Seminar

Cluster algebras are certain combinatorially defined algebras that have been shown to relate to a myriad of areas in math and physics. One can define a cluster algebra structure on orientable surfaces where the generators correspond to certain curves on the surface and the relations are given by skein relations. In this talk, we will focus on an extension of this construction to non-orientable surfaces. We will discuss various joint work developing non-orientable analogues of many results from cluster theory. Time permitting, we will define a partitioned quiver associated to the surface, give expansion formulae for the generators, give an algebraic interpretation of the mapping class group and state some enumerative results about these algebras. This will be based on various joint works: one project with Véronique Bazier-Matte and our students: Fenghuan He, Ruiyan Huang, Hanyi Luo; another with Cody Gilbert and McCleary Philbin; and lastly, work in progress with James Hughes.