- Course of Michel Brion:
By a result of Grothendieck, the commutative algebraic
groups over a field k are the objects of an abelian category C,
with morphisms being the homomorphisms of algebraic k-groups.
If k is algebraically closed, then by work of Serre and Oort,
the homological dimension of C is 1 in characteristic 0 and 2
in positive characteristics; here the homological dimension is
the smallest integer n such that all Ext groups vanish in degrees > n.
The lectures will address the category of commutative algebraic
groups up to isogeny, defined as the quotient of C by the Serre
subcategory F of finite groups. In particular, we will show that
the homological dimension of C/F is 1 for any field k. The proof
of this uniform result is based on structure theorems for commutative
algebraic groups, which take very different forms in characteristic
0 and in positive characteristics.
- Course of Bill Jacob:
Cohomological tools have been crucial to the development of the theory of quadratic forms and division algebras. The cohomological interpretation of the Brauer group and the related descriptions of the graded Witt group as Galois cohomology are basic examples. In 1982 Kazuya Kato was able to resolve the Milnor problem in characteristic two when he computed the graded Witt group as cohomology using a careful analysis of differential forms and this opened up a new techniques. Voevodsky’s subsequent solution to the Milnor problem away from characteristic two required development of substantially different tools. Depending upon the question, the case of finite characteristic can be easier or more difficult, and work in finite characteristic has led to interesting examples and approaches.
These lectures will focus on finite characteristic and start by examining recent applications of differential forms to problems involving quadratic forms in characteristic two and division algebras in characteristic p. They will then turn to use of the de Rham Witt complex and groups introduced by Izhboldin in determining Witt and Cohomological kernels under algebraic extensions. The emphasis will be on developing the basic theory, where the machinery gives groups are no longer p-torsion and this feature is extremely helpful to capture arithmetic that otherwise washes out when only using differential forms or cohomology mod p. The lectures will assume familiarity with the basic algebraic theory of quadratic forms, division algebras, the Brauer groups as well as Witt vectors. Much of the recent work described in the lectures is joint with Roberto Aravire and Manuel O’Ryan.
- Course of Max Karoubi:
Hermitian K-theory has at least two origins.
Surgery on non simply connected manifolds yields
C.T.C. Wall to introduce subtle invariants
associated to group algebras, usually
noncommutative. The second origin, much older, is
the rich theory of quadratic forms over fields.
The first part of the course is devoted to basic
definitions of these invariants and the "higher"
ones associated to Quillen's + construction
applied to classifying spaces of suitable
Afterwards, we shall make a digression in
topological K-theory in order to show that
"Hermitian K-theory" is a nice unified framework
for the formulation of the 10 fundamental
homotopy equivalences of Bott which in turn imply
his famous periodicity theorems. Note that the
first proof of these equivalences were relying on
Our more algebraic viewpoint leads to what we
call "the fundamental theorem of Hermitian
K-theory" which has many applications, besides
Bott periodicity. For instance, we generalize to
any ring, even noncommutative, the classical
Hasse-Witt invariant for quadratic forms over
At the end we shall give two recent applications of the theory
1) The analog of Quillen-Lichtenbaum conjecture
in the Hermitian framework which is related to a
description of the "Hermitian spectrum" as an
homotopy fixed spectrum of ℤ/2 acting of the
2) A description of the Witt ring associated to a real algebraic variety.