Triangular decoupling of systems of differential equations, with application to separation of variables on Schwarzschild spacetime

Certain tensor wave equations admit a complete separation of variables on the Schwarzschild spacetime (asymptotically flat, static, spherically symmetric black hole in 4d), resulting in complicated systems of radial mode ODEs. Almost none of the important questions about these radial mode equations can be answered in their original form. I will discuss a drastic simplification of these ODE systems to sparse upper triangular form, which uncovers their general properties. Essential to this simplification are geometric properties of the original tensor wave equations, ideas from homological algebra and from the theory of ODEs with rational coefficients. Based on arXiv:1711.00585, arXiv:1801.09800, arXiv:2004.09651.