Mathematical Sciences: Number Theory, Arithmetic Geometry, and Transcendence

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This research is in many areas of number theory. Montgomery will pursue a number of basic questions in analytic number theory and related areas of harmonic analysis, diophantine approximation and the geometry of numbers. Most of these questions involve sets of numbers or vectors which are almost independent. The object is to exploit the almost independence appropriately in order to achieve the desired aim. Masser will study some diophantine problems which lend themselves into the following groupings: (a) using transcendence techniques to examine isomorphism classes of elliptic curves; (b) to obtain upper bounds for the number of points of small height on a fixed abelian variety; (c) to establish precise conditions for the algebraic independence of values of certain Mahler-type power series at algebraic points; and (d) to study the multiplicities of linear recurrences for a finite field. Milne's research object is to produce a comprehensive set of reciprocity laws describing how automorphisms of the complex numbers act on automorphic functions automorphic forms, their special values, their "Fourier-Jacobi series, and the Eisenstein series attached to cusp forms on boundary components. Keating's research object is to understand Igusa curves better by studying their function fields including applications to coding theory and factorization theory. He proposes to use Drinfeld's theory of extensions of function fields to produce a more explicit characterization of the function fields. This research is in very broad areas of number theory, the study of the properties of the integers. Montgomery's focus is on that part of the subject where classical analysis is used to understand these deep questions. Masser's research concentrates on using transcendence techniques (again analytic but of a very different type than Montgomery's techniques) to study problems concerning solutions of equations in integers. Milne combines algebraic and modern analytic techniques to study special functions that arise in number theory. Keating studies special domains that are important in number theory.