Demazure flags: A combinatorial journey

In this talk, we will briefly review the theory of characters of Demazure modules which are modules for the standard maximal parabolic subalgebra of an affine Lie algebra. The discussion will be followed by some connections that I have discovered(with collaborators) in my own research between algebraic combinatorics and other areas of mathematics such as representation theory and number theory. For instance, we show that the graded multiplicities of higher level Demazure modules in Demazure flags can be expressed in terms Dyck paths. I will describe some results and further questions in this direction.