Topics in Quasiconformal mappings and in PDE

本研究项目研究的问题以几何泛函的临界点及其梯度流的定性研究为共同主题。更具体地说，首席研究员将继续研究一类非线性偏微分方程（PDE）的弱解，这些方程是平均曲率流PDE的亚黎曼类似物，最小表面PDE和保形n-拉普拉斯方程。主要研究者还将研究强伪凸集间生物全纯映射边界正则性的几何函数理论/非线性PDE方法，并对空间中的极值拟共形映射进行研究。无论是在自然科学还是在工程中，研究复杂系统的平衡态都是很重要的。数学允许对能量泛函的临界点等状态进行建模，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取决于“背景几何”，它模拟了现实生活中的特征，如材料的非均匀性，或约束的存在（如机器人手臂的运动等，. ...）。这种“背景几何”最普遍的例子之一是亚黎曼几何，它对空间进行建模，其中运动只能沿着给定的一组方向进行，如在复杂分析、机械臂运动和控制理论、量子计算、卫星、量子力学以及哺乳动物视觉皮层的结构功能中的应用。PI将与他的学生和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“现实世界”问题的推动，PI讲座面向从研究生水平到高中水平的各种各样的受众。

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.