Mathematical Sciences: Properties of p-adic complex powers

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This research will study the properties of p-adic complex powers, more specifically the connections between the poles of the p-adic complex power, eigenvalues of monodromy and roots of the b-function. The first problem is to show that in generic cases of arbitrary dimension every eigenvalue of monodromy must be associated to a pole of the p-adic complex power. The second problem is to examine the relationship between poles of the p- adic complex power and eigenvalues of monodromy in the low dimensional case of surfaces with quasi-ordinary singularities. A third problem related to this would be a geometrical interpretation of roots of the b-function for this class of surfaces. This research is in the exciting area which bridges number theory and algebraic geometry. The powerful tools of the latter subject are brought to bear in solving problems in the former. This research is in a new and erudite area with lots of promise. Meuser is one of the real experts here and one can only wait to see what important and unexpected result she uncovers.