In this talk, we will discuss several results on noncommutative Hardy spaces. In his PhD thesis, Mei introduced the definition of the operator-valued Hardy space $\mathrm{H}_1(\mathbb{R}^n,\mathcal{M})$ associated with a von Neumann algebra $\mathcal{M}$ as a space of functions $f : \mathbb{R}^n \to \mathcal{M}$. The construction of these spaces relies on the noncommutative analog of the Lusin integral, which comes along with the definition of the Hardy space as the sum of a column part $\mathrm{H}_1^c(\mathbb{R}^n,\mathcal{M})$ and a row part $\mathrm{H}_1^r(\mathbb{R}^n,\mathcal{M})$. As it occurs in the classical setting, numerous definitions of the Hardy space can be introduced and shown to be equivalent. For instance, a recently introduced new atomic decomposition has allowed us to obtain estimates for Calderón-Zygmund operators on $\mathrm{H}_1^c(\mathbb{R},\mathcal{M})$. Also, we will discuss how different characterizations may yield new results regarding the boundedness of Fourier multipliers and Schur multipliers. The content of this talk is part of joint work with Éric Ricard and work in progress with Tao Mei.

One of the fascinating phenomena of spin glasses is the dramatic change in behavior that occurs between the high and low temperature regimes. The free energy of the spherical Sherrington-Kirkpatrick (SSK) model, for example, has Gaussian fluctuations at high temperature, but Tracy-Widom fluctuations at low temperature. A similar phenomenon holds for the bipartite SSK model, and we show that, when the temperature is within a small window around the critical temperature, the free energy fluctuations converge to an independent sum of Gaussian and Tracy-Widom random variables (joint with Han Le). Our work follows two recent papers that proved similar results for the SSK model (by Landon and by Johnstone, Klochkov, Onatski, Pavlyshyn). Our techniques involve random matrix theory and asymptotic analysis of contour integrals. Beyond the physics motivation for these results, there are statistical applications related to hypothesis testing in critically spiked matrix models.