Parking functions, Catalan combinatorics, and algebra

Classical parking functions can be defined in terms of cars with preferred parking spaces attempting to park in open spaces along a one-way, linear parking lot. However, parking functions arise naturally in numerous other combinatorial, algebraic, and geometric settings. In fact, their increasing rearrangements are enumerated by the Catalan numbers, which are known to enumerate over 200 other combinatorial objects! In this talk we will first focus on a specific generalization of parking functions that allows backward movement and exhibit bijections to other Catalan objects in order to enumerate these generalized parking functions. Then we will turn our attention to how parking functions appear in the study of a family of infinite groups known as Artin groups as well as the study of matroids via the algebraic combinatorics of the Tutte polynomial.