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Common features of known quantum gravity theories may hint at the generalnature of quantum gravity. The absence of continuous global symmetries is onesuch feature. This inspired the Weak Gravity Conjecture, which bounds masses ofcharged particles. We propose the