Geometry, Analysis, and Variational Methods

几何、分析和变分方法

• 批准号: 2105557
• 负责人: Fernando Marques
• 金额: $ 44.88万
• 依托单位: Princeton University
• 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 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2105557&HistoricalAwards=false
• 关键词: Geometry; Analysis; Variational; Methods

This project will investigate questions related to the variational theory of minimal surfaces and its applications. Minimal surfaces, of which soap bubbles are an illustrative example, are among the most natural objects in differential geometry. They have applications in many areas, such as three-dimensional topology, mathematical physics, complex and conformal geometry, and materials science. In general relativity, minimal surfaces appear as models for the apparent horizons of black holes. The minimal surface equation plays a very important role as a model for several kinds of nonlinear phenomena. Minimal surfaces have also been recently used in the design of materials with applications in biology and in chemistry. The project also includes training of PhD students and junior researchers. The PI will also disseminate his work through lectures, conferences, and workshops.This project will advance our basic understanding of minimal surfaces and their general existence theory. It concerns foundational questions about when these objects exist and how their properties relate to features of the ambient. The aim is to investigate the Morse-theoretic properties of the space of minimal varieties in a given Riemannian manifold. The idea is to use a combination of min-max methods, with the Almgren-Pitts min-max theory, and topological methods with the existence of homotopically nontrivial families of varieties. PI will study several questions related to this theme, including constructions of minimal hypersurfaces in higher dimensions and in the noncompact case. The project also includes plans for continued training of PhD students and post-doctoral researchers.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.

该项目将研究与最小曲面变分理论及其应用相关的问题。 最小曲面（肥皂泡就是一个说明性的例子）是微分几何中最自然的物体之一。 它们在许多领域都有应用，例如三维拓扑、数学物理、复杂和共形几何以及材料科学。 在广义相对论中，最小表面作为黑洞视界的模型出现。 最小曲面方程作为多种非线性现象的模型起着非常重要的作用。 最近，最小表面也被用于材料设计，并应用于生物学和化学领域。该项目还包括对博士生和初级研究人员的培训。 PI 还将通过讲座、会议和研讨会传播他的作品。该项目将增进我们对最小曲面及其一般存在理论的基本理解。 它涉及有关这些对象何时存在以及它们的属性如何与环境特征相关的基本问题。 目的是研究给定黎曼流形中最小簇空间的莫尔斯理论性质。这个想法是结合使用最小-最大方法、Almgren-Pitts 最小-最大理论以及存在同伦非平凡簇的拓扑方法。 PI 将研究与该主题相关的几个问题，包括更高维度和非紧情况下的最小超曲面的构造。该项目还包括继续培训博士生和博士后研究人员的计划。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。