Random polynomials and random Kähler geometry

随机多项式和随机凯勒几何

• 批准号: 444030945
• 负责人: Professor Dr. Alexander Drewitz
• 金额: --
• 依托单位: Abteilung Mathematik
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/444030945?language=en
• 关键词: Random; polynomials; random; hler; geometry

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.

该计划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，我们的目标是结合联合收割机方法从复杂的几何和几何分析与概率技术，研究几个问题的局部和整体的统计性质的全纯线丛的Kähler流形上的全纯截面。这种设置的一个特别重要的例子是随机多项式的情况。我们感兴趣的多项式/节合奏协方差核的渐近性，其分布的普遍性，中心极限定理和大偏差原则。一方面，我们计划研究的问题从数学的角度来看是有趣和具有挑战性的。另一方面，近几十年来，它们也显示出与理论物理学的重要联系，其中随机多项式作为量子混沌哈密顿本征函数的基本模型。