Recorded 18 April 2024. Pavel Galashin of the University of California, Los Angeles, presents "Move-reduced graphs on a torus vs positroid Catalan numbers" at IPAM's Integrability and Algebraic Combinatorics Workshop. Abstract: I will discuss how the same combinatorial dynamics on the affine symmetric group has applications to two seemingly different problems: (i) Classifying which bipartite graphs on a torus are reduced under square/spider moves, and which reduced graphs are move-equivalent; (ii) computing positroid Catalan numbers, which are generalizations of Catalan numbers motivated by the geometry of positroid varieties in the Grassmannian, knot theory. Based on joint works with T. George and T. Lam. Learn more online at: https://www.ipam.ucla.edu/programs/workshops/workshop-ii-integrability-and-algebraic-combinatorics/