 He introduced the notion of a mixed Hodge structure, developed techniques for working with them, gave their construction (based on Heisuke Hironaka's resolution of singularities) and related them to the weights on l-adic cohomology, proving the last part of the Weil conjectures.To motivate the definition, consider the case of a reducible complex algebraic curve "X" consisting of two nonsingular components "X"1 and "X"2, which transversally intersect at the points "Q"1 and "Q"2.Further, assume that the components are not compact, but can be compactified by adding the points "P"1, ..., "Pn".