In response to [2], the classification of topological insulators relies on moduli spaces, see for example this tutorial by Daniel Freed. (For a more formal description, see here.)

In response to [2], the classification of topological insulators (or more generally, of any gapped condensed matter, so also superconductors) relies on moduli spaces, see for example this tutorial by Daniel Freed. (For a more formal description, see here.)