The Hodge Conjecture, one of the Millennium Prize Problems, is one of the most profound unsolved mysteries in maths. It aims to bridge the two major mathematical fields of topology, the study of shapes and data structures, and algebraic geometry, the study of shapes defined by polynomial equations.
Due to its complexity, the Clay Mathematics Institute named the conjecture one of the seven Millennium Prize Problems in 2000. Proving or disproving it therefore carries a $1 million reward. The conjecture remains unproved and it is often considered the most abstract and difficult to visualise of the seven sister problems.
In the 20th century, mathematicians developed mathematical tools to study shapes in higher dimensions that cannot be seen directly. Topology as a field measures a space by looking at its ‘holes’ and ‘shadows’. While highly flexible, this can often introduce abstract ‘Hodge classes’ that lack any clear geometric meaning. Algebraic geometry builds shapes rigidly from the ground up using ‘algebraic cycles’ – sub-shapes defined by polynomial formulae.
In 1941, Scottish mathematician William Vallance Douglas Hodge proposed that for a particular class of ‘well-behaved’ shapes, any abstract topological piece exhibiting a specific internal symmetry