Non-crossing Partitions and Combinatorics: From Random Matrices to Free Probability Theory - Part 1

Explore a mathematics lecture that delves into free cumulants and their definition using non-crossing partitions, examining their natural emergence from random matrix theory and integration on unitary groups. Learn about the fundamental relationship between symmetric groups and unitary groups through Schur-Weyl duality, while discovering the foundations of free probability and its connection to random matrix theory through non-crossing partition combinatorics. Part of a three-lecture series from the Workshop on "Recent Perspectives on Non-crossing Partitions through Algebra, Combinatorics, and Probability," this 41-minute presentation sets the groundwork for understanding the relationship between non-crossing partitions, parking functions, and the Tamari lattice.