Jacobian conjecture
Definition
A legendary open problem in algebraic geometry asserting that a polynomial mapping of affine n-space to itself has a polynomial inverse if and only if the determinant of its Jacobian matrix is a non-zero constant.
A mathematical 'Holy Grail' that serves as the ultimate test of patience for graduate students and a reliable source of 'I almost solved it last night' excitement at three in the morning.
Examples
The Jacobian conjecture is the mathematical equivalent of a blind date that promises to be a life-changing encounter but usually just leaves you questioning your own existence in a library basement.
I told my advisor I had a breakthrough on the Jacobian conjecture, but it turned out I had just accidentally doodled a set of parametric equations for a very complicated coffee mug.
Solving the Jacobian conjecture is the academic version of trying to find a matching pair of socks that have been through a black hole, except with more notation and significantly less fashion sense.