Topics in quasiconformal mappings and subelliptic PDE

利用亚黎曼几何和亚椭圆型偏微分方程对具有约束动力学的真实的生命系统进行建模。这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。通过研究空间到自身的某些变换族的行为，以定量的方式捕获这种空间的几何和分析性质。本项目旨在研究此类变换的精细性质。特别是，拟议的研究将提供一个理论基础，实现现实生活中的系统的数值模拟。在更广泛的影响方面，PI将让研究生和本科生参与研究和设计推广活动的几个方面，以吸引K12学生学习数学。拟议研究的技术重点是解决具有最小失真的准共形映射的Liouville定理。主要目的是证明这种映射的光滑性和刚性。所提出的方法是基于解决长期开放的问题的正则性的一类次椭圆拟线性偏微分方程。这类偏微分方程的研究本身就很重要，而且有着广泛的应用。该项目包括海森堡群之外的次椭圆p-拉普拉斯算子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.