Regularity and Singularity Issues in Geometric Variational Problems

几何变分问题被用来描述由表面张力能驱动的系统的行为，如晶体或肥皂泡。经典的例子是等周问题(寻找包围给定体积的最小周长的实体)和平台问题(寻找跨越给定边界曲线的最小面积的曲面)。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，可以描述广义相对论中黑洞的行为，也可以在微分拓扑学中的同调和同伦类中提供首选代表。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的拓扑障碍的集中有关。对正则集和奇异集的精细描述对于我们理解根本问题是非常重要的。这个项目旨在通过解决一系列关于几何变分问题解的规律性和奇异性的描述的基本问题来加强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开发新的方法和技术，这在数学的其他领域也是有价值的。该项目将为研究生和本科生提供研究培训机会。尽管有大量的研究致力于几何变分问题，但关于其解的正则行为和奇异行为的几个基本问题仍然知之甚少。该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：质量最小化电流的边界正则性、各向异性表面张力临界点的正则性、谱优化和自由边界问题解的正则性以及偏微分方程约束措施的结构。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、拓扑学和调和分析，以及引入新的技术。该奖项反映了NSF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。

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.