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Title: Hilbert's 14th problem, invariant theory and algebraic homogeneous spaces

Abstract: In this talk we intend to give a short survey of some of the develpments --old and new-- concerning the three problems mentioned above. A counterexample to the original 14th problem, was found by Nagata in 1958. The problem was then reformulated and as such it remains open. We will show the close relationship of the reformulated problem, with the study of algebraic homogeneous spaces of the form G/H (here G is an affine algebraic group and H a closed subgroup). We touch along the way on some problems of invariant theory, in particular we discuss invariants of unipotent groups.

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