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.

当将框架浸入肥皂泡沫溶液中时，从数学上讲，该物体是一个恒定平均曲率表面，它与高原问题的解决方案密切相关，该问题需要找到跨越一个区域的最小面积的表面。这个问题是变分法中的一个经典问题，该区域是一个能量泛函，并且期望最小化它会导致稳定的配置。在这个项目中，PI 解决了有关最小化某些区域的问题。考虑到所建模现象的噪声和小随机波动的能量泛函，期望该理论将更适合反映自然界中出现的实际最小化问题。面积泛函的基本特征是它在旋转下保持不变。空间（如果该空间是均匀的）。 PI 将解决均匀和晶体状空间中的几何和分析问题，提供更准确的自然模型。该项目将通过对研究生和导师的培训和指导，为美国劳动力发展做出贡献。博士后的目标之一是证明“几乎最小化器”，即噪声变分问题的最小化器继承了无噪声的相同函数的最小化器的一些属性。预计在此过程中开发的新思想将在其他具有自由边界的变分问题中得到应用，该项目有关进一步发展非光滑域的分析的目的是表征域的几何形状。欧几里得空间的规范（各向异性）算子的解的性质有关可修正性的项目有望揭示类晶体空间上定义的测度的精细结构，该项目的总体主题汇集了几何测度理论的工具。 、偏微分方程、势理论、调和分析和变分微积分，在这些领域之间架起桥梁，同时通过新思想的涌入来改变它们。该奖项反映了 NSF 的法定使命，并通过评估被认为值得支持利用基金会的智力优势和更广泛的影响审查标准。