Arithmetic Algebraic Geometry

Nicholas Katz98 01633这是一个有三名研究人员参与的数学领域的项目，称为数论。Katz教授将研究定义在有限域上的簇，特别是它们的上同调，它们的Zeta函数和L函数的零点的分布以及单调性。余教授将从Drinfeld模的角度研究与Sato-Tate猜想类似的问题。Vakil教授将集中研究某些模空间的Chow群，包括稳定的n点亏格g曲线的模空间。这个项目的主要重点可以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到这些类型的解决方案。有限域上的簇是一组多项式方程的所有模解的集合。Katz教授将考虑这样的解的数目，这个数目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的问题。这些数学对象的许多抽象版本可以帮助阐明这些问题。Drinfeld模可以用来作为簇的推广，模空间可以用来作为整体来研究簇的集合。在这个项目中，三名研究人员将尝试将几种不同的技术结合起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.