The fruitful interplay between mathematics and physics has a long history going back to the very beginnings of both subjects. In recent times quantum field theory has been at the forefront of novel directions in this interplay. In particular, work by Witten, Segal, Atiyah and many others beginning in the 1980s and inspired by quantum field theory has lead to new insights into low dimensional topology, knot theory and their relations to other areas such as category theory, quantum groups and operator algebras. While this development, also known as topological quantum field theory, has lead to a whole new branch of algebraic topology, its impact back on physics has been more limited. While it plays an important role in two-dimensional conformal field theory, its potential for elucidating the mathematical foundations of the type of quantum field theories at the basis of our modern understanding of nature is largely unexplored. This is a main aspect of research in quantum field theory at the CCM.
It is well known that quantum field theory in its present form is incompatible with key principles of general relativity. This may be traced back to the prominent role of a non-relativistic conception of spacetime build into the formalism of quantum theory at its very inception. One of the aims of research at the CCM is to seek a formulation of the foundations of quantum theory that does away with any reference to an external classical notion of time, while still embracing the overwhelming empirical success of quantum field theory in fundamental physics as we know it.