Condensed matter seminar. Two-dimensional quantum Hall (QH) systems are seminal examples of topological phases of matter. As recognised by the 2016 Physics Nobel Prize, these systems exhibit a remarkably robust quantization of conductance due to the nontrivial topological invariants of the filled electronic energy bands. Interestingly, it has been shown that QH physics can be generalised mathematically to higher dimensions with the next new QH effect arising in four dimensions. While this was previously thought to be of purely theoretical interest, Dr. Hannah Price, Royal Society University Research Fellow and a Birmingham Fellow in the Theoretical Physics group, will present in this talk how recent advances are allowing us to explore this effect with ultracold atoms. After introducing the physics of the 4D QH effect, she will present the results of her recent experimental collaboration to realise a dynamical version of the 4D QH for the first time by implementing a 2D topological charge pump for ultracold atoms. She will then propose how current experiments on so-called “synthetic dimensions” could be extended so as to allow atoms to effectively explore topological lattice models with four or more spatial dimensions. This sets the stage for the exploration of many novel topological phenomena in higher dimensions.