Jonathan Pila, speaking at the Institute for Advanced Study as part of a workshop on Hodge theory and o-minimality, presents joint work with Jacob Tsimerman on the rank one Riemann-Hilbert correspondence. He starts from a result of Carlos Simpson that characterizes which subspaces appear as irreducible algebraic varieties on both sides of this correspondence, linking local systems to their associated flat connections. Pila argues that Simpson's theorem points toward an underlying Ax-Schanuel statement, a transcendence result in the style of the functional Ax-Schanuel theorems that have driven recent progress in unlikely intersections and o-minimality. He walks through how establishing this Ax-Schanuel statement lets him and Tsimerman recover Simpson's bi-algebraicity result from a different direction, tying together model theory, Hodge theory, and the geometry of local systems. The talk assumes graduate-level familiarity with these areas and is aimed at specialists following the workshop's themes rather than a general audience.