Mathematics
The Hodge Conjecture for Smooth Projective Varieties
Quick fact
The Hodge conjecture is one of the Clay Millennium Prize Problems, yet it remains unproven even after decades of effort. It asserts that every Hodge class on a smooth projective variety can be expressed as a rational linear combination of algebraic cycles.
Why this is interesting
You might think that the shape of a geometric object is fully captured by its holes and tunnels. But what if some hidden structure, invisible to topology, could be traced back to something as tangible as curves and surfaces? That is the mystery at the heart of the Hodge conjecture.