10 h 11h Leep I
11h 15 12h15 Leep II
12h15 14h repas
14h 15h30 A. Merkurjev I
15h45 17h15 Morel I
Le mercredi 25 juin.
10h 11h Morel II
11h15 12h15 Morel III
12h15 14h repas
14h- 15h30 A. Merkurjev II
15h45 17h15 Leep III
Le jeudi 26 juin.
10h 11h A. Merkurjev III
11h15 12h15 A. Merkurjev IV
12h15 14h repas
14h 15h30 F. Morel IV
15h45 17h15 D. Leep IV
Le vendredi 27 juin.
9h –
10h
Stefan Gille : "The Koszul complex and Witt groups"
10h15 – 11h15 A. Laghribi : "Sur les formes quadratiques a deploiement maximal en
caracteristique 2".
11h30 –
12h30 TBA
12h30 –
14h repas
14h –
15h T. De Medts
: "Quadratic forms of type E_6, E_7 and E_8"
15h15 –
16h15 G.
Berhuy : “Essential dimension
of cubics”.
Steenrod
operations in algebraic geometry (A. Merkurjev)
a) Refined Gysin homomorphisms
b) Excess formula
c) Equivariant Chow groups of a cyclic group
a) Operations of homological and cohomological type
b) Functorial properties
c) Adem relations
a) Definition of the modulo p Steenrod algebra
b) Presentation by generators and relations
a) Formulas arising from Steenrod operations
b) Applications
the Steenrod operations in motivic cohomology.
He used these operations in his proof of the
Milnor Conjecture. Later P.Brosnan gave an
elementary construction of the Steenrod
operations in the Chow theory, the special
case of motivic cohomology.
I am going to give an exposition of the Brosnan's
work. It includes review of the equivariant
Chow groups theory (due to Edidin and Graham),
definition of the reduced power operations,
description of the Steenrod algebra structure.
The Steenrod operations will be applied to prove
various degree formulas.
(D.Leep)
I plan to
talk about the method of exponential sums to study systems of
homogeneous
forms defined over Q_p, the field of p-adic
numbers.
Special emphasis will be given to systems of quadratic forms.
n defined
over Q_p. One motivation for this work is the
attempt to
compute the u-invariant of K, or at least to show that u(K)
is
finite. It is known that u(K) \geq 2^{n+2} and that equality
holds when
n = 0. More recently, Parimala and Suresh showed that when
n = 1 and p
is odd, then u(K) \leq 10.
over an
arbitrary field k is through the study of systems of quadratic
forms
defined over k. A field k is a quadratic C_i-field if every
system of r
quadratic forms defined over k in more than r\cdot 2^i
variables
has a common nontrivial zero defined over k. The standard
methods of
the Tsen-Lang theory show that if K is a function field of
transcendence
degree n defined over k, and if k is a quadratic
C_i-field,
then u(K) \leq 2^{i+n}. Thus, if Q_p is a
quadratic
C_2-field, and K is a finitely generated function field of
transcendence
degree n defined over Q_p, then it would follow
that u(K) =
2^{n+2}.
have
conjectured, or at least speculated, that this should be the case at
least when
p is odd. Systems of r quadratic forms defined over
Q_p have
been studied in detail when r =1,2,3, with only
partial
results known in the case r =3. The famous Ax-Kochen theorem
applied to systems
of quadratic forms defined over Q_p states
the
following. For a given positive integer r , there exists an integer
N (that
depends on r) such that if p > N, then every system of r
quadratic
forms defined over Q_p in more than 4r variables has
a
nontrivial common zero defined over Q_p. There is no
(reasonable)
bound known for N. Since N might increase quickly as r
increases,
the Ax-Kochen theorem does not help with our problem.
of
quadratic forms defined over Q_p.
Other
related topics might be covered if time permits.
K-theorie de
Milnor-Witt, puissances de l'ideal fondamental et Sq^2 (F.Morel)
Milnor sur le
gradué de l'anneau de Witt qui repose sur la suite spectrale d'Adams
en cohomologie
motivique modulo 2 et les résultats de Voevodsky. On s'attachera surtout
a expliciter le
role fondamental joué par l'opération de Steenrod Sq^2 , qui fournit
entre autre des
extensions fonctorielles non triviales de la K-théorie de Milnor
par les
puissances de l'idéal fondamental, appelées K-théorie de Milnor-Witt.