Computer algebra for geometric evolution equations

几何演化方程的计算机代数

• 批准号: 171126814
• 负责人: Professor Dr. Oliver Schnürer
• 金额: --
• 依托单位: Arbeitsgruppe Differentialgeometrie
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2010
• 资助国家: 德国
• 起止时间: 2009-12-31 至 2014-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/171126814?language=en
• 关键词: Computer; algebra; geometric; evolution; equations

Many challenging problems in geometry concern flow equations like e. g. mean curvature flow or Ricci flow. The behavior of solutions to such flow equations is often controlled as follows: For a geometrically significant quantity, the evolution equation is computed. It is proved that this quantity is monotone, i. e. a Lyapunov function. This allows to control solutions of the flow equation. The computation of evolution equations for prospective Lyapunov functions is purely algebraic but usually quite tedious. We propose to develop a program that does these algebraic computations in many different situations. We also wish to use algebraic and experimental methods to select prospective Lyapunov functions and to check, whether the resulting evolution equations allow to deduce monotonicities. Based on these Lyapunov functions we wish to provide a tool to systematically prove new theorems for geometric evolution equations. We want to focus on the behavior of solutions for large times or near singularities.

几何学中许多具有挑战性的问题都涉及流动方程，如e。G.平均曲率流或Ricci流。这种流动方程的解的行为通常被控制如下：对于一个几何上有意义的量，计算演化方程。证明了这个量是单调的，即i。e.李雅普诺夫函数这允许控制流动方程的解。未来李雅普诺夫函数的演化方程的计算是纯代数的，但通常是相当繁琐的。我们建议开发一个程序，在许多不同的情况下进行这些代数计算。我们还希望使用代数和实验的方法来选择未来的李雅普诺夫函数和检查，所得到的演化方程是否允许推导单调性。基于这些李雅普诺夫函数，我们希望提供一个工具，系统地证明几何发展方程的新定理。我们想把重点放在大时间或奇点附近的解的行为。