In the first two lectures I will define free cumulants, using non-crossing partitions, and explain how this definition emerges naturally from the theory of random matrices and integration on unitary groups. The Schur-Weyl duality between symmetric groups and unitary groups plays an important role. I will then explain the basics of free probability and how the combinatorics of non-crossing partitions and free cumulants allows to relate it to random matrix theory.
In the last lecture I will take on another topic, the combinatorics of maximal chains in the non-crossing partition lattice, which relates non-crossing partitions, parking functions, and the Tamari lattice.