Mathematical Conversations

In 2017, Miller conjectured, based on computational evidence, that for any fixed prime $p$ the density of entries in the character table of $S_n$ that are divisible by $p$ goes to $1$ as $n$ goes to infinity.  K. Soundararajan and I proved this conjecture, but there are still many statistical questions about the character values of the symmetric group where basically no progress has been made.  I'll talk about a few of them, including the problem of determining the asymptotic density of zeros in the character table of $S_n$, where it is not even clear from computational data what one should expect.