Multi-gap topological physics: geometrical notions, physical phases and novel responses.

In this talk I will review recent work on multi-gap topological states. These phases are characterized by topological structures that cannot be captured by advances in more conventional symmetry-based topological classifications schemes. Upon utilizing new insights into connections with general geometric identities I will elucidate the structure of these phases and highlight physical signatures in both equilibrium and out-of-equilibrium settings. As a highlight I will address how this understanding vice versa also relates to new takes on formulating quantum geometrical notions that can be probed by physical responses.