Seminar on Free Probability
and Random Matrices

Winter 2016

This is a continuation of my previous talk.

This is a continuation of my previous talk.

This is a continuation of my previous talk.

In this talk we will extend a theorem of Mingo and Nica to compute the matricial second-order Cauchy transform of matricial-valued semicircular elements. Then, the fluctuations of block Gaussian random matrices will be obtained using this transform. This is joint work with Serban Belinschi and Jamie Mingo.

The wrapping transformation W is a homomorphism from the semigroup of probability measures on the real line, with the convolution operation, to the semigroup of probability measures on the circle, with the multiplicative convolution operation. We show that on a large class L of measures, W also transforms the three non-commutative convolutions—free, Boolean, and monotone—to their multiplicative counterparts. Moreover, the restriction of W to L preserves various qualitative properties of measures and triangular arrays. We use these facts to give short proofs of numerous known, and new, results about multiplicative convolutions. This is joint work with Michael Anshelevich.

In this talk we will present some analytic properties of certain two-variables Cauchy transforms. We will also discuss how these properties are related to questions about the existence of certain second-order Cauchy transforms.

This talk will conclude my presentation of my results on second order freeness.

In this talk, we first show how to calculate the joint distribution of the entries of a uniformly distributed signed permutation matrix. Then, we explore the idea of using Hadamard matrices and uniformly distributed signed permutation matrices to deliver asymptotic freeness of second order. This is a continuation from March 10.

In this talk, we first show how to calculate the joint distribution of the entries of a uniformly distributed signed permutation matrix. Then, we explore the idea of using Hadamard matrices and uniformly distributed signed permutation matrices to deliver asymptotic freeness of second order.

Free independence is an analogue of classical independence where tensor products are replaced by free products. The parallel between the two theories run quite deep and extends to both the analytic and combinatorial approaches to independence. Recently a model was found where free and tensor independence can co-exist on the same space. Moreover other notions of independence make sense in this model. This is a continuation from January 21.

We extend existing results that study when Jordan derivations are derivations. Under a mild condition, it is shown that a Jordan $(α,β)$-derivation on a CSL subalgebra of a von Neumann algebra is an $(α,β)$-derivation. It is shown that a Jordan $(α,α)$-derivation on a CSL subalgebra of a von Neumann algebra is an $(α,α)$-derivation, where $α,β$y are automorphisms on a CSL subalgebra of a von Neumann algebra. We also investigate the generalized $n$-th power maps on CSL subalgebras of von Neumann algebras.

We will review some of the results given in a previous talk. Then we will discuss recent developments and further applications.

Free independence is an analogue of classical independence where tensor products are replaced by free products. The parallel between the two theories run quite deep and extends to both the analytic and combinatorial approaches to independence. Recently a model was found where free and tensor independence can co-exist on the same space. Moreover other notions of independence make sense in this model. I will review some of my recent results and indicate where further progress might be made.