Braided algebras in representation theory

Methods originating in theoretical physics have proven to be extremely useful in mathematical research. In particular, mathematical apparatus developed in quantum theory led, in mid-1980s, to the discovery of quantum groups which were one of the most exciting developments in contemporary mathematics. It turns out that the use of quantum methods in pure mathematics should not be limited to quantum groups, and they may have new and unexpected applications in modern algebra, for instance in representation theory which aims to describe highly abstract algebraic notions in more elementary terms such as matrices consisting of numbers. The proposed research will devemop and explore the new applications of these quantum methods to solve important open problems in algebra, which will lead to progress in other areas of mathematics such as geometry.