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04/06/2019, 15:00 — 16:00 — Room P3.10, Mathematics Building

Brian Hall, *University of Notre Dame*

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Large-$N$ Segal-Bargmann transform with application to random matrices

I will describe the Segal-Bargmann transform for compact Liegroups, with emphasis on the case of the unitary group $U(N)$. In this case, the transform is a unitary map from the space of $L^2$ functions on $U(N)$ to the space of $L^2$ holomorphic functions on the "complexified" group $\operatorname{GL}(N;\mathbb{C})$. I will then discuss what happens in the limit as $N$ tends to infinity. Finally, I will describe an application to the eigenvalues of random matrices in $\operatorname{GL}(N;\mathbb{C})$. The talk will be self-contained and have lots of pictures.