Past Applied Topology Seminar

I will review notions of categorical complexity, and the more recent work of Biran, Cornea and Zhang on fragmentation in triangulated persistence categories (TPCs), then go on to discuss applications of this to filtered topology. In particular, we will consider a suitable category of filtered topological spaces and detail some constructions and properties, before showing that an associated 'filtered stable homotopy category' is a TPC. I will then give some interesting results relating to this.

Studying the topology of zero sets of maps is a central topic in many areas of mathematics. Classical homological invariants, such as Betti numbers, are not always suitable for this purpose due to the fact that they do not distinguish between topological features of different sizes. Topological data analysis provides a way to study topology coarsely by ignoring small-scale features. This approach yields generalizations of a number of classical theorems, such as Bézout's theorem and Courant’s nodal domain theorem, to a wider class of maps. We will explain this circle of ideas and discuss potential directions for future research. The talk is partially based on joint works with L. Buhovsky, J. Payette, I. Polterovich, L. Polterovich and E. Shelukhin.

In this talk, we present a topological framework for interpreting the latent representations of Multilayer Perceptrons (MLPs) [1] using tools from Topological Data Analysis. Our approach constructs a simplicial tower, a sequence of simplicial complexes linked by simplicial maps, to capture how the topology of data evolves across network layers. This construction is based on the pullback of a cover tower on the output layer and is inspired by the Multiscale Mapper algorithm. The resulting commutative diagram enables a dual analysis: layer persistence, which tracks topological features within individual layers, and MLP persistence, which monitors how these features transform across layers. Through experiments on both synthetic and real-world medical datasets, we demonstrate how this method reveals critical topological transitions, identifies redundant layers, and provides interpretable insights into the internal organization of neural networks.

An arrangement of hypersurfaces in projective space is strict normal crossing if and only if its Euler discriminant is nonzero. We study the critical loci of all Laurent monomials in the equations of the smooth hypersurfaces. These loci form an irreducible variety in the product of two projective spaces, known in algebraic statistics as the likelihood correspondence and in particle physics as the scattering correspondence. We establish an explicit determinantal representation for the bihomogeneous prime ideal of this variety.

Joint work with T. Kahle, B. Sturmfels, M. Wiesmann

The cover of a dataset is a fundamental concept in computational geometry and topology. In TDA (topological data analysis), it is especially used in computing persistent homology and data visualisation using Mapper. However only rudimentary methods have been used to compute a cover. In this talk, we formulate the cover computation problem as a general optimisation problem with a well-defined loss function, and use gradient descent to solve it. The resulting algorithm, ShapeDiscover, substantially improves quality of topological inference and data visualisation. We also show some preliminary applications in scRNA-seq transcriptomics and the topology of grid cells in the rats' brain. This is a joint work with Luis Scoccola and Heather Harrington.

Alesker's theory of generalized valuations unifies smooth measures and constructible functions on real analytic manifolds, extending classical operations on measures. Therefore, operations on generalized valuations can be used to define integral transforms that unify both classical Radon transforms and their topological analogues based on the Euler characteristic, which have been successfully used in shape analysis. However, this unification is proven under rather restrictive assumptions in Alesker's original paper, leaving key aspects conjectural. In this talk, I will present a recent result obtained with A. Bernig that significantly closes this gap by proving that the two approaches indeed coincide on constructible functions under mild transversality assumptions. Our proof relies on a comparison between these operations and operations on characteristic cycles.

Geometry and topology call tell us about the shape of data. In this talk, I will give an introduction to my work on learning stratified spaces from samples, look at the use of persistent homology in cell biophysics, and apply persistence in understanding material structures.

The Morse-Conley complex is a central object in information compression in topological data analysis, as well as the application of homological algebra to analysing dynamical systems. Given a poset-graded chain complex, its Morse-Conley complex is the optimal chain-homotopic reduction of the initial complex that respects the poset grading.  In this work, we give a purely algebraic derivation of the Morse-Conley complex using homological perturbation theory. Unlike Forman’s discrete Morse theory for cellular complexes, our algebraic formulation does not require the computation of acyclic partial matchings of cells.  We show how this algebraic perspective also yields efficient algorithms for computing the Conley complex.  This talk features joint work with Álvaro Torras Casas and Ulrich Pennig in "Computing Connection Matrices of Conley Complexes via Algebraic Morse Theory" (arXiv:2503.09301). 
 

