Geometric Flows and Applications

几何流有许多实际应用，包括材料科学，生物学和图像处理。从数学上讲，它们是抛物型偏微分方程，可以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，它们在其他数学学科中也有潜在的应用，如数学物理和低维拓扑。该奖项支持几何流的两个基本例子，平均曲率流和里奇流的调查。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，监督本科生，研究生和年轻学者，并组织研讨会和会议，非常重视微分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参与STEM以促进社会多样性和公平方面发挥重要作用。该项目的第一部分是关于低熵闭超曲面的性质。它涉及到的平均曲率流的渐近锥形自膨胀的模空间的整体特征的探索。首要目标是验证光滑四维Schoenflies猜想的超曲面与低熵。第二部分涉及渐近圆锥自扩张器新例子的变分构造。第三部分研究了平均曲率流和Ricci流孤子解的渐近结构。PI的目的是显示这些孤子解的几何条件下，温和的拓扑限制是有界的各种sension.This奖项反映了NSF的法定使命，并已被认为是值得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的支持。

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.