Random polynomials and random Kähler geometry

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The research in this proposal focuses on the interplay between complex geometry and probability theory. More specifically, we aim to combine methods from complex geometry and geometric analysis with probabilistic techniques to study several problems concerning local and global statistical properties of zeros of holomorphic sections of holomorphic line bundles over Kähler manifolds. A particularly important example of this setting is the case of random polynomials. We are interested in the asymptotics of the covariance kernels of the polynomial/section ensembles, the universality of their distributions, central limit theorems and large deviation principles. On the one hand, the questions we plan to investigate are interesting and challenging from a mathematical point of view. On the other hand, in recent decades they have also shown important connections to theoretical physics, where random polynomials serve as a basic model for the eigenfunctions of quantum chaotic Hamiltonians.