Seminar Algebra and Geometry: Vladimir Bavula (University of Sheffield)

Two 45 minutes talks: "An analogue of the Conjecture of Dixmier is true for the algebra of polynomial integro-differential operators" and "The group of automorphisms of the algebra of one-sided inverses of a polynomial algebra"

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

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' as a  finite field extension. In more detail, the first problem/conjecture of Dixmier asks: is it true that an algebra endomorphism of the Weyl algebra is 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.

The group of automorphisms of the  algebra of one-sided inverses of a polynomial algebra

The algebra S_n of one-sided inverses of a polynomial algebra P_n in n variables is obtained from P_n by adding commuting, left (but not two-sided) inverses of its canonical generators. Ignoring non-Noetherian property, the algebra S_n belongs to a family of algebras like the n'th Weyl algebra  and the polynomial algebra in 2n variables (e.g., the Gelfand-Kirillov dimension of S_n is 2n).
The group of automorphisms G_n of S_nis huge. Recently, I found the group G_n and its explicit generators. In my talk,  I will explain the ideas behind the proofs and I will do an overview of the basic properties of these algebras and their connections with polynomials, differential and integro-differential operators.

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