From critical phenomena to quantum gravity, conformal field theories describe the universal scale-invariant structures that lie at the heart of theoretical physics. The conformal bootstrap is the powerful idea, dating back to the 70’s, that one can use fundamental consistency conditions to constrain, solve, and map out the space of conformal field theories. In this talk I will explain how one can use the conformal bootstrap in a variety of dimensions to perform rigorous and precise calculations in strongly-interacting theories without reference to a microscopic Lagrangian, summarizing recent progress at learning about the 3D Ising and O(N) vector models, which are the universal structures describing phase transitions in fluids, magnets, superconductors, and numerous other systems in nature.
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