Motives, Motivic Cohomology and Values of Zeta-Functions

Lichtenbaum教授将试图表明，他和T。Goodwille满足人们对一个范畴所期望的所有性质。 例如，他希望表明，该类别有一个对象，对应于泰特动机。Lichtenbaum教授还将研究zeta函数的特殊值。在早期的工作中，他已经证明了某些公式描述的行为zeta函数的品种在有限领域的积分值广义欧拉特征。 这些广义欧拉特征线是相关的（假设的）动机上同调复合体的etale层。他将根据德利涅和J。S。米尔恩。这个项目在数学领域被称为代数几何。 从世纪初开始，数学家们一直在将19世纪世纪的解析几何转化为越来越多的代数背景。 其结果是一个复杂的，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和直觉是更多的情况。这种几何观点导致了其他领域的重大进展，如数论，现代分析和数学物理。Lichtenbaum教授的工作集中在这个抽象的方法几何的一些基本对象。 随着他继续揭示这些物体的性质，代数几何将成为数学和物理学其他部分的更有价值的工具。

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.