Sobolev spaces on metric measure spaces have been extensively studied since the foundational works of Cheeger and Shanmugalingam in the 1990s. However, their framework does not yield a suitable notion of Sobolev space for fractals such as the Sierpinski carpet. We develop an alternate approach to Sobolev spaces on the Sierpinski carpet, which originated in the study of diffusion on fractals. We also highlight the relevance of Sobolev spaces and the corresponding energy measures to the attainment problem for conformal dimension. The attainment problem for conformal dimension is closely related to major conjectures in geometric group theory such as Cannon’s conjecture and the Kleiner-Kapovich conjecture. This is joint work with Ryosuke Shimizu.