Stephen Lichtenbaum - US grants

This research is concerned with problems in algebraic geometry. The principal investigator will continue his study of motivic cohomology and establishing connections with algebraic K-theory and the values of Dedekind zeta functions. There are possible applications to the Soule-Beilinson conjecture and related topics which will also be explored. This is research in the field of algebraic geometry, one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origins, it treated figures that could be defined in the plane by the simplest equations, namely polynomials. Nowadays the field makes use of methods not only from algebra, but from analysis and topology, and conversely is finding application in those fields as well as in theoretical computer science and robotics.

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.

This three-year award will provide a group of six U.S. graduate students and a faculty advisor from Brown University with support for annual summer research visits to the University of Paris VI (Universite Pierre et Marie Curie) and University of Paris VII. The purpose is to introduce students to mathematical research in analysis, partial differential equations, number theory and scientific computing. The summer program is led by David Gottlieb of Brown University and Yvon Maday of the University of Paris VI.

Intellectual Merit

The goal of this international research and education effort is to engage students in mathematics research within the context of active research collaborations between Brown University and French mathematicians. U.S. and French faculty members will develop programs in scientific computing (high order accuracy methods for time dependent problems), functional analysis, and number theory.

Broader Impacts

The proposed project will provide research training to graduate students and advance their careers through new connections to leading French mathematicians. Designed for second year Ph.D. students, the summer program incorporates opportunities to engage students in a wide variety of mathematics research. After this initial experience in France, U.S. students become eligible to participate in a joint Ph.D. program between Brown University and the University of Paris VI. This component is a significant and unique aspect to summer research abroad. Students continue their connections with France until completion of their Ph.D. with the possibility of maintaining their collaboration with French colleagues through their career.

The proposer has constructed a Grothendieck topology for varieties over finite fields
such that, modulo standard conjectures, special values of zeta-functions can be
computed as Euler characteristics (in a suitable sense) of certain motivic sheaves or
complexes of sheaves. He now intends to work on the construction of a similar
Grothendieck topology for schemes over number rings.

The Riemann zeta-function (discovered by Riemann in the nineteenth century) and its
generalizations express many deep relations between analysis and number theory.
These relations should be consequences of a deep underlying topological structure.
This project is an investigation into the properties which that structure would have to possess.

This project will support six semiannual AGNES weekend workshops in Algebraic Geometry, the first of which is to be held at Stony Brook University, April 25 - 27, 2014. Algebraic geometry is a dynamic subject with vital connections to physics, applied mathematics, and fields such as number theory, differential geometry and representation theory. The Northeastern United States are a particular nexus, with a density of researchers and an output rivaling any region in the world. AGNES weekend workshops bring together a spectrum of algebraic geometers from the region and beyond, as well as representatives from allied fields invigorating algebraic geometry. There is diversity both in speakers and participants, from senior experts to burgeoning young researchers. There is a special emphasis on students: AGNES hosts "students-only" introductory pre-talks, as well as professional development sessions. Through lectures, panels, problem sessions, poster sessions, and informal discussion periods, AGNES informs the community of developments, AGNES fosters new research collaborations, and AGNES trains junior algebraic geometers.

Algebraic geometers study the geometry of shapes defined by the polynomial equations of Algebra. In fact, the geometric shapes occurring in nature -- in science, in engineering, etc. -- are frequently "algebraic", explaining the vital importance of algebraic geometry in physics and applied mathematics. This grant promotes research, dissemination and training in algebraic geometry through a series of conferences held throughout the Northeastern United States. There is an emphasis on supporting graduate students, highlighting the broad array of work in the area and its many connections to allied areas, and fostering communication in our broad research community.

AGNES website is available at http://www.agneshome.org/