Just like the punctual Hilbert scheme of a smooth surface parametrizes ideals supported at a given point, the flag Hilbert scheme parametrizes such ideals together with a full flag to the structure sheaf. It can be thought of as an lci counterpart of the variety of commuting upper triangular matrices, and as such is relevant for partially resolving the singularities of the punctual Hilbert scheme. We recently found applications of this variety to torus knot invariants and the representation theory of the Hilbert scheme, and a number of people are looking into using it to better understand categorical braid invariants. Joint work with Eugene Gorsky and Andrei Okounkov.