RÉSUMÉ

The nature of subfields of rational function fields has long fascinated mathematicians, and their study has motivated many developments in classical algebraic geometry. More recently, subrings of polynomial rings have been investigated in connection with thorny problems in "affine algebraic geometry", such as the cancellation problem, or linearization for algebraic group actions on affine spaces. I will discuss some progress, and a lot of non-progress, on these "affine" Lueroth type problems, and, in general terms, the algebraic geometry behind them.

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