CAREER: Motives: Voevodsky versus Kontsevich

Massachusetts Institute Of Technology, Cambridge MA

Investigators

Abstract

In the sixties, Grothendieck had the motivic idea that one should replace all the different cohomology theories of algebraic varieties (Hodge, de Rham, etale, crystalline, etc) by the "universal" one. This insight led to major breakthroughs and to several categories of motives. Noncommutative algebraic geometry, a recent branch of mathematics, is concerned with the study of noncommutative algebraic varieties, i.e., differential graded (=dg) categories. Recently, Kontsevich had also the motivic idea of replacing the different homology theories of noncommutative algebraic varieties (K-theory, cyclic homology, topological Hochschild homology, etc) by the "universal" one. Similarly, this led to several categories of noncommutative motives. The main objective of this project is to bridge the gap between the above two motivic ideas. Concretely, the Principal Investigator (PI) will construct appropriate symmetric monoidal functors relating the categories of motives and noncommutative motives. Among the manifold uses of such functors, the PI will exploit them to address the Schur/Kimura finiteness of motives as well as the rationality of motivic zeta functions. The theory of motives was envisioned by Grothendieck in the sixties and greatly developed since then by Beilinson, Deligne, Voevodsky, and others. In contrast, the theory of noncommutative motives was only recently introduced by Kontsevich. The PI proposes to construct precise bridges between motives and noncommutative motives. These will enable the interchange of results, techniques, ideas, and insights between the commutative and the noncommutative world. In order to make these results accessible to the mathematical community, the PI will lecture graduate courses, disseminate them online, organize working seminars, and write the first book on the subject. Complementing these educational activities, the PI will also mentor undergraduate students through research projects on applications of motives to several areas of mathematics.