## Geometry and Representation Theory Seminar

### Monday, September 17th, 2018

**Time:** 4:30-5:30 p.m. **Place:** Jeffery Hall 319

**Speaker:** Tianyuan Xu (Queen's University)

**Title:** Broken lines and the topological ordering of the alternating quiver of type A

**Abstract: **The Positivity Conjecture in cluster algebra theory states that the coefficients of the Laurent expansion of any cluster variable in a cluster algebra are always positive integers. In 2014, Gross, Hacking, Keel and Kontsevich constructed a so-called Theta function basis to prove the conjecture for all cluster algebras of geometric type. A key ingredient in the construction of the Theta functions is the broken line model. In this talk, we will discuss the broken lines associated to the alternating quiver of type A, with an emphasis on relating its combinatorial properties to the topological ordering of the quiver, the partial order obtained by taking the transitive and reflexive losure of the relation “v<w if v->w is an edge” on the vertices of the quiver.

The talk is based work in progress with Ba Nguyen, David Wehlau and Imed Zaguia.