250130 VO Algebraic Theory of Differential Equations (2020S)

In this course we will look at linear ordinary differential equations with polynomial, holomorphic or meromorphic coefficients from an algebraic point of view. As such, a rich panorama of techniques and results unfolds:

The Euler-Gauss hypergeometric differential equation, hypergeometric series, regular and irregular singular points, normal forms of ode's, Picard-Fuchs equations, Apéry's differential equation, irrationality of odd values of the zeta-function, modular forms and elliptic curves, the monodromy representation, the Riemann-Hilbert correspondence, differential Galois theory, fundamental groupds and analytic continuation, indicial equations, D-modules and the Weyl algebra, linear recursions, lattice walks, enumerative combinatorics.

Nevertheless, the approach will be mostly elementary and does not require a huge amount of prerequisites.