Timur Sadykov: Dessins d’enfants and differential operators for generic algebraic curves

My talk will be based on a joint work with F. Larusson.

We state and solve a discrete version of the classical Riemann-Hilbert problem. In particular, we associate a Riemann-Hilbert problem to every dessin d’enfants and show how to compute the solution for a dessin that is a tree. This amounts to finding a Fuchsian differential equation satisfied by the local inverses of a Shabat polynomial.

We classify those plane trees that have a representation by Moebius transformations and those that have a linear representation of dimension at most two. As a corollary, we obtain a computationally efficient method for constructing the linear differential operator with polynomial coefficients whose space of holomorphic solutions is spanned by all the branches of a function defined by a generic algebraic curve. The proposed method does not require solving the algebraic equation and can be applied in the case when its Galois group is not solvable.