In this series of lectures we will explore the combinatorics of finite lattices, their connection to Coxeter-Catalan combinatorics, representation theory, and dynamics. We will play special attention to the interplay between the lattice of noncrossing partitions and c-Cambrian lattices.
My lectures will be self-contained. While some basic familiarity with posets will be helpful, I will not assume knowledge about lattices and their properties. We will begin with a survey of helpful definitions, useful theorems/tools, and classical constructions. Our second lecture will focus directly on the lattice theory of the weak order, c-Cambrian lattices, and non-crossing partitions.
Finally, we will end with very recent results and open questions, including connections to representation theory and dynamics.