Abstract or Additional Information
The Jacobian Conjecture (O. Keller, 1939) is one of the oldest unsolved problems in algebraic geometry. It states that every polynomial self-map of the complex space \(\mathbb{C}^n\) that is everywhere locally one-to-one (i.e. has constant nonzero Jacobian) is globally one-to-one (which implies that it has a polynomial inverse). This is trivially true in dimension one, but even the two-dimensional case is still wide open. This conjecture has got a well deserved reputation of being much trickier than it seems, with many incorrect proofs produced by respected mathematicians.
In this talk I will give a brief overview of some known results and approaches to this conjecture. I feel that the explosive development of the birational algebraic geometry during the last three decades created the tools necessary to finally resolve this open question, at least its two-dimensional case. I will discuss some of the relevant results and outline my approach and its relation to other approaches. Most of the talk will be accessible to general mathematical audience with no special algebraic geometry background.