer and Luis Scoccola, Generic multi-parameter persistence modules are nearly indecomposable, 2022.

[3] Ulrich Bauer, Magnus Bakke Botnan, and Benedikt Fluhr, Structure and interleavings of relative interlevel set cohomology, 2022.

[4] Ulrich Bauer and Benedikt Fluhr, Relative interlevel set cohomology categorifies extended persistence diagrams, 2022.

 

Tue, 21 Nov 2023
11:00
L1

Singularity Detection from a Data "Manifold"

Uzu Lim
(Mathematical Institute)

Note: we would recommend to join the meeting using the Teams client for best user experience.

Abstract

High-dimensional data is often assumed to be distributed near a smooth manifold. But should we really believe that? In this talk I will introduce HADES, an algorithm that quickly detects singularities where the data distribution fails to be a manifold.

By using hypothesis testing, rather than persistent homology, HADES achieves great speed and a strong statistical foundation. We also have a precise mathematical theorem for correctness, proven using optimal transport theory and differential geometry. In computational experiments, HADES recovers singularities in synthetic data, road networks, molecular conformation space, and images.

Paper link: https://arxiv.org/abs/2311.04171
Github link: https://github.com/uzulim/hades
 

Fri, 10 Nov 2023

15:00 - 16:00
L5

Topological Data Analysis (TDA) for Geographical Information Science (GIS)

Padraig Corcoran
(Cardiff University)
Further Information

Dr Padraig Corcoran is a Senior Lecturer and the Director of Research in the School of Computer Science and Informatics (COMSC) at Cardiff University.

Dr Corcoran has much experience and expertise in the fields of graph theory and applied topology. He is particularly interested in applications to the domains of geographical information science and robotics.

Abstract

Topological data analysis (TDA) is an emerging field of research, which considers the application of topology to data analysis. Recently, these methods have been successfully applied to research problems in the field of geographical information science (GIS). This includes the problems of Point of Interest (PoI), street network and weather analysis. In this talk I will describe how TDA can be used to provide solutions to these problems plus how these solutions compare to those traditionally used by GIS practitioners. I will also describe some of the challenges of performing interdisciplinary research when applying TDA methods to different types of data.

Fri, 03 Nov 2023

15:00 - 16:00
L5

The Expected Betti Numbers of Preferential Attachment Clique Complexes

Chunyin Siu
(Cornell)
Further Information

Chunyin Siu (Alex) is a PhD candidate at Cornell University at the Center for Applied Mathematics, and is a Croucher scholar (2019) and a Youde scholar (2018).

His primary research interests lie in the intersection of topological data analysis, network analysis, topological statistics and computational geometry. He is advised by Prof. Gennady Samorodnitsky. Before coming to Cornell University, he was a MPhil. student advised by Prof. Ronald (Lokming) Lui at the Chinese University of Hong Kong.

Abstract

The preferential attachment model is a natural and popular random graph model for a growing network that contains very well-connected ``hubs''. Despite intense interest in the higher-order connectivity of these networks, their Betti numbers at higher dimensions have been largely unexplored.

In this talk, after a brief survey on random topology, we study the clique complexes of preferential attachment graphs, and we prove the asymptotics of the expected Betti numbers. If time allows, we will briefly discuss their homotopy connectedness as well. This is joint work with Gennady Samorodnitsky, Christina Lee Yu and Rongyi He, and it is based on the preprint https://arxiv.org/abs/2305.11259

Fri, 27 Oct 2023

15:00 - 16:00
L5

Universality in Persistence Diagrams and Applications

Primoz Skraba
(Queen Mary University, Mathematical Sciences)
Further Information

 

Primoz Skraba is a Senior Lecturer in Applied and Computational Topology. His research is broadly related to data analysis with an emphasis on topological data analysis. Generally, the problems he considers span both theory and applications. On the theory side, the areas of interest include stability and approximation of algebraic invariants, stochastic topology (the topology of random spaces), and algorithmic research. On the applications side, he focuses on combining topological ideas with machine learning, optimization, and  other statistical tools. Other applications areas of interest include visualization and geometry processing.

He received a PhD in Electrical Engineering from Stanford University in 2009 and has held positions at INRIA in France and the Jozef Stefan Institute, the University of Primorska, and the University of Nova Gorica in Slovenia, before joining Queen Mary University of London in 2018. He is also currently a Fellow at the Alan Turing Institute.

Abstract

In this talk, I will present joint work with Omer Bobrowski:  a series of statements regarding the behaviour of persistence diagrams arising from random point-clouds. I will present evidence that, viewed in the right way, persistence values obey a universal probability law, that depends on neither the underlying space nor the original distribution of the point-cloud.  I will present two versions of this universality: “weak” and “strong” along with progress which has been made in proving the statements.  Finally, I will also discuss some applications of this phenomena based on detecting structure in data.

Fri, 20 Oct 2023

15:00 - 16:00
L5

Euler characteristic in topological persistence

Vadim Lebovici
(Mathematical Institute, University of Oxford)
Further Information

Vadim Lebovici is a post-doc in the Centre for Topological Data Anslysis. His research interests include: 

  • Multi-parameter persistent homology
  • Constructible functions and Euler calculus
  • Sheaf theory
  • Persistent magnitude
Abstract

In topological data analysis, persistence barcodes record the
persistence of homological generators in a one-parameter filtration
built on the data at hand. In contrast, computing the pointwise Euler
characteristic (EC) of the filtration merely records the alternating sum
of the dimensions of each homology vector space.

