The European Mathematical Society

The Nash problem was formulated in the sixties in the attempt to understand the relation between the structure of resolution of a singularities of an algebraic variety over a field of characteristic 0 and the space of arcs (germs of parametrized curves) in the variety. In 2003, two scientists from the United States and Japan (J. Kollar and S. Ishii) showed that this relation does not apply to objects with four or more dimensions. However, Dr Fernández de Bobadilla, in a joint work with Dr María Pé Pereira (both Madrid, Spain), has now demonstrated that there is indeed a direct correspondence between the two as regards singularity of surfaces. This correspondence is known in Algebraic Geometry as the ‘bijectivity of the Nash mapping for surfaces’.
Information: http://erc.europa.eu/succes-stories/solution-nobel-prize-winner-john-nas...
http://annals.math.princeton.edu/wp-content/uploads/FernandezdeBobadilla...