This research will concentrate on the refined p-adic Stark conjectures of B. Gross, using computational methods over number fields of low degree as a base. Also the second order zero case of Stark's conjectures over a real quadratic number field or function field as a base will be studied. Finally the implications of the conjectures for the existence of Hecke characters of algebraic type will be investigated. This research will concentrate on some of the analytic invariants that play such an important role in number theory. These invariants are both in the usual real domain and in the more arithmetic "p-adic" domain. The computer will be a major tool in the investigation. Much of fundamental importance can be expected from this research.
