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Explore advanced geometric structures and their special submanifolds through the lens of unlikely intersections and functional transcendence in this research seminar from the Institute for Advanced Study. Delve into how natural collections of special submanifolds encode internal symmetries in various geometric spaces, including abelian varieties with their sub-abelian varieties, locally symmetric spaces with totally geodesic subspaces, period domains with sub-period domains, and strata of abelian differentials with affine invariant submanifolds. Examine recent major advances in understanding these structures, particularly their application to studying non-arithmetic complex hyperbolic lattices and affine invariant submanifolds as complements to existing dynamical approaches. Investigate the proof of the geometric Zilber-Pink conjecture and discover general results concerning the distribution of the Hodge locus, gaining insight into cutting-edge research at the intersection of geometry, dynamics, and number theory.
