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Title: The Circular Beta Ensemble: Magic, Mystery, and the Extremes of Random Matrices.

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The circular beta ensemble is a one-parameter family of random matrices that, at beta equal to two, reduces to the most elementary object in random matrix theory: a Haar-distributed unitary matrix, the uniform distribution on unitary matrices. And yet this single family sits at a remarkable crossroads — it is the source of miraculous exact identities, deep connections to Gaussian multiplicative chaos, a canonical random fractal, and, through the Keating–Snaith philosophy, a probabilistic model for the Riemann zeta function on the critical line.

n 2013, Fyodorov, Hiary, and Keating made a pair of celebrated conjectures that brought these threads together: they predicted that the maximum of the logarithm of the characteristic polynomial, suitably recentered, converges to a sum of two independent Gumbel random variables — a classical law from extreme value theory. They made the same prediction for the Riemann zeta function at a random height on the critical line. Their argument drew on replica methods and a physically motivated ansatz from statistical physics — deeply suggestive, but far from rigorous. The conjecture became one of the central open problems at the interface of probability, random matrix theory, and number theory.

In recent joint work with Ofer Zeitouni, Gautier Lambert, and Joseph Najnudel, we resolve the conjecture completely on the random matrix side, for all values of beta. The proof exploits the very magic that makes the circular beta ensemble special — in particular, exact structural connections to multiplicative chaos revealed by identities going back to Diaconis and Shahshahani and Matsumoto-Jiang. In this talk, I will introduce the audience to this magical world: the circular beta ensemble and its many faces, the conjecture and the ideas behind its resolution, and the mysteries that remain.

Venue: Hybride - CRM, Salle / Room 5340, Pavillon André Aisenstadt