Geometry of Dyck Paths and Their Symmetric Function Theory

Explore a lecture from Kyoto University's Syu Kato at the Institute for Advanced Study examining the geometric foundations of Dyck paths and their relationship to symmetric functions. Delve into two major research trends: the theory of Catalan symmetric functions with its geometric realization, and the study of chromatic symmetric functions in graphs. Learn about smooth algebraic varieties that demonstrate Catalan symmetric functions and their role in validating geometric predictions from Broer, Shimozono-Weyman, Chen-Haiman, and Blasiak-Morse-Pun. Discover how these varieties represent chromatic symmetric functions of unit interval graphs and potentially serve as geometric realizations of Dyck paths, distinct from Hessenberg varieties. Based on research from arXiv:2301.00862 and arXiv:2410.12231, gain insights into recent developments in this mathematical domain.