Noncommutative $n$-simplex maps and integrable systems

In this talk I will give an introduction to the theory of $n$-simplex maps, which are set-theoretical solutions to the n-simplex equation. The $n$-simplex equations are a family of fundamental equations of mathematical physics which generalise the famous Yang-Baxter (2-simplex) equation.
I will demonstrate methods for constructing $n$-simplex maps for $n=2, 3, 4$, and will show their relation to matrix refactorisation problems. Then, I will explain the relation between Yang-Baxter (2-simplex) maps and discrete integrable systems, and I will show how to construct Yang-Baxter maps together with their associated discrete integrable systems.
Finally, I will present examples of Yang-Baxter (2-simplex) and Zamolodchikov tetrahedron (3-simplex) maps on noncommutative groups and rings, that are related to KdV, NLS and Boussinesq type equations.