Winter 2022/2023

Title: A structural view of maximal green sequences.

Abstract: We initiate a new approach to maximal green sequences, whereby they are considered under an equivalence relation. Doing this reveals extra structure on the set of maximal green sequences of an algebra, namely hidden partial orders. We prove that there are several different ways of defining the equivalence relation. Similarly, there are several ways of defining a partial order on the equivalence classes, but as yet these partial orders are only conjecturally equivalent. We prove this conjecture in the case of Nakayama algebras. This is joint work with Mikhail Gorsky. 

Title: The fan of g-vector cones of tau-rigid pairs.

Abstract: We give a proof of the fact that cones of g-vectors associated to tau-rigid pairs form a fan using stability conditions and their associated torsion and torsion-free classes. As a consequence of this we obtain a completely algebraic proof that every tau-rigid pair is determined by its g-vector. 

Title: Chains of torsion classes and weak stability conditions.

Abstract: Based on joint work-in-progress with Hipolito Treffinger. Joyce introduced the concept of weak stability conditions for an abelian category as a generalisation of Rudakov's stability conditions. In this talk, we show an explicit relation between chains of torsion classes and weak stability conditions in an abelian category. Consequently, we give a new characterisation of torsion classes, discuss the structure of the space of chains of torsion classes and its relation to the stability manifold.  

Title: Species and semilinear clannish algebras from surfaces with orbifold points (II) 

Abstract: A classical theorem of Gabriel asserts that every finite-dimensional algebra over an algebraically closed field is Morita-equivalent to a quotient of the path algebra of a quiver. To obtain a similar statement over non-algebraically closed fields one needs to recourse to 'species' instead of quivers. In this talk, I will present a concrete construction of some species, see that their representations are given by linear maps that are 'semilinear' with respect to elements of Galois groups, and describe some Morita-equivalences between these species, which Geuenich and I constructed in 2015-2017, and 'semilinear clannish algebras' as introduced this year by Bennett-Tennenhaus and Crawley-Boevey.

Title: Species and semilinear clannish algebras from surfaces with orbifold points (I) 

Abstract: A classical theorem of Gabriel asserts that every finite-dimensional algebra over an algebraically closed field is Morita-equivalent to a quotient of the path algebra of a quiver. To obtain a similar statement over non-algebraically closed fields one needs to recourse to 'species' instead of quivers. In this talk, I will present a concrete construction of some species, see that their representations are given by linear maps that are 'semilinear' with respect to elements of Galois groups, and describe some Morita-equivalences between these species, which Geuenich and I constructed in 2015-2017, and 'semilinear clannish algebras' as introduced this year by Bennett-Tennenhaus and Crawley-Boevey.

Title: From formal to actual deformations of associative algebras (II) 

Abstract: The formal deformation theory of associative algebras can be described via "higher structures" on the Hochschild complex, this higher structure being given by the Gerstenhaber bracket. In these two talks I will give a short review of deformation theory via DG Lie and L∞ algebras and explain how to use this framework of homotopy algebras to obtain more tractable descriptions of the deformation theory of associative algebras by using the language of reduction systems. This latter approach allows one to address the fundamental problem of passing from a formal deformation to an "actual" deformation, where the deformation parameters are evaluated to constants. This allows one to obtain interesting applications in a range of subject areas, including representation theory, algebraic geometry, deformation quantization and symplectic geometry and their intersections.  

Title: From formal to actual deformations of associative algebras (I) 

Abstract: The formal deformation theory of associative algebras can be described via "higher structures" on the Hochschild complex, this higher structure being given by the Gerstenhaber bracket. In these two talks I will give a short review of deformation theory via DG Lie and L∞ algebras and explain how to use this framework of homotopy algebras to obtain more tractable descriptions of the deformation theory of associative algebras by using the language of reduction systems. This latter approach allows one to address the fundamental problem of passing from a formal deformation to an "actual" deformation, where the deformation parameters are evaluated to constants. This allows one to obtain interesting applications in a range of subject areas, including representation theory, algebraic geometry, deformation quantization and symplectic geometry and their intersections.  

Title: Group actions on Cluster algebras and Cluster categories (II). 

Abstract: In these series of talks, we are going to introduce the rest part of Paquette-Schiffler’s paper “Group actions on cluster algebras and cluster categories”,  we are going to introduce the admissible group actions on the surfaces with marked points and the corresponding cluster algebras. As a result, we will prove the orbit space of the cluster algebra has the structure of Paquette-Schiffler generalized cluster algebra structure which could be defined by a given triangulated orbiforld. In order to study the categorification of this Paquette-Schiffler generalized cluster algebras, we are also going to introduce the cluster character which gives all the cluster variables of  Paquette-Schiffler generalized cluster algebras.  This talk will be based on the paper: Group actions on cluster algebras and cluster categories,  you can find the paper here: https://arxiv.org/abs/1703.06174v2 

Title: Group actions on Cluster algebras and Cluster categories (I). 

Abstract: In these series of talks, we are going to introduce the rest part of Paquette-Schiffler’s paper “Group actions on cluster algebras and cluster categories”,  we are going to introduce the admissible group actions on the surfaces with marked points and the corresponding cluster algebras. As a result, we will prove the orbit space of the cluster algebra has the structure of Paquette-Schiffler generalized cluster algebra structure which could be defined by a given triangulated orbiforld. In order to study the categorification of this Paquette-Schiffler generalized cluster algebras, we are also going to introduce the cluster character which gives all the cluster variables of  Paquette-Schiffler generalized cluster algebras.  This talk will be based on the paper: Group actions on cluster algebras and cluster categories,  you can find the paper here: https://arxiv.org/abs/1703.06174v2 

Title: Homological dimensions and the finitistic dimension conjecture (II)

Abstract: I am going to present an approach to the study of the finitistic dimension conjecture. This approach is based on the so called Igusa-Todorov functions, which constitue homological measures that can be used to obtain bounds for the finitistic dimension. In a recent work together with Marcelo Lanzilotta we have used the Igusa-Todorov functions to define GLIT algebras, a vast family of artin algebras that satisfy the finitistic dimension conjecture. I will present said definition, together with the main properties that we have been able to show for GLIT algebras, as well as some open questions.

Title: Homological dimensions and the finitistic dimension conjecture (I)

Abstract: I am going to present an approach to the study of the finitistic dimension conjecture. This approach is based on the so called Igusa-Todorov functions, which constitue homological measures that can be used to obtain bounds for the finitistic dimension. In a recent work together with Marcelo Lanzilotta we have used the Igusa-Todorov functions to define GLIT algebras, a vast family of artin algebras that satisfy the finitistic dimension conjecture. I will present said definition, together with the main properties that we have been able to show for GLIT algebras, as well as some open questions.