Geometric Flows and Applications

几何流在现实世界中有许多应用，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，可以将几何物体变形为最佳形状。除了它们在几何分析中的重要性外，它们也有潜在的应用于其他数学学科，如数学物理和低维拓扑。该奖项支持对几何流的两个基本例子的研究，即平均曲率流和利玛窦流。PI将开发新的思想和强大的技术，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，研究生和青年学者，以及组织研讨会和会议，强调微分几何和相关主题的教育。该计划亦将在促进女性及其他在STEM领域未获充分代表的群体方面发挥重要作用，以加强社会的多样性和公平。项目的第一部分是关于低熵闭超曲面的性质。研究了平均曲率流的渐近锥形自展开器的模空间的全局特征。一个总体目标是验证具有低熵的超曲面的光滑四维舍恩菲猜想。第二部分是关于渐近圆锥自展开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。π的目的是表明在温和拓扑限制下这些孤子解的几何形状在各种意义上是有界的。该奖项反映了美国国家科学基金会的法定使命，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。

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.