In 1968, Dixmier posed six problems for the algebra of polynomial

differential operators, i.e. the Weyl algebra. In 1975, Joseph

solved the third and sixth problems and, in 2005, I solved the

fifth problem and gave a positive solution to the fourth problem

but only for homogeneous differential operators. The remaining three problems are still open. The first problem/conjecture of Dixmier (which is equivalent to the Jacobian Conjecture as was shown in 2005-07 by Tsuchimito, Belov and Kontsevich) claims that the Weyl algebra `behaves'

like a finite field. The first problem/conjecture of

Dixmier: is it true that an algebra endomorphism of the Weyl

algebra an automorphism? In 2010, I proved that this question has

an affirmative answer for the algebra of polynomial

integro-differential operators. In my talk, I will explain the main

ideas, the structure of the proof and recent progress on the first problem/conjecture of Dixmier.