*event*Friday, April 28, 2023

*access_time*3:45pm (CDT)

*room*PHSC 1105

*free_breakfast*Tea at 3:15pm (CDT) in PHSC 424

**Abstract:** Let kQ/I be a finite representation type gentle algebra. Two modules M and N are called almost rigid if they do not have any nonsplit extensions or if any extension between M and N is indecomposable. A module T is called maximal almost rigid (mar) if its indecomposable summands form a maximal collection of pairwise almost rigid modules. First, we use a modified version of the surface model developed by Baur and Coelho Simões to show that each mar module T corresponds bijectively to a permissible triangulation of a surface. We then show that the endomorphism algebra of a mar module over kQ/I is the endomorphism algebra of a tilting module over a bigger gentle algebra. Finally, we define an oriented flip graph of the mar modules and conjecture that it is acyclic.
For the hereditary type A case, the mar modules correspond to triangulations of a polygon, and thus the type A mar modules give us a new class of Catalan objects. In this case, the endomorphism algebras of the mar modules are tilted algebras of type A. Furthermore, their oriented flip graph is the oriented exchange graph of a smaller type A cluster algebra which is known to define a poset which is a Tamari or Cambrian lattice.
This talk is based on joint work with Emily Barnard, Raquel Coelho Simões, and Ralf Schiffler.