Jeongwan Haah, a physicist at Stanford University, presents this Institute for Advanced Study seminar challenging a widely held belief in topological order. The standard view holds that gapped local Hamiltonians built from commuting terms in 2+1 dimensions cannot realize a chiral anyon theory, a claim backed by a formula expressing the chiral central charge as an integral of commutators of the Hamiltonian's local terms. Haah works through the construction that breaks this belief: a locally generated, locally finite dimensional operator algebra in two dimensions, paired with a local commuting Hamiltonian, that realizes the three-fermion anyon theory despite the supposed prohibition. He then shows the boundary of this three-fermion theory can be gapped out, though not through the usual boson condensation mechanism, instead relying on a different choice of local operator algebra. The talk is aimed at researchers in condensed matter and mathematical physics already fluent in anyon theory, tensor categories, and lattice Hamiltonian methods, delivered at IAS's Bloomberg Lecture Hall as part of the Joint SNS/SOM High Energy Theory Seminar series.