The metric entropy of spaces of holomorphic functions

Speakers: Siarhei Finski (Ecole Polytechnique)\n\nHow much information is needed to determine a function up to a given accuracy? Pontryagin-Schnirelmann and Kolmogorov made this question precise with the notion of metric entropy: the logarithm of the minimal number of balls of a fixed radius needed to cover a compact subset of a metric space. This talk is devoted to a problem posed by Kolmogorov: computing the metric entropy of spaces of holomorphic functions.\nMore precisely, for a compact subset of a complex manifold, consider the continuous functions on it that extend holomorphically to the whole manifold with a uniform bound. By Montel’s theorem, this family is compact in the topology of uniform convergence. Our aim is to make this compactness quantitative by determining the asymptotic behavior of its metric entropy. For domains in a complex vector space, this was determined through the work of Zakharyuta and Nivoche, and later by Bandtlow-Nivoche. I will explain a new proof that extends this result to arbitrary complex manifolds, drawing on complex and algebraic geometry, approximation theory, and pluripotential theory.\n\nhttps://indico.math.cnrs.fr/event/17064/