Motives, Motivic Cohomology and Values of Zeta-Functions

Lichtenbaum教授将试图证明他和T.Goodwille先前构造的混合动机范畴满足人们期望的范畴的所有性质。例如，他希望证明这一类别有一个与泰特动机相对应的对象。Lichtenbaum教授还将研究Zeta函数的特殊值。在早期的工作中，他猜想了某些公式，用广义欧拉特征来描述有限域上整值变元的Zeta函数的行为。这些广义的欧拉特征是相关的(假设的)基元上同调复形。他将根据P.Deligne和J.S.Milne的最新思想，通过用“Weil上同调”代替etale上同调来提炼这些思想。这个项目在数学领域被称为代数几何。从本世纪初开始，数学家们一直在将19世纪解析几何的大部分内容转化为越来越多的代数背景。其结果是一种复杂但强大的方法来研究曲线、曲面和其他几何对象。这种现代的几何方法允许数学家使用几何技术，直觉是更多的情况。这种几何观点导致了数论、现代分析和数学物理等其他领域的重大进步。Lichtenbaum教授的工作集中了几何这一抽象方法中的一些基本对象。随着他继续揭示这些物体的性质，代数几何作为数学和物理的其他部分的工具将变得更加有价值。

Professor Lichtenbaum will attempt to show that the category of mixed motives previously constructed by him and T. Goodwille satisfies all the properties one expects of a category. For example, he expects to show that the category has an object that corresponds to the Tate motive. Professor Lichtenbaum will also study special values of zeta functions. In earlier work, he had conjectured certain formula describing the behavior of the zeta function of varieties over finite fields at integral values in terms of generalized Euler characteristics. These generalized Euler characteristics are associated (hypothetical) motivic cohomology complexes of etale sheaves. He will refine these ideas by replacing etale cohomology with a "Weil coholomology" based on recent ideas of P. Deligne and J. S. Milne.This project in the mathematical area known as algebraic geometry. Starting from the beginning of the century, mathematicians have been translating much of 19-th century analytic geometry into a more and more algebraic setting. The result is a complicated but powerful method for studying curves, surfaces and other geometric objects. This modern approach to geometry allows mathematicians to use geometric technique and intuition is many more situations. This geometric point of view has led to major advances in such diverse other fields as number theory, modern analysis, and mathematical physics. Professor Lichtenbaum's work concentrates some of the basic objects in this abstract approach to geometry. As he continues to uncover the properties of these objects, algebraic geometry will become even more valuable as a tool in other parts of mathematics and physics.