Geometry of Measures and Applications

测量几何与应用

• 批准号: 1954545
• 负责人: Tatiana Toro
• 金额: $ 22.81万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1954545&HistoricalAwards=false
• 关键词: Geometry; Measures; Applications

When dipping a frame in a solution of soap suds one produces a thin soap film. Mathematically this object is a constant mean curvature surface. It is closely related to the solution of the Plateau problem, which requires finding a surface of minimal area that spans a given shape in space. This problem is a classical question in the Calculus of Variations. The area is an energy functional, and the expectation is that minimizing it will lead to a stable configuration. In this project the PI addresses questions concerning the minimization of certain energy functionals that take into account noise and small random fluctuations of the phenomena being modeled. The expectation is that this theory will be better suited to reflect actual minimization questions arising in nature. A fundamental feature of the area functional is that it is invariant under rotations of space (if that space is homogeneous). The PI will address geometric and analytic questions in inhomogeneous and crystal-like spaces providing a model that reflects nature more accurately. This project will contribute to US workforce development through training and mentoring of graduate students and post-docs. One of the PI’s goals is to show that “almost minimizers”, which are minimizers to noisy variational problems inherit some of the properties of minimizers of the same functional without noise. This study requires using tools from calculus of variations, harmonic analysis and geometric measure theory. The expectation is that the new ideas developed along the way will find applications in other variational problems with free boundaries. The aim of the project concerning further developing analysis on non-smooth domains is to characterize the geometry of domains in Euclidean space in terms of the properties of solutions to canonical (anisotropic) operators. The project concerning the rectifiability of measures promises to reveal the fine structure of measures defined on crystal-like spaces. The overarching theme of this project brings together tools from Geometric Measure Theory, PDE, Potential Theory, Harmonic Analysis and Calculus of Variations, building bridges between these areas while transforming them by the influx of new ideas.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.

将框架浸入肥皂泡的溶液中时，会产生薄肥皂膜。从数学上讲，该对象是恒定的平均曲率表面。它与高原问题的解决方案密切相关，该解决方案需要找到跨越空间形状的最小区域的表面。这个问题是变化计算中的一个经典问题。该区域是一个能量功能，期望是最小化它将导致稳定的配置。在该项目中，PI解决了有关某些能量功能的最小化问题，这些功能考虑了所建模现象的噪声和少量随机波动。期望这一理论将更适合反映自然界中出现的实际最小化问题。该区域功能的一个基本特征是它在空间旋转（如果空间是均匀的话）下是不变的。 PI将在不均匀和晶体状的空间中解决几何和分析问题，提供更准确地反映自然的模型。该项目将通过培训和心理化研究生和培训来为美国的劳动力发展做出贡献。 PI的目标之一是表明“几乎最小化的人”是噪声变异问题的最小化，从而继承了同一功能的最小化物的某些属性，而无需噪声。这项研究需要使用变化，谐波分析和几何测量理论的计算中的工具。期望的是，沿途开发的新想法将在其他自由边界的其他变化问题中找到应用。该项目涉及对非平滑域的进一步开发分析的目的是，根据规范（各向异性）操作员的溶液特性来表征欧几里得空间中域的几何形状。关于措施的纠正性，该项目有望揭示在晶体样空间上定义的措施的精细结构。该项目的总体主题汇集了几何测量理论，PDE，潜在理论，和声分析和变化的微积分，在这些领域之间建立桥梁的工具，同时通过新思想的影响来改变它们。该奖项反映了NSF的法定任务，并以评估的评估来表现出珍贵的支持。