Topics in quasiconformal mappings and subelliptic PDE

用次黎曼几何和次椭圆偏微分方程来模拟有约束动力学的实际系统。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。特别是，所提出的研究将为实现现实系统的数值模拟提供理论基础。就更广泛的影响而言，PI将涉及研究生和本科生的几个方面的研究和设计外展活动，以吸引K12学生学习数学。提出的研究的技术重点是具有最小畸变的拟共形映射的Liouville定理。主要目标是证明这种映射的平滑性和刚性。提出的方法是基于求解一类亚椭圆拟线性偏微分方程的长期开放的正则性问题。这种PDE的研究本身就很重要，具有更广泛的应用。本课题包含了超越海森堡群的次椭圆p-拉普拉斯的正则性和次黎曼流形的p-调和坐标的构造。

SubRiemannian geometry and subelliptic Partial Differential Equations are used to model real life systems 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 particular, the proposed research will provide a theoretical basis for implementing numerical simulations of real-life systems. In terms of broader impacts, the PI will involve graduate and undergraduate students in several aspects of the research and design outreach activities to attract K12 students to mathematics.The technical focus of the proposed research addresses the Liouville Theorem for quasiconformal mappings with minimal distortion. The main goal is to prove smoothness and rigidity of such mappings. The proposed approach is based on the solution of longstanding open problems of regularity for a class of subelliptic quasilinear PDE. The study of such PDE is important on its own and has broader applications. This project encompasses regularity for subelliptic p-laplacian beyond the Heisenberg group and construction of p-harmonic coordinates for subRiemannian manifolds.