The Jacobian Conjecture is one of the most well-known open problems in algebraic geometry. It now seems that a proof has been found by Carolyn Dean of the University of Michigan, for the case of polynomials in two complex variables (for more variables, many people believe it is not even true). For more information about this, see Graham Leuschke’s weblog.
Dean hasn’t published any papers in almost 15 years and is nominally a lecturer in mathematics education at Michigan. There have been many false proofs of this conjecture over the years, and if this one holds up it will be quite a story. The paper doesn’t seem to be publicly available yet, but Dean will be lecturing on the proof at Michigan next month. One of the experts in the field, Mel Hochster, has gone over it carefully and is convinced it is correct. The rumor I hear is that it has been submitted for publication to the Journal of the American Mathematical Society.
Update: There’s an announcement of Dean’s talks posted on sci.math.research.
Update: Someone wrote in with a comment to another post pointing out that Dean has found a hole in her proof. For some more information about this, go here.