Number theory in function fields

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Classically, number theory is the study of arithmetic properties of integers. My research focuses on the interplay between this theory, and the theory of a related object, the set of polynomials over a finite field. The set of integers, and the aforementioned set of polynomials have ostensibly different structures. For example, given a non-zero integer, no multiple of it is zero. However, a fixed finite field always admits a prime number p for which any p-multiple of an element is zero, hence this property is carried over to polynomials on this field. Despite these apparent differences, it is a striking theme in arithmetic that there are remarkable similarities between these two sets. These similarities have many analogues that, in general, relate number fields to function fields. Since there are more structures in polynomials than in integers, the study of function fields has served as a stimulus to research in other branches of number theory. On the other hand, since finite fields admit multiples which are zero, many methods used to solve integer number theoretic problems fail to provide viable answers to their polynomial analogues. As new approaches are often required, number theory in function fields is of interest for its own sake. In addition, the theory of function fields is but another guise for the theory of algebraic curves. The latter plays an important role in the study of modern number theory, including the proof of Fermat's Last Theorem. The study of function fields and algebraic curves has many applications, a prominent one being the use of elliptic curves in cryptography, which facilitates secure transmissions of data.