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.

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