In this talk, we will show that despite losing the classical
"signal/noise" dichotomy, EC tools are powerful descriptors, especially
when combined with new integral transforms mixing EC techniques with
Lebesgue integration. Our motivation is fourfold: their applicability to
multi-parameter filtrations and time-varying data, their remarkable
performance in supervised and unsupervised tasks at a low computational
cost, their satisfactory properties as integral transforms (e.g.,
regularity and invertibility properties) and the expectation results on
the EC in random settings. Along the way, we will give an insight into
the information these descriptors record.

This talk is based on the work [https://arxiv.org/abs/2111.07829] and
the joint work with Olympio Hacquard [https://arxiv.org/abs/2303.14040].

 

 

Fri, 13 Oct 2023

15:00 - 16:00
L5

What do we want from invariants of multiparameter persistence modules?

Luis Scoccola
(Mathematical Institute, University of Oxford)
Further Information

Luis Scoccola is a post-doc in the Centre for Topological Data Analysis, Mathematical Institute. He is a mathematician and computer scientist working in computational topology and geometry, and applications to machine learning and data science.

Abstract

Various constructions relevant to practical problems such as clustering and graph classification give rise to multiparameter persistence modules (MPPM), that is, linear representations of non-totally ordered sets. Much of the mathematical interest in multiparameter persistence comes from the fact that there exists no tractable classification of MPPM up to isomorphism, meaning that there is a lot of room for devising invariants of MPPM that strike a good balance between discriminating power and complexity of their computation. However, there is no consensus on what type of information we want these invariants to provide us with, and, in particular, there seems to be no good notion of “global” or “high persistence” features of MPPM.

With the goal of substantiating these claims, as well as making them more precise, I will start with an overview of some of the known invariants of MPPM, including joint works with Bauer and Oudot. I will then describe recent work of Bjerkevik, which contains relevant open questions and which will help us make sense of the notion of global feature in multiparameter persistence.

 

Fri, 16 Jun 2023

15:00 - 16:00
Lecture room 5

Topology of Artificial Neuron Activations in Deep Learning

Bei Wang
Abstract

Deep convolutional neural networks such as GoogLeNet and ResNet have become ubiquitous in image classification tasks, whereas
transformer-based language models such as BERT and its variants have found widespread use in natural language processing. In this talk, I
will discuss recent efforts in exploring the topology of artificial neuron activations in deep learning, from images to word embeddings.
First, I will discuss the topology of convolutional neural network activations, which provides semantic insight into how these models
organize hierarchical class knowledge at each layer. Second, I will discuss the topology of word embeddings from transformer-based models.
I will explore the topological changes of word embeddings during the fine-tuning process of various models and discover model confusions in
the embedding spaces. If time permits, I will discuss on-going work in studying the topology of neural activations under adversarial attacks.
 

Fri, 02 Jun 2023

15:00 - 16:00
Lecture room 5

Projected barcodes and distances for multi-parameter persistence modules

Francois Petit
Abstract

In this talk, I will present the notion of projected barcodes and projected distances for multi-parameter persistence modules. Projected barcodes are defined as derived pushforward of persistence modules onto R. Projected distances come in two flavors: the integral sheaf metrics (ISM) and the sliced convolution distances (SCD). I will explain how the fibered barcode is a particular instance of projected barcodes and how the ISM and the SCD provide lower bounds for the convolution distance. 

Furthermore, in the case where the persistence module considered is the sublevel-sets persistence modules of a function f : X -> R^n, we will explain how, under mild conditions, the projected barcode of this module by a linear map u : R^n \to R is the collection of sublevel-sets barcodes of the composition uf . In particular, it can be computed using software dedicated to one-parameter persistence modules. This is joint work with Nicolas Berkouk.

Fri, 26 May 2023

15:00 - 16:00
Lecture room 5

DREiMac: Dimensionality Reduction with Eilenberg-Maclane Coordinates

Jose Perea
Abstract

Dimensionality reduction is the machine learning problem of taking a data set whose elements are described with potentially many features (e.g., the pixels in an image), and computing representations which are as economical as possible (i.e., with few coordinates). In this talk, I will present a framework to leverage the topological structure of data (measured via persistent cohomology) and construct low dimensional coordinates in classifying spaces consistent with the underlying data topology.

Fri, 19 May 2023

15:00 - 16:00
Lecture room 5

Some recent progress in random geometric graphs: beyond the standard regimes

Xiaochuan Yang
Abstract

I will survey on the cluster structure of random geometric graphs in a regime that is less discussed in the literature. The statistics of interest include the number of k-components, the number of components, the number of vertices in the giant component, and the connectivity threshold. We show LLN and normal/Poisson approximation by Stein's method. Based on recent joint works with Mathew Penrose (Bath).

Fri, 12 May 2023

15:00 - 16:00
Lecture room 5

TBC

Abhishek Rathod
Abstract

TBC