9409775 Dean This project is concerned with a solution of the Jacobian Conjecture. The principal investigator describes a new approach for verying the conjecture in dimension two. The ideas eminate from work on the relationship between the Jacobian Conjecture and the Kirillov-Dixmier Conjecture for the first Weyl algebra. An extension of this approach to higher dimensions is also discussed. It is well known that an invertible polynomial map on a complex affine space has a jacobian which is a nonzero constant. The Jacobian Conjecture asserts the converse: a polynomial map with a jacobian which is a nonzero constant is invertible. In spite of the vast amount of attention which it has attracted over the last fifty years, the Jacobian Conjecture remains open in every dimension greater than one. This award supports work on a novel approach to this problem.