The central topic of the thesis is a recently proposed form of matter called altermagnetism. In common with simple antiferromagnets these are magnetic materials supporting long-range magnetic order with no net moment. In simple antiferromagnets the up and down spin electronic bands are degenerate. But in altermagnets they are not. One way of thinking about these materials is that they are nonmagnetic in real space but magnetic in momentum space thus combining features of ferromagnets and antiferromagnets. These materials have generated a great deal of interest in the spintronics community. Roughly speaking this community has, for a long time, been interested in antiferromagnets that support spin currents because antiferromagnets are insensitive to stray fields and can support faster device switching than in typical ferromagnets. Altermagnets have the potential to realize the dreams of antiferromagnetic spintronics. At the same time, altermagnets are of fundamental interest in condensed matter physics. It turns out that altermagnetism is grounded in a peculiar type of symmetry breaking described by the theory of spin groups.

The goal of this thesis project is to extend our understanding of spin groups in condensed matter especially in the direction of altermagnetism and topological materials.

The last few decades of condensed matter research have seen the emergence of a rich new physics, based on the notion of "spin liquids". Interest in these new states of matter stems from the fact that they exhibit large-scale quantum entanglement, a property that is fundamental to quantum computation. By directly exploiting this notion of entanglement, a quantum computer would enable revolutionary approaches to certain classes of problems, compared with conventional computers.

The study of spin liquids is therefore a key technological issue, and the aim of this thesis project is to contribute to this fundamental research effort.