CAREER: Fine Structure of the Singular Set in Some Geometric Variational Problems

几何变分问题的研究是数学中最古老和最吸引人的话题之一。这些问题的解描述了物理系统的平衡构型，并为研究流形的几何和拓扑提供了规范的工具。物理上可以观察到这一点，例如，当三个肥皂泡在一条公共线上合并形成一个角，或者研究冰山融化成水的过渡区的结构。这个项目的目标是研究这种奇异解的结构，这通常是它们在几何、拓扑和物理中应用的主要绊脚石。该项目的核心是教育活动的综合计划。这包括组织REU项目和加州大学圣地亚哥分校冬季研究生院关于几何分析的最新趋势。该项目将邀请该领域的专家在加州大学圣迭戈分校停留五天，以增加研讨会的整体活动，让研究生接触到最有趣的结果和开放的问题，并鼓励合作。这个项目将专注于两个最经典和最有影响力的几何变分问题：极小曲面和自由边界问题。极小曲面为研究流形的拓扑提供了规范对象，是肥皂膜和划分问题的一个很好的模型。自由边界问题是模拟一系列物理现象的基本问题，例如相变(如冰融化成水)、喷流和空泡流动、形状优化类型问题和美式期权的定价。几何变分问题的解具有奇异性。在极小曲面的背景下，PI将研究面积最小化超曲面和任何余维的曲面的奇异集的正则性，在整数和模p的情况下都是如此。对于自由边界问题，焦点将集中在两相问题和向量Alt-Caffarelli问题的所谓分支点集的结构上，以及它与西诺里尼问题解的高频点集的关系。这将通过改进PI和他的合作者最近引入的一些新技术来实现，并通过开发新的技术来实现，这些新技术将对几何分析中的许多其他问题产生影响。REU项目和冬季研究生院的主要目标之一是向本科生和研究生介绍这些问题和技术。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。

The study of Geometric Variational problems is one of the oldest and most fascinating topics in Mathematics. Solutions to these problems describe equilibrium configurations of physical systems and provide canonical tools to study the geometry and topology of manifolds. Physically this can be observed for instance when three soap bubbles merge on a common line forming a corner, or studying the structure of the transition region of an iceberg melting into water. The goal of this project is to investigate the structure of such singular solutions, which is often the major stumbling block in their application to Geometry, Topology and Physics. Central to the project is an integrated plan of educational activities. This consists in the organization of a REU program and a winter Graduate School at UCSD on recent trends in Geometric Analysis. The PI will invite experts in the field for five days stays at UCSD to increase the overall activity of the seminar, expose graduate students to the most interesting results and open questions, and encourage collaborations.This project will focus on two of the most classical and influential Geometric Variational problems: Minimal Surfaces and Free-Boundary problems. Minimal surfaces provides canonical objects to study the topology of manifolds and are a good model for soap films and partition problems. Free-Boundary problems are fundamental in modeling a wide range of physical phenomena, such as phase transition (e.g. the melting of ice into water), flows with jets and cavities, shape optimization type problems and the pricing of American options. Solutions to geometric variational problems are known to exhibit singularities. In the context of Minimal Surfaces, the PI will investigate the regularity of the singular set for Area Minimizing hypersurfaces and for surfaces in any codimension, both in the integer and modulo p cases. For Free-Boundary problems, the focus will be on the structure of the set of so-called branching points for the Two-Phase problem and the Vectorial Alt-Caffarelli problem, and its relation with the set of points of high frequency of solutions to the Signorini problem. This will be achieved refining some new techniques recently introduced by the PI and his collaborators, and by developing new ones, which will have an impact in many other problems in Geometric Analysis. One of the major goals of the REU program and the winter Graduate School is to introduce undergraduate and graduate students to these problems and techniques.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.