Recent Developments on Geometric Measure Theory and its Applications

该奖项提供了部分参与者支持会议“几何测量理论及其应用的最新发展”，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。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。大自然会设计一片叶子来捕捉阳光和运输养分，但要受到植物位置的限制;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，研究自然和人工设计问题的最佳形状一直是数学家日益复杂的研究焦点，涉及微积分，几何和偏微分方程的基础技术。 在过去几年中，在这一主题的一些不同领域取得了突破，但在美国，很少有会议将来自该主题各个领域的专家聚集在一起，在一次会议上交流观点。 本次会议的目的就是要集思广益，我们列出了一些最近取得重大进展的领域。首先，在一般度量空间（包括有限维和无限维）中，几何测度理论的基础有了重要的发展。在几何变分问题上也有了巨大的进展，包括正则性理论的一些领域和极小超曲面的极大极小理论。我们也看到了节点集的结构，平均曲率流和调和测度的深刻结果。更多细节可在www.example.com上获得https://math.rice.edu/NewsEvents/Conferences/BobHardtGMTConference/index.html.This奖项反映了NSF的法定使命，并被认为值得通过使用基金会的知识价值和更广泛的影响审查标准进行评估来支持。

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.