Geometric Flows and Applications

几何流及其应用

• 批准号: 2141529
• 负责人: Lu Wang
• 金额: $ 17.78万
• 依托单位: Yale 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 2023-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2141529&HistoricalAwards=false
• 关键词: Geometric; Flows; Applications

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，可将几何对象变形为其最佳形状。除了在几何分析中的重要性之外，它们还具有在其他数学学科中的潜在应用，例如数学物理和低维拓扑。该奖项支持对几何流的两个基本示例的研究，即平均曲率流和里奇流。 PI 将开发新的想法和强大的技术，这将有利于其他几何偏微分方程和相关应用的研究。此外，PI将通过教学、指导本科生、研究生和年轻学者以及组织研讨会和会议，重点关注微分几何及相关主题的教育。 PI 还将在促进女性和其他代表性不足的群体参与 STEM 方面发挥重要作用，以增强社会的多样性和公平性。该项目的第一部分是研究低熵封闭超曲面的性质。它涉及对平均曲率流渐近圆锥自扩张器模空间的全局特征的探索。首要目标是验证低熵超曲面的平滑四维 Schoenflies 猜想。第二部分涉及渐近圆锥自扩张器新例子的变分构造。第三部分探讨了平均曲率流和Ricci流的孤子解的渐近结构。该 PI 旨在展示这些孤子解在轻度拓扑限制下的几何形状在各种意义上都是有界的。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。