"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 $A$.
"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.
"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 Csaba Schneider and 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 is doing his doctorate in Brasilia with Pavel Zalesskii now.
"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 is doing his doctorate now with Viktor Bekkert here at the UFMG, and I'm 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 is doing his doctorate now 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. Diogo is doing his doctorate with me now.
"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 is doing is doctorate with me now.
"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 continues in Samuel's doctorate, which he's doing with Kostia at the University of Sao Paulo, and which I'm 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 is doing his doctorate with me now.
"Polinômios, Corpos de Decomposição e uma Introdução à Teoria de Galois": Leandro learned some Galois Theory and wrote a friendly introduction to the subject.
"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.
Marlon Stefano Fernandes Estanislau, Samuel Amador dos Santos Quirino (cosupervisor), Júlio César Magalhães Marques (cosupervisor)
Matheus Johnny Caetano
Enzo Marques Lavezzo, Thiago Fernando de Souza
Follows a list of topics I would like very much to understand but which I probably won't manage to learn by myself. Any one of them could probably be a (not at all easy) Master's project -- if you're interested let me know.
Natural operations on modules for finite groups give you direct sums of modules. When you do the same thing with profinite groups and modules you probably get a sort of "profinite sheaf". These should be dual to "normal sheaves" (which I also don't understand). I'd like to study these objects and the duality between them.
The algebras I'm interested in have two reasonably well behaved kinds of modules: discrete and pseudocompact. The category of
either of these is abelian (which is v important!). But sometimes it would be useful to have a category containing BOTH these
types of modules at once, but when you do this abelian-ness fails. A fancy new type of maths called "condensed mathematics"
has fancy ways to deal with this type of problem (more sheaves!). I'd like to learn a bit about it and I'd like to know if
it's helpful for my modules. We should probably learn the prerequisites and then try and read
UPDATE: André Contiero and I are organizing a seminar to learn about this stuff. Let me know if you'd like to participate!
I really want to understand this article, but there is a lot of prerequiste algebraic geometry that I don't know. So I'm asking for a volunteer to learn enough commutative algebra and algebraic geometry with me that we can try to understand this article.