There are two prominent sources of groups in mathematics: symmetries and topological spaces, and they are related through the theory of differential equations, or more generally, gauge theory. Namely, by solving a differential equation, a path in a topological space can be represented as a symmetry of a vector space. In this way, the concatenation structure of paths is reflected in the composition of symmetries. The universal example of this correspondence is the path signature, a construction, originally due to Chen, that assigns a non-commutative power series to a path in Euclidean space. Remarkably, the signature completely characterizes a path up to translation, reparametrization, and cancellation, and can be used to construct the solutions to a family of differential equations. In this talk, I will give an overview of thin homotopy and the path signature. If time permits, I will discuss ongoing work, joint with Camilo Arias Abad and Darrick Lee, which aims to extend the signature so that it can be used to encode two-dimensional surfaces.