If $G$ is a connected semisimple group over a local field $k$ then $G(k)$ is compact for its analytic topology if and only if $G$ does not contain GL$_1$ as a $k$-subgroup. The latter condition makes sense over any field, so one may seek an algebro-geometric generalization: is there a group-theoretic characterization of when an algebraic group $G$ occurs as an open subvariety of a projective $k$-variety $X$ such that $X(k) = G(k)$ (thereby forcing compactness of $G(k)$ when $k$ is a local field)?
There is a simple necessary condition conjectured to be sufficient. Sufficiency in the semisimple and commutative cases was settled affirmatively long ago (providing canonical $X$ with wonderful properties), but this is far from the general case when $k$ is not perfect (such as local and global function fields). I will discuss ideas of Gabber and joint work with Gille and Prasad that has led to much progress in the general case, resting on recent progress in the structure theory of pseudo-reductive groups (which will be summarized).