In these lectures we shall introduce motives and present results in Jannsen's paper, which say that the "conjectural" category of motives is semisimple abelian iff the adequate equivalence relation taken is numerical equivalence. We shall also explain what is still "conjectural" About this.

Koszul algebras are the algebras over which the resolution of the residue class field is given entirely by linear matrices. This series of talks will be a survey on results obtained about Koszul algebras since they were introduced by Priddy in 1970. In the first talk, We shall see lots of examples of Koszul algebras, and discuss several characterizations of Koszul algebras.

The Rost nilpotence principle is an important tool in the study of motivic decompositions of smooth projective varieties over a field. We will introduce this principle and briefly survey the cases in which it is known to hold. We will then outline a new approach to the question using etale motivic cohomology, which helps us to give a simpler and more conceptual proof of Rost nilpotence for surfaces, generalize the known results over one-dimensional bases and sheds more light on the situation in higher dimensions. The talk is based on joint work with Andreas Rosenschon.