Multi-gap topological phases: geometrical properties and realizations.*

With the recent advent of powerful classification schemes a significant step has been made towards characterizing and classifying topological band theory. Within this talk I will address a new direction, entailing phases that go beyond these wisdoms, thriving on the concept of multi-gap topology. In particular, such phases conspire into new types of invariants and a rich novel interplay with geometry [1-3]. I will highlight some of these new aspects that we have obtained this year and point out a variety of experimental settings, including phonon spectra, meta-materials, cold atoms and electronic band systems, where these concepts culminate in readily observable experimental signatures [4,5].