Geometric Flows and Applications

几何流在材料科学、生物学和图像处理等领域有着广泛的应用。从数学上讲，它们是抛物型偏微分方程式，可以将几何物体变形成最佳形状。除了它们在几何分析中的重要性外，它们还有可能应用于其他数学学科，如数学物理和低维拓扑学。该奖项支持对几何流的两个基本例子的研究，即平均曲率流和Ricci流。PI将发展出新的思想和稳健的技术，这将有助于其他几何偏微分方程组的研究和相关应用。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，高度重视微分几何和相关主题的教育。PI还将在促进STEM中的妇女和其他代表性不足的群体方面发挥重要作用，以增强社会的多样性和公平性。该项目的第一部分是关于具有低熵的封闭超曲面的性质。它涉及到平均曲率流的渐近锥形自扩张器的模空间的全局特征的探索。一个重要的目标是验证光滑的四维Schoenfly猜想，用于低熵的超曲面。第二部分是渐近锥形自扩张器新实例的变分构造。第三部分讨论了平均曲率流和Ricci流的孤子解的渐近结构。PI旨在展示这些孤子解在温和的拓扑限制下的几何在各种意义上是有界的。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.