One of the goals of Hodge theory is to measure and describe what happens if you introduce extra ``parameters'' in the equations. In our example, one might want to vary the radius of the circle. A related question is to describe the subsets of a given variety which themselves are given by supplementary polynomial equations; these are called ``subvarieties, or more generally ``algebraic cycles''. The theory has been named after Sir William Hodge (1903-1975) who boosted research in this area after he formulated a conjecture. In fact it is one of the millenium prize questions posed by the Clay Foundation.
Hodge theory has links to other fields. It has for instance been discovered that Hodge theory is a natural tool to attack the so called ``mirror conjecture'', a conjecture posed by mathematical physicists. See for instance this book.