Video which introduces the inner product as an operation that allows us to solve the problem of "finding coefficients" in finite-dimensional vector spaces in a simple and scalable manner through the use of orthogonal bases, and then uses this result to give a geometric interpretation of the inner product.
Video which introduces the notion of norm and its relationship with inner products, as well as that of orthonormal bases, which simplify the solution to the problem of "finding coefficients" even further. (Currently in production in collaboration with Animathica; the video script and animation code can be found in this repository.)
Video which illustrates the orthogonalization and orthonormalization processes in finite dimensional inner product spaces and explains the Gram-Schmidt Theorem.
Published in Universidad Nacional Autónoma de México, 2022
Recommended citation: D. A. Barceló Nieves. "Una introducción a las categorías extrianguladas". Universidad Nacional Autónoma de México, October 2022. http://132.248.9.195/ptd2022/septiembre/0831398/Index.html
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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.
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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).
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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.
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Gave a 4 hour minicourse based on the article Hereditary extriangulated categories: silting objects, mutation, negative extensions by M. Gorsky, H. Nakaoka and Y. Palu. Links: slides.
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Gave the second lecture in a series of lectures by Jan Šťovíček on (co)silting modules and torsion pairs at the Winter School on Large Modules and Endofiniteness. Links: slides and lecture notes (found at the bottom of the linked site).
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Links: abstract and (expanded) slides.
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Links: abstract.
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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.
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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.
Undergraduate course, National Autonomous University of Mexico, Faculty of Sciences, 2020
I taught several Linear Algebra courses, the first one–which started on January 27th, 2020–being initially on-site and the rest being fully online, up until the end of 2021. The course notes and exams (in Spanish) are available in this repository. Additionally, I also led a team of undergraduate students with whom I created educational Linear Algebra videos as part of the student animation project Animathica, working as creative director, writer, animator, narrator, composer and editor. All of our video scripts (in Spanish) and animation code can be found in this repository.
Undergraduate course, National Autonomous University of Mexico, Faculty of Sciences, 2022
I taught Computational Algorithms courses (online) during the Spring semesters of 2022 and 2023, mainly using the Julia programming language. Course materials such as the syllabus, interactive notebooks, and project assignments are available (in Spanish) at this repository, whereas lecture recordings may be found here.
Undergraduate course, National Autonomous University of Mexico, Faculty of Sciences, 2023
I taught a course on Group Theory. Course notes (in Spanish) will be made publicly available via this link once they are ready for widespread sharing.
Undergraduate and graduate courses, University of Verona, Department of Informatics, 2023
During my PhD studies, I have served as a TA for a graduate course in Homological Algebra, as well as several undergraduate courses in Abstract Algebra and Linear Algebra.