Recently there has been a great deal of interaction between model theorists and number theorists on topics around `unlikely intersections', see for example . One outcome of this is that there are now various functional transcendence results known for certain covering maps. The original example of this is Ax's functional version  of Schanuel's conjecture. This result and its more recent descendants have been used to study interdefinability of Weierstrass elliptic functions  and the initial aim of this project is to extend this to abelian functions. This would involve a mixture of model theory, differential algebra and number theory, although these are not all required to get started. It should also lead naturally to further interesting questions in these areas.
 Ax, James On Schanuel's conjectures. Ann. of Math. (2) 93 1971 252–268. See http://www.jstor.org/stable/1970774
 Jones, G., Kirby, J. and Servi, T., Local interdefinability of Weierstrass elliptic functions. Journal of the Institute of Mathematics of Jussieu, To appear. See http://dx.doi.org/10.1017/S1474748014000425
 Zannier, Umberto, Some problems of unlikely intersections in arithmetic and geometry. Annals of Mathematics Studies, 181. Princeton University Press, Princeton, NJ, 2012