The interplay between convexity and Euclidean harmonic analysis has been very fruitful. While it is less clear that notions of convexity should even appear in abstract harmonic analysis, Stanley observed in the 1980’s that representation theory can be powerfully used to prove unimodality of certain combinatorial sequences, and Okounkov observed in the 1990’s that notions of convexity and log-concavity also arise naturally in the representation theory of various Lie groups. Inspired by, but completely unrelated to these historical bridges, we show that natural discrete probability measures that arise in harmonic analysis of the symmetric group (to be precise, the distributions of row lengths of Young diagrams drawn from the “poissonized" Plancherel measure) are log-concave. This has several significant implications, including the verification of a variant of a 2008 conjecture of W.Y.C.Chen about the longest increasing subsequence of a random permutation, and the fact that the Tracy-Widom laws arising in probability and mathematical physics are log-concave. The talk is based on joint work with Jnaneshwar Baslingker and Manjunath Krishnapur.