LEAPS-MPS: Sharp Inequalities in Probability and Analysis

该项目将通过研究不等式来研究概率，分析和偏微分方程（PDE）-理解物理现象的基本工具-的数学领域之间的联系。 数学中的一个经典研究课题，不等式描述了可能不精确但可以估计或近似的量之间的关系。 在这个项目中要考虑的不等式在膜的基本频率的研究中有物理和工程应用;随机粒子的运动;和扭转刚度，弹性，静电容量和材料的热含量。 这些发现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存在于日常生活中。 该项目将让本科生参与动手数学研究，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与一个会议，这将促进知识，同时创造一个更具包容性的文化和归属感，特别是在数学代表性不足的群体。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。该项目将集中在两个主要的研究方向：1）证明尖锐的不等式，涉及的期望寿命的扩散开始在一个区域和主要的Dirichlet特征值;和2）解决退化扩散的功能不等式。第一个研究方向的重点是证明尖锐的不等式涉及的基本频率和扭转刚度域通过出口时间的扩散。扭转刚度测量具有由特定域给定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对工程问题有实际应用。PI将通过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梯度估计和其他功能的不平等的调和函数有关退化扩散。PI的目标将集中在退化亚椭圆情形下的问题，其中没有规范的潜在亚黎曼结构，称为弱霍曼德扩散。所使用的主要概率工具之一涉及发展扩散过程的急剧耦合。此外，该项目还包括第三个研究方向，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价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.