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We develop notions of "almost complex analytic subsets" of almost complex manifolds, modelled after complex analytic subsets of complex manifolds. Basic analytic-geometric results are presented, including an identity principle for almost complex maps, and a proof that the singular locus of an almost complex analytic set is itself an "equational" almost complex analytic set, under certain conditions.This work is partly motivated by geometric model theory. B. Zilber observed that a compact complex manifold, equipped with the logico-topological structure given by its complex analytic subsets, satisfies the axioms for a so-called "Zariski geometry", kick-starting a fruitful model-theoretic study of complex manifolds. Our results point towards a natural generalization of Zilber’s theorem to almost complex manifolds, using our notions of almost complex analytic subset. We include a discussion of progress towards this goal.Our development draws inspiration from Y. Peterzil and S. Starchenko’s theory of nonstandard complex analytic geometry. We work primarily in the real analytic setting.

Expanding on the ideas of o-minimality, we study three kinds of expansions of the real field and discuss certain tameness properties that they possess or lack. We prove that the ring of integers is definable in the expansion of the real field by an infinite convex subset of a finite-rank additive subgroup of the reals. We give structure theorem for expansions of the real field by families of restricted complex power functions and apply it to classify expansions of the real field by families of locally closed trajectories of linear vector fields. We examine certain polynomially bounded o-minimal structures over the real field expanded by multiplicative subgroups of the reals. The main result is that any nonempty, bounded, definable d-dimensional submanifold has finite d-dimensional Hausdorff measure if and only if the dimension of its frontier is less than d.