Additive and probabilistic number theory in function fields

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Elementary number theory is concerned with the arithmetic properties of integers. As the ring of polynomials over a finite field, called a function field, bears a close resemblance to the ring of integers, many results which hold for integers have analogues for polynomials. My research is to study additive and  probabilistic number theory in function fields. Since there are more structures in polynomials than in integers which one can play with, work in function fields has served as a stimulus to research in other branches of number theory. On the other hand, since the characteristic of the base field is positive, many methods used for integers cannot be applied directly to polynomials. As new approaches are often required, number theory in function fields is of interest for its own sake. Moreover, the theory of function fields is but another guise for the theory of algebraic curves. The latter plays an essential role  in the proof of Fermat's Last Theorem. It also has many real-world applications, such as the use of elliptic curves in modern cryptography. I would like to further explore and elaborate upon the connections of  number theory in function fields to other areas of mathematics.