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

高维随机结构：矩阵、多项式和点过程

• 批准号: 2246624
• 负责人: Marcus Michelen
• 金额: $ 26万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-09-01 至 2026-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2246624&HistoricalAwards=false
• 关键词: Random; structures; dimensions; Matrices; polynomials

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.

该项目的主要重点是使用概率论、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这三类模型的核心问题进行合作，并指导本科生研究。随机多项式和随机矩阵研究的一个共同主题是对普遍性现象的理解。 这种形式的结果具有数学意义和实际意义，因为它们表明某些行为并不取决于给定模型的细节，而仅取决于模型的少数一般属性。 吉布斯点过程的工作包括理解和扩展这些模型的唯一性机制，以及理论和算法方面的结果和提出的问题。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值进行评估，被认为值得支持以及更广泛的影响审查标准。