Multiparameter persistent homology is a generalization of persistent homology that allows for more than a single filtration function. Such constructions arise naturally when considering data with outliers or variations in density, time-varying data, or functional data. Even though its algebraic roots are substantially more complicated, several new invariants have been proposed recently. In this talk, I will go over such invariants, as well as their stability, vectorizations and implementations in statistical machine learning.

In this talk, I will discuss Figalli and Gigli’s formulation of optimal transport between non-negative Radon measures in the setting of metric pairs. This framework allows for the comparison of measures with different total masses by introducing an auxiliary set that compensates for mass discrepancies. Within this setting, classical characterisations of optimal transport plans extend naturally, and the resulting spaces of measures are shown to be complete, separable, geodesic, and non-branching, provided the underlying space possesses these properties. Moreover, we prove that the spaces of measures 
equipped with the $L^2$-optimal partial transport metric inherit non-negative curvature in the sense of Alexandrov. Finally, generalised spaces of persistence diagrams embed naturally into these spaces of measures, leading to a unified perspective from which several known geometric properties of generalised persistence diagram spaces follow. These results build on recent work by Divol and Lacombe and generalise classical results in optimal transport.

Informally, monodromy captures the behavior of objects when one circles around a singularity. In persistent homology, non-trivial monodromy has been observed in the case of biparameter filtrations obtained by sublevel sets of a continuous function [1]. One might consider the fundamental group of an admissible open subspace of all lines defining linear one-parameter reductions of a bi-parameter filtration. Monodromy occurs when this fundamental group acts non-trivially on the persistence space, i.e. the collection of all the persistence diagrams obtained for each linear one-parameter reduction of the bi-parameter filtration. Here, under some tameness assumptions, we formalize the monodromy behavior in algebraic terms, that is in terms of the persistence module associated with a bi-parameter filtration. This allows to translate monodromy in terms of persistence module presentations as bigraded modules. We prove that non-trivial monodromy involves generators within the same summand in the direct sum decomposition of a persistence module. Hence, in particular interval-decomposable persistence modules have necessarily trivial monodromy group.

Dey and Xin (J. Appl.Comput.Top. 2022) describe an algorithm to decompose finitely presented multiparameter persistence modules using a matrix reduction algorithm. Their algorithm only works for modules whose generators and relations are distinctly graded. We extend their approach to work on all finitely presented modules and introduce several improvements that lead to significant speed-ups in practice.

Our algorithm is FPT with respect to the maximal number of relations with the same degree and with further optimisation we obtain an O(n³) algorithm for interval-decomposable modules. As a by-product to the proofs of correctness we develop a theory of parameter restriction for persistence modules. Our algorithm is implemented as a software library aida which is the first to enable the decomposition of large inputs.

This is joint work with Tamal Dey and Michael Kerber.

Topological data analysis, and in particular persistent homology, has provided robust results for numerous applications, such as protein structure, cancer detection, and material science. In the field of neuroscience, the applications of TDA are abundant, ranging from the analysis of single cells to the analysis of neuronal networks. The topological representation of branching trees has been successfully used for a variety of classification and clustering problems of neurons and microglia, demonstrating a successful path of applications that go from the space of trees to the space of barcodes. In this talk, I will present some recent results on topological representation of brain cells, with a focus on neurons. I will also describe our solution for solving the inverse TDA problem on neurons: how can we efficiently go from persistence barcodes back to the space of neuronal trees and what can we learn in the process about these spaces. Finally, I will demonstrate how algebraic topology can be used to understand the links between single neurons and networks and start understanding the brain differences between species. The organizational principles that distinguish the human brain from other species have been a long-standing enigma in neuroscience. Human pyramidal cells form highly complex networks, demonstrated by the increased number and simplex dimension compared to mice. This is unexpected because human pyramidal cells are much sparser in the cortex. The number and size of neurons fail to account for this increased network complexity, suggesting that another morphological property is a key determinant of network connectivity. By comparing the topology of dendrites, I will show that human pyramidal cells have much higher perisomatic (basal and oblique) branching density. Therefore greater dendritic complexity, a defining attribute of human L2 and 3 neurons, may provide the human cortex with enhanced computational capacity and cognitive flexibility.

