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

职业：一些几何变分问题中奇异集的精细结构

基本信息

批准号：
2044954
负责人：
Luca Spolaor
金额：
$ 55万
依托单位：
University of California-San Diego
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-07-01 至 2026-06-30
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2044954&HistoricalAwards=false
关键词：
CAREER Fine Structure Singular Set

项目摘要

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.

几何变异问题的研究是数学中最古老，最迷人的主题之一。这些问题的解决方案描述了物理系统的平衡配置，并提供了研究歧管的几何形状和拓扑的规范工具。从物理上讲，这可以观察到，例如，当三个肥皂气泡在形成角落的公共线上合并，或研究冰山的过渡区域融化成水中的结构。该项目的目的是研究这种奇异解决方案的结构，这通常是其应用于几何，拓扑和物理学的主要绊脚石。该项目的核心是一项综合教育活动计划。这在于，关于几何分析的最新趋势，在UCSD的REU计划和冬季研究生院的组织组成。 PI将邀请该领域的专家五天留在UCSD，以增加研讨会的整体活动，使研究生接触到最有趣的结果和开放的问题，并鼓励合作。该项目将重点放在两个最经典和最有影响力的几何变异问题上：最小的表面和自由问题。最小表面提供了研究歧管拓扑的规范对象，并且是肥皂膜和分区问题的良好模型。自由边界问题对于建模广泛的物理现象是基本的，例如相变（例如，冰熔化到水中），用喷气机和空腔流动，形状优化类型问题以及美国选择的定价。已知对几何变异问题的解决方案表现出奇异性。在最小表面的背景下，PI将调查在整数和模量P病例中，在任何编成的奇迹中最小化的奇异设置的规律性，以最大程度地减少任何编成曲折的表面。对于自由结合的问题，重点将放在两相问题和矢量alt-carfarelli问题的所谓分支点的结构上，及其与Signorini问题的高频解决方案点的关系集。这将是通过PI及其合作者最近引入的一些新技术以及开发新技术的方法来实现的，这将对几何分析中的许多其他问题产生影响。 REU计划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的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛的影响来通过评估来支持的。