As a physical theory quantum mechanics explains the physics at the atomic scale. It developed into a well defined mathematical theory a large part of which consists of various problems of analysis and geometry related to the Schoedinger operator. Quantum field theory was developed as a physical theory describing the interaction of elementary particle. Its mathematical complexity surpasses quantum mechanics by a magnitude.
On the other hand the existence of a structure, of the framework of quantum field theory is undisputable. Constructing objects with such structure can be a problem, and it is a serious problem most theories which are most interesting for physics, such as the Yang-Mills theory. But there are models where the framework of quantum field theory can actually be implemented as a mathematical theory. Among them is the Chern-Simons theory, conformal field theories, and integrable quantum field theories.
The goal of this talk is to explain the framework of quantum field theory, outline the progress and challenges in this direction.