In this talk I use applied mathematics to understand emergent multiscale phenomena arising in two fundamental problems in fluids and biology.

In the first part, I discuss an overarching question in developmental biology: how is it that cells are able to decode spatio-temporally varying signals into functionally robust patterns in the presence of confounding effects caused by unpredictable or heterogeneous environments? This is linked to the general idea first explored by Alan Turing in the 1950s. I present a general theory of pattern formation in the presence of spatio-temporal input variations, and use multiscale mathematics to show how biological systems can generate non-standard dynamic robustness for ‘free’ over physiologically relevant timescales. This work also has applications in pattern formation more generally.

In the second part, I investigate how the rapid motion of 3D microswimmers affects their emergent trajectories in shear flow. This is an active version of the classic fluid mechanics result of Jeffery’s orbits for inert spheroids, first explored by George Jeffery in the 1920s. I show that the rapid short-scale motion exhibited by many microswimmers can have a significant effect on longer-scale trajectories, despite the common neglect of this motion in some mathematical models, and how to systematically incorporate this effect into modified versions of Jeffery’s original equations.