Computer algebra for 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.

DFG-Verfahren Schwerpunktprogramme

Teilprojekt zu SPP 1489:  Algorithmic and Experimental Methods in Algebra, Geometry and Number Theory