In this talk, I will motivate and present some results in the algebraic
theory of differential equations with an emphasis on their history.
Starting from the classical formulation of Hilbert's 21st problem, we
will visit some basic results in the field of D-modules that have been
inspired by this question: A crucial notion in modern Algebraic Analysis
is that of a Riemann‑Hilbert correspondence in its various incarnations,
building bridges between analytic and topological categories. I will in
particular describe the important steps towards understanding irregular
singular points in dimension one. The key ingredient for their study is
a phenomenon already described by Stokes in the 19th century.
Finally, we will see some applications of recent groundbreaking works in
this area, which allow for a purely topological study of certain data
encoded in a differential system.