Algebra/Topology Seminar

Speaker: Dani Kaufmann

Title: Decorated Teichmüller Theory and Cluster Algebra Invariants

Abstract: Penner's decorated Teichmüller theory provides a simple and concrete approach to Teichmüller theory by way of studying hyperbolic structures on surfaces via triangulations and lengths of geodesics. I will give a brief introduction to these ideas, and use them as topological/geometric motivation for the notion of a cluster algebra. I can then introduce the concept of a mutation invariant function on a cluster algebra and show that traces of monodromy operators give examples of such functions - this is the basis of recent work of mine on the classification of mutation invariants on affine cluster ensembles.  If time, I will discuss other appearances of these functions e.g. as Diophantine equations, symbols of scattering amplitudes, and canonical bases of cluster algebras.