"Propriedades cohomológicas de álgebras de incidência de posets": An efficient algorithm is constructed that calculates minimal projetive resolutions of simple modules for finite dimensional incidence algebras. The same algorithm calculates the Hochschild cohomology groups of these algebras. New reduction techniques are presented, allowing simplifications of the partially ordered set before the cohomology groups are calculated. Used together with previously known reductions, the simplifications can be drastic.
Projective resolutions of simple modules and Hochschild cohomology for incidence algebras
"A functorial approach to Gabriel quiver constructions": The thesis has two parts. The first and largest part is a continuation of work of Iusenko and myself, defining the Gabriel quiver of a pointed coalgebra and the path coalgebra as functors, in such a way that they form an adjoint pair. Further generalizations, for example allowing cases where the (co)algebra is basic but not pointed, are considered. In the second part, it is shown that the algebra of fixed points under a continuous homogeneous group action on a completed path algebra is again a completed path algebra.
A functorial approach to Gabriel $k$-quiver constructions for coalgebras and pseudocompact algebras
"Adjunções entre categorias de álgebras e extensões de quociente bifinito": The thesis consists of two essentially independent parts. Firstly, the adjunctions studied by Iusenko, myself and Quirino are generalised and refined, presenting a left adjoint to the functor sending a basic algebra $A$ to $A/J^n(A)$. In the case where $n=2$ this is a direct refinement of the adjunctions mentioned above, whereas for $n>2$ the conditions are more technical. In the second part, it is proved that if $B$ is a subalgebra of the finite dimensional algebra $A$ such that $A/B$ has finite projective dimension as a $B$-bimodule, then if $A$ has finite finitistic dimension, so does $B$. Examples show how such $B$ can be substantially more complex than the corresponding $A$.
Quotient bifinite extensions and the finitistic dimension conjecture
"Linearly topologized representations of algebras and coalgebras and their applications": vector spaces with a linear topology are studied from a foundational perspective, giving "tensor-hom type" adjunctions for many such classes of these spaces. The results are applied to classes of topological (co)modules for coalgebras and pseudocompact algebras. These results are in turn applied in the construction of Auslander-Reiten sequences for comodules, generalizing results of Chin, Kleiner and Quinn, and to tilting theory, giving a "pseudocompact version" of a theorem of Angeleri-Hügel and Coelho.
"Block Theory for Profinite Groups": the defect group of a block of a profinite group is defined and studied, giving several classifications. A Brauer Correspondence is proved for relatively pro-$p$ groups (we've since generalized the correspondence to profinite groups). Blocks of profinite groups with cyclic defect group are classified using a profinite analogue of Brauer Tree Algebras. Ricardo's thesis was awarded the best thesis of the UFMG mathematics department in 2021.
Block theory and Brauer's first main theorem for profinite groups
Blocks of profinite groups with cyclic defect group
Lucas worked on the "underlying vector spaces" that may be required when you want to do representation theory of pseudocompact algebras over a discrete field: there are direct sums of the field (discrete vector spaces) and direct products of the field (linearly compact vector spaces), and these categories are abelian. But when you want a category containing both sums and products, new techniques are required to get abelian categories.
Working with Marlon Stefano and me, Anderson classified a natural class of semidirect products of group using the theory of integral representations of $p$-groups, and in particular of the integral representations of the cyclic group of order $p^2$.
"Matemática Condensada": Matheus read some extremely dense lecture notes due to Peter Scholze wherein the concepts of a new, categorial approach to topological algebra are presented. Matheus made sense of some very complicated arguments and presented them in detail.
"Módulos de permutação $p$-ádicos para $p$-grupos abelianos elementares": Marlon studied $p$-adic representations of abelian $p$-groups, using a correspondence due to Butler to find an explicit example which answers a question left open in a paper of Zalesskii and I. Marlon is doing his doctorate with me now.
