The Hodge conjecture is one of the Clay Mathematics Institute's Millennium Prize Problems, with a prize of $1,000,000 for whoever can prove or disprove the Hodge conjecture. It is a problem that arises in the algebraic geometry that relates the algebraic topology of a non-singular complex algebraic variety and the sub-varieties of that variety. In short it is to prove whether we can approximate the shape of a given object by gluing together simple geometric building blocks of increasing dimension. This was widely useful in the present day, the problem arose when the geometric origins of the procedure became obscured in this generalization, meaning that it became necessary to add pieces that did not have any geometric interpretation. This was the Hodge conjecture.
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