А Вы надолго в ИХЕСе? Может быть, приедете к нам с докладом?

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В Ихесе до конца марта. К вам - это в Бордо? Я непрочь, спасибо. Сейчас занимаюсь вот чем:
Short abstract:

We describe the Voevodsky category $DM$ of motives in terms of Suslin complexes of smooth projective varieties. We give a description of any triangulated subcategory of $DM$. We descibe
'truncation' functors $t_N$ for $N>0$. $t_0$ generalizes the weight complex of Soule and Gillet; its target is the homotopy category of Chow motives; it calculates $K_0(DM)$, it checks whether a motive is a mixed Tate one. $t_N$ give a weight filtration and a 'motivic descent spectral sequence' for 'standard' (and more general) realizations.

Longer (and more precise) version:

The main classification result of the talk is an explicit
description of the category of effective geometric motives defined
by Voevodsky. We construct explicitly a category $H$ that is only
slightly different from the category defined by Voevodsky using
{\it twisted complexes} (the notion defined by Bondal and
Kapranov) over a certain differential graded category $J$; the
objects of $J$ are Suslin complexes of smooth projective
varieties. This gives a description of the Voevodsky's category
$DM$ that is similar to the category defined by Hanamura and yet
works on the integral level. In particular, for any motivic
complex $M$ (for instance, the Suslin complex of an arbitrary
variety) there exists a complex $M'$ 'constructed from' the Suslin
complexes of smooth projective varieties; $M'$ is unique up to a
homotopy. This fact can be used for the calculation of cohomology
of motives (with coefficients in any complex of sheaves with
transfers with homotopy invariant cohomology).

As an application we give a general description of any subcategory
of $DM$ that is generated by a fixed set of objects and of localizations of $DM$. We also
consider the question of constructing realizations of motives. We
construct a family of canonical {\it truncation} functors $t_N$
for $N>0$. The target of $t_0$ is just the homotopy category of
$Cho$ (that is 'almost' the category of Chow motives). $t_0$
induces an isomorphism of $K_0$ for the category of effective
geometric Voevodsky motives with $K_0(Chow)$. It turns out that
the $N$-th weight filtration of the \'etale and De Rham
realizations can be factorized through $t_N$. It seems very
probable that this result could be extended to other 'standard'
realizations of motives. The weight complex of Gillet and Soul\'e
is (essentially) the restriction of $t_0$ to motives with compact
support of varieties. Weight complex is extended to a true motive (in Voeovodsky's sense).

We describe a vast generalization of the
method of constructing cohomological functors described by Gillet
and Soul\'e. We construct a spectral sequence converging to the
cohomology of an arbitrary motive $X$. This spectral sequence
could be called the {\it spectral sequence of motivic descent}
(note that the usual cohomological descent spectral sequences
compute cohomology of varieties only). Its $E_1$-terms are
cohomology of smooth projective varieties; its $E_{n}$-terms
depend only on $t_{2n-4}(X)$, $n\ge 2$. This spectral sequence gives a
canonical weight filtration on a wide class of cohomological
functors; for the 'standard' realizations this filtration
coincides with the usual one. A smooth correspondence $f$ induces
a zero morphism on cohomology modulo torsion if $t_0(f)=0$.

В принципе, мог бы также вспомнить предыдущую деятельность, но половину ее я в Бордо уже докладывал:

A complete classification
of finite local flat commutative group schemes over mixed
characteristic complete discrete valuation rings in terms of their
Cartier modules (defined by Oort) is described. These Cartier
modules could be studied using the 'explicit theory of Cartier
modules for formal groups' developed by the author.
The properties of
the generic fibre functor for group schemes are discussed. We
also state several properties of tangent space of these schemes.
These results are applied to the study of reduction of Abelian
varieties. A finite $p$-adic semistable reduction for Abelian
varieties is formulated. It looks especially nice for the ordinary
reduction case.

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Поговорил с органайзерами. Увы, весь март забит-перезабит. Жалко, что я раньше не узнал о Вашем приезде.

Вы впредь заранее сообщите мне, что во Фр. появитесь.

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