Diego Alberto Barceló Nieves

Diego Alberto Barceló Nieves

PhD student in Mathematics, UniVr

Talks and presentations

Mutation of torsion pairs of small modules and silting subcategories

November 14, 2025

Outreach talk, Chata Sport Ski, Kořenov, Czech Republic

The goal of the representation theory of algebras is to study the richness of the abstract algebraic structures known as “algebras” via their “modules” (or “representations”), each of which partially models the structure of the algebra. Torsion pairs of small modules over an algebra allow one to decompose its category of small modules into a pair of simpler “disjointed” subcategories from which the entire category can then be reconstructed. Since each torsion pair provides a unique decomposition, being able to control them yields a powerful tool for the study of this category. In this talk, we give a brief overview of the theory for mutation of torsion pairs of small modules, with a particular focus on their relation to silting subcategories—including “large” (i.e. either product or coproduct-closed) silting subcategories, which we note can be used to mutate bounded (co)silting complexes. Links: TBD.

Mutación de complejos cosilting de dos términos usando categorías extrianguladas

November 13, 2025

Outreach talk, Instituto de Matemáticas (IMATE), Universidad Nacional Autónoma de México, Mexico City, Mexico

English translation of the abstract: A torsion pair is a way of “twisting” a category into two “orthogonal” subcategories which we can further “untwist” in order to recover the whole category, allowing us to study it in its entirety via simpler portions of it. Each torsion pair provides a unique way of decomposing the category, by which the ability to change or “mutate” from one to another can be of great use. For an associative unital ring $R$, torsion pairs in $\text{mod}(R)$ are in bijection with two-term cosilting complexes, a special class of complexes of injective $R$-modules. Recently, a general theory of mutation of cosilting objects in triangulated categories with products was introduced in [ALSV25]; however, when applying it to the bounded homotopy category of injective $R$-modules in order to mutate a two-term cosilting complex, there are no guarantees that the resulting cosilting object is of the same type. In this talk, we present recent work in which we show how, by changing our focus to the extriangulated category of large injective copresentations $\mathcal{K}^{[0,1]}(\text{Inj}R)$, we can restrict the previous operation to one of mutation between two-term cosilting complexes. Links: slides.

Una introducción a las categorías extrianguladas

November 11, 2022

Outreach talk, Instituto de Matemática y Estadística Rafael Laguardia (IMERL), Universidad de la República, Montevideo, Uruguay

Shared the work done for my Master’s thesis Una introducción a las categorías extrianguladas (An Introduction to Extriangulated Categories, in English) with researchers and students from Uruguay, Chile, Venezuela, Switzerland and Mexico. Links: master’s thesis; animated slides; regular slides; recording; English translation of the abstract.

Una introducción a las categorías extrianguladas

September 19, 2022

Outreach talk, Instituto de Matemáticas (IMATE), Universidad Nacional Autónoma de México, Mexico City, Mexico

English translation of the abstract: Currently, the basic structures used in the study of homological algebra are abelian categories, exact categories and triangulated categories. A particular relationship between exact and triangulated categories is well-known–given by Frobenius categories and their associated stable categories–, and many results of homological nature are valid in both contexts. However, the processes for transfering results between these two types of categories are quite complex. In order to overcome these difficulties, in 2019 H. Nakaoka and Y. Palu introduced a simultaneous generalization of exact categories and triangulated categories, axiomatizing the properties of the Ext1 bifunctors in both contexts which are relevant for the definition of cotorsion pairs, which they called extriangulated category. In this talk, we will give an introduction to this recently defined type of category and discuss some of its fundamental results. Links: abstract (in Spanish).

Telegram como herramienta de educación a distancia

September 11, 2020

Workshop, Online (remote), Mexico City, Mexico

Near the beginning of the SARS-CoV-2 global pandemic–which forced teaching personnel from all around the world to rapidly adapt to teaching at a distance–I gave a short workshop on how to use the Telegram Messenger app as a powerful and versatile tool for online teaching to educators of the National Autonomous University of Mexico’s Faculty of Sciences. Links: recording.