Free probability, free convolution and associated combinatorics of non-crossing partitions

Abstract: This is an expository talk on free probability, which is a part of operator theory with very strong parallels to probability theory. In particular, we shall focus on free convolution, which is a binary operation on measures on the real line that is different from the usual convolution that arises when one adds independent random variables. Getting rid of analysis and expressing everything algebraically, the difference between the two forms of convolution arises from the difference between the lattice of all set partitions of a finite set and the lattice of all non-crossing set partitions of the same. We would also like to explain how free convolution arises in random matrix theory (what are the eigenvalues of the sum of two large matrices?) and in asymptotic representation theory of symmetric groups (what are the Littlewood-Richardson coefficients of large partitions?). 

Not much knowledge of probability will be assumed. We shall refer to Bernoulli and Gaussian random variables, independence and convolution, central limit theorem, mainly to explain the analogies. The talk should be accessible to advanced undergraduates.

combinatoricsoperator algebrasprobabilityrings and algebrasrepresentation theory

Audience: advanced learners

( slides | video )

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ARCSIN - Algebra, Representations, Combinatorics and Symmetric functions in INdia

Series comments: Timings may vary depending on the time zone of the speakers.

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