Satellite mini-conference "Analysis and Probability"
Kontakt: yacin [dot] ameur [at] math [dot] lu [dot] se
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1. 14.00-14.55: Tomas Sjödin (Linköping): "Some Extremal Problems for Harmonic Vector Fields"
2. 15.15-16.10: Maurice Duits (KTH), "Universality of global fluctuations in random matrix theory".
1. Sjödin: Harmonic vector fields is a natural generalization of analytic (or, to be more precise, anti-analytic) functions to higher dimensions.
The aim of this talk is to discuss recent progress on upper and lower bounds of the distance $\lambda_p(\Omega)$ between the space of these harmonic vector fields on a smooth bounded subset $\Omega$ of $n$-dimensional Euclidean space and the identity in the $L^p$ norm. The quantity $\lambda_p(\Omega)$ is called the Bergman analytic content (simply analytic content if $p=\infty$).
We will survey the results, with a particular focus on the case $p=\infty$, and give some ideas of the proofs, which involve both a potential-theoretic construction called partial balayage, as-well as results from the theory of optimal transport.
The talk is based on joint work with S.J. Gardiner and M. Ghergu.
2. Duits: An important theme in the study of random matrices is that their eigenvalue statistics give rise to a variety of patterns that are expected to be universal. This talk will be an introduction to the universality of the global fluctuations. Such fluctuations are typically described by the Gaussian free field. Starting from the Gaussian Unitary Ensemble and Dyson’s Brownian motion, we will see that for a large class of processes the characterization of the global fluctuations can be reduced to determining the asymptotic behavior of large finite sections of (products of) exponentials of banded matrices.