Random partitions pop up in many areas of mathematics, sometimes directly but more often in disguise. There are some very natural probability measures on partitions, such as the Plancherel measure, whose behavior we understand very well thanks to the rich algebraic structure that the partitions possess. The Plancherel measure underpins the classical Ulam problem about the longest increasing subsequence in a random permutation and plays a key role in Nekrasov's theory of instanton counting. The aim of my lectures will be to explain what is the Plancherel measure, how did we learn what we know about it, and which conclusions can we draw from this for various applications.
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