CAREER: Algebraic Topology and Exterior Calculus in Numerical Analysis

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

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.