Hilbert's twenty-first problem

Hilbert's twenty-first problem concerns finding solutions to certain types of equations called Fuchsian differential equations, which involve complex variables and have regular singular points. The question asks whether, for a given set of these points and prescribed behaviors (called monodromy), it’s possible to construct an equation that exactly matches these conditions. Essentially, it's about understanding when and how we can design differential equations with specific properties around singularities, ensuring we can realize desired behaviors in complex systems. The problem was affirmatively solved in the early 20th century, confirming that such equations can indeed be constructed to match given monodromy data.