9214772 Beilinson This project deals with two subjects. The first one is the theory of mixed motives. This, yet conjectural, theory is aimed to unfold the hidden topological pattern of arithmetic geometry. It helps to explain various phenomena in K-theory and the theory of algebraic cycles and to provide information about the values of the special functions of arithmetic origin. Several results in that direction have already been obtained, e.g., in a joint paper with P. Deligne, a proof of part of D. Zagier's conjecture on the values of the polylogarithm function. At the moment, the problem of the actual construction of the category of mixed motives (at least modulo Grothendieck's Standard Conjectures) seems to be quite tractable. The second subject of the project is the study of the geometry of conformal field theory. This theory has been developing rapidly during the last decade. The investigator's first modest aim was to write down (jointly with B. Feigin and B. Mazur) an account of an algebro-geometric setting. The next thing to pursue (work in progress with V. Drinfeld and V. Ginzburg) is a theory, as foreseen by Drinfeld, relating the representations of Kac-Moody algebras at the critical level with the geometric version of Langlands' correspondence. The ultimate aim of the project is to tie together the two subjects above, which remains a tantalizing hope. The two subjects of this project, conformal field theory and the theory of motives, have somewhat different origins. Conformal field theory came first to physics approximately ten years ago. It was observed there that often objects at the critical temperature (the melting ice cube in your glass) suddenly acquire infinitely greater inner symmetry than before; this symmetry governs the intricate pattern that singles out the object. The mathematical structures that described this soon became - after the string theory upheaval - the most popular structures of theoretical physics, a dev elopment that also helped to connect previously unrelated but intensively studied mathematical areas such as representation theory of infinite-dimensional groups, geometry of moduli spaces, and the Langlands' program. The first part of this project dwells here. The theory of motives, discovered by Grothendieck in the mid-60's, has, in a sense, a similar flavor. It starts with an old insight that number theory suggests that one consider the whole numbers as if they were functions on a certain complicated space. This space, with its inner symmetries, replaces the ordinary geometric notion of a point; morally, the ordinary geometry acquires an extra arithmetic dimension. The theory of motives studies such a geometry. As an application, it may provide a range of yet unpredictable results about the values of the classical arithmetic functions. ***
