University of Minnesota Combinatorics Seminar: 2021-2022

Date: 10/29/2021

Speaker: Chris Fraser

Cluster Combinatorics of $SL_k$ Skein Algebras in the Presence of Punctures
Fock and Goncharov showed that the moduli space of decorated $SL(k)$-local systems on a bordered marked surface is a cluster variety. When $k$ is 2, the resulting cluster algebras are the cluster algebras from surfaces studied by Fomin, Shapiro, and Thurston. In this case, clusters are in bijection with tagged triangulations of the surface. We propose a conjectural way of extending these combinatorial structures to higher $k$, focusing on the combinatorics which happens "nearby punctures" in the surface, and present results in support of our conjectures. The main inputs to our work are due to Goncharov-Shen and Fomin-Pylyavskyy. This is joint with Pavlo Pylyavskyy.

Cluster Combinatorics of $SL_k$ Skein Algebras in the Presence of Punctures

Fock and Goncharov showed that the moduli space of decorated $SL(k)$-local systems on a bordered marked surface is a cluster variety. When $k$ is 2, the resulting cluster algebras are the cluster algebras from surfaces studied by Fomin, Shapiro, and Thurston. In this case, clusters are in bijection with tagged triangulations of the surface. We propose a conjectural way of extending these combinatorial structures to higher $k$, focusing on the combinatorics which happens "nearby punctures" in the surface, and present results in support of our conjectures. The main inputs to our work are due to Goncharov-Shen and Fomin-Pylyavskyy. This is joint with Pavlo Pylyavskyy.