This research will focus mainly on two areas. The first concerns congruences for values of L-functions predicted by the theory of multiplicative Galois structures. The second concerns the connections between Arakelov theory and adelic capacity theory. Work will also continue on determining the smallest arithmetic hyperbolic three manifold with a given number of cusps and to study whether the minimal volume of all hyperbolic three manifolds with a given number of cusps is achieved by an arithmetic manifold. Finally, the connections between Mahler measures, L-series and K-groups of rings of integers will be studied. This research will encompass broad areas of number theory and related areas. It will move into areas of analysis, algebra and geometry. The connections between all these areas will be focused upon giving this research a particular appeal. Much of great interest will come from these endeavors.
