Mathematical Sciences: Topics in Algebra and Geometry

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This award supports the research in K-Theory and Algebraic Geometry of Professor Henri Gillet of the University of Illinois at Chicago. In continuing joint work with C. Soule, he plans to investigate the relationship between algebraic and differential- geometric invariants of arithmetic varieties, and possibly to apply these results to questions in number theory. Another project of Dr. Gillet is to continue joint work with Dr. Peter Shalen in studying the actions of groups on generalized trees, with the idea of using these actions to obtain information about the cohomology of the groups involved. 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 origin, 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 in Robotics.