Random structures in high dimensions: Matrices, polynomials and point processes

这个项目的主要焦点是使用概率论、组合学、分析和理论计算机科学的工具来识别大型随机结构中的可预测行为。所考虑的模型的具体示例包括用于对噪声数据集、气体和其他类型的空间数据进行建模的模型。从数学上讲，该项目主要关注的三类模型是随机矩阵、随机多项式和吉布斯点过程。该项目包括与研究生就涉及这三类模型的核心问题进行合作，并指导本科生的研究。关于随机多项式和随机矩阵的这项研究的一个共同主题是对普遍性现象的理解。这种形式的结果既有数学意义，也有实际意义，因为它们表明，某些行为并不取决于给定模型的细节，而只取决于该模型的少数一般性质。吉布斯积分程序的工作包括理解和扩展这些模型的独特性机制，结果和提出的问题既有理论上的，也有算法上的。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The main focus of this project is to identify predictable behavior in large random structures using tools from probability theory, combinatorics, analysis, and theoretical computer science. Specific examples of models considered consist of those used to model noisy data sets, gases, and other types of spatial data. Mathematically, the three classes of models that are the primary focus of the project are random matrices, random polynomials, and Gibbs point processes. The project includes collaboration with graduate students on core problems concerning these three classes of models and mentoring of undergraduate research. A common theme to this research on random polynomials and random matrices is the understanding of universality phenomena. Results of this form are of mathematical interest as well as practical interest, as they show that certain behavior does not depend on the particulars of a given model but only on a small number of general properties of the model. The work on Gibbs point processes consists of understanding and expanding the uniqueness regime of these models, with results and proposed problems both theoretical as well as algorithmic.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.