# Ayoub:

**Barbieri Viale: **Consider
the derived category of 1-motives (up to isogeny) along with its fully-faithful
embedding Tot into Voevodsky's triangulated category of motives. Regarding Tot
as a universal realisation functor we show (jointly with B. Kahn) that it has a
left adjoint LAlb, the motivic Albanese triangulated functor. Dually, composing
with (motivic) Cartier duality, we obtain the functor RPic. Note that the
counit also provide a universal map, the motivic Albanese map, which 'contains'
the classical Albanese map.

These
functors provide natural complexes of 1-motives (up to isogeny) LAlb(X) and
RPic(X) of an algebraic variety X over a perfect field. Their 1-motivic
homology and cohomology would recover the Picard and Albanese 1-motives (introduced
jointly with V. Srinivas) as well as the 1-motives predicted by Deligne's
conjecture (proven jointly with A. Rosenschon and M. Saito).

**Beauville: **The Chow ring
CH(X) of a (smooth, projective) variety X is a fundamental invariant,
unfortunately rather poorly understood. Some light is shed by the deep
conjectures of Bloch and Beilinson, which predict the existence of a functorial
ring filtration of CH(X) with a (conjectural) description of the associated
graded ring. In some exemples the filtration actually splits, i.e. is the
filtration associated to a graduation: this is the case for K3 surfaces and,
conjecturally, for abelian varieties. I will discuss for what kind of varieties
one can expect such behaviour, in particular why it might be the case for (holomorphic)
symplectic manifolds.

**Bloch: **

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**Colliot-Th****élčne:
**It is conjectured that the reduced zero-dimensional Chow group
of a rationally connected variety over a p-adic field is a finite group. This
is known in dimension 2, and in the good reduction case. For smooth
compactifications of linear algebraic groups, I shall prove the finiteness up
to p-torsion. A new algebraic tool is the notion of flasque resolution attached
to a connected reductive group.

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# Conte:

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**Deglise: **In the theory
of mixed motives of Voevodsky, the triangulated category of effective mixed
motives is enlarged in a triangulated category of motivic complexes, following
the approach of Beilinson to define motivic cohomology. We will show how this
later category is related to the theory of cycle modules of Rost, a notion
directly inspired by the work of Kato on unramified Milnor K-theory. More
precisely, we will relate the category of cycle modules with the heart of the
triangulated category of motivic complexes with respect to the natural
t-structure (called the homotopy t-structure). As a subsequent development of
this line of thought, we will then present the definition of a spectral
sequence in motivic cohomology which looks like the Serre spectral sequence in
classical singular cohomology.

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**Deninger: **In joint work
with Annette Werner we define functorial isomorphisms of parallel transport
along etale paths for a class of vector bundles on a p-adic curve. All bundles
of degree zero whose reduction is strongly semistable belong to this class. In
particular, they give rise to representations of the algebraic fundamental
group of the curve. This may be viewed as a partial analogue of the classical
Narasimhan-Seshadri theory of vector bundles on compact Riemann

surfaces.

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**Esnault: **The
philosophy of motives, as developed by P. Deligne, predicts a link between the
Hodge type of varieties defined over the field of complex numbers and
congruences for the number of rational points of varieties defined over finite
fields. We show that if a smooth projective variety defined over a p-adic field
has its etale cohomology supported in codimension 1 (which according to the
Hodge conjecture is equivalent to saying that the Hodge type is at least one),
then the mod p reduction of a regular model has one rational point modulo the
cardinality of the finite field.

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**van Geemen:**

**Griffiths:**

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# Hanamura: Grothendieck's theory of pure motives
studies smooth projective varieties over a field. This has been generalized to
(a) the theory triangulated category of mixed motives over a field (by M.
Levine, V. Voevodsky and myself) and (b) the theory of pure motives over a base
variety. For my approach to (a), see M. Hanamura: Mixed motives and algebraic
cycles II, Invent. Math, 2004. For (b) see A. Corti and M. Hanamura: Motivic
decomposition and intersection Chow groups I, Duke Math. J. 103. The theory of
relative motives has been applied to decomposition problems
(Chow-K\"unneth decompositions proposed by Jacob Murre): (1) Hilbert
modular varieties (work with B. Gordon and J. Murre), and (2) Lefschetz pencil
of a surface.

In this talk I explain how (a) and (b) can be
further generalized to the theory of mixed motivies over a base variety.

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**Jannsen:**

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**Künnemann:**

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**Migliorini:**

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# Mueller-Stach:

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**Oort: **Some aspects of the
scientific work of Jaap Murre will be presented. The contents of some papers
from the period 1957 - 1975 will be discussed.

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**Ramakrishnan:**

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**Saito:**

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**Shioda:**

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**Srinivas:**

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**Tommasi: **The moduli
space of complex non-singular curves of genus 4 admits a stratification such
that each stratum is the geometric quotient of the complement of a discriminant
in complex projective space. This allows us to compute the rational cohomology
of each stratum. The tools used are Vassiliev-Gorinov's method for the
cohomology of the complement of a discriminant, and a theorem of Peters and
Steenbrink on the cohomology of geometric quotients.

In this way we can determine the rational
cohomology of the whole moduli space, with its mixed Hodge structure.

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**Verra:**