Eskin, Mirzakhani, and Mohammadi showed that GL(2, R) orbit closures of translations surfaces are affine invariant submanifolds of strata of abelian differentials.;The first chapter of this thesis discusses the field of definition of an affine invariant submanifold M, which is the smallest subfield of R such that M can be defined in local period coordinates by linear equations with coefficients in this field. We show that the field of definition is equal to the intersection of the holonomy fields of translation surfaces in M, and is a real number field of degree at most the genus.;We show that the projection of the tangent bundle of M to absolute cohomology H1 is simple, and give a direct sum decomposition of H1 analogous to that given by Moller in the case of Teichmuller curves.;Applications include explicit full measure sets of translation surfaces whose orbit closures are as large as possible, and (only briefly mentioned in this thesis, joint work with Matheus) finiteness of algebraically primitive Teichmuller curves in the minimal stratum in prime genus at least 3.;The second chapter of this thesis discusses certain deformations of translation surfaces M supported on the set of all cylinders in a given direction. We show that these cylinder deformations remain in the GL(2, R)-orbit closure of M.;Applications are given concerning complete periodicity, field of definition, the number of parallel cylinders which may be found on a translation surface in a given orbit closure, and (only briefly mentioned in this thesis, joint work with Aulicino and Nguyen) the classification of higher rank orbit closures in the minimal stratum in genus 3.
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