In this talk, I will give an overview of recent joint work on Topological Data Analysis (TDA). The first one is an application of TDA to quantify porosity in pathological bone tissue. The second is an extension of persistent homology to directed simplicial complexes. Lastly, we present an evaluation of the persistent Laplacian in machine learning tasks. This is joint work with Ysanne Pritchard, Aikta Sharma, Claire Clarkin, Helen Ogden, and Sumeet Mahajan; David Mendez; and Tom Davies and Zhengchao Wang, respectively.
 

Persistent homology is infeasible to compute when a dataset is very large. Inspired by the bootstrapping method, Chazal et al. (2014) proposed a multiple subsampling approach to approximate the persistence landscape of a massive dataset. In this talk, I will present an extension of the multiple subsampling method to a broader class of vectorizations of persistence diagrams and to persistence diagrams directly. First, I will review the statistical foundation of the multiple subsampling approach as applied to persistence landscapes in Chazal et al. (2014). Next, I will talk about how this analysis extends to a class of vectorized persistence diagrams called Hölder continuous vectorizations. Finally, I will address the challenges in applying this method to raw persistence diagrams for two measures of centrality: the mean persistence measure and the Fréchet mean of persistence diagrams. I will demonstrate these methods through simulation results and applications in estimating data shapes. 

Multiparameter persistence is an area of topological data analysis that synthesises the geometric information of a topological space via filtered homology. Given a topological space and a function on it, one can consider a filtration given by the sublevel sets of the space induced by the function and then take the homology of such filtration. In the case when the filtering function assumes values in the real plane, the homological features of the filtered object can be recovered through a "curved" grid on the plane called the extended Pareto grid of the function. In this talk, we explore how the computation of the biparameter matching distance between regular filtering functions on a regular manifold depends on the extended Pareto grid of these functions. 
 

To neural activity one may associate a space of correlations and a space of population vectors. These can provide complementary information. Assume the goal is to infer properties of a covariate space, represented by ochestrated activity of the recorded neurons. Then the correlation space is better suited if multiple neural modules are present, while the population vector space is preferable if neurons have non-convex receptive fields. In this talk I will explain how to coherently combine both pieces of information in a bifiltration using Dowker complexes and their total weights. The construction motivates an interesting extension of Dowker’s duality theorem to simplicial categories associated with two composable relations, I will explain the basic idea behind it’s proof.

Discrete Morse theory serves as a combinatorial tool for simplifying the structure of a given (regular) CW-complex up to homotopy equivalence, in terms of the critical cells of discrete Morse functions. In this talk, I will introduce a refinement of this theory that not only ensures homotopy equivalence with the simplified CW-complex but also guarantees a Whitehead simple homotopy equivalence. Furthermore, it offers an explicit description of the construction of the simplified Morse complex and provides bounds on the dimension of the complexes involved in the Whitehead deformation.
This refined approach establishes a suitable theoretical framework for addressing various problems in combinatorial group theory and topological data analysis. I will show applications of this technique to the Andrews-Curtis conjecture and computational methods for inferring the fundamental group of point clouds.

This talk is based on the article: Fernandez, X. Morse theory for group presentations. Trans. Amer. Math. Soc. 377 (2024), 2495-2523.

State-of-the art experimental data promises exquisite insight into the spatial heterogeneity in tissue samples. However, the high level of detail in such data is contrasted with a lack of methods that allow an analysis that fully exploits the available spatial information. Persistent Homology (PH) has been very successfully applied to many biological datasets, but it is typically limited to the analysis of single species data. In the first part of my talk, I will highlight two novel techniques in relational PH that we develop to encode spatial heterogeneity of multi species data. Our approaches are based on Dowker complexes and Witness complexes. We apply the methods to synthetic images generated by an agent-based model of tumour-immune cell interactions. We demonstrate that relational PH features can extract biological insight, including the dominant immune cell phenotype (an important predictor of patient prognosis) and the parameter regimes of a data-generating model. I will present an extension to our pipeline which combines graph neural networks (GNN) with local relational PH and significantly enhances the performance of the GNN on the synthetic data. In the second part of the talk, I will showcase a noise-robust extension of Reani and Bobrowski’s cycle registration algorithm  (2023) to reconstruct 3D brain atlases of Drosophila flies from a sequence of μ-CT images.

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.

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. 

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.

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.

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.

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.

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.

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
 

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.

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

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.

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].

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.

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.
 

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.

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.

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).

TBC