A standard tenet of condensed matter physics is that when a crystal with a partially filled energy band is forced to insulate, interesting physics must arise. As band insulators appear only when the filling – the number of electrons per unit cell and spin projection – is an integer, at fractional filling an insulating phase that preserves all symmetries is a Mott insulator, i.e. it is either gapless or, if gapped, displays fractionalized excitations and topological order. Remarkably, the little-studied inverse question – whether a 'trivial' band insulator is always possible at integer filling -- has a rich answer involving basic ideas of crystallography. In my talk, I will show that lattice symmetries may forbid a band insulator even at certain integer ﬁllings, if the crystal is *non-symmorphic* – a property of the majority of three-dimensional crystal structures. In these cases, one may infer the existence of topological order and the presence of fractionalized excitations, if the ground state is gapped and fully symmetric -- in other words, these are Mott insulators, but with fully filled bands! This is demonstrated using a non-perturbative ﬂux threading argument, and has immediate applications to quantum spin systems and bosonic insulators in addition to electronic band structures in the absence of spin-orbit interactions . Along the way, we are led naturally to systematic methods for constructing symmetric gapped ground states for interacting boson and spin models on generic symmorphic lattices [2, 3]. This work also has implications for the understanding of topological crystalline insulators, which I will comment on briefly.
 S.A. Parameswaran, A. M. Turner, D.P. Arovas & A. Vishwanath, Nature Physics 9, 299 (2013)
 I. Kimchi, S.A. Parameswaran, A. M. Turner, F. Wang & A. Vishwanath, Proc. Nat. Acad. Sci 110, 16378 (2013).
 S.A. Parameswaran, I. Kimchi, A. M. Turner, D. M. Stamper-Kurn & A. Vishwanath, Phys. Rev. Lett. 110, 125301 (2013).