The cohomology groups of a complex algebraic manifold enjoy an extremely rich structure, traditionally the subject of Hodge theory. A key piece of this structure stems from Grothendieck's purely algebraic description of the de Rham cohomology of smooth varieties. But what about singular varieties?

I will begin by reviewing the story for smooth varieties, and then explain how a modern perspective provides an answer to the preceding question. More precisely, we will see why derived algebraic geometry — specifically, derived de Rham cohomology — resolves a conjecture of Grothendieck concerning a purely algebraic description of the cohomology groups of an arbitrarily singular complex algebraic variety.