.Clay
Mathematics Institute |

## The Hodge Conjencture |
## millennium prize problems / the hodge conjecture problem |

In the twentieth century mathematicians discovered powerful ways to investigate the shapes of complicated objects. The basic idea is to ask to what extent we can approximate the shape of a given object by gluing together simple geometric building blocks of increasing dimension. This technique turned out to be so useful that it got generalized in many different ways, eventually leading to powerful tools that enabled mathematicians to make great progress in cataloging the variety of objects they encountered in their investigations. Unfortunately, the geometric origins of the procedure became obscured in this generalization. In some sense it was necessary to add

piecesthat did not have any geometric interpretation. The Hodge conjecture asserts that for particularly nice types of spaces calledprojective algebraic varieties, the pieces calledHodge cyclesare actually (rational linear) combinations of geometric pieces calledalgebraic cycles.