# RTAA

30 July, 2021

## Topics on the Igusa-Todorov functions

One of the most important conjecture in the representation theory of Artin algebras is the finitistic dimension conjecture. It states that $sup\{pd(M) : M\ \text{is a f.g module of finite projective dimension}\}$ is finite. In an attempt to prove the conjecture this conjecture Igusa and Todorov defined in “On finitistic global dimension conjecture for artin algebras”, two functions from the objects of modA to the natural numbers, which generalize the notion of projective dimension. Nowadays they are known as the Igusa-Todorov functions. In this talk I will introduce the Igusa-Todorov functions and their main properties. We will also see how to use these functions to prove the finitistic dimension conjecture in several contexts.

23 July, 2021

## Feixes instantons e representações de aljavas

O seminário terá como tema apresentar uma nova compactificação para o espaço de módulos (moduli spaces) de feixes intantons de carga $1$ sobre $\mathbb P^3$. Isto será feito identificando estes feixes com representações de uma aljava com três vértices e oito flechas que atendem certas propriedades e olhando essas representações no espaço de módulos de King de representações teta-estáveis sobre esta aljava.

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16 July, 2021

## On the modular automorphism of twisted Calabi-Yau algebras

A twisted Calabi-Yau algebra $A$ is, by definition, endowed with an automorphism $\sigma:A\to A$, well-defined only up to inner automorphisms, that controls the duality theory for representations of $A$. The algebra is Calabi-Yau, and not only twistedly so, precisely when $\sigma$ can be taken to be inner and even the identity map of $A$, but in general it cannot. The objective of the talk will be to present some results that give information about this automorphism in the general situation, and show how it can be used to construct interesting and non-trivial invariants of the algebra, of automorphisms and derivations of the algebra, and of related objects. Some applications of these invariants to the problem of computing Hochschild cohomology and the automorphism groups of our algebras will be discussed.

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25 June, 2021

## Variety defined by the Groebner basis of a module

This is a joint project with Ed. Green and Schroll Sibylle.
Our set up is an algebra of the form $kQ/I$, ($I$ is admissible, for simplification).
Given a right module M we define, using its right groebner basis, a variety. Each point in this variety corresponds to a module which has the same tip set. Each module corresponding to a point in the variety has the same dimension vector of the original module. We denote this variety by $V(M)$. There is a natural injection of sets $V(M) \to V_d(KQ)$, where $V_d(kQ)$ is the usual variety of representations Q representations of dimension vector $d$. We do not know if the injection is a morphism of varieties ( I conjecture it is). We also do not know which subgroup of $Gl_d$ let the subset of the variety be invariant, so we would have a grupo acting on it. Some curious results, for instance, is that if the module is an indecomposable projective then our variety consists of one point. I am open to suggestions etc...

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18 June, 2021

## Álgebra de Árvore de Brauer e Blocos de Grupos Profinitos

Sejam $G$ um grupo profinito , $k$ um corpo de característica $p> 0$. A álgebra de grupo completa de G, $k[[G]]$, possui uma decomposição como um produto direto de álgebras pseudocompactas indecomponíveis, que chamaremos de blocos. Cada bloco de $k[[G]]$ tem associado um pro-$p$ subgrupo de $G$ chamado grupo de defeito. O grupo de defeito proporciona uma medida que tão próximo estará um bloco de ser uma álgebra semisimples. Nesta palestra explicaremos como associar a cada bloco um grupo de defeito e descreveremos os blocos cujo grupo de defeito é um grupo cíclico usando a estrutura de álgebra de árvore de Brauer.

Este trabalho foi feito junto com o John Macquarrie.

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11 June, 2021

## Preprojective algebras: classical and higher

I will give an introduction to representations of quivers, aimed at non-experts, and then introduce preprojective algebras of quivers. Iyama showed how many nice aspects of the representations of quivers generalise to certain algebras of higher global dimension, and the preprojective algebra can be defined in this setting. I will explain some results on higher preprojective algebras obtained in joint work with Osamu Iyama.

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04 June, 2021

## Moduli of flags of sheaves: a quiver description

In [1], the moduli spaces of framed flags of sheaves on $\mathbb P^2$ were described by means of representations of the so-called enhanced ADHM quiver. We first review those results, with a recent refinement concerning the chamber structure in the space of stability parameters. We shall then discuss the obstruction theory for these moduli spaces, showing in general that they have a perfect obstruction theory, and providing for specific choices of invariants a class of unobstructed points. Finally, open problems and possible further developments will be presented. This is a joint work with Rodrigo von Flach and Marcos Jardim.

1. R. A. von Flach and M. Jardim, Moduli spaces of framed flags of sheaves on the projective plane. Journal of Geometry and Physics 118 (2017), 138–168.

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14 May, 2021

## Higher Auslander-Reiten theory: what is it and how to understand it

Higher Auslander-Reiten theory was introduced by Osamu Iyama in the 2000s as a generalization of the theory developed by Maurice Auslander and Idun Reiten in the 1970s. In this talk, we will discuss what is this theory and how to understand it. In doing so, we will examine both the definitions of $n$-almost split sequence and of $n$-cluster tilting subcategory. In the end, we will also look at some open problems of the theory.

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07 May, 2021

## Propriedades homológicas de álgebras preservadas por certas extensões

Uma extensão de álgebras associativas é simplesmente uma álgebra associativa $A$ com subálgebra $B$. Em uma sequência de artigos recentes, Cibils, Lanzilotta, Marcos e Solotar colocaram condições sobre a extensão (extensões "limitadas") e mostraram que, sobre essas condições, duas propriedades homológicas valem para $B$ se, e somente se, valem para $A$: nomeadamente "dimensão global finita" e "suporte da homologia de Hochschild finita". Em particular, segue disso que a "Conjectura de Han" vale para $B$ se, e somente se, vale para $A$.

Kostiantyn Iusenko e eu temos interesse numa classe de álgebras que generaliza naturalmente a classe de álgebras de dimensão finita: as Álgebras Pseudocompactas. Para estender os resultados de CLMS, a gente generalizou as condições de "extensões limitadas" e provou que as propriedades nomeadas acima continuam sendo preservadas por nossa classe mais geral de extensões. Explicarei todas as palavras desse resumo e darei umas das ideias da generalização.

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