Multipoint conformal integrals

The key input data required for the application of conformal bootstrap methods are conformal partial waves, which are represented through conformal integrals within the shadow formalism. However, the lack of analytical expressions for multipoint conformal integrals significantly complicates the further investigation of conformal correlation functions. In this talk, I will discuss an approach known as the reconstruction conjecture, which aims to provide such expressions. This approach is based on the expansion of conformal integrals over a set of basis functions, the construction of which relies on the combination of the Mellin-Barnes representation, formulas for the analytic continuation of multivariate hypergeometric functions, and the analysis of symmetries associated with the action of the permutation group.