The realizations of Bose-Einstein condensation and Fermi degeneracy in dilute atomic gases over twenty years ago marked the start of a new era for creating designer many-body quantum systems. Quantum matter based on dilute gases of atoms and molecules offer a number of unique features compared to solid-state systems - we can write down an exact Hamiltonian to describe them, we can utilize many internal states with unique and useful properties, and we can manipulate them on fine and fast scales with lasers and other laboratory fields. Based on these latter two points, the kinds of systems open to study with cold atoms and molecules is truly expansive. In the context of trying to study new aspects of topological physics, I'll describe how new tools for Hamiltonian design can be achieved when one studies transport not in real-space lattices, but in so-called "synthetic lattices" based on alternative degrees of freedom (e.g., internal or momentum states). I'll discuss some capabilities that have been opened up so far in neutral atoms based on this technique, as well as planned experiments based on Rydberg atoms. Finally, time permitting, I'll describe a totally different type of "synthetic lattice" that we're building in my group for the study of topological mechanics, based on networks of mechanical oscillators with "synthetic" couplings.
