FRG: Collaborative Research: New Challenges in Geometric Measure Theory

FRG：协作研究：几何测度理论的新挑战

• 批准号: 1854344
• 负责人: Francesco Maggi
• 金额: $ 14.15万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-09-01 至 2023-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1854344&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; New; Challenges

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.

在数学、计算机科学和运筹学中，优化问题无处不在。这些问题通常是这样表述的：在一组可用的替代方案中，是否存在一个最佳元素（就某些标准而言）？解决这类问题的尝试在现代数学的发展中发挥了根本性的作用。1760年，约瑟夫·路易斯·拉格朗日提出了一个问题：是否存在一个面积最小且有规定边界的曲面。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，他对肥皂膜的实验得出了一个类似的数学问题。这种曲面的存在性和规律性是几何测量理论（GMT）的一部分。更一般地说，GMT结合了数学分析的方法，从微分几何的概念，开发适当的设置，研究在偏微分方程和在变分法的关键现象，往往从优化问题产生。该领域最近的发展预示着一个即将到来的繁荣。ppi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产生将在未来几年塑造该领域的发展。pi团队的垂直整合结构确保该项目将对人力资源产生相当大的影响。pi的主要目标之一是教育下一代几何测量理论的研究人员。该学科在30年代由贝西科维奇开创，在50年代和60年代随着德乔吉、费德勒、弗莱明、赖芬贝格和阿尔姆格伦对多维高原问题的研究而蓬勃发展。在那个极具创造性的时期发展起来的思想深刻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以及调和分析和势理论中产生了显著的影响。当前的项目集中于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pi及其同伙将领导解决这些问题的努力。在这个项目中研究的挑战是：-理解最小曲面和自由边界的奇异集合；-发展简并椭圆或非光滑表面能的正则性和刚性定理；——量化局部到全局几何刚度的结果。这些问题具有以下共同特点：(i)它们都是该领域最有趣的开放性问题；（ii）这些学科最近有显著的发展，增加了成功学习的机会；（iii）其决议承诺在格林尼治标准时间以外产生相关影响；（iv）需要一个广泛的方法，包括三个专业人士的专业知识。该奖项反映了美国国家科学基金会的法定使命，并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，被认为值得支持。

项目成果

Rigidity theorems for best Sobolev inequalities

最佳索博列夫不等式的刚性定理

• DOI: 10.1016/j.aim.2023.109330
• 发表时间: 2023
• 期刊: Advances in Mathematics
• 影响因子: 1.7
• 作者: Maggi, Francesco;Neumayer, Robin;Tomasetti, Ignacio
• 通讯作者: Tomasetti, Ignacio

Collapsing and the convex hull property in a soap film capillarity model

皂膜毛细管模型中的塌陷和凸包特性

• DOI: 10.1016/j.anihpc.2021.02.005
• 发表时间: 2021
• 期刊: Analyse non linéaire
• 作者: King, Darren;Maggi, Francesco;Stuvard, Salvatore
• 通讯作者: Stuvard, Salvatore

Smoothness of Collapsed Regions in a Capillarity Model for Soap Films

皂膜毛细管模型中塌陷区域的平滑度

• DOI: 10.1007/s00205-021-01727-3
• 发表时间: 2022
• 期刊: Archive for Rational Mechanics and Analysis
• 影响因子: 2.5
• 作者: King, Darren;Maggi, Francesco;Stuvard, Salvatore
• 通讯作者: Stuvard, Salvatore

Symmetry and Rigidity of Minimal Surfaces with Plateau-like Singularities

具有高原状奇点的最小曲面的对称性和刚度

• DOI: 10.1007/s00205-020-01593-5
• 发表时间: 2021
• 期刊: Archive for Rational Mechanics and Analysis
• 影响因子: 2.5
• 作者: Bernstein, Jacob;Maggi, Francesco
• 通讯作者: Maggi, Francesco

Plateau's Problem as a Singular Limit of Capillarity Problems

作为毛细管问题的奇异极限的高原问题

• DOI: 10.1002/cpa.22048
• 发表时间: 2022
• 期刊: Communications on Pure and Applied Mathematics
• 影响因子: 3
• 作者: King, Darren;Stuvard, Salvatore;Maggi, Francesco
• 通讯作者: Maggi, Francesco