Ax-Schanuel Conjectures and the Geometry of Exceptional Loci in O-Minimal Expansions of the Real Field

Mathematical Applications of Science Fiction We establish a transfinite generalization of the Ax-Schanueltheorem for variations of mixed Hodge structures definablein o-minimal expansions of the real field. By constructing ahyper-definable stratification of the exceptional loci Eα asso-ciated with the pre-dimension operator δ, we prove that thefunctional transcendence degree of the uniformizing mapsis bounded below by the dimension of the ambient Shimuravariety modulo the geodesic flow. We introduce the notionof transfinite defect, ∆ω , and show that for any definable setV ⊂ S, the intersection of V with the special subvarietiesof S is governed by the analytic rank of the Jacobian JV .This resolves the ”Unlikely Intersection” conjecture for abroad class of non-algebraic flows in Ran, exp.