Recent Developments on Geometric Measure Theory and its Applications

该奖项为将于2020年3月19-21日在莱斯大学(德克萨斯州休斯敦)举行的“几何测量理论及其应用的最新发展”会议提供部分参与者支持。自然界和人类工程学的一个共同趋势是在解决问题时寻求最大效率。大自然将设计一片树叶来捕捉阳光并运输养分，但受植物位置的限制；肥皂膜将使用最少的材料来跨越边界；人类将设计一种道路系统，以最高效地将人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们日益复杂的研究的重点，涉及微积分、几何和偏微分方程式的基础技术。在过去的几年里，在这一主题的一些不同领域取得了突破，但在美国，很少有来自这一主题范围内的专家聚集在一起在一次会议上会面和交流观点的会议。这次会议旨在提出这样一种混合的想法。我们列出了最近取得重大进展的一些领域。首先，在一般度量空间中，无论是有限维的还是无限维的，几何测度论的基础都有了重要的发展。几何变分问题在正则性理论和极小超曲面的极大极小理论的许多领域也取得了巨大的进展。我们还看到在节点集的结构、平均曲率流和调和度量方面的深刻结果。更多细节可在https://math.rice.edu/NewsEvents/Conferences/BobHardtGMTConference/index.html.This网站获得，该奖项反映了国家科学基金会的法定使命，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。

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.