| Thursday, January 30 |
| 13:15 - 15:30 |
Registration
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| 13:30 - 15:30 |
Lunch
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| TBA |
| TBA |
| 15:30 - 16:15 |
Sergio Ardanza-Trevijano - Topological data analysis of particulate materials: from statics to dynamics
Abstract ▼
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| The typical topological data analysis pipeline using persistent homology involves the choice of a simplicial complex filtration, a vectorization (or kernel) method, and a statistical/machine learning procedure. We will explore the benefits and challenges of aggregating persistent curves applied in a granular material context and how some of these and other tools coming from persistent homology may be used in a dynamic setting.
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| 16:15 - 16:45 |
Coffee Break
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| 16:45 - 17:30 |
Rubén Ballester - Topological transformers: Learning with high-order attention
Abstract ▼
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The last few years have been key for the development of advanced AI methods in modalities such as text or images (e.g., ChatGPT, Claude, Gemini…). Moreover, there has been increasing interest in graph deep learning, with the objective of developing neural networks whose tasks involve annotated graphs (i.e., graphs whose vertices—and possibly edges—have associated data). However, graphs are naturally limited to binary relationships between vertices, whereas much real-world data exhibits higher-order relationships—for example, rings in molecules or simplices in a co-authorship network representing papers written concurrently by a set of authors.
In this talk, we will focus on an increasingly relevant field known as topological deep learning, which addresses the problem of learning in higher-order domains—in our case, cell complexes. Specifically, we will dive into topological transformers, a transformer-like architecture that adapts state-of-the-art methods from text and vision to cell-complex-based learning. We will also demonstrate how these methods can be applied to molecular tasks, outperforming other specialized neural networks for molecular learning. |
| 17:45 - 18:30 |
Víctor Toscano - Interpolation and Function Approximation Using Neural Networks and Barycentric Coordinates
Abstract ▼
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| This talk introduces an innovative approach to Bayesian Neural Networks (BNNs) where neuron weights and biases are derived from barycentric coordinates instead of traditional linear function parameters. This methodology, implemented through the MultiSegmentNetwork architecture, addresses the problem of piecewise linear interpolation for a given set of points (xi, yi), where xi represents the input and yi its associated value, simulating a function f(x). Furthermore, this approach extends to the approximation of continuous piecewise linear and strictly continuous functions, offering an efficient and accurate alternative for applications in data analysis, modeling, and prediction. By the end, we use persistent entropy in order to calculate the minimum number of points we need to create the network that approximate a continuous function veryfing a specific degree of approximation. |
| 21:00 |
Social Dinner - Restaurante El Rincón del Vino
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| Friday, January 31 |
| 10:00 - 10:45 |
Elena Camacho - Quantitative Approaches to Early Human Embryonic Development: Data, Models, and Open Challenges
Abstract ▼
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Understanding how organisms develop and grow is crucial for deciphering life’s diversity and informing medical advances, including the bioengineering of tissues and organs. However, the complexity of the underlying biological processes and their interactions across scales leaves many questions unanswered. In particular, how complex networks of genes, proteins, and cells interpret information to produce tissues and organs of defined sizes, shapes, and patterns remains an active area of research.
Recent experimental advances—for instance, CRISPR-Cas9 fluorescent reporters or single-cell omics technology—now enable the collection of rich quantitative data on cell signalling and fate decisions. However, integrating and leveraging these data to understand underlying mechanisms remains challenging. In this talk, I will discuss our work combining live-cell imaging of signalling and cell-fate reporters with mathematical modelling to uncover the decision-making processes underlying early human embryonic development. I will highlight key open questions along with novel directions we are pursuing, emphasising opportunities for topological data analysis to contribute new methods and insights to this rapidly evolving field. |
| 10:45 - 11:30 |
Coffee Break and Poster Session
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| 11:30 - 12:15 |
Pablo Munarriz Senosiain - Quantum Approaches to Topological Data Analysis
Abstract ▼
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| In this talk, we will explore how TDA can be implemented on quantum computers and identify cases where quantum computation could outperform classical computation. More precisely, we will discuss how quantum computers can estimate Betti numbers and examine the computational cost of the algorithms involved in the process, which will allow us to understand the constraints required to efficiently apply these algorithms. |
| 12:30 - 13:15 |
Manuel Mellado - Decoding patterns of meaning: reassessing construal clusters and TDA perspectives
Abstract ▼
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| Empirical knowledge about construals—social affinity groups that share similar patterns of meaning to navigate social life—has grown significantly in recent years. This progress is largely due to the development of Construal Clustering Methods (CCMs), which analyze representative survey data to classify respondents into construal classes based on the similarity of their response patterns. In this talk, I will review existing CCMs and highlight their limitations, introducing a novel CCM called Bipolar Class Analysis (BCA), designed specifically for bipolar question data. Additionally, I will compare all CCMs through simulations, demonstrating the strengths of BCA as well as its limitations, which point to directions for future research. Finally, I will explore how this class of problems can be enriched using TDA and how BCA could support TDA tools in uncovering new insights in the social sciences. |
| 13:30 - 15:30 |
Lunch
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| 15:30 - 16:15 |
Adrián Inés - A Topological Approach for Semi-Supervised Learning: measuring distances with partial matchings
Abstract ▼
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| Nowadays, Machine Learning and Deep Learning methods have become the state-of-the-art approach to solve data classification tasks. In order to use those methods, it is necessary to acquire and label a considerable amount of data; however, this is not straightforward in some fields, since data annotation is time consuming and might require expert knowledge. This challenge can be tackled by means of semi-supervised learning methods that take advantage of both labelled and unlabelled data. In this work, we present new semi-supervised learning methods based on techniques from Topological Data Analysis (TDA). In particular, we have created two semi-supervised learning methods following two topological approaches. In the former, we have used a homological approach that consists in studying the persistence diagrams associated with the data using the bottleneck and Wasserstein distances. In the latter, we have considered the connectivity of the data. In addition, we have carried out a thorough analysis of the developed methods and a metric based on partial matchings proposed by R. González-Díaz et al. using 9 tabular datasets with low and high dimensionality. The results show that the developed semi-supervised methods outperform the results obtained with models trained with only manually labelled data, and are an alternative to other classical semi-supervised learning algorithms. |
| 16:30 - 17:15 |
Andrea Guidolin - Algebraic Wasserstein distances and stable homological invariants of data
Abstract ▼
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| Wasserstein distances between persistence diagrams are a common way to compare the outputs of an analysis pipeline based on persistent homology. In this talk, I will explain how a notion of p-norm for persistence modules leads to an algebraic version of Wasserstein distances which fit into a general framework for producing distances between persistence modules. I will then present stable invariants of persistence modules which depend on Wasserstein distances and can be computed efficiently. The use of these invariants in a machine learning context will be illustrated with some examples. |
