Galois cohomology of reductive groups over global fields

Let $F$ be a number field (say, the field of rational numbers Q) or a p-adic
field (say, the field of p-adic numbers $Q_p$), or a global function field
(say, the field of rational functions of one variable over a finite field
$F_q$). Let $G$ be a connected reductive group over $F$ (say, $SO(n)$ ). One needs
the first Galois cohomology set $H^1(F,G)$ for classification problems in
algebraic geometry and linear algebra over $F$. In the talk, I will give
closed formulas for $H^1(F,G)$ when $F$ is as above, in terms of the algebraic
fundamental group $\pi_1(G)$ introduced by the speaker in an MPIM preprint of
1989. All terms will be defined and examples will be given.

The talk is based on a joint work with Tasho Kaletha  arXiv:2303.04120.