Topological materials can exhibit robust properties that are protected

  against disorder even in the absence of quantum effects. In

  mechanics and 

  optics, topological protection has been primarily

  applied to linear waves and 

  non-interacting systems at zero temperature. In this talk, we demonstrate

  how to construct topologically protected states that arise from the

  combination of strong interactions and thermal fluctuations inherent to soft

  matter. Specifically, we consider fluctuating lines under tension, subject to a class of

  spatially modulated substrate potentials. At equilibrium, the lines acquire

  a collective tilt proportional to an integer topological invariant called the

  Chern number. This quantized tilt is robust against substrate disorder,

  as verified by classical Langevin dynamics simulations. We establish the

  topological underpinning of this pattern via a mapping that we develop 

  between the line fluid and Thouless pumping of an imaginary-time Mott insulator in which

  excitations are gapped by interactions. Our work points to a new class of classical 

  topological phenomena in which the topological signature manifests itself in a

  structural property rather than a transport measurement.