# Álvaro Torras Casas

I am a Doctoral Student in the Geometry, Algebra, Mathematical Physics and Topology Research Group at Cardiff University. My supervisor is Dr. Ulrich Pennig.

I expect to complete my PhD in January 2021.

### Research Interests:

Computational Topology, Homological Algebra

## Preprints

### Distributing Persistent Homology via Spectral Sequences

Álvaro Torras Casas

Abstract: We set up the theory for a distributive algorithm for computing persistent homology. For this purpose we develop linear algebra of persistence modules. We present bases of persistence modules, and give motivation as for the advantages of using them. Our focus is on developing efficient methods for the computation of homology of chains of persistence modules. Later we give a brief, self contained presentation of the Mayer-Vietoris Spectral sequence. Then we study the Persistent Mayer-Vietoris Spectral Sequence and present a solution to the extension problem. Finally we outline applications of our results for Vietoris-Rips complexes, cubical complexes and $$\alpha$$-complexes.

## Talks

Input-Distributive Persistent Homology, ECTR 2019, University of Sheffield, 6th of June 2019.

Input-Distributive Persistent Homology, Guest talk at Topology Seminar, Universitat de Barcelona, 29th of May 2019.

Input-Distributive Persistent Homology, 2019 Welsh Mathematical Colloquium, Gregynog Hall, 21st of May 2019.

Input-Distributive Persistent Homology, GAPT seminar, Cardiff University, 21st of March 2019.

Distributing Persistent Homology using Spectral Sequences, Postgraduate Seminar, University of Leicester, 5th of December 2018.

Distributing Cohomology Computations via Sheaf Cohomology, 2018 Welsh Mathematical Colloquium, Gregynog Hall, 22nd of May 2018.

## Teaching at Cardiff University

Spring 2019, Tutor in Foundations II.

Spring 2019, Tutor in Finance I.

Autumn 2018, Tutor in Geometry.

Autumn 2017, Tutor in Linear Algebra.

Additionally, I am a frequent tutor in Maths Support. This is a drop-in service where students with mathematical questions can get help. This is provided by the Cardiff University School of Mathematics.