Regularity and Singularity Issues in Geometric Variational Problems

几何变分问题用于描述由表面张力能驱动的系统的行为，如晶体或肥皂泡。经典的例子是等长问题（寻找包围给定体积的最小周长的固体）和高原问题（寻找跨越给定边界曲线的最小面积的表面）。在过去的50年里，几何变分问题在纯数学和应用数学中都有许多令人惊讶的应用。它们的解可以描述数学物理中物理系统的平衡构型，广义相对论中黑洞的行为，或者它们可以提供微分拓扑中同调和同伦类的首选表示。对于几何变分问题的解，奇点的存在是不可避免的，它与它们所模拟的系统的物理行为或几何问题中拓扑障碍物的集中有关。对正则集和奇异集的精细描述对于我们理解潜在问题具有根本的重要性。本项目旨在通过解决一系列关于几何变分问题解的正则性和奇点描述的基本问题，提高我们对几何变分问题解的定性和定量行为的理解。这些问题的答案将需要新的方法和技术的发展，这在其他数学领域也将是有价值的。该项目将为研究生和本科生提供研究培训机会。尽管对几何变分问题进行了大量的研究，但关于其解的正则性和奇异性的几个基本问题仍然知之甚少。本项目旨在解决这些问题，并开发新的工具和技术来解决这些问题。该项目涉及四个相互关联的研究方向：质量最小电流的边界规则、各向异性表面张力临界点的规则、谱优化和自由边界问题解的规则以及PDE约束措施的结构。他们的研究将需要来自不同数学领域的技术的相互作用，如偏微分方程（PDE）、几何分析、几何测量理论、拓扑学和谐波分析，以及新技术的引入。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。

Geometric variational problems are used to describe the behavior of systems driven by surface tension energies, like crystals or soap bubbles. Classical examples are the isoperimetric problem (find the solid of minimal perimeter enclosing a given volume) and the Plateau problem (finding the surface of minimal area spanning a given boundary curve). In the last 50 years, geometric variational problems have found a number of surprising applications in both pure and applied mathematics. Their solutions can describe the equilibrium configurations of physical systems in Mathematical Physics, the behavior of black holes in General Relativity, or they can provide preferred representatives in homology and homotopy classes in Differential Topology. For solutions of geometric variational problems, the presence of singularities is unavoidable and it is linked to the physical behavior of the systems they model or to the concentration of topological obstructions in geometric problems. A fine description of the regular and singular set is of fundamental importance in our understanding of the underlying problem. This project aims to enhance our understanding of the qualitative and quantitative behavior of solutions of geometric variational problems by addressing a series of basic questions concerning their regularity and the description of their singularities. The answer to these questions will require the development of new methods and techniques, which will also be valuable in other areas of mathematics. The project will provide research training opportunities for graduate and undergraduate students. Despite the great amount of study dedicated to geometric variational problems, several basic questions concerning the regular and singular behavior of their solutions are still poorly understood. This project is intended to address these questions and to develop new tools and techniques to tackle them. The project involves work on four deeply interconnected directions of research: boundary regularity for mass minimizing currents, regularity of critical points of anisotropic surface tensions, regularity of solutions to spectral optimization and free boundary problems, and the structure of PDE constrained measures. Their study will require the interaction of techniques coming from different areas of mathematics, such as Partial Differential Equations (PDE), Geometric Analysis, Geometric Measure Theory, Topology, and Harmonic Analysis, as well as the introduction of new ones.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.