Geometric Flows and Applications

几何流及其应用

基本信息

批准号：
2018220
负责人：
Lu Wang
金额：
$ 17.78万
依托单位：
California Institute of Technology
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-09-01 至 2021-09-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2018220&HistoricalAwards=false
关键词：
Geometric Flows Applications

项目摘要

Geometric flows have many real-world applications including material sciences, biology and image processing. Mathematically they are parabolic partial differential equations that deform geometric objects to their optimal shapes. In addition to their importance in geometric analysis, they also have potential applications to other mathematical disciplines, such as mathematical physics and low-dimensional topology. This award supports the investigation of two fundamental examples of geometric flows, mean curvature flow and Ricci flow. The PI will develop new ideas and robust techniques that will benefit the study of other geometric partial differential equations and related applications. In addition, the PI will place a strong emphasis on education in differential geometry and related topics through teaching, supervising undergraduate, graduate students and young scholars, and organizing seminars and conferences. The PI will also play an important role in the promotion of women and other underrepresented groups in STEM to enhance diversity and equity in the society.The first part of the project is on the properties of closed hypersurfaces with low entropy. It involves an exploration of global features of the moduli space of asymptotically conical self-expanders of mean curvature flow. An overarching goal is to verify the smooth four-dimensional Schoenflies conjecture for hypersurfaces with low entropy. The second part concerns the variational construction of new examples of asymptotically conical self-expanders. The third part probes the asymptotic structure of soliton solutions to mean curvature flow as well as Ricci flow. The PI aims to show the geometry of these soliton solutions under mild topological restrictions is bounded in various senses.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.

几何流量有许多现实世界的应用，包括材料科学，生物学和图像处理。从数学上讲，它们是抛物线偏微分方程，将几何对象变形为最佳形状。除了在几何分析中的重要性外，它们还对其他数学学科（例如数学物理学和低维拓扑结构）也有潜在的应用。该奖项支持对几何流量，平均曲率流和RICCI流的两个基本示例的研究。 PI将开发新的想法和强大的技术，从而使其他几何部分微分方程和相关应用的研究受益。此外，PI将通过教学，监督本科，研究生和年轻学者以及组织研讨会和会议来强烈重视差异几何和相关主题的教育。 PI还将在促进妇女和其他代表性不足的群体中发挥重要作用，以增强社会的多样性和公平性。该项目的第一部分是在距离熵较低的封闭性超表面的特性上。它涉及对平均曲率流的渐近圆锥体自扩散器的模量空间的整体特征的探索。一个总体目标是验证熵低的侧面表面的平滑四维schoenflies猜想。第二部分涉及渐近锥形自扩张者的新示例的变分结构。第三部分探测孤子溶液的渐近结构，以使平均曲率流以及RICCI流动。 PI的目的是在温和的拓扑限制下显示这些孤子解决方案的几何形状具有各种意义。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和更广泛影响的评估审查标准来通过评估来支持的。