Conformal field theories (CFT) are ubiquitous in physics: they describe phase transitions of magnets and superfluids that can be studied in the lab, as well as provide the only known complete description of quantum gravity via the AdS/CFT conjecture. These CFTs are often strongly coupled, however, so for precise calculations we need exact methods such as the conformal bootstrap, supersymmetric localization, and harmonic analysis. I will explain how these new methods have been used to resolve long standing experimental questions about the superfluid Helium phase transition, provide the first rigorous derivation of a specific model of AdS/CFT, and in general probe strongly coupled systems across theoretical physics.
