Mathematical Conversations

Solutions to some differential equations are related to geometric structures on the underlying manifold. For instance certain hypergeometric equations are related to the uniformization of Riemann surfaces. I will start by recalling some classical theorems from the 19th century and explain how they are related to Hodge theory. I will then present some more recent results about higher-dimensional manifolds. It turns out that the key questions reduce to understanding groups generated by concrete matrices, and I will end with a few open questions.