The Plancherel measure of the symmetric group gives the most basic
random Young diagram. The limit shape of Plancherel Young diagrams is
well known as the Logan--Shepp--Vershik--Kerov curve. Philippe Biane
obtained results on the derivation of limit shapes using free cumulants
for a wider class of random Young diagrams. In this colloquium, Biane's
result will be the main focus of the presentation. Furthermore, we
touch upon the limit shapes of random Young diagrams determined from
spin representations of symmetric groups, obtained in collaboration with
Piotr \'Sniady and the speaker.

(16:15- tea)