Recorded 18 April 2024. Colin Defant of Harvard University presents "Toric Promotion with Reflections and Refractions" at IPAM's Integrability and Algebraic Combinatorics Workshop. Abstract: Inspired by recent work on refraction billiards in dynamics, we introduce a notion of refraction for combinatorial billiards. This allows us to define a generalization of toric promotion that we call \emph{toric promotion with reflections and refractions}, which is a dynamical system defined using a graph G whose edges are partitioned into a set of \emph{reflection edges} and a set of \emph{refraction edges}. This system is a discretization of a billiards system in which a beam of light can pass through, reflect off of, or refract through each toric hyperplane in a toric arrangement. Vastly generalizing the main theorem known about toric promotion, we give a simple formula for the orbit structure of toric promotion with reflections and refractions when G is a forest. We also completely describe the orbit sizes when G is a cycle with an even number of refraction edges; this result is new even for ordinary toric promotion (i.e., when there are no refraction edges). When G is a cycle of even size with no reflection edges, we obtain an interesting instance of the cyclic sieving phenomenon. This is joint work with Ashleigh Adams and Jessica Striker. Learn more online at: https://www.ipam.ucla.edu/programs/workshops/workshop-ii-integrability-and-algebraic-combinatorics/