Mathematical Sciences: Regulators, Class Groups and Iwasawa Theory

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This research will concentrate on three important problems in number theory. The first will be to sharpen the lower bounds known for regulators of number fields. The second will be the study of the distribution of the Frobenius map acting on the Jacobians of hyperelliptic curves over a finite field and thereby study the distribution of the divisor class groups of such curves. This is connected with the Cohen-Lenstra conjecture in number fields. The third problem is the study of the purely ring theoretic properties that may govern the vanishing of the Iwasawa mu-invariant. This research will consider various problems in number theory. Number theory is the study of the properties of the integers. The proposer brings to bear a wide variety of tools to perform this research. Using analysis and geometry the study of lower bounds of one of the crucial invariants in algebraic number theory, the regulator, will be studied. The study of the distribution of divisor class groups involves a study in probability concerning these geometric objects. The problem on the Iwasawa involves a heavy use of algebra techniques. The principle has a firm command of these diverse tools and will contribute a great deal to the theory.