Hodge Conjecture

Hodge conjecture, in algebraic geometry, assertion that for certain “nice” spaces (projective algebraic varieties), their complicated shapes can be covered (approximated) by a collection of simpler geometric pieces called algebraic cycles. The conjecture was first formulated by British mathematician William Hodge in 1941, though it received little attention before he presented it in an address during the 1950 International Congress of Mathematicians, held in Cambridge, Mass., U.S. In 2000 it was designated one of the Millennium Problems, seven mathematical problems selected by the Clay Mathematics Institute of Cambridge, Mass., for a special award. The solution for each Millennium Problem is worth $1 million. In 2008 the U.S. Defense Advanced Research Projects Agency (DARPA) listed it as one of the 23 DARPA Mathematical Challenges, mathematical problems for which it was soliciting research proposals for funding—“Mathematical Challenge Twenty-one: Settle the Hodge Conjecture. This conjecture in algebraic geometry is a metaphor for transforming transcendental computations into algebraic ones.”