Traditionally, states of matter are classified by their symmetry. The breaking of a symmetry drives the system through a phase transition. For example, when translational symmetry is broken in a liquid it makes a phase transition to the crystalline state, and when magnetic moments lose their rotational symmetry a ferromagnetic state can appear. Half a century after the discovery of superconductivity it was found that also this phase transition is associated with a broken “gauge” symmetry. It appeared for a while that we had completed the classification of phase transitions, but we were wrong.

 We have now learned that Nature offers an altogether different type of phase transition, in which no symmetry is broken. The different states of matter are distinguished by topology rather than symmetry. Two states that are topologically distinct cannot be transformed into each other by smooth deformations.

 Because topology is intrinsically a mathematical concept, theoretical physics (the branch of physics having mathematics as its mother tongue) plays a central role in the discovery and development of this new territory. We identify three key questions that we plan to address in a concentrated effort:

- Which topologically distinct states of matter are produced by electron-electron interactions?

- How can the topological structure of matter, supporting Majorana fermions and other non-Abelian anyons, be used to protect a quantum computation from decoherence?

- How can the properties of soft materials be made stable and tunable by topological defects?