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I will revisit the spectrum of single-trace operators in N=4 SYM, and describe how to classify the irreducible representations of the superconformal symmetry that they form, in terms of a non-compact generalisation of Young diagrams. These multiplets are in one-to-one correspondence to “rational” solutions to a Q-system. Traditionally, this Q-system is defined by the algebra in question, but I will argue that it lives a more natural, algebra-independent life on the corresponding Young diagram. Using this intuition, I will present a simple algorithm to solve such systems, providing a cleaner and more powerful alternative to Bethe equations. These solutions form the starting point for perturbative calculations of anomalous dimensions via the Quantum Spectral Curve. I will sketch a conceptually very simple method to do such calculations for general states, discuss patterns in the results, and speculate about how to get more info out of the QSC.
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