Computer algebra for geometric evolution equations

许多具有挑战性的几何问题涉及流动方程，如平均曲率流或里奇流。这类流动方程解的行为通常控制如下：对于几何上显著的量，计算演化方程。证明了该量是单调的，即李雅普诺夫函数。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，但通常相当繁琐。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方法来选择预期的李雅普诺夫函数，并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来系统地证明几何演化方程的新定理。我们要关注的是大时间点或接近奇点时解的行为。

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.