## Speakers

Aravind Asok
Joseph Ayoub
Spencer Bloch
Mikhail Bondarko
Frédéric Déglise
Thomas Geisser
Henri Gillet
Bruno Kahn
Rahul Pandharipande
Ivan Panin
Charles Weibel
Jerzy Weyman

## Titles of the talks, schedule

### Monday, June 11, 2012

 09:30 Spencer Bloch Marc Levine's work on motives and motivic cohomology 11:00 Ivan Panin On Levine-Morel algebraic cobordism 13:30 Jerzy Weyman Vinberg theta groups and structure theorems for the finite free resolutions of length three 15:00 Bruno Kahn Birational motives and birational invariants 16:15 Frédéric Déglise Integral motives: results and prospects

### Tuesday, June 12, 2012

 09:30 Aravind Asok Homotopy theory of projective modules on smooth affine algebras of dimension 3 11:00 Joseph Ayoub A relative version of the Kontsevich-Zagier conjecture 13:30 Henri Gillet Higher derivations and descent in characteristic $p$ 15:00 Mikhail Bondarko On mixed motivic sheaves and weights for them

### Wednesday, June 13, 2012

 09:30 Thomas Geisser Results on p-parts of motivic cohomology theories 11:00 Vladimir Voevodsky Univalent fibrations 13:30 Charles Weibel $K$-theory of toric varieties in positive characteristic 15:00 Rahul Pandharipande On some applications of algebraic cobordism

## Abstracts

#### Aravind Asok: Homotopy theory of projective modules on smooth affine algebras of dimension 3

I will explain how to show that cancellation holds for rank 2 projective modules on a smooth affine threefold over an algebraically closed field having characteristic unequal to 2.  To this end, I will recall some aspects F. Morel's $\mathbb A^1$-homotopy classification of vector bundles and discuss some new information about the $\mathbb A^1$-homotopy type of the general linear group. These results are joint work with J. Fasel.

#### Mikhail Bondarko: On mixed motivic sheaves and weights for them

It was conjectured by Beilinson that for any scheme $S$ there exists an abelian category $MM(S)$ (generalizing the conjectural category of mixed motives over a field $k$) that is a 'motivic analogue' of the category of perverse etale sheaves over $S$. $MM(S)$ should possess a weight filtration that is strictly compatible with morphisms. When the triangulated category of motives $DM(k)$ was very successfully defined by Voevodsky, Levine, and Hanamura, people conjectured that $MM(k)$ should be the heart of (a conjectural) motivic t-structure $t_M$ for $DM(k)$. The latter can (conjecturally) be defined in terms of the etale realization of motives. Actually, Hanamura proved that the 'standard' motivic conjectures together with Murre's filtration conjectures yield the existence of $t_M$ over $k$. On the other hand, Bondarko defined weight structures for triangulated categories; the Chow weight structure for $DM(k)$ (that is a modification of a construction of Hanamura) yields certain 'weights' for it. Recently a 'nice' $DM(S)$ was defined by Cisinski and Deglise; Hebert and Bondarko defined the Chow weight structure for it. In this talk the following result is formulated: the 'motivic conjectures' over universal domains yield the existence of $MM(S)\subset DM(S)$ and all of the properties expected for this category (when $S$ is equi-characteristic; the Chow weight structure and the general theory of weight structures is crucial here).

#### Henri Gillet: Higher derivations and descent in characteristic $p$

In characteristic $0$ the Kodaira-Spencer class is the obstruction to a family of curves being isotrivial, and more generally one knows that if $X$ a projective variety over an algebraically closed field $K$ of characteristic zero, then it is defined over the fixed field of all the derivations of $K$ which lift to $X$. I shall describe how one may use higher derivations, in the sense of Hasse-Schmidt, to obtain a similar result in positive characteristic.

#### Jerzy Weyman: Vinberg theta groups and structure theorems for the finite free resolutions of length three

In this lecture I will describe the new connection between root systems and structure of free resolutions over commutative rings. An important problem in the study of free resolutions is the existence of the generic free resolutions of a given format over some Noetherian ring. For resolutions of length two the generic rings were known classically for cyclic modules (Hilbert-Burch theorem) and they were constructed by Hochster for arbitrary length two format. For resolutions of length three the situation is more complicated but can be completely described. For the format $0\rightarrow F_3\rightarrow F_2\rightarrow F_1\rightarrow F_0$ with $d_i:F_i\rightarrow F_{i-1}$, $r_i=\mathop{\rm rank} d_i$, we associate to the format the triple $(p, q, r)= (r_1+1, r_2-1, r_3+1)$. Consider the graph $T_{p,q,r}$ with three arms of lengths $(p-1), (q-1) (r-1)$ centering at a point. Let $g(T_{p,q,r})$ be a Kac-Moody Lie algebra associated to the graph $T_{p,q,r}$. Then the generic ring alwas exists, has a multiplicity free action of Lie algebra $gl(F_2)\times gl(F_0)\times g(T_{p,q,r})$. The generic ring is Noetherian if and only if $T_{p,q,r}$ is a Dynkin graph.