Each complex semisimple Lie algebra carries an adjoint quotient mapping, and this connects representation theory to matters of complete integrability. Such connections are particularly apparent if one considers the so-called "universal centralizer", a symplectic variety on which the adjoint quotient manifests as a completely integrable system. While this system has non-compact fibres, Balibanu's recent work shows it to admit a certain fibrewise compactification. The total space of her fibrewise compactification is a log-symplectic variety, and its geometry is intimately related to the De Concini--Procesi wonderful compactification of the adjoint group $G$.

On the other hand, each choice of regular element produces a Mishchenko--Fomenko map on the given Lie algebra. This map turns out to manifest as a completely integrable system on a certain Kostant--Whittaker reduction of $T^*G$, and one can likewise seek a fibrewise compactification. I will discuss some partial results in this direction, emphasizing a role played by the log-cotangent bundle of the wonderful compactification.

This represents joint work with Markus R\"oser