Mathematical Sciences: Geometric Measure Theory and the Calculus of Variations

首席研究员将继续进行一项正在进行的关于几何变分及其应用的项目。这将包括研究曲面的几何结构，最小化几何积分，如面积。此外，他将研究曲面向最小化方向的演变，并分析其计算算法。他将开始对一般进化过程进行理论研究，比如模拟晶体从融化到冻结的生长过程。几何测量理论为这些分析及其在连续介质力学中的应用提供了数学基础。本项目涉及几何测度理论的各种应用以及理论本身的扩展。晶体生长和肥皂膜是这一理论已经应用到的两种物理现象。首席研究员将研究几何测量理论的理论方面和开发用于计算机图形学的软件的方法。

The principal investigator will continue an ongoing project in the geometric calculus of variations and its applications. This will include an investigation into the geometric structure of surfaces minimizing integrals with geometric integrands such as area. Additionally, he will study the evolution of surfaces towards minimization and analyze the algorithms for their computation. And he will begin a theoretical study of general evolution processes such as would model the growth of a crystal as it freezes from its melt. Geometric measure theory provides the mathematical setting for these analyses and their applications in continuum mechanics. This project involves a variety of applications of geometric measure theory as well as extensions of the theory itself. Crystal growth and soap films are two physical phenomena to which this theory has already been applied. The principal investigator will study both the theoretical aspects of geometric measure theory and ways of developing software for its use in computer graphics.