Random polynomials and random Kähler geometry

这项建议的研究重点是复几何和概率论之间的相互作用。更具体地说，我们的目的是将复几何和几何分析的方法与概率论相结合，研究Kähler流形上全纯线丛的全纯截面零点的局部和整体统计性质的若干问题。这种设置的一个特别重要的例子是随机多项式的情况。我们感兴趣的是多项式/分段系综协方差核的渐近性、其分布的普适性、中心极限定理和大偏差原理。一方面，从数学的角度来看，我们计划调查的问题很有趣，也很有挑战性。另一方面，近几十年来，它们也显示出与理论物理的重要联系，在理论物理中，随机多项式作为量子混沌哈密顿的本征函数的基本模型。

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.