Higher Auslander algebras of type $\mathbb{A}$ and the higher Waldhausen $\operatorname{S}$-constructions

These notes are an expanded version of my talk at the ICRA 2018 in Prague, Czech Republic; they are based on joint work with Tobias Dyckerhoff and Tashi Walde. In them we relate Iyama's higher Auslander algebras of type $\mathbb{A}$ to Eilenberg--Mac Lane spaces in algebraic topology and to higher-dimensional versions of the Waldhausen $\operatorname{S}$-construction from algebraic $K$-theory.
Comment: 16 pages. The author's contribution to the Proceedings of the ICRA 2018, v.2 minor edits following referee report