ИСТИНА

The field of quantum invariants of knots, which is an example of a very interesting modern crossdisciplinary mathematical theory, was motivated purely by topological problems, but stimulated significant progress in algebra and the theory of integrable systems. In addition to the theory of quantum groups based on the Yang-Baxter equation, the problem of quantum invariants has higher analogues associated with the higher n-simplex equations. In a broader sense, the theory of quantum invariants can be considered as a program for the algebraization of topological invariants. Along this path, a large number of interesting algebraic structures have already emerged, finding their place in algebra, topology and the theory of integrable systems. Such structures include: quandles, racks, braces, rack bialgebras, infinitesimal bialgebras. In this overview report, I will talk about generalizations of the problem of quantum invariants to higher dimensions, as well as to the listed alternative algebraic structures. Some of the results presented were obtained jointly with V.G. Bardakov.