Some functional aspects of the conformal bootstrap

The conformal bootstrap is a powerful tool in the study of CFTs.  It uses crossing-symmetry of the Operator Product Expansion (OPE) in a 4-point function. In numerical approaches of bootstrap one obtains strong bounds in the operator spectrum (conformal dimensions). The CFTs that saturate these bounds are very interesting but hard to locate. In a modern formulation of bootstrap, the Analytic functional approach, such solutions become manifest giving a far better analytic handle. It relies on the analytic and Regge boundedness properties of CFT correlators, and has an interesting connection to AdS Witten diagrams ('Polyakov blocks') and CFT dispersion relations.  I will give a general overview of these ideas and discuss some of its generalizations. I will also discuss some exciting new analytical and numerical results.