Topological phenomena arise in a wide range of systems, with fascinating physical consequences. There is great interest in finding new ways to measure such consequences in ultracold atomic gas experiments. These experiments have significant advantages over the solid-state as ultracold atoms are controllable, tuneable and clean. They can also be used to investigate properties which are inaccessible in other quantum systems.

We explore some of the novel features of topological energy bands and topological solitons in ultracold gases. Topological energy bands have important geometrical properties described by the Berry curvature. Bands with nonzero Berry curvature arise in key areas of current research, such as optical lattices with more than one band; strong artificial magnetic fields and 2D spin-orbit coupling. Topological solitons are also relevant to cutting-edge experiments as they can be created and studied with high temporal and spatial resolution.

In this thesis, we investigate the consequences of Berry curvature for the semiclassical dynamics of a wavepacket and the collective modes of an ultracold gas. We also study theoretically the dynamics of skyrmion-antiskyrmion pairs in a Bose Einstein condensate.

Firstly, we propose a general method by which experiments can map the Berry curvature across the Brillouin zone, and thereby determine the topological properties of the energy bands of optical lattices. The Berry curvature modifies the semiclassical dynamics and hence the trajectory of a wavepacket undergoing Bloch oscillations. Our general protocol allows a clean measurement from the semiclassical dynamics of the Berry curvature over the Brillouin zone. We discuss how this protocol may be implemented and explore the semiclassical dynamics for three relevant systems. We discuss general experimental considerations for observing Berry curvature effects before reviewing some of the progress in the field since the publication of our work.

Secondly, we show that the Berry curvature changes the hydrodynamic equations of motion for a trapped Bose-Einstein condensate, and causes significant modifications to the collective mode frequencies. We illustrate our results for the case of two-dimensional Rashba spin-orbit coupling in a Zeeman field, where we also apply both a sum rule and an operator approach to the dipole mode. Extending the operator method, we derive the effects of Berry curvature on the dipole mode in very general settings. We show that the sizes of these effects can be large and readily detected in experiment. Collective modes therefore provide a sensitive way to measure geometrical properties of topological energy bands.

Lastly, we study theoretically the dynamics of two-component Bose-Einstein condensates in two dimensions, which admit topological excitations related to the skyrmions of nuclear physics. We explore a branch of uniformly propagating solitary waves, which, at high momentum, can be viewed as skyrmion-antiskyrmion pairs. We study these solitary waves for a range of interaction regimes and show that, for experimentally relevant cases, there is a transition to spatially extended spin-wave states at low momentum. We explain how this can be understood by analogy to the two-dimensional ferromagnet and discuss how such solitary waves can be generated and studied in experiment.