Gregorio Baldi, speaking at the Institute for Advanced Study's special year research seminar, lays out how the theory of unlikely intersections and functional transcendence illuminates the special submanifolds that sit inside geometric spaces such as abelian varieties, locally symmetric spaces, period domains, and strata of abelian differentials. He opens by applying this framework to non-arithmetic complex hyperbolic lattices and affine invariant submanifolds, setting it alongside existing dynamical methods used to study the same objects. The second half outlines his proof of the geometric Zilber-Pink conjecture and presents general results on how the Hodge locus is distributed inside these spaces. The talk assumes graduate-level familiarity with algebraic and arithmetic geometry and moves through definitions, prior approaches, and the architecture of the new arguments rather than offering an introductory survey. It is a working researcher's account of a current proof, not a popularization.