Schedule and Abstracts

Mini Course 2006


  • 10h-11h: Kahn I
  • 11h-11h15: coffee
  • 11h15-12h15: Kahn II
  • 12h30-14h: Lunch
  • 15h-16h: Hoffmann I
  • 16h-16h30: coffee
  • 16h30-17h30: Zainoulline
  • 10h-11h: Hoffmann II
  • 11h-11h15: coffee
  • 11h15-12h15: Hoffmann III
  • 12h30-14h: Lunch
  • 15h-16h: Garibaldi I
  • 16h-16h30: coffee
  • 16h30-17h30: Shripad
  • 9h-10h: Kahn III
  • 10h-10h30: coffee
  • 10h30-11h30: Garibaldi II
  • 11h30-11h45: coffee
  • 11h45-12h45: Garibaldi III
  • 13h-14h: lunch
    The wednesday afternoon is free!
  • 10h-11h: Hoffmann IV
  • 11h-11h15: coffee
  • 11h15-12h15: Kahn IV
  • 12h30-14h: lunch
  • 15h-16h: Garibaldi IV
  • 16h-16h30: coffee
  • 16h30-17h30: Calmès
  • 9h-10h: Garibaldi V
  • 10h-10h30: coffee
  • 10h30-11h30: Kahn V
  • 11h30-11h45: coffee
  • 11h45-12h45: Hoffmann V
  • 13h-14h: lunch.


  1. Course of Skip Garibaldi

    • Title: Cohomological invariants
    • Abstract: The determinant and Hasse-Witt invariant of quadratic forms are two examples of cohomological invariants. The interesting fact is that one can--sometimes!--explicitly describe all the cohomological invariants. This was done for quadratic forms and several other examples in Serre's portion of the book "Cohomological invariants in Galois cohomology". These lectures will summarize the necessary tools and apply them in various cases. We will emphasize examples not covered in Serre's lectures, including Rost's results on invariants of Spin groups.

  2. Course of Detlev W. Hoffmann

    • Title: Quadratic forms in characteristic 2
    • Abstract: For many years, much of the research on quadratic forms has focussed up to sporadic exceptions on the case of characteristic not 2. However, over the past few years, the theory of quadratic forms in the case 2=0 has experienced a resurgence in the work of Arason, Aravire, Baeza, Hoffmann, Jacob, Knebusch, Laghribi etc. In this lecture series, we present some of these recent developments in the algebraic theory of quadratic forms in characteristic 2. Topics may include (if time permits) specialization and generic splitting of quadratic forms, exact sequences of Witt groups, the behaviour of quadratic forms and cohomology (Ã la Kato) under field extensions, in particular extensions given by function fields of quadratic forms.

  3. Course of Bruno Kahn

    • Title: "1-motifs"
    • Abstract: Le mini-cours, concernant un travail en collaboration avec Luca Barbieri-Viale, couvrira les thèmes suivants:
      1. 1-motifs de Deligne (sur un corps parfait).
      2. 1-motifs avec torsion (Barbieri-Viale-Rosenschon-Saito).
      3. La catégorie dérivée des 1-motifs.
      4. Lien avec les motifs étales de Voevodsky.
      5. Applications: faisceaux 1-motiviques, théorème de Roitman, conjecture de Deligne.


  1. Talk of Baptiste Calmès

    • Title: A push-forward for the coherent Witt groups of schemes (joint work with Jens Hornbostel)
    • Abstract: Anyone studying quadratic forms knows the Witt group of a field. Witt groups of schemes are less well known, though they probably can be quite useful, even to provide information about the Witt group of a field. Paul Balmer has given a very workable definition of higher Witt groups in terms of triangulated categories with dualities. He proved important properties such as a long exact sequence of localization in this abstract framework. He applied this mainly to the derived category of locally free sheaves on a scheme, recovering the classical definition of Knebush in this case. One useful tool that he did not provide (in the framework of schemes) is a push forward, along a certain class of morphisms. We will explain how this can be defined in an abstract framework, and then apply it to the coherent Witt groups of schemes using results from the theory of Grothendieck duality. We will give a few examples of how it can be useful.

  2. Talk of Shripad Garge

    • Title: On excellence of $F_4$
    • Abstract: An algebraic group $G$ defined over a field $k$ is said to be excellent if for any extension $L$ of $k$ the anisotropic kernel of the group $G \otimes_k L$ is defined over $k$. This notion was introduced by Kersten and Rehmann. The excellence properties of some groups of classical type have been studied so far and it follows easily that a group of type $G_2$ is excellent over any field. We prove that a group of type $F_4$ is also excellent over any field $k$ of characteristic other than 2 and 3.

  3. Talk of Kirill Zainoulline

    • Title: On canonical p-dimensions of algebraic groups
    • Abstract: We compute all p-canonical dimensions of (split) linear algebraic groups using p-exceptional degrees invented by V.Kac in the beginning of 80-s.