Applications of Quasiconformal Geometry and Partial Differential Equations

偏微分方程（PDE）被用于对各种应用领域的重要系统进行建模；特别是，亚椭圆PDE在有约束动力学的情况下是有用的。此类系统的例子包括机器人手臂的运动、哺乳动物视觉皮层第一层的结构功能、金融市场的布莱克-斯科尔斯模型和量子计算。这些空间的几何和解析性质是通过研究空间本身的某些变换族的行为以定量的方式捕获的。本项目旨在研究此类变换的精细性质。就更广泛的影响而言，首席研究员将让研究生和本科生参与研究的几个方面，并将设计外展活动，以吸引K-12学生学习数学。提出的研究的技术重点是卡诺群中的曲线收缩流，某些Gromov双曲空间边界间拟共形映射的调和扩展的研究，以及某些非线性退化抛物型PDE的正则性。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。

Partial differential equations (PDE) are used to model important systems in various application areas; in particular, subelliptic PDE are helpful in settings where there is a constrained dynamics. Examples of such systems include the motion of robot arms, structural functions of the first layer of the mammalian visual cortex, the Black-Scholes model for financial markets, and quantum computing. Geometric and analytic properties of such spaces are captured in a quantitative fashion by studying the behavior of certain families of transformations of the space into itself. This project aims at studying fine properties of such transformations. In terms of broader impacts, the principal investigator will involve graduate and undergraduate students in several aspects of the research and will design outreach activities to attract K-12 students to mathematics.The technical focus of the proposed research addresses a curve-shrinking flow in Carnot groups, the study of harmonic extensions of quasiconformal mappings between boundaries of certain Gromov hyperbolic spaces, and regularity of certain nonlinear, degenerate parabolic PDE.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.