Noncrossing Partitions of a Marked Surface

This 47-minute lecture by Nathan Reading explores noncrossing partitions of marked surfaces as part of the Workshop on "Recent Perspectives on Non-crossing Partitions through Algebra, Combinatorics, and Probability" held at the Erwin Schrödinger International Institute for Mathematics and Physics (ESI). Delve into the mathematical concept of marked surfaces—compact surfaces with a finite set of distinguished points—and how their triangulations serve as models for associated cluster algebras. Learn how noncrossing partitions of a cycle model the interval [1,c] in the absolute order on the symmetric group, and discover joint work with Brestensky that models the interval [1,c] in the affine symmetric group using noncrossing partitions of an annulus. Explore how these concepts connect to lattices constructed by McCammond and Sulway for proving properties of Euclidean Artin groups. The lecture also covers symmetric noncrossing partitions of surfaces with double points and their relationship to classical affine Coxeter groups, concluding with mentions of ongoing work with Eric Hanson on realizing McCammond and Sulway's lattice in the representation theory of hereditary algebras of affine type.