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.

这个项目将通过研究不等式来研究概率、分析和偏微分方程(PDE)等数学领域之间的联系--这些是理解物理现象的基本工具。作为数学研究的经典课题，不等式描述了量之间的关系，这些量可能不是精确知道的，但可以估计或近似。本项目中要考虑的不等式在物理和工程中的应用包括膜的基频、随机粒子的运动以及材料的扭转刚性、弹性、静电容量和热含量的研究。所发现的见解将与快速变化的社会相关，在这个社会中，数据和随机性越来越多地出现在日常生活中。该项目将让本科生参与实践数学研究，目标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个会议，该会议将促进知识的发展，同时创造一种更具包容性的文化和归属感，特别是在数学领域代表性不足的群体中。一个杰出的系列讲座将向普通观众开放，从而通过介绍概率及其对社会的影响来提高公众的科学素养和对科学的参与。这个项目将集中在两个主要的研究方向：1)证明涉及区域内开始的扩散的预期寿命和主要Dirichlet特征值的尖锐不等式；2)解决退化扩散的泛函不等式。第一个研究方向是通过扩散的出口时间证明涉及区域的基频和扭转刚度的尖锐的不等式。扭转刚度衡量的是具有特定区域的横截面的杆抵抗扭转力的程度。因此，获得扭转刚度的精确界限将在工程问题中有实际应用。PI将应用概率和分析方法研究潜在的PDE。第二个方向将集中于证明与退化扩散有关的调和函数的梯度估计和其他函数不等式。PI的目标将集中在退化的亚椭圆型情况下的问题，在这种情况下没有典型的底层次黎曼结构，这被称为弱Hörmander扩散。使用的主要概率工具之一涉及开发扩散过程的尖锐耦合。此外，该项目的第三个研究方向将与本科生研究人员一起探索，重点是随机奇异矩阵乘积的显式极限定理理论。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。