# Thursday, 17 November 2016: Joint Geometry/Topology-NODES Seminar, Durham

**Peter Jorgensen (Newcastle, Maths):** SL_2-tilings, infinite triangulations, and continuous cluster categories

## Time and Location:

13:00 in CM221 (Durham)

## Abstract:

An SL_2-tilings is an infinite grid of positive integers such that each adjacent 2×2-submatrix has determinant 1. These tilings were introduced by Assem, Reutenauer, and Smith for combinatorial purposes.

We will show that each SL_2-tiling can be obtained by a procedure called Conway-Coxeter counting from certain infinite triangulations of the circle with four accumulation points. We will see how properties of the tilings are reflected in the triangulations. For instance, the entry 1 of a tiling always gives an arc of the corresponding triangulation, and 1 can occur infinitely often in a tiling. On the other hand, if a tiling has no entry equal to 1, then the minimal entry of the tiling is unique, and the minimal entry can be seen as a more complex pattern in the triangulation.

The infinite triangulations also give rise to cluster tilting subcategories in a certain cluster category with infinite clusters related to the continuous cluster categories of Igusa and Todorov. The SL_2-tilings can be viewed as the corresponding cluster characters.

This is a report on joint work with Christine Bessenrodt and Thorsten Holm.

*Snapshots from the event*