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Explore a groundbreaking approach to understanding S-duality's physical implications in four-dimensional N=4 super Yang-Mills (SYM) theory in this hour-long lecture. Delve into the theory of harmonic analysis on the fundamental domain of SL(2,Z) and discover how this mathematical tool imposes strong constraints on N=4 SYM observables' analytic structure. Examine a specific set of integrated correlators in detail, observing their dramatic simplification when expressed in the SL(2,Z)-invariant eigenbasis. Investigate the statistics of CFT data in N=4 SYM theory ensembles and their implications for holography at large N. Uncover the equivalence between observables in the strongly coupled planar theory, dual to type IIB supergravity on AdS5 x S5, and their ensemble average over the N=4 SYM conformal manifold.
