Recent Developments on Geometric Measure Theory and its Applications

几何测度理论及其应用的最新进展

• 批准号: 2001095
• 负责人: Michael Wolf
• 金额: $ 3万
• 依托单位: William Marsh Rice University
• 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: 2020-02-01 至 2023-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2001095&HistoricalAwards=false
• 关键词: Recent; Developments; Geometric; Measure; Theory

This award provides partial participant support of the conference "Recent Developments on Geometric Measure Theory and its Applications", to be held at Rice University (Houston, Texas) on 19-21 March 2020. A common tendency of both nature and human engineering is to seek maximum efficiency in solving a problem. Nature will design a leaf to catch the sun and transport nutrients, subject to the constraining influence of the location of the plant; a soap film will use the least amount of material to span a boundary; humans will design a road system to most efficiently move people and goods from place to place. The study of what the optimal shapes are for natural and artificial design problems has been a focus of increasingly sophisticated study by mathematicians for centuries, involving techniques from the foundations of calculus, geometry and partial differential equations. In the past few years, there have been breakthroughs in a number of disparate areas of this subject, but few conferences in the United States that bring together experts from across the range of the subject to meet and exchange perspectives in a single meeting. This conference aims for such a mixture of ideas.We list a number of areas where there has been deep recent progress. First, there have been important developments in the basics of geometric measure theory in general metric spaces, both finite and infinite dimensional. There has also been enormous progress in geometrical variational problems, both in a number of areas of regularity theory and in minimax theory for minimal hypersurfaces. We also see deep results in the structure of nodal sets, mean curvature flows and in harmonic measure. Further details are available at https://math.rice.edu/NewsEvents/Conferences/BobHardtGMTConference/index.html.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.

该奖项为将于 2020 年 3 月 19 日至 21 日在莱斯大学（德克萨斯州休斯顿）举行的“几何测度理论及其应用的最新进展”会议提供部分参与者支持。自然和人类工程学的共同趋势是寻求解决问题的最大效率。大自然会设计一片叶子来吸收阳光和输送养分，但受到植物所处位置的制约影响；肥皂膜将使用最少的材料来跨越边界；人类将设计一个道路系统，以最有效地将人员和货物从一个地方运送到另一个地方。 几个世纪以来，对自然和人工设计问题的最佳形状的研究一直是数学家日益复杂的研究焦点，涉及微积分、几何和偏微分方程基础技术。 在过去的几年中，该主题的许多不同领域都取得了突破，但在美国，很少有会议将来自该主题范围的专家聚集在一起，在一次会议上会面并交换观点。 这次会议的目的是融合各种想法。我们列出了一些最近取得了深刻进展的领域。首先，在有限维和无限维的一般度量空间中，几何测度理论的基础已经取得了重要的发展。在几何变分问题方面也取得了巨大进展，无论是在正则理论的许多领域还是在最小超曲面的极小极大理论方面。我们还看到了节点集结构、平均曲率流和调和测度方面的深层结果。更多详细信息，请访问 https://math.rice.edu/NewsEvents/Conferences/BobHardtGMTConference/index.html。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。