Non-Abelian radon transform and its applications

Considerations of the non-Abelian Radon transform were started in [Man-akov, Zakharov, 1981] in the framework of the theory of solitons in dimension 2 +1. On the other hand, the problem of inversion of transforms of such a type arises in different tomographies, including emission tomographies, polarization tomographies, and vector field tomography. In this article, we give a short review of old and recent results on this subject. This article is an extended version of the talk given at the conference "100 Years of the Radon Transform, " Linz, 27-31 March 2017.