- 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.
- 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.)
- 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.
- 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.