Nils Dencker

The 2005 Clay Research Award was made to Nils Dencker for his complete resolution of a conjecture made by F. Trèves and L. Nirenberg in 1970.  This conjecture posits an essentially geometric necessary and sufficient condition, Psi, for a pseudo-differential operator of principal type to be locally solvable, i.e., for the equation Pu = ƒ to have local solutions given a finite number of conditions on ƒ.

Dencker’s work provides a full mathematical understanding of the surprising discovery by Hans Lewy in 1957 that there exists a linear partial differential operator — a one-term, third-order perturbation of the Cauchy-Riemann operator — which is not locally solvable in this sense. The necessity of condition “Psi” was shown for operators in dimension 2 by R. Moyer in 1978 and in general by L. Hormander in 1981. The sufficiency of the condition has resisted many previous attacks.