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Geometry of Surfaces and Four-Dimensional Manifolds

曲面几何和四维流形

基本信息

批准号：
2104988
负责人：
Hung Tran
金额：
$ 21.19万
依托单位：
Texas Tech University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-08-01 至 2024-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2104988&HistoricalAwards=false
关键词：
Geometry Surfaces Four Dimensional Manifolds

项目摘要

The goal of this project is to explore critical points of natural geometric functionals, particularly in the fields of minimal surfaces with boundaries and four-dimensional manifolds. These concepts arise naturally such as when one dips a wireframe into a soap solution, forming a soap film, which is an example of a minimal surface. Similarly, the world we are living in can be modeled as a four-dimensional manifold with three spatial and one time directions. Thus, advancements in this area will have applications in physics, biology, and applied sciences. As a consequence, the project's objective is to advance knowledge by exploring the following research directions. The project also includes mentoring of students (at both the undergraduate and graduate levels) and the organization of a mini-school aimed at broadening participation of under-represented minorities in STEM fields.The first research direction in this project aims to study stationary surfaces with boundaries, examining the first and second variations to draw out classification and uniqueness results. In particular, motivated by Lawson and Willmore's conjectures for minimal surfaces in a sphere, the PI will conduct a parallel study for free boundary minimal surfaces in a ball. By introducing creative ideas (Jacobi-Steklov eigenvalue, interpreting constraints as linear functionals), the PI will develop an approach that gives a framework for further investigations. The second thrust studies the connection between the geometry and topology of a four-dimensional manifold. The PI aims to classify these manifolds under natural conditions, particularly resolving a differentiable sphere conjecture of the second kind and providing insights on one of Hopf's conjectures and a folklore conjecture about Einstein metrics. The guiding principle is that the Hodge star operator gives rise to several elliptic identities in dimension four. The project also includes several mentoring activities as well as impactful activities aimed at broadening participation of under-represented minorities in STEM fields.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.

这个项目的目标是探索自然几何泛函的临界点，特别是在具有边界的极小曲面和四维流形领域。这些概念是自然产生的，例如当人们将线框浸入肥皂溶液中时，形成肥皂膜，这是最小表面的一个例子。类似地，我们生活的世界可以被建模为具有三个空间方向和一个时间方向的四维流形。因此，这一领域的进展将在物理学、生物学和应用科学中得到应用。因此，该项目的目标是通过探索以下研究方向来推进知识。该项目还包括对学生（本科生和研究生）进行辅导，并组织一个小型学校，旨在扩大在STEM领域代表性不足的少数群体的参与。该项目的第一个研究方向旨在研究有边界的静止表面，检查第一和第二种变化，以得出分类和独特性结果。特别是，受Lawson和Willmore的球面极小曲面的启发，PI将对球中的自由边界极小曲面进行平行研究。通过引入创造性的想法（Jacobi-Steklov特征值，将约束解释为线性泛函），PI将开发一种方法，为进一步的研究提供一个框架。第二个推力研究四维流形的几何和拓扑之间的联系。 PI的目的是在自然条件下对这些流形进行分类，特别是解决第二类可微球猜想，并提供对Hopf的一个猜想和关于爱因斯坦度量的民间猜想的见解。指导原则是霍奇星星运营商产生了几个椭圆身份在第四个维度。该项目还包括几项指导活动以及旨在扩大STEM领域代表性不足的少数群体参与的有影响力的活动。该奖项反映了NSF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。