Hodge Conjecture: The Hodge Conjecture, formulated by W.V.D. Hodge in 1950, sits at the intersection of algebraic geometry and topology. It proposes that certain cohomology classes (called Hodge classes) on non-singular complex projective varieties can be expressed as rational linear combinations of algebraic cycles—subvarieties defined by polynomial equations. In essence, it suggests that topological features of complex shapes can be captured by algebraic geometry. While proven in special cases (e.g., for curves and surfaces), the general conjecture remains elusive. It is deeply tied to the structure of manifolds and the classification of geometric objects. Progress has been hindered by the abstract nature of Hodge theory and the difficulty of constructing algebraic cycles. As a Clay Millennium Prize problem, its resolution would illuminate the hidden algebraic structure within continuous shapes and reshape modern geometry.
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