Motives, Motivic Cohomology and Values of Zeta-Functions

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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.