Abstracts

Mini Course 2013

Courses:

  1. Course of Tom De Medts:

    During the last few years, there has been a growing interest in Moufang sets, which were introduced in 1990 by Jacques Tits. The notion of a Moufang set is equivalent to that of a group with a split BN-pair of rank one. Moufang sets are of great importance: On the one hand they are important for purely group-theoretical purposes (and in particular, for classification purposes of both simple algebraic groups and finite simple groups). On the other hand, there are many connections with algebraic structures. It was known since some time that there is a deep connection between Jordan division algebras and Moufang sets. It has recently become apparent that this connection is much more general: every structurable division algebra gives rise to a Moufang set. It turns out that many (and perhaps all) of the simple linear algebraic groups of k-rank 1 can be described by a Moufang set arising from such a structurable algebra. This approach gives a lot of new insight, in particular in the rank one forms of the exceptional groups. In this series of lectures, we will introduce the theory of Moufang sets and the theory of structurable algebras in a self-contained way. We will then make the connection with linear algebraic groups of k-rank 1. We will give many examples along the way.

  2. Course of Manfred Knebusch:

    I intend to give an outline of the specialization theory of quadratic forms over a field K of any characteristic with respect to a place λ : K → L ∪ ∞ and to apply this to find relations between the generic splitting behaviour of a quadratic form φ over K and of its specialization λ*(φ) over L. {Main question: How many hyperbolic planes split off from λ*(φ)⊗L' for a given field extension L'/L?} The standard hypothesis here is that φ has “good reduction” under λ, but for many questions it is important to assume only that φ has so called “fair reduction” under λ.
    There exists also a full fledged parallel specialization theory, where λ is replaced by a so called “quadratic place” Λ : K → L ∪ ∞. Here proofs are much more tricky than for ordinary places. Probably the time schedule will only allow hints at this still mysterious part of specialization theory.

  3. Course of Joël Riou:

    Dans ce cours, j'introduirai la théorie de l'homotopie motivique définie par Fabien Morel et Vladimir Voevodsky, que ce dernier utilisa de façon essentielle dans sa démonstration de la conjecture de Milnor, et plus récemment de la conjecture de Bloch-Kato (démontrée avec Markus Rost) reliant la K-théorie de Milnor d'un corps et sa cohomologie galoisienne. Cette théorie donne un sens à la notion de type d'homotopie motivique. J'illustrerai certains des résultats fondamentaux de la théorie en donnant des exemples de variétés algébriques ayant le même type d'homotopie motivique. Je discuterai aussi de la définition dans ce contexte du classifiant d'un groupe algébrique, ce qui permettra de donner une interprétation de la K-théorie algébrique dans cette théorie de l'homotopie, et si le temps le permet, j'esquisserai la construction des opérations de Steenrod motiviques.

Talks:

  1. Talk of Adam Chapman:

    Analyzing square-central elements in central simple algebras of degree $4$, we show that every two elementary abelian Galois maximal subfields are connected by a chain of nontrivially-intersecting pairs. Similar results are proved for non-central quaternion subalgebras, and for central quaternion subalgebras when they exist. Along these lines we classify the maximal square-central subspaces. We also show that every two standard quadruples of generators of a biquaternion algebra are connected by a chain of basic steps, in each of which at most two generators are being changed.

  2. Talk of Uriya First:

    Let (H,*,w) be a hermitian category. I call H non-reflexive if w : id --> ** is only assumed to be a natural transformation, rather than a natural isomorphism. Most results about hermitian categories only apply to the reflexive case (i.e. when w is an isomorphism). In this talk I show that given a non-reflexive category (H,*,w), there exists a reflexive category (H',*',w') such that the category of arbitrary bilinear forms over (H,*,w) (even non-symmetric forms) is equivalent to the category of symmetric regular (=unimodular) bilinear forms over (H',*',w'). Next, I show how systems of bilinear forms can be understood as a single bilinear form in an appropriate non-reflexive hermitian category. Combining both observations leads to many application regarding systems of bilinear forms and also regarding hermitian forms over rings which are defined over non-reflexive modules. Among the applications are Witt's Cancellation Theorem and various results about isometry of (systems of) bilinear forms. (Joint work with E. Bayer-Fluckiger and D. Moldovan.)

  3. Talk of Ronan Flatley:

    We give the key definitions leading to Bourbaki's definition of the symmetric powers of a class of symmetric bilinear forms in the Witt-Grothendieck ring of a field. We compute the symmetric powers of hyperbolic forms over a field $K$ of characteristic different from $2$, irrespective of whether $K$ is formally real or not. Also, we derive formulae for the symmetric powers of quadratic trace forms of a symbol algebra and relate these to earlier results on the exterior powers of such forms.

  4. Talk of Marcus Zibrowius:

    Consider complex vector bundles over a homogeneous variety of the form G/P, where G is a complex linear algebraic group and P is a parabolic subgroup. By a classical result of Atiyah and Hirzebruch, all such vector bundles arise from representations of P, at least up to stable isomorphism. For symmetric vector bundles, i.e. vector bundles equipped with non-degenerate quadratic forms, the analogous statement is false: in general, one can find symmetric vector bundles over G/P that do not arise from symmetric representations of P. We will explain in what sense such vector bundles are captured by the Witt ring of G/P, and show how this leads to a description of the Witt ring and (hence) of the real K-theory of full flag varieties.