Dynamical constraints on RG flows and cosmology

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In QFT, the renormalization group is usually formulated in Euclidean signature. I will discuss time-dependent probes of the RG, in Lorentzian signature, and derive new dynamical constraints that govern the spread of local operators. Through a chain of Wick rotations and dualities, the same methods lead to new sum rules for inflationary correlators, which relate observable quantities like the inflationary speed of sound to properties of the UV.