Mathematical Sciences: Algebraic K-Theory and Motivic Cohomology

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This award supports the study of motivic cohomology of schemes and its relationships to algebraic K-theory. Recent developments concerning a spectral sequence relating algebraic K-theory to higher Chow groups and the computation of the higher Chow groups of complex varieties of sufficiently high codimension in terms of etale cohomology give a complete understanding of algebraic K-theory with finite coefficients of smooth complex varieties of dimension less than or equal to 2. The principal investigator will pursue further this line of investigation in order to obtain an understanding of algebraic K-theory of higher dimensional complex varieties and of more general schemes. In addition to this study of motivic cohomology, he will investigate the relationship between polyrelative algebraic K-theory and cyclic homology, thus completing a series of works on excision in algebraic K-theory. 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.