Hodge Conjecture

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That for projective algebraic varieties, Hodge cycles are rational linear combinations of algebraic cycles.

Simply: Last century, mathematicians discovered powerful ways to investigate the shapes of complicated objects.
They asked to what extent we can approximate the shape of such objects by gluing together simple geometric building
blocks of increasing dimension. This technique was generalised in many ways, obscuring its geometric origins and
sometimes adding pieces with no geometric interpretation. Cycles refer to Hodge’s suggestion that all objects may be
built from smaller parts being repeatedly projected. The conjecture asserts that certain complicated forms in algebraic
geometry can be reduced to combinations of much simpler forms.

Originator: American mathematician William Hodge, 1903-1975, in 1950.

Discipline: Algebraic geometry.

Incentive: $US1million, one of the seven Millennium Prize Problems of the USA-based Clay Mathematics Institute.

Interesting aspects: The conjecture uses visualisation to investigate mathematical results and associated functions,
which are studied as discrete objects.

Progress: The strongest evidence in favor of the Hodge conjecture is the 1995 algebraicity result of Cattani,
Deligne and Kaplan. While mathematicians agree that the conjecture is important, they have not been able to find a
resolution or even agree on the best way to do this.


Published in IMAges 9 - October 2010