FRG: Collaborative Research: New Challenges in Geometric Measure Theory

在数学、计算机科学和运筹学中，最优化问题无处不在。这些问题通常表述如下：是否存在一个最好的元素（关于一些标准）从一些可用的替代品？ 解决这类问题的尝试在现代数学的发展中发挥了重要作用。在1760年约瑟夫路易斯拉格朗日问是否存在一个表面的最小面积和规定的边界。这个问题被称为高原问题，是以物理学家约瑟夫·高原的名字命名的，他用肥皂膜做的实验产生了一个类似的数学问题。这些曲面的存在性和正则性是几何测度理论（GMT）的一部分。更一般地说，GMT将数学分析方法与微分几何的概念相结合，为研究偏微分方程和变分法中的关键现象提供适当的设置，这些关键现象通常来自优化问题。该领域最近的发展预示着一个即将到来的繁荣。PI的目标是利用他们独特的优势来利用这个非凡的机会。他们的科学目标的实现将屈服于未来几年将塑造该领域的发展。项目负责人团队的纵向一体化结构确保了该项目将对人力资源产生重大影响。PI的主要目标之一是教育下一代的研究人员在几何测量Theory.Pioneered在工作的贝西科维奇在三十年代，这个问题蓬勃发展，在五六十年代的工作与多维高原问题的德乔治，费德勒，弗莱明，赖芬贝格和Almgren。在这一极具创造性的时期发展起来的思想深深地影响了偏微分方程理论和变分法的进一步发展，在几何分析和数学广义相对论、调和分析和势能理论中产生了显著的影响。当前项目的重点是GMT中的三个主要挑战，所有这些挑战都将对其他分析领域产生重大影响。预计首席执行官及其同事将领导解决这些问题的努力。本项目研究的挑战是：-理解极小曲面和自由边界的奇异集;-发展退化椭圆或非光滑曲面能的正则性和刚性定理;-量化局部到全局的几何刚性结果。（ii）它们是最近引人注目的事态发展的主题，这增加了成功研究它们的机会;（iii）它们的解决有望在格林尼治标准时间以外产生相关影响;（四）该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的知识产权进行评估来支持。优点和更广泛的影响审查标准。

In mathematics, computer science and operations research, optimization problems are ubiquitous. The questions are often formulated as follows: does there exist a best element (with regard to some criterion) from some set of available alternatives? Attempts to address this type of questions have played a fundamental role in the development of modern mathematics. In 1760 Joseph-Louis Lagrange asked whether there exists a surface with minimal area and prescribed boundary. This problem known as the Plateau problem was named after the physicist Joseph Plateau whose experiments with soap films yield a similar mathematical problem. The existence and regularity of such surfaces are part of Geometric Measure Theory (GMT). More generally, GMT combines methods of mathematical analysis with concepts from differential geometry, to develop the appropriate setting for studying critical phenomena in Partial Differential Equations and in the Calculus of Variations, often arising from optimization questions. Recent developments in the field forecast an imminent boom. It is the PIs' objective to capitalize on this extraordinary opportunity they are uniquely positioned to take advantage of. Fulfillment of their scientific goals will yield to developments that will shape the field for years to come. The vertically integrated structure of the PIs' teams ensures that this project will have a considerable impact in human resources. One of the PIs' main goals is to educate the next generation of researchers in Geometric Measure Theory.Pioneered in the work of Besicovitch in the thirties, the subject boomed in the fifties and sixties with the work on the multidimensional Plateau problem by De Giorgi, Federer, Fleming, Reifenberg and Almgren. The ideas developed in that extremely creative period have deeply influenced the further development of the theory of Partial Differential Equations and of the Calculus of Variations, with noticeable effects in Geometric Analysis and Mathematical General Relativity, and Harmonic Analysis and Potential Theory. The current project focuses on three major challenges in GMT, all of which are poised to have a significant impact in other areas of analysis. The PIs and their associates are expected to lead the efforts to address these problems. The challenges investigated in this project are: - Understanding singular sets of minimal surfaces and free boundaries;- Developing regularity and rigidity theorems for degenerate elliptic or non-smooth surface energies;- Quantifying local-to-global geometric rigidity results.These problems share common traits of: (i) they exemplify the most interesting open questions in the area; (ii) they have been the subject of striking recent developments, which increase the chances of their successful study; (iii) their resolution promises to have relevant impacts outside of GMT; (iv) requiring a broad approach which is encompassed by the expertise of the three PIs.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.