# An analogue of the Conjecture of Dixmier is true for the algebra of polynomial integro-differential operators''

24 May 2011
17:00
Prof. V. Bavula
Abstract

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