"The Kurosh Subgroup Theorem for profinite groups": Mattheus read Luis Ribes' new (at the time book) "Profinite Graphs and Groups", about profinite graphs and how they help to understand constructions in the combinatorial theory of profinite groups, and taught me a bit about it. Mattheus did his doctorate in Brasilia with Pavel Zalesskii.
"The finitistic dimension conjecture": Júlio César read two articles about the intoxicatingly easy-to-state-hard-to-prove finitistic dimension conjecture. The famous paper of Igusa and Todorov wherein the Igusa-Todorov functions are defined and a new (at the time) paper of Rickard relating the conjecture to seemingly unrelated questions in the derived category. Júlio César then his doctorate with Viktor Bekkert here at the UFMG, with me cosupervising.
"Definable Subcategories and the Ziegler Spectrum": João Vitor read Mike Prest's book "Purity, Spectra and Localisation", about what you can say about the modules for a ring by looking at the first order logical structure, and taught me a bit about it. João Vitor then did his doctorate with George Willis at the University of Newcastle, Australia.
"Quivers com potenciais": Diogo read the hugely influential article "Quivers with potentials and their representations I: Mutations" by Derksen, Weyman and Zelevinsky. What I most loved about the article is how it uses without apology the fact that the completed path algebra of a finite quiver (a pseudocompact algebra!) is a far better behaved object than the abstract path algebra.
"Álgebras de caminhos generalizadas e uma representação para K-álgebras de dimensão finita": Fernando read the article "Generalized path algebras" by Coelho and Liu. He used generalized path algebras to prove a sort of generalization to arbitrary finite dimensional algebras of Gabriel's famous theorem that every finite dimensional pointed algebra is isomorphic to a well-behaved quotient of a path algebra. Fernando then did his doctorate with me.
"Path coalgebra as a right adjoint functor": Samuel read the first article Kostia and I wrote, generalizing some results we proved essentially for finite dimensional algebras to arbitrary pointed coalgebras. This research continued in Samuel's doctorate, which he did with Kostia at the University of Sao Paulo, with me cosupervising.
"Teoria de Morita para coalgebras": Ricardo read diverse articles about coalgebras, with focus on understanding for coalgebras and pseudocompact algebras a version of Morita's fundamental theorem that (roughly speaking) every finite dimensional algebra has a "basic version" that can be treated combinatorially. Ricardo then did his doctorate with me.
"Polinômios, Corpos de Decomposição e uma Introdução à Teoria de Galois": Leandro learned some Galois Theory and wrote a friendly introduction to the subject.
"Álgebra Comutativa": After getting some background in rings and modules, we read the start of Miles Reid's book "Undergraduate Commutative Algebra", which is arguably not really an undergraduate book.
"Representações de posets e grupos abelianos": Douglas read David Arnold's book "Abelian groups and representations of finite partially ordered sets" and taught me a bit about representations of posets and how they help us understand, using a technique due to Butler, certain classes of torsion free abelian groups. Douglas will begin his Master's here soon.
"Grupos topológicos": André studied topological and profinite groups. He found an example showing that the topological space of double cosets of a profinite group can be surprisingly complicated (in particular, not homogenous). This has implications for a profinite version of the famous Double Coset Formula due to Mackey, but I haven't got round to looking at this in detail yet. André went on to do a master's at the Federal University of Rio de Janeiro (UFRJ).
"Grupos abelianos infinitos": If you think abelian groups are easy it's because you don't know abelian groups. João Vitor read and taught be about (the whole of!) Irving Kaplansky's fantastic but dense little book "Infinite abelian groups", which gives an overview of the theory of just how weird infinitely generated abelian groups can be. João Vitor went on to do his Master's with me.
Ana Twayene Pereira, Marlon Stefano Fernandes Estanislau
Lucas Giraldi Almeida Coimbra, Pedro Henrique de Oliveira Neves
Arthur Freiman da Silva, Enzo Marques Lavezzo, Gustavo Neves da Cruz, Ludovica Dias Zuppo Morães