GENERAL DESCRIPTION
The session will present recent advances in Topological Data Analysis, a rapidly developing area at the interface of algebraic topology, geometry, and data science. The talks will cover current theoretical developments in persistent homology, including multiparameter persistence, descriptors of persistence modules, sheaf-theoretic methods, and the topology of geometric networks. They will also highlight how these techniques are being applied to the analysis of complex data arising in a variety of scientific disciplines. Applications will include medicine, neuroscience, and the study of complex systems, among other areas. The program is intended to provide both an overview of recent mathematical progress and a broad perspective on emerging applications, aiming to be accessible to participants with interests in topology, geometry, and data analysis.
____________________________________________________________________
SCHEDULE
Dijous 8 d’octubre (Pi i Sunyer)
16:00h Chain modularity, by Wojciech Chachólski (KTH Stockholm)
16:45h When networks degrade, what data survive? A sheaf-theoretic view of robustness in simplicial complex networks, by Daniel Hernández (University of Salamanca)
17:30h Topological analysis of human movement trajectories, by Laura Mestres (University of Barcelona)
Divendres, 9 d’octubre (Pere i Joan Coromines)
11:00h Investigating directionality and bias using quadratic forms, by Yossi Bokor Bleile (Institute of Science and Technology Austria)
11:45h Matchings induced by persistence maps: results and future questions, by Álvaro Torras (University of Seville)
12:30h Multiparameter persistence for clinical data analysis, by Marta Mariscal (University of Barcelona)
____________________________________________________________________
ABSTRACTS
Investigating directionality and bias using quadratic forms
by Yossi Bokor Bleile (ISTA Austria)
Abstract
Exploring morphology of tree-like and pile-of-stick-like structures is instrumental to under-standing both their functional properties and the processes that generate them. Quantifying and comparing such branching structures across different contexts remains a central challenge. We will look at how quadratic forms can be used to summarise and compare this directionality.
____________________________________________________________________
When networks degrade, what data survive? A sheaf-theoretic view of robustness in simplicial complex networks
by Daniel Hernández Serrano (University of Salamanca)
Abstract
Complex networks are commonly said to be robust when they preserve relevant structural or functional properties under failures or targeted attacks. Topological robustness usually asks whether connectivity or homological features survive failures in a network. In higher-order networks modeled by simplicial complexes, however, degradation may affect simplices of any dimension. Classical Betti numbers detect connected components and holes, but do not describe the higher-dimensional structure that supports them.
In this talk, we introduce thickness and cohesion as complementary refinements of simplicial cohomology. Thickness measures whether a topological feature depends on sufficiently high-dimensional interactions, while cohesion measures the role of higher-order adjacency patterns in holding that feature together. Persistence then provides a dynamic notion of resilience by tracking how these forms of support evolve during a degradation process.
We subsequently extend this framework to cellular sheaves, which encode local data and the maps through which those data are transmitted or compared across a network. The resulting theory studies not only whether the network remains topologically intact, but also which globally compatible data and cohomological obstructions survive. For discourse sheaves, for example, this amounts to asking whether generalized forms of agreement remain possible when communication channels or collective interaction contexts are progressively lost.
____________________________________________________________________
Multiparameter persistence for clinical data analysis
by Marta Mariscal (University of Barcelona)
Abstract
During the last years, multiparameter persistence has emerged as an active research area in topological data analysis. In this work, we present how some of its invariants can be used to characterize clinical images and train machine learning models for their classification.
____________________________________________________________________
Topological analysis of human movement trajectories
by Laura Mestres (University of Barcelona)
Abstract
Movement trajectories are time-dependent objects, making their comparison across experi-mental conditions a challenging problem. We propose an analysis pipeline in which trajectories in R³ are encoded using Echo State Networks, transforming each trajectory into a point in a common high-dimensional representation space. Point clouds formed from these represen-tations can then be studied using tools from Topological Data Analysis.
Persistent homology is used to obtain topological summaries of the resulting point clouds, and statistical differences between experimental conditions are assessed through distances between persistence diagrams and the Robinson-Turner permutation test. The approach is demonstrated on an existing dataset of hand trajectories recorded during a reaching task, involving different spatial distances and movement goals. The results illustrate how the combination of temporal encoding and topological summaries can characterize condition-dependent differences in the structure of movement data.
____________________________________________________________________
Matchings induced by persistence maps: results and future questions
by Álvaro Torras Casas (University of Seville)
Abstract
Persistent homology is a topological data analysis technique that quantifies features (such as connected components, cycles, or voids) across multiple scales into a single barcode invariant. These invariants can be derived from sources such as CT scans, finite metric spaces, and signals. In addition, modern techniques (such as fibered or projected barcodes) enable combining several parameters into one, while current algorithms handle computations efficiently. Finally, to compare barcodes in a meaningful way, one can use combinatorial metrics (such as bottleneck or Wasserstein distances) or vectorisation methods (such as landscapes or images).
In this talk, I will present induced matchings as a framework for comparing persistent homology barcodes based directly on the underlying data. First, I will provide a brief literature review covering recent results such as the Wasserstein stability theorem and the topological bootstrap, as well as matrix-based representations of morphisms between persistence modules (persistence maps). Next, I will motivate the need for a well-defined, additive, and stable invariant for persistence maps. I will then introduce our approaches to constructing such an invariant —namely, the block function and the induced matching. Finally, I will discuss recent work connecting persistence maps to multi-parameter persistence.
This is joint work with M. Delgado-Garrido, R. González-Díaz, L. Molberg-Salbu and M. Soriano-Trigueros.
____________________________________________________________________
Chain modularity
by Wojciech Chachólski (KTH Stockholm)
Abstract
In this talk I will characterise all chain modular functions over arbitrary posets and use it to shed some light on chain modular vector space representations. Why should anyone care? Well, I will explain how such functions might lead to Topological Data Analysis invariants.