turkmath.org

The theory of motives was conceived as a universal cohomology theory for algebraic varieties. Today it is a vast subject systematically developed in many directions spanning algebraic geom- etry, arithmetic geometry, higher category theory, and abstract homotopy theory. After offering a brief historical commentary on the origins of the theory of motives, we will go through salient aspects of the triangulated categories of mixed motives as developed by Voevodsky and others. We will then discuss the construction of so-called noncommutative motives of dg-categories following Kontsevich, Tabuada, and Robalo, which formalizes some motivic themes that emerge in derived algebraic geometry. Since there is a great deal of abstract machinery involved, we will try to keep the discussion conceptual.