I will discuss new positivity bounds for scattering amplitudes in theories with a massless graviton in the spectrum in four spacetime dimensions. The bounds imply that extremal black holes are self-repulsive, M/|Q|<1 in suitable units, and that they are unstable to decay to smaller extremal black holes, hence providing an S-matrix proof of the weak gravity conjecture. I will also present applications of these bounds to modified gravity theories, such as weakly broken galileons, showing that they have a very low cutoff, much smaller than their strong coupling scale.
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