This award supports research in three areas of number theory. The first involves the relationship between numbers of small height and the arithmetic properties of the fields that they generate. This may yield some insight into Lehmer's conjecture, as well as into the phenomenon that powers of small Salem numbers frequently turn out to be exceptional units. The second project is to study the variation of the multiplicative order of an integer modulo m for varying m, and to investigate generalizations of these results to number fields, function fields, and abelian varieties. The third project concerns the arithmetic theory of dynamical systems. Problems being studied in this area include bounds for periods of rational periodic points, finiteness of integral points in orbits, fields of moduli for rational maps, and analytic continuation of canonical height zeta functions. This research is in the general area of number theory. Number theory has its historical roots in the study of the whole numbers, addressing such questions as those dealing with the divisibility of one whole number by another. It is among the oldest branches of mathematics and was pursued for many centuries for purely aesthetic reasons. However, within the last half century it has become an indispensable tool in diverse applications in areas such as data transmission and processing, and communication systems.
