Schedule and Abstracts

Mini Course 2010


  • 10h-11h:   Hartmann I
  • 11h-11h15:  coffee
  • 11h15-12h15:  Hartmann II
  • 12h30-14h:  Lunch
  • 15h-16h:  Karpenko I
  • 16h-16h30:  coffee
  • 16h30-17h30:  Karpenko II

  • 10h-11h:  Weiss I
  • 11h-11h15:  coffee
  • 11h15-12h15:  Hartmann III
  • 12h30-14h:  Lunch
  • 15h-16h:  Karpenko III
  • 16h-16h30:  coffee
  • 16h30-17h30:  Karpenko IV

  • 9h15-10h15:  Karpenko V
  • 10h15-10h30:  coffee
  • 10h30-11h30:  Weiss II
  • 11h30-11h45:  coffee
  • 11h45-12h45:  Hartmann IV
  • 13h-14h:  lunch
    The Wednesday afternoon is free!

  • 10h-11h:  Hartmann V
  • 11h-11h15:  coffee
  • 11h15-12h15:   Weiss III
  • 12h30-14h:   lunch
  • 15h-16h:   Weiss IV
  • 16h-16h30:   coffee
  • 16h30-17h15:   Auel

  • 9h30-10h15:   Stavrova
  • 10h15-10h30:   coffee
  • 10h30-11h15:   Petrov
  • 11h15-11h30:   coffee
  • 11h30-12h30:   Weiss V
  • 13h-14h: lunch.

Abstracts of courses:

  1. 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.

  2. 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 be discussed.

  3. 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 dimension.
    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.

Abstracts of talks:

  1. Talk of A. Auel:

    We will describe classes in the "involutive" Brauer group of Azumaya algebras with involution, arising from (generalized) Clifford algebras of line bundle-valued quadratic forms over schemes (with 2 invertible). In particular, when the form has trivial discriminant and rank divisible by 4, the two pieces of the even Clifford algebra are equivalent in the Brauer group, but have involution classes differing by the 1st Chern class of the value line bundle.

  2. Talk of V. Petrov:

    Despite the current progress in the study of motives of projective homogeneous varieties, there were little known about varieties of outer type until recently. Daniel Krashen showed that varieties of outer type 2An are closely related to those of inner type Dn+1. We study a similar relationship for varieties of types 2E6 (compared to E7) and 3D4 (compared to F4). Using a recent result of Nikita Karpenko, we are able to compute the motives of all generically quasi-split homogeneous varieties.

  3. Talk of A. Stavrova:

    The Serre---Grothendieck conjecture states that if a principal $G$-bundle of a reductive group scheme G over a regular local ring R is trivial over the field of fractions of R, then it is trivial already over R. This conjecture was previously settled in the cases where R contains a field k
    and G is defined over k ("the constant case") by J.-L. Colliot-Thélène, M. Ojanguren and M.S. Raghunathan, where G is a torus by
    J.-L. Colliot-Thélène and J.-J. Sansuc, and for certain classical groups by M. Ojanguren, A. Suslin, I. Panin and K. Zainoulline. In 2009 in a series of papers by I. Panin, N. Vavilov, V. Petrov, V. Chernousov and myself the conjecture has been proved for all isotropic reductive group schemes, as well as for several classes of anisotropic reductive group schemes of exceptional type, under the only assumption that R contains an infinite perfect field. I will give a brief survey of these results.