Hodge theory is both beautiful and extremely powerful; it was used in ingenious ways by another prominent mathematician, the Fields medalist Maryam Mirzakhani, to study the deformations of flat surfaces. These smooth surfaces, not obviously connected to algebraic geometry, were found woven into a tapestry of rich algebraic structures by work of Eskin, Mirzakhani and Wright.

The Hodge conjecture has a similar philosophy: you define certain data in terms of "harmonic analysis" on the geometric object, and the conjecture says that they all correspond to the algebraic data out of which the object can be rebuilt.