The iTHEMS Math seminar was held on 24 Oct., inviting Ken Shiozaki from Kyoto university. The title of the talk was "Atiyah-Hirzebruch spectral sequence in band theory". The main subject of the talk was topological K-theory, a branch of topology in mathematics, from the viewpoint of physics. In the theory of condensed matter physics, it is well-understood that a classification of topological phases is given by topological (twisted equivariant) K-theory. Conversely, some physical intuitions help us understanding (or even calculating) K-theory. In the talk, the speaker illustrated "physical meanings" of basic mathematical notions such as the axioms of cohomology theory, the Mayer-Vietoris exact sequence and the Bott periodicity.