Random polynomials and random Kähler geometry
The research in this proposal focuses on the interplay between complex geometry and probability theory. More precisely, we aim to combine methods from complex geometry and geometric analysis with probabilistic techniques in order 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 instance of this setting is given by the case of random polynomials. We are interested in the asymptotics of the covariance kernels of the polynomial / sections ensembles, universality of their distributions, central limit theorems and large deviation principles in this context.
On the one hand, the questions we are planning to investigate are interesting and challenging from a mathematical point of view. On the other hand, during the last couple of decades, they have also been exhibiting important connections to theoretical physics, where random polynomials serve as a basic model for eigenfunctions of quantum chaotic Hamiltonians.