- Fatma Kader Bingöl (Scuola Normale Superiore di Pisa)
- Title: Classification theorems for involutions by cohomological invariants.
- Abstract: Quadratic forms over a field F are classified up to isometry by dimension, discriminant, and
Clifford invariant when I^3F, the third power of the fundamental ideal in the Witt ring of F, vanishes, or
in different terms, when the norm form of every quaternion algebra over F is surjective. Quadratic pairs on
central simple algebras can be seen as twisted forms of quadratic forms, as in the split case, the quadratic pair
is given explicitly by a quadratic form.
The classification theorem has been extended to the setting of quadratic pairs, as well as to involutions, under a
stronger condition that the base field has 2-separable dimension at most 2. A field F has 2-separable dimension at
most 2 if and only if I^3L=0 for all finite separable extensions L/F. We show that the classification theorems hold
under the weaker condition that the base field F satisfies I^3F=0, if the underlying central simple algebra has exponent
at most 2. This is joint work with Anne Quéguiner-Mathieu and Karim Johannes Becher.
- Stephen Scully
(University of Victoria) To be confirmed.
- Thomas Unger (University College Dublin)
- Title: Pfister's local-global principle for Azumaya algebras with involution.
- Abstract: This is joint work with Vincent Astier. We prove Pfister's local-global principle
for hermitian forms over Azumaya algebras with involution over semilocal rings,
and show in particular that the Witt group of nonsingular hermitian forms is
2-primary torsion. Our proof relies on a hermitian version of Sylvester's law of
inertia, which is obtained from an investigation of the connections between a
pairing of hermitian forms extensively studied by Garrel, signatures of hermitian
forms, and positive semidefinite quadratic forms.
- A fourth speaker To be announced.
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