Topological soft matter: from metamaterials to polymers

The classification of physical systems using topology has ushered in a new paradigm for condensed matter physics. Although topologically nontrivial states, with their accompanied features of exotic edge behaviour and robustness to disorder, were first predicted and observed in electronic systems, they have since been realized in a variety of classical systems. Soft matter provides a particularly rich playground for topological physics across diverse length scales. At the macroscopic scale, topology can be used to control tangible features such as vibrational response and mechanical rigidity, bringing elegant mathematical concepts literally to our fingertips. At microscopic scales, the messiness inherent to thermally fluctuating matter provides challenges as well as opportunities for realizing topologically nontrivial behaviour. Across these varied scales, topology holds promise as a tool to create artificial materials with desirable features that are robust to disorder and imperfections. I will demonstrate the breadth and richness of topological soft matter by describing two examples from either end of the spectrum. At the macroscale, I'll survey a class of mechanical metamaterials in which rigidity can be controlled both locally and globally using topology. At the microscale, I'll how strong interactions and thermal fluctuations can be harnessed to generate robust equilibrium patterns in a system of fluctuating lines, with potential applications in polymer patterning and manipulation of magnetic fields in superconductors.