Arithmetic Algebraic Geometry

Nicholas Katz98 01633这是一个有三个研究人员的项目在数学领域叫做数论。Katz教授将研究在有限域上定义的变量，特别是它们的上同调，它们的ζ和L函数的零点分布以及单态。余教授将以德林菲尔德模研究佐藤-塔特猜想的类比。Vakil教授将专注于某些模空间的Chow群，包括稳定n点g曲线的模空间。这个项目的主要焦点可以被描述为研究多项式方程的某些类型的解，称为模解。计算机科学家对数论的许多应用都涉及这类解决方案。有限域上的变量是一组多项式方程的所有模解的集合。Katz教授将考虑的问题涉及到这样的解的数量，这个数量如何随着多项式的变化而变化，以及不同多项式的解集如何相关。这些数学对象的许多抽象版本可以帮助阐明这些问题。Drinfeld模可以作为一个品种的推广，模空间可以作为一个整体来研究品种的集合。在这个项目中，三位研究者将尝试结合几种不同的技术来解决一个基本的数学问题。

Nicholas Katz98 01633This is a project with three investigators in the area of mathematics called number theory. Professor Katz will investigate varieties defined over finite fields in particular their cohomology, the distribution of the zeros of their zeta and L functions and monodromy. Professor Yu will study the analogue of the Sato-Tate conjecture in terms of Drinfeld Modules. Professor Vakil will concentrate on Chow groups of certain moduli spaces including the moduli space of stable n-pointed genus g curves.The main focus of this project can be described as the study of certain kinds of solutions to polynomial equations called modular solutions. Many of the applications computer scientists have for number theory involve these kinds of solutions. A variety over a finite field is a collection of all the modular solutions to a set of polynomial equations. Professor Katz will consider questions involving the number of such solutions, how this number changes as the polynomials change, and how solution sets of different polynomials are related. Many abstract versions of these mathematical objects can help shed light on these questions. Drinfeld modules can be used as a generalization of a variety, and moduli spaces can be used to study collections of varieties as a whole. In this project, three investigators will try to combine several different techniques to a basic problem in mathematics.