Workshop on Topology: Identifying Order in Complex Systems

Critical mechanical structures are structures at the verge of mechanical instability. These structures are characterized by their floppy modes, which are deformations costing little energy. On the one hand, numerous interesting phenomena in soft matter are governed by the physics of critical mechanical structures, because they capture the critical state between solid and liquid. On the other hand, the design of mechanical metamaterials (i.e., engineered materials that gain their unusual mechanical properties, such as negative Poisson's ratio, from their structures) often rely on floppy modes to realize novel properties, and the floppy modes in this situation are called "mechanisms". This talk focuses on the intersection between the research of soft matter and mechanical metamaterials. I will propose a new design principle, for mechanical metamaterials that are transformable between states with dramatically different properties. These different properties are highly robust because they are topologically protected. Then I will discuss entropic effects on floppy modes (mechanisms), the interplay of which with topology leads to fascinating phenomena that need be considered when machines and metamaterials are made at small scales.