Esther Banaian, Monica Garcia, Emily Gunawan

Symposium

**13.30 Monica Garcia (Paris-Saclay University)**

*g-finiteness in the category of projective presentations*

An algebra is said to be g-finite if it admits finitely many isomorphism classes of tau-tilting pairs. This notion was introduced and thoroughly studied by L. Demonet O. Iyama and G. Jasso, who showed that this property is equivalent to the module category admitting finitely many isomorphism classes of bricks (which is equivalent to having finitely many wide subcategories), finitely many functorially finite torsion classes, and equivalent to all torsion classes being functorially finite. Many of these concepts and their relationships have been shown to have counterparts in the extriangulated category of two-term complexes of projective modules. In this talk, we introduce new equivalent conditions to an algebra being g-finite in the context of the category of 2-term complexes. Namely, we establish that being g-finite is equivalent to the category of 2-term complexes admitting finitely many thick subcategories, finitely many complete cotorsion pairs and equivalent to all cotorsion pairs being complete.

**14.45 Esther Banaian (Aarhus University)**

*Orbifold Markov Numbers*

The Markov numbers are a family of positive integer solutions to a certain Diophantine equation, originally studied in the context of Diophantine approximation. They have been of considerable interest for the past century, due to their connections to various fields as well as Frobenius's famous (and still open) Uniqueness Conjecture. Markov numbers come in triples, and the set of all Markov triples is connected via an arithmetic rule that resembles mutation in a cluster algebra. This observation allows one to study Markov numbers in the context of both cluster theory and representation theory. After reviewing highlights of this study, we move on to discuss a related equation which is related to Chekhov and Shapiro's generalization of a cluster algebra arising from the study of Teichmüller spaces on orbifolds. This is partially based on joint work with Archan Sen (arXiv 2210.07366).

**16.00 Emily Gunawan (University of Massachusetts Lowell)**

*Pattern-avoiding polytopes and Cambrian lattices via the Auslander–Reiten quivers*

There is a bijection between type A Coxeter elements c and type A Dynkin quivers Q. For each type A Coxeter element c, we define a pattern-avoiding Birkhoff subpolytope whose vertices are the permutation matrices of the c-singletons. We show that the (normalized) volume of our polytope is equal to the number of longest chains in a corresponding type A Cambrian lattice. Our work extends a result of Davis and Sagan which states that the volume of the convex hull of the 132 and 312 avoiding permutation matrices is the number of longest chains in the Tamari lattice, a special case of a type A Cambrian lattice. Furthermore, we prove that each of our polytopes is unimodularly equivalent to the (Stanley's) order polytope of the heap H of the longest c-sorting word. The Hasse diagram of H is given by the Auslander—Reiten quiver for the quiver representations of Q. This talk is based on joint work with Esther Banaian, Sunita Chepuri, and Jianping Pan.

*Tea, coffee and cake will be provided.*

Organised by: AarHomAlg