This talk links algebraic geometry, noncommutative ring theory, and the Virasoro Lie algebra, known for its importance in conformal field theory. We begin with a question in ring theory. Let X be a projective surface with an automorphism a, and consider the skew polynomial extension k(X)[t; a]. Which finitely generated graded subalgebras are noetherian?
It turns out that, morally, if a has dense orbits on X there are many noetherian subalgebras, and if not there are few. In the first part of the talk we explain this. In the second, we show how this leads to a proof that the enveloping algebra of the Virasoro algebra is not noetherian.