The Hodge conjecture: The complications of understanding the shape of geometric spaces


The Hodge conjecture is one of the seven millennium problems, and is framed within differential geometry and algebraic geometry. It was proposed by William Hodge in 1950, and is currently a stimulus for the development of several theories based on geometry, analysis and mathematical physics. It proposes a natural condition for the existence of complex subvarieties within a complex variety. Varieties are the spaces in which geometric objects can be considered. In complex varieties, the structure of the space is based on complex numbers, instead of the most intuitive structure of geometry, based on real numbers.


complex geometry; topology; homology; Hodge theory; varieties


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