In mathematical models of opinion dynamics, individuals interact with each other and adjust their opinions based on their interactions. In opinion models, network structures determine which individuals can interact with each other and thereby affect how opinions change with time. In this talk, I will introduce opinion models and why scientists study them. I will also discuss several variants of bounded-confidence models (BCMs), in which opinions take continuous values in a region, and I will examine how network structure affects the formation of consensus, polarization, and fragmentation of populations.

Internal gravity waves (IGWs) are oscillatory modes of motion that exist in the interior of a stably stratified fluid with gravity as the restoring force. In Earth's oceans, these waves play a fundamental role in the transport and mixing of energy, momentum, heat, carbon and nutrients. Much like ocean surface waves, IGWs become unstable and break when their amplitude is large and density interfaces become steeply inclined. This talk will explore the evolution and fate of such waves in two different scenarios.

I will first focus on IGWs that propagate into horizontally sheared currents, explaining how the fate of these waves depends both on their geometry and the strength of the shear flow, and why the total energy dissipated by ensuing turbulence is unexpectedly large. Second, I will discuss the dynamics of internal bores – long, large-amplitude solitary waves that propagate along sharp density interfaces. I will outline the mathematical links between internal bores and gravity currents and, by studying observations off the coast of California, demonstrate their relevance to understanding transport and dissipation on the continental shelf.

Aperiodic (quasicrystalline) tilings, such as Penrose's tiling, can be built up from (for example) kites and darts, squares and triangles, rhombi or shield-shaped tiles and can have a variety of different symmetries. However, almost all quasicrystals occurring in soft matter are of the dodecagonal (12-fold rotation symmetry) type, and many can be described in terms of square and equilateral triangular tiles. Here, we explore what contributes to the thermodynamic stability of soft-matter quasicrystals, both in two dimensions and in three, and how the details of how soft-matter particles interact leads to different kinds of aperiodic tilings. Although dodecagonal quasicrystals are the most common, this work points to how more general (beyond dodecagonal) quasicrystals can be designed in soft matter.