Details of Grant 

EPSRC Reference:	EP/J009342/1
Title:	The Conjecture of Dixmier
Principal Investigator:	Bavula, Professor V
Other Investigators:
Researcher Co-Investigators:
Project Partners:
Department:	Pure Mathematics
Organisation:	University of Sheffield
Scheme:	Standard Research
Starts:	09 April 2012	Ends:	08 October 2014	Value (£):	55,282
EPSRC Research Topic Classifications:	Algebra & Geometry
EPSRC Industrial Sector Classifications:	No relevance to Underpinning Sectors
Related Grants:
Panel History:	Panel Date Panel Name Outcome 30 Jan 2012 Mathematics Prioritisation Panel Meeting January 2012 Announced
Summary on Grant Application Form
In Mathematics there are two old open problems: the Jacobian Conjecture (open since 1938) for the polynomial algebras in n variables and the Conjecture of Dixmier (open since 1968) for the algebras A(n) of polynomial differential operators, the so-called Weyl algebras, that claims that the Weyl algebras behave like the finite fields. More precisely, every algebra endomorphism of the Weyl algebra is an automorphism. In 1982, Bass, Connell and Wright proved that the Conjecture of Dixmier implies the Jacobian Conjecture. In 2005-07, Tsuchimoto, Belov-Kanel and Kontsevich proved that these two conjectures are equivalent. The Weyl algebra A(n) is a subalgebra of the algebra I(n) of polynomial integro-differential operators. At the end of 2010, I proved that an an analogue of the Conjecture of Dixmier holds for the algebra I(1) (V. Bavula, ``An analogue of the Conjecture of Dixmier is true for the algebra of polynomial integro-differential operators,'' Arxiv:math.RA: 1011.3009), and conjectured that the same result is true for all algebras I(n). The aim of this project is to prove this conjecture and as a result to have a progress on the Conjecture of Dixmier. Another goal of the project is to find the K-groups for the algebras I(n) and to answer the question of whether or not the Bott periodicity holds. The most interesting (and difficult) is the case of the K(1)-groups for the algebras I(n) since it leads to finding explicit generators for the automorphism groups of the algebras I(n). The groups of automorphisms of the algebras I(n) are infinite dimensional algebraic groups. Little is known about their structure in general. In the polynomial case there are several papers by Shafarevich (1966, 1981) and more recently by Kambayashi (1996, 2003, 2004). We are going to obtain generalizations of these results for the Weyl algebras A(n) and I(n).
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Organisation Website:	http://www.shef.ac.uk