Sergey Gorchinsky had a 2-hour seminar from 10:30 - 12:30. It started from the Chow groups of smooth irreducible variety $X$ over a field $k$. This was a story on the algebraic cycles modulo rational equivalence on $Z^p (X)$. Then it was on the conjecture by Bloch in algebraic K-theory; if $h^{(0,2)} > 0$, then the kernel of the Albanese map $T (X) = 0$. In the 70's, it is known for $X$ NOT of general type (with $h^{0,2}=0$). He also explained that (by Beilinson?) for $X$-smooth projective surface over $\mathbb{C}$, suppose $\alpha \in CH^2 (X \times X)$ the Hodge realization $h (\alpha) = 0$. Then $\alpha_{\ast}|_{T (X)} = 0$. The remark was on the preservation of the filtration.
After a break, $X$ will be more generally an arbitrary variety over a field $k$. The corresponding categories are the de Rham, e'tale, and Betti (mixed motives like onions (by Bloch)). He then worked on the mixed motive version $\mathcal{MM}_k$ as a right tensor abelian $\mathbb{Q}$-linear category from $Vect_k$. Of course we assumed the vanishing conjecture of Beilinson-Soule' for the existence of the category of mixed motives. Well, he then functored $Vect_k$ to a triangulated category $DM_k$, which has a non-degenerated $t$-structure s.t. its heart is the image of the inclusion of $\mathcal{MM}_k$ to $DM_k$. The story came to the Chow motive $CHM_k$ to pure motive $\mathcal{PM}_k$. And there was a "believable" conjecture of Beilinson. Then Sergei showed some examples of smooth projective $X$ of Murre. Finally he explained the theorem of he and Guletskii of when $X$ is a smooth projective 3-folds over $\mathbb{C}$, then there exists a split (or decomposition?) of $M(X)$.
After the talk, I asked him whether there exists a relation between Kontsevich's motivic integration and the conjectural category of mixed (Tate) motives after Bloch-Kriz. He informed me of the paper by Lunts, which showed that there is not a homemorphism from a motivic measure category $M_k$ to the $K_0$ group of smooth scheme $X$. I am not sure about the case of "refined" motivic integration, which has the value not on the Grothendieck group, but rather a trianglated category.
