- Course of J. Hartmann:
Patching is a method that until recently had mostly found use in inverse Galois theory. The underlying idea is to construct global objects by building them locally on "patches".
A new variant of this technique developed by D. Harbater and myself appears to be more suitable for applications outside Galois theory (e.g. in joint work also with D. Krashen).
The minicourse explains the method of patching over fields. One of the main ingredients is a factorization result for the general linear group. It turns out that the generalization of this result to other classes of linear
algebraic groups is the key to understanding rational points on homogeneous spaces under such groups in a local way.
Special emphasis will be put on applications, the guiding example being that of quadratic forms (where the associated orthogonal group acts on the projective quadric defined by the form). The local global principle for quadratic forms obtained from patching was used by Colliot-Thélène, Parimala, and Suresh to obtain a more classical local global principle in terms of valuations. Whether a similar result can be obtained for homogeneous spaces is still open.
- Course of N. Karpenko:
The main objective of the mini-course is to prove that an orthogonal
involution on a central simple algebra that
becomes isotropic over any splitting field of the algebra, also
becomes isotropic over an odd degree
extension of the base field. In particular, any non-hyperbolic
involution remains non-hyperbolic over some
splitting field of the algebra. The following two results needed in
the proof will also be treated:
a structure theorem for the motives of the projective homogeneous varieties
under a semisimple algebraic group and a computation of the canonical
dimension of the generalized
Severi-Brauer varieties. Symplectic and unitary involutions will also
- Course of R. Weiss:
In his Lecture Notes from 1974 Tits classified (irreducible thick)
spherical buildings of dimension l ≥ 2. At the heart of this
classification is his result that a spherical building of dimension
l ≥ 2 is uniquely determined by certain subbuildings of dimension 1.
A spherical building of dimension 1 is a generalized polygon, that
is, simply a bipartite graph whose diameter equals half the length of a
shortest circuit. Generalized polygons themselves are too numerous to classify,
but Tits observed that all the generalized polygons that occur in
spherical buildings of higher dimension are what he called
Moufang polygons, that is, generalized polygons satisfying a certain
group theoretical property. Subsequently, a classification of Moufang
polygons was obtained which, in turn, allowed for a much simpler
proof of the classification of spherical buildings of higher
In this course we will give a brief introduction to the theory of
spherical buildings and Moufang polygons emphasizing
the various algebraic structures which arise in
their classification: anisotropic quadratic forms and pseudo-quadratic spaces
as well as octonion division algebras, certain Jordan division algebras and more exotic things. We
will then focus on the Moufang quadrangles (i.e. Moufang polygons of diameter 4)
of type E6, E7, E8 and F4.
The Moufang quadrangles of type E6, E7 and E8 are the
spherical buildings associated to certain forms of E6, E7 and E8;
they owe their existence to the extraordinary properties of certain anisotropic
quadratic forms of dimension 6, 8 and 12. The Moufang quadrangles of type F4 are buildings defined over a pair of imperfect fields K
and F of characteristic 2 such that K2 Ì F Ì K; they owe their existence to a
pair of anisotropic (but defective!) quadratic forms, one defined over K and one defined over F.
We will also describe two important open problems, one having to do with the
automorphism group of a Moufang quadrangle of type E8 (equivalent to one case
of the Kneser-Tits conjecture) and the other related
to the existence of a corresponding affine building when the field of definition
of a Moufang quadrangle of type E8 is complete with respect to a discrete valuation.