Hodge conjecture

In mathematics, the Hodge conjecture is a major unsolved problem in the field of algebraic geometry that relates the algebraic topology of a non-singular complex algebraic variety and the subvarieties of that variety. More specifically, the conjecture says that certain de Rham cohomology classes are algebraic, that is, they are sums of Poincaré duals of the homology classes of subvarieties.

Quotes

 * The twin conjectures of Hodge and Tate have a status in algebraic and arithmetic geometry similar to that of the Riemann hypothesis in analytic number theory.


 * The Hodge conjecture postulates a deep and powerful connection between three of the pillars of modern mathematics: algebra, topology, and analysis. Take any variety. To understand its shape (topology, leading to cohomology classes) pick out special instances of these (analysis, leading to Hodge classes by way of differential equations). These special types of cohomology class can be realised using subvarieties (algebra: throw in some extra equations and look at algebraic cycles). That is, to solve the topology problem 'what shape is this thing?' for a variety, turn the question into analysis and then solve that using algebra. Why is that important? The Hodge conjecture is a proposal to add two new tools to the algebraic geometer's toolbox: topological invariants and Laplace's equation. It's not really a conjecture about a mathematical theorem; it's a conjecture about new kinds of tools.