Grant: Discovery Grants Program - Individual

This research proposal is concerned with various questions resulting from the recent progress in the study of `motives'. These are mathematical objects which were introduced by Grothendieck and Manin in the 1960's, initially, to unify several cohomolog theories and to study a long-standing conjecture by Weil. Over the years it has become clear that motives have many applications in different branches of mathematics, for example, algebraic geometry, topology, number theory and representation theory. In fact, it turned out that the language of motives provides a common framework in which many fundamental problems in different branches of mathematics can be explained. Whereas for a long time the theory of motives was largely conjectural, in recent years there has been dramatic progress in making this theory precise. These developments created new theories as well as powerful computational tools. In this proposal we address theoretical and concrete computational questions whose resolution would significantly improve our understanding of this fundamental theory.