Many cohomological invariants of algebraic groups are related to trace forms of algebras, including all the invariants of PGL4, G2, and F4 in mod 2 Galois cohomology. In the talk, I will discuss some examples of nonassociative (structurable) algebras with involution that play a role in constructing groups of type E6, E7, and E8, and show how to extract cohomological invariants from their trace forms. The hope is for these invariants to survive the constructions and produce invariants of exceptional algebraic groups. I will present some partial progress in that direction, as well as a surprising discovery of a degree 5 cohomological invariant of PGSp8 (or symplectic involutions of degree 8). This is based on joint work with Victor Petrov.
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