Mathematical Sciences: Number Theory, Arithmetic Geometry, and Transcendence

这项研究涉及到数论的许多领域。蒙哥马利将在解析数论以及调和分析、丢番图近似和数几何的相关领域中探讨一些基本问题。这些问题中的大多数都涉及几乎相互独立的数字或向量集合。其目的是适当地利用几乎独立，以达到预期的目的。Masser将研究以下几类丢番图问题：(A)利用超越技术研究椭圆曲线的同构类；(B)得到固定交换簇上小高度点的个数的上界；(C)建立某些Mahler型幂级数在代数点的值的代数无关性的精确条件；以及(D)研究有限域的线性递归的重数。Milne的研究目标是产生一套全面的互易定律，描述复数的自同构如何作用于自同构函数的自同构形式、它们的特定值、它们的“傅立叶-雅可比级数”以及附加在边界分量上的尖点形式上的Eisenstein级数。基廷的研究目的是通过研究伊古萨曲线的函数域，包括在编码理论和因式分解理论中的应用，来更好地理解伊古萨曲线。他建议使用Drinfeld的函数域扩张理论来对函数域进行更明确的刻画。这项研究是在非常广泛的数论领域，研究整数的性质。蒙哥马利的重点是这一主题的那部分，在这部分中，经典分析被用来理解这些深刻的问题。马瑟的研究集中在使用超越技术(同样是解析的，但与蒙哥马利的技术非常不同的类型)来研究与整数方程的解有关的问题。米尔恩结合了代数和现代分析技术来研究数论中出现的特殊函数。基廷研究在数论中很重要的特殊领域。

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.