This talk will begin by discussing phase transitions in high-dimensional statistical inference problems. Some effort will be made to distinguish between problems with random structure (e.g., random codes and sparse PCA) and problems with deterministic structure (e.g., highly symmetric codes such as Reed-Muller codes). For problems with deterministic structure, we will observe that symmetry can sometimes play a key role in characterizing their phase transitions. In particular, I will describe my recent work with Galen Reeves that proves Reed-Muller (RM) codes achieve capacity on binary memoryless symmetric (BMS) channels with respect to bit-error rate. This result resolves a long-standing open problem that connects information theory and error-correcting codes. Our approach generalizes some elements of an earlier proof for the binary erasure channel but also derives new tools to avoid previous steps that do not generalize. The new idea is to combine a nesting property of RM codes with new information inequalities relating the derivative of the conditional entropy (as a function of the channel parameter) with minimum mean-squared error estimation.