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(1) MAGNETISM at the MICROSCOPIC SCALE: The microscopic basis of magnetic phenomena, hierarchies of effective Hamiltonians in strongly-correlated systems, including well-known models like the Anderson and Kondo Hamiltonians (and lattice versions of these), the Hubbard Hamiltonian, and refinements of these like the ZSA model. There will be some emphasis on the chemistry of interesting magnetic systems, including magnetic molecules, and on new 'quantum materials' showing magnetic properties. The physics of strongly-correlated systems requires both analytic and powerful numerical methods, and the results of some of these latter methods will be discussed in some detail, notably density functional, dynamical cluster, and dynamical mean field theories, and also Quantum Monte Carlo methods.

(2) EXOTIC ORDER in QUANTUM MAGNETS: The standard classical magnetic ordering theory fails to describe ordering in genuine quantum magnets. This is particularly clear in lower dimensions, where one can get many exotic kinds of ordering, both local and non-local. This even happens in 3 dimensions, with systems of particular interest being He-3 (solid and superfluid) and quantum spin glasses like the LiHoYF system. A key feature of many of the novel magnetic states is their non-trivial topological properties. New kinds of quantum liquids range from simple spin liquids to more exotic systems like the Quantum Hall ferromagnets or spin Bose-Einstein condensates- of key interest are spin and charge fractionalisation, and exotic quasiparticle statistics. Some of the most interesting states occur in 1-dimensional spin systems, which are also of great current interest in the context of quantum computation.

(3) DISORDERED MAGNETS: Many remarkable critical phenomena (including the clearest examples of quantum critical phenomena) occur in disordered magnetic systems. In recent years novel features of these have been discovered in the low-T quantum regime of these systems, including quantum spin glass phases, as well as novel ordering in systems having random fields and/or positional disorder. There are also very interesting connections to phenomena in other systems, such as dipolar glasses at very low T.

(4) QUANTUM NANOMAGNETISM: At very small scales, or near surfaces, magnetic systems can behave very differently from at macroscopic scales. Three are of particular interest, viz., (i) Magnetic molecules, which show a variety of quantum tunneling phenomena, and which have been the object of many studies in a search for coherent tunneling in the search for spin qubit systems; (ii) mesoscopic and nanoscopic conductors, which not only show interesting classical 'spintronic' phenomena, but also are predicted to show a variety of interesting quantum effects, including 'spin Hall' effects, which often depend on interesting topological properties of the underlying quantum states; and (iii) spins on metallic surfaces, which show fascinating many-body effects, and which can can be investigated directly using STM imaging techniques.

(5) LARGE-SCALE QUANTUM PHENOMENA IN MAGNETS: Tunneling of large-spin molecules and of magnetic solitons has been seen already at the nanoscopic and mesoscopic scale, and further more exotic phenomena of this kind are predicted. Perhaps the most dramatic part of this field is the search for large-scale entanglement, and the application of this to quantum computation. The key problem here, as in other kinds of quantum computation, is the understanding of both the mechanisms of decoherence, how to suppress decoherence, and how to understand its dynamics. Magnetic systems are offering a rather unique window on these very general questions. One exciting new possibility has appeared in the idea of topological quantum computation, where the computation is embodied in the topological properties of a spin wave-function, and is almost immune to decoherence.