LEAPS-MPS: Sharp Inequalities in Probability and Analysis

LEAPS-MPS：概率和分析中的尖锐不平等

• 批准号: 2316968
• 负责人: Phanuel Mariano
• 金额: $ 22.72万
• 依托单位: Union College
• 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 2025-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2316968&HistoricalAwards=false
• 关键词: LEAPS; MPS; Sharp; Inequalities; Probability

This project will investigate connections between the mathematical fields of probability, analysis, and partial differential equations (PDEs) – fundamental tools in understanding physical phenomena – through the study of inequalities. A classical topic of study in mathematics, inequalities describe relationships between quantities which may not be known precisely but can be estimated or approximated. The inequalities to be considered in this project have applications to physics and engineering in the study of the fundamental frequency of membranes; the motion of random particles; and the torsional rigidity, elasticity, electrostatic capacity, and heat content of materials. The insights uncovered will be relevant to a rapidly changing society, in which data and randomness are increasingly present in daily life. The project will engage undergraduate students in hands-on mathematical research with the goal of increasing the mathematical talent pool in the United States. Graduate students, postdoctoral researchers, and early career mathematicians will be involved in a conference that will advance knowledge while creating a more inclusive culture and sense of belonging, particularly among under-represented groups in mathematics. A distinguished lecture series will be accessible to a general audience, thereby increasing public scientific literacy and engagement with science by introducing probability and its impacts on society. This project will focus on two main research directions: 1) proving sharp inequalities involving the expected lifetime of a diffusion started inside a domain and the principal Dirichlet eigenvalue; and 2) addressing functional inequalities for degenerate diffusions. The first research direction focuses on proving sharp inequalities involving the fundamental frequency and the torsional rigidity of domains through exit times of diffusions. The torsional rigidity measures how much a rod with cross-sections given by a particular domain is resistant to twisting forces. Thus, obtaining sharp bounds for the torsional rigidity will have physical applications to engineering problems. The PI will study the underlying PDEs by applying probabilistic and analytic methods. The second direction will focus on proving gradient estimates and other functional inequalities for the harmonic functions related to degenerate diffusions. The PI’s goal will be to focus on problems in the degenerate hypoelliptic case where there is no canonical underlying sub-Riemannian structure, which are called weak Hörmander diffusions. One of the main probabilistic tools that is used involves developing sharp couplings of diffusion processes. Additionally, the project features a third research direction, to be explored with undergraduate researchers, that focuses on the theory of explicit limit theorems for the products of random singular matrices.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.

该项目将通过研究不平等的研究来研究概率，分析和部分微分方程（PDES）的数学领域（PDES） - 理解物理现象的基本工具。不平等的数学研究主题描述了数量之间的关系，而数量可能不确定，但可以估计或近似。在研究膜的基本频率中，该项目中要考虑的不平等已应用于物理和工程；随机颗粒的运动；以及材料的扭转刚度，弹性，静电能力和热含量。发现的见解将与一个快速变化的社会有关，在日常生活中，数据和随机性越来越多。该项目将吸引本科生进行动手数学研究，目的是增加美国的数学人才库。研究生，博士后研究人员和早期职业数学家将参加一次会议，该会议将提高知识，同时创造更具包容性的文化和归属感，尤其是在数学领域代表性不足的群体中。一般受众将可以访问一个杰出的讲座系列，从而通过引入概率及其对社会的影响来提高公共科学素养和对科学的互动。该项目将集中在两个主要研究方向上：1）提供涉及域内预期寿命的急剧不平等，从域内和主要的迪里奇特征值开始； 2）解决差异差异的功能不平等。第一个研究方向的重点是提供涉及基本频率和通过差异时间的域刚性的尖锐不平等。扭转刚度可以测量特定域给出的带有横截面的杆的杆，对扭曲力具有抵抗力。这是为了获得扭转刚度的尖锐界限，将在工程问题上有物理应用。 PI将通过采用有问题的分析方法来研究基本PDE。第二个方向将集中于证明与退化扩散有关的谐波功能的梯度估计和其他功能不平等。 PI的目标是将重点放在堕落的低纤维化案例中，而该案例中没有规范的基础亚riemannian结构，这些结构被称为弱的Hörmander差异。所使用的主要概率工具之一是开发差异过程的尖锐耦合。此外，该项目的特点是与本科研究人员一起探索的第三个研究方向，重点是对随机奇异物品的产品的明确限制定理理论。该奖项反映了NSF的法定任务，并被视为值得通过基金会的知识分子和更广泛影响的评估审查审查标准来通过评估来获得支持。