[Com-top_seminars] Talk: Connection Matrices via Algebraic Morse Theory - Álvaro Torras Casas

[MANUEL SORIANO TRIGUEROS]

Buenas tardes:

Vuelven los seminarios de topología computacional tras el parón veraniego. Esta vez Álvaro Torras Casas nos hablará el viernes de la semana que viene sobre matrices de conexión y teoría de Morse algebraica, a raíz de un trabajo publicado recientemente en Homology, Homotopy and Applications junto a Ambrose Yim (University of Oxford) y Ulrich Pennig (Cardiff University).

Además, el seminario incluirá una pequeña introducción sobre el rol que juegan las matrices de conexión en sistemas dinámicos,  impartida por Manuel Soriano Trigueros, añadiendo contexto al trabajo de Álvaro.

El seminario tendrá lugar el viernes 16 de octubre a las 12:00, por lo que invitamos a todos/as los/as participantes a quedarse después a tomar algo 🍻.

Título: Connection Matrices via Algebraic Morse Theory
Lugar: Seminario del departamento de Matemática Aplicada I, ETSII (B2.85)
Hora: 12:00
Fecha: 16 de octubre de 2026 (viernes)

Resumen
In topological data analysis, several algorithms and methods exist for the efficient computation of summaries that account for a single parameter:  a cubical filtration obtained from a CT scan, a Vietoris-Rips filtration from a large distance matrix, or an alpha filtration from millions of points. However, in applications, it is often natural to consider the relations between several parameters: e.g., voxel position and Hounsfield units for CT scans, or distance and sample densities for Vietoris-Rips complexes. Unfortunately, existing algorithms in this context are less efficient, and their complexity often increases rapidly with the number of variables.
With this motivation, one may ask whether it is possible to reduce the size of input multi-parameter filtrations without losing topological information. One can frame this problem from the perspective of chain complexes graded over a poset (the parameter space). In this context, one wishes to compute the complex of minimum size that is homotopy-equivalent to a given poset-graded complex; this is known as the Conley Complex, and the matrix associated with its differentials is the Connection Matrix.
A general approach involves using Discrete Morse Theory, where an acyclic partial matching of cells is used to formulate the gradient flow between critical cells. This idea can be reproduced at an algebraic level—matching isomorphic subspaces instead of cells—and is known as Algebraic Morse Theory.
In particular, I will show how this approach can be adapted to our problem to obtain explicit formulas for a contraction from any poset-graded complex to its Conley Complex. Our strategy uses an elementary contraction which is then modified via the Homological Perturbation Lemma. As a byproduct, this approach leads to an algorithm for computing the Connection Matrix that is analogous to the standard matrix reduction algorithm for one-parameter persistence computation. Finally, I will present example applications and future directions.

Saludos,

Manuel Soriano Trigueros
Prof. Ayudante Doctor / Assistant Professor

Dpto de Ciencias de la Computación e Inteligencia Artificial
Universidad de Sevilla


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