Topics in Quasiconformal mappings and in PDE

本研究课题的共同主题是对几何泛函的临界点及其梯度流的定性研究。更具体地说，主要研究人员将继续研究一族非线性偏微分方程组(PDE)的弱解，这些偏微分方程组是平均曲率流PDE、最小曲面PDE和共形n-拉普拉斯方程的次黎曼类似。主要研究者还将研究强伪凸集之间双全纯映射的边界正则性的几何函数理论/非线性偏微分方程法，并继续研究空间中的极值拟共形映射。在自然科学和工程中，研究复杂系统的平衡态都是非常重要的。数学允许将这样的状态建模为能量泛函的临界点，并通过分析与这些泛函相关的某些微分方程来研究它们的性质。这些方程的解的性质依赖于对诸如材料的非均质性或约束的存在(例如在机器人手臂的运动中等)等现实生活特征进行建模的‘背景几何’。这种“背景几何”最普遍的例子之一就是亚黎曼几何，它模拟的空间只能沿着一组给定的方向运动，例如在复杂分析、机器人手臂的运动和控制理论、量子计算、卫星、量子力学和哺乳动物视觉皮质的结构功能中的应用。PI将与他的学生以及他的合作者一起进行这项工作。亚黎曼几何的研究自其起源以来，一直受到“真实世界”问题和圆周率讲座的推动，从研究生水平到高中水平，各种听众都有参与。

The problems studied in this research project have as common theme the qualitative study of critical points of functionals arising from geometry and of their gradient flows. More specifically the principal investigator will continue to study weak solutions to a family of nonlinear partial differential equations (PDE) that are the sub-Riemannian analogues of the mean curvature flow PDE, the minimal surface PDE, and the conformal n-Laplacian. The principal investigator will also study a geometric function theory/nonlinear PDE approach to the boundary regularity of biholomorphic mappings between strongly pseudoconvex sets and pursue the study of extremal quasiconformal mappings in space. Both in the natural sciences and in engineering it is important to study equilibrium states of complex systems. Mathematics allows to model such states as critical points of energy functionals and to study their properties by analyzing certain differential equations associated to these functionals. The properties of the solutions to these equations depend on a 'background geometry' that models such real-life features as the non-homogeneity of materials, or the presence of constraints (such as in the motion of robot arms, etc. etc. ...). One of the most ubiquitous instances of such 'background geometry' is sub-Riemannian geometry, modeling spaces where motion is possible only along a given set of directions, as in applications to complex analysis, motion of robot arms and control theory, quantum computing, satellites, quantum mechanics and in the structural functions of the mammalian visual cortex. The PI will pursue this work together with his students as well as his collaborators. The study of sub-Riemannian geometry has been driven since its origins by "real world" problems and the PI lectures to a wide variety of audiences from the post-graduate level to the high school level.