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Integrability in Gauge/String Dualities. Beyond the Spectrum

I will survey how some seemingly new and exotic objects in integrable models can actually be put to use in a rather sharp way to tame Gauge/String theories at finite coupling. In particular, we will discuss how a particular class of so-called pentagon and hexagon twist operators provide one with a window into structure constants and scattering amplitudes in planar N=4 SYM at any loop order.