The Teichmüller space of a surface is a fundamental construction in low-dimensional geometry and topology. It is a space parametrizing hyperbolic metrics on the surface up to isotopy. The goal of quantum Teichmüller theory is to construct a quantum deformation of the algebra of functions on the classical Teichmüller space. 

One approach to the quantization of Teichmüller space uses the Kauffman bracket skein algebra. This is a noncommutative algebra generated by knots and links, modulo the Kauffman skein relations. In this talk, I will explain how the same noncommutative algebra appears in a different context in geometric representation theory. Namely, I will explain how skein algebras arise as quantized K-theoretic Coulomb branches in the sense of Braverman, Finkelberg, and Nakajima. This is joint work with Peng Shan.