This proposal aims at setting up an international network of theoretical physicists, developing the methods of quantum integrability and conformal field theory and applying them to the emerging fields of low-dimensional quantum physics. These theoretical studies present, at once, fascinating convergences between exactly solvable statistical models and field theories, such as the topological theories of non-Abelian anyons, and recent developments of modern mathematics, such as infinite dimensional algebras, topology and combinatorics. These topics find timely applications to present and forthcoming experimental results concerning quantum spin chains, strongly-correlated electron systems, cold atoms in optical lattices and interferometry. This proposal also deals with the fast-developing topic of quantum topological computation, which exploits the topological properties of many-body quantum systems to encode and manipulate quantum information in a manner that is unaffected by decoherence. Another, related subject is the off-equilibrium dynamics of extended quantum systems, their thermalization properties and ergodicity after being subjected to a quantum quench. This research aims at opening new perspectives in the attractive area of exactly solved models by applying its sophisticated theoretical tools to the growing experimental area of low-dimensional quantum systems.