This talk is prompted by the long standing question of whether it is possible for the universal enveloping algebra of an in finite dimensional Lie algebra to be noetherian. To address this problem, we answer a 23-year-old question of Carolyn Dean and Lance Small; namely, we prove that the universal enveloping algebra of the Witt (or centerless Virasoro) algebra is not noetherian. We employ algebro-geometric techniques from Sue Sierra's classification of (noncommutative) birationally commutative projective surfaces to accomplish this task. As a consequence of our main result, we show that the enveloping algebras of many other infinite dimensional Lie algebras also are not noetherian. These Lie algebras include the Virasoro algebra and all Z-graded simple Lie algebras of polynomial growth. This is joint work with Sue Sierra.