This award supports the research in analytic number theory of Professor Julia Mueller of Fordham University. Dr. Mueller will be conducting studies in diophantine approximation over both function fields and number fields. In the number-field case, she hopes to reformulate Siegel's conjecture so that all the parameters will be invariant under the special linear group, and thereby to get much more precise statements on the relationship between the coefficients and the number of solutions of the equations under consideration. The field of analytic number theory applies to the discrete realm of the whole numbers the techniques of analysis, dependent on the notions of continuity and limit, originating in calculus. The idea of using continuous methods to investigate the discrete is two centuries old, but with the work of the modern analytic number theorists such as Professor Mueller, the field has had a new rebirth.