Unlikely Intersections in the Irregular-singular Context

I'll talk about some classical problems around orthogonal polynomials and harmonic analysis that fall into the "unlikely intersection" paradigm: they can be reformulated as questions about counting integers in definable sets whose algebraic parts are controlled by suitable Ax-Schanuel type theorems. Some interesting new features arise in this context: The "uniformizing maps" are more analytically involved: their study requires the steepest descent or other asymptotic methods, making the analysis more challenging than in the classical setting. Accordingly, the relevant sets are not definable in R_{an,exp} but rather in the larger o-minimal structure R_{G,exp} of multisummable functions. Functional transcendence is still controlled by differential Galois theory, but now for irregular-singular systems. This brings new concepts (exponential tori, Stokes phenomena) into the description of the weakly special subvarieties. Based on joint work with Avner Kiro and Jonathan Pila.