On a noncommutative cross-ratio

the cross-ratio of four points on a projective line is one of the main projective invariants that finds the most unexpected applications in modern mathematics, from geometry and topology to the theory of integrable systems. I will talk about how we can extend the definition of the cross-ratio to the case when the projective line is replaced by its “noncommutative analog”, that is, instead of projecting pairs of real or complex numbers, we consider "projectivization of a noncommutative algebra". It turns out that there is a way to do this so that most of the useful properties are preserved. Exploring the applications of the resulting expression is an interesting open problem. The report is based on joint work with V. Retakh and V. Rubtsov https://arxiv.org/abs/1905.01366.