Soft matter is a broad class of materials with many examples found in everyday life: foods, crude oil, many biological materials, granular materials, liquid crystals, plastics. All of these are unified by the property that they're readily deformable because the elastic energy is of the same order of magnitude as the ambient thermal energy. Moreover, they spontaneously assemble into richly ordered structures that respond to many different kinds of external stimuli. Soft materials are therefore ideal candidates for advanced engineering applications including soft, biomimetic robots, self-building machines, shape-shifters, artificial muscles, new high-performance all-optical switches and chemical delivery packages. In each of these, the material must make a dramatic change in shape with an accompanying re-ordering of the material. To optimize the materials and structures, it is necessary to have a detailed understanding of how the microstructure and macroscopic shape co-evolve.

In this talk, I will therefore discuss the interactions between order and shape, as well as the role of the dynamics in determining the final state, with examples primarily drawn from my group's work on liquid crystals and emulsions. To develop the description, we draw upon differential geometry, topology, optimization theory and computer simulations, revealing beautiful and profound connections between mathematics and superficially mundane things in the world immediately around us.