Our group is developing techniques to create and understand programmable matter in both real space and in so-called synthetic dimensions. One approach uses optical potentials in real space to program arbitrary lattice geometries for fermions, allowing experiments to perform “quantum simulations” of Hubbard models in regimes that have never before been accessible. Another, perhaps more surprising, approach uses synthetic dimensions, systems with degrees of freedom that behave in a way that can mimic motion in space, using for example Rydberg levels or molecules' rotational levels. I will describe a collaboration with experimentalists that used ultracold Rydberg atoms that created and observed topological edge states in these systems. I will show how interactions, being studied in ongoing experiments, can lead to novel phenomena such as quantum strings and membranes, and to parastatistical quasiparticles, a type of particle beyond fermions and bosons, which, unlike anyons, can persist to 3D.