What is topological quantum matter? Today I saw the answer to this question in a video by TED-Ed on YouTube. But I couldn't make any sense of it (for I am still in high-school). Could someone provide me the answer in the simplest way a high-school-er may understand (please don't tell me that this particular topic is beyond my scope, I am curious!)?
A "topological invariant" is something that you can calculate about an object that stays the same if you morph or distort that object. For example, the number of holes an object has is a topological invariant, because if you morph a donut, it will still have one hole unless you do something drastic like ripping it apart. Objects are "topologically trivial" in this case if they have zero holes.
In matter, how electrons travel and occupy the solid is complicated, but they often have their own topological invariants. If the topological invariant is nontrivial, then we have examples like topological insulators, topological superconductors, Weyl semimetals, etc., which have unique properties. I'm guessing these are what the video you mention is referring to by "topological quantum matter". For example, "topological insulators" are insulators which have a different topological invariant from trivial insulators, and as a result they can be conducting on the surface.
This isn't a complete answer, but explaining what it means for a material to have a topologically nontrivial electronic structure would take a lot more explanation.