Recent problems in partitions and other combinatorial functions

In this talk, I will discuss recent work, joint with a number of
collaborators, on analytic and combinatorial properties of the partition
and related functions. This includes work on recent conjectures of Stanton,
which aim to give a deeper understanding into the "rank" and "crank"
functions which "explain" the famous partition congruences of Ramanujan. I
will describe progress in producing such functions for other combinatorial
functions using the theory of modular and Jacobi forms and recent
connections with Lie-theoretic objects due to Gritsenko-Skoruppa-Zagier. I
will also discuss how analytic questions about partitions can be used to
study Stanton's conjectures, as well as recent conjectures on partition
inequalities due to Chern-Fu-Tang and Heim-Neuhauser, which are related to
the Nekrasov-Okounkov formula.