CAREER: Algebraic Topology and Exterior Calculus in Numerical Analysis

PI将致力于离散外部微积分的数值分析方面。 这是一类用于求解偏微分方程（PDE）的数值方法，这些方法试图保留被建模物理的几何和代数结构。 该项目的一个组成部分将是使用代数拓扑和微分几何来创建和分析出现在重要偏微分方程中的对象和外部演算算子的离散化。 几个离散化将派生和由此产生的数值方法的收敛性和稳定性进行了研究。 除此之外，PI还将开发算法和软件，以便更容易地进行涉及此类离散化的实验研究。 偏微分方程的数值求解是数值分析及其在工程和科学中应用的核心部分。PI将研究的一些方程可应用于地下水污染、石油勘探、天气建模、核聚变、星星和行星形成以及太阳耀斑和风暴的形成。在这个项目中的研究将使用纯数学的主题，如微分几何和代数拓扑以及计算数学。因此，它将是一个独特的车辆，介绍这些主题的学生在数学以及计算机科学。由于上述应用的广泛吸引力，PI将使用这项研究中产生的动画，图像和概念，以培养小学生对数学和科学的兴趣。

The PI will work on the numerical analysis aspects of discrete exterior calculus. This is a class of numerical methods for solving partial differential equations (PDEs) and these methods attempt to preserve the geometric and algebraic structures of the physics being modeled. One ingredient of this project will be the use of algebraic topology and differential geometry to create and analyze discretizations of objects and operators of exterior calculus that appear in important PDEs. Several discretizations will be derived and the convergence and stability of the resulting numerical methods will be studied. In addition to this, the PI will develop algorithms and software to make it easier to conduct experimental studies involving such discretizations. All of this will be done in the context of several physical problems.Numerical solution of partial differential equations is a core part of numerical analysis and its applications in engineering and science. Some of the equations that the PI will work on have applications in ground water contamination, oil exploration, weather modeling, nuclear fusion, star and planet formation and formation of solar flares and storms. The research in this project will use pure mathematics topics like differential geometry and algebraic topology as well as computational mathematics. Thus it will be a unique vehicle with which to introduce these topics to students in mathematics as well as computer science. Due to the wide appeal of the applications mentioned above, the PI will use animations, images and concepts produced in this research to create interest in mathematics and science amongst primary school students.