One of the most classical subjects in mathematics is the study of algebraic varieties, which are subsets of the projective space cut out by polynomial equations. If the variety is smooth and hence topologically a manifold, its cohomology acquires additional structure such as the Hodge decomposition, which puts heavy restrictions on what kind of topological spaces can be made into an algebraic variety. It was discovered in the 1970s by Deligne that even if the variety is singular and hence not a manifold, its algebraic structure leaves a shadow in its topology in the form of the so-called weight filtration on cohomology. I will survey this topic and mention recent joint work with Peter Haine in which we show these filtrations exist at a "spectral" level, in the realm of higher algebra.