- Course of Jean Fasel:
In these lectures, we will explain how to classify vector bundles on smooth affine schemes using the machinery of the A1-homotopy category. We will start with a quick introduction on the A1-homotopy category, and then explain the results of Morel on the classification of vector bundles. We will then explain how to extend these results using Postnikov towers, together with the known computations of the second non trivial A1-homotopy sheaf of the affine punctured space of a given dimension. Time permitting, we will also explain how to classify symplectic or orthogonal bundles.
- Course of Ulf Rehmann:
The so called "lower" algebraic K-theory is very closely interrelated
with the theory of Chevalley groups and - more general - with
the theory of almost simple algebraic groups. The course will give an
introduction into these topics.
Tentative Lecture Titles:
- The group structure of SL_n over a field
- Chevalley groups over fields
- K_1, K_2 related to Chevalley groups
- Almost simple algebraic groups over arbitrary fields
- K-theoretic results related to simple algebraic groups
[BT] A. Borel, J. Tits, Groupes réductifs
Publ. Math. IHES, 27 (1965), p. 55-151
Compléments à l'article "Groupes réductifs".
Publ. Math. IHES, 41 (1972), p. 253-276
[HO] Hahn, Alexander J.; O'Meara, O. Timothy, The classical groups and $K$-theory
Grundlehren der Mathematischen Wissenschaften 291, 1989.
[KMRT] M.-A. Knus, A. Merkurjev, M. Rost, J.-P. Tignol, The book of involutions, AMS Colloquium Publications, Vol. 44, 1998.
[Mi] J. Milnor, Introduction to Algebraic K-Theory,
Annals of Math. Study 72, 1971, Princeton University Press
[Sp] T.A. Springer, Linear Algebraic Groups, 2nd edition, BirkhÃ¤user,
[Ti] J. Tits, Classification of algebraic semisimple groups.
Algebraic Groups and Discontinuous Subgroups
(Proc. Sympos. Pure Math., Boulder, Colo., 1965) pp. 33â€“62
Amer. Math. Soc., Providence, R.I., 1966 Proc. Symp. Pure Math
18, 1966, 33-62
- Course of Yoav Segev:
My interest in maps on division rings that resemble valuations
started with a conjecture in algebraic group theory. In fact
given an algebraic group (or any group G for that matter),
the first question to ask is what are the normal subgroups
of G. The conjecture that I got interested in is the
Margulis-Platonov conjecture which is a conjecture about the
normal subgroup structure of certain algebraic groups. The resolution of this
conjecture in one important case (the inner forms of type An) was reduced to showing that
if D is a finite dimensional division algebra over a global field
and N ◁ D is a normal subgroup of finite index, then the group D×/N
is not (abstractly) simple.
In trying to show this I
was led to consider the commuting graph of the group D×/N,
and this graph (miraculously) led to a certain order (via inclusion)
of what I sometimes call the shadow of elements of D\ N
on N. In trying to better understand this ``miracle'' one is
led to the notions of partially preordered and partially ordered
groups, the notions of leveled, strongly-leveled, valuation-like
and strong-valuation-like -- maps. These are certain maps
from N to ``almost'' ordered groups. In my lecture series I will
discuss all these notions and show how they reveal certain hidden
treasures inside D (where now D can be any division algebra).
I will also mention some generalizations, and
a certain (notoriously difficult) related open problem.