Nondefinability results for restrictions of the exponential maps of abelian varieties

This thesis is concerned with definability questions for structures given by expanding the ordered real field by various classical functions. Initial results in this area concern the structure given by expanding the ordered real field by the real exponential function. The rich model theory of this structure immediately shows that the sine function is not definable in this structure. Bianconi showed that no restriction of the sine function is definable in this structure. In this work we consider similar definability questions for a function that is similar to the exponential function, which is known as the Weierstrass P-function. The proofs of these results rely on a version of a functional transcendence result known as Ax's Theorem for the Weierstrass P-function. A corresponding theorem for the modular j-function is due to Pila and Tsimermann and by using this theorem we also obtain a definability result for the j-function. Definability questions for expansions of the real field by several P-functions were considered and answered in work of Jones, Kirby and Servi. The P-function arises in the exponential map of elliptic curves, which are the abelian varieties of dimension 1. In this thesis we give a corresponding result for the exponential maps of all abelian varieties. This is joint work with Jones and Kirby.