Involutions on central simple algebras give rise to algebraic groups and to a particular type of varieties. Many concepts and statements from the theory of quadratic forms over fields can be extended to involutions. But involutions can also be used to study quadratic forms and central simple algebras. For example, composition formulae for quadratic forms can be interpreted as relations between Clifford algebras and the adjoint involutions of quadratic forms. Classical invariants of quadratic forms such as the discriminant, the Clifford algebra and signatures can be extended to invariants of algebras with involution. Whether involutions are classified by certain invariants depends on the base field, the degree of the algebra and the type of the involution.
Tensor products of very small algebras which carry a canonical involution are a natural generalisation of so-called Pfister forms. Algebras with involution of this form are said to be totally decomposable, and they typically have trivial invariants. It turns out, however, that decomposability cannot always be detected by the known invariants. The construction of examples disproving some motivated conjectures often involve field valuations and gauges on algebras.
The aim of the summer school is to introduce young researchers to this modern research area and to expose them to some challenging open problems.
Abstracts of the different sessions and talks can be found here (pdf-185 kB).