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.
- Algebra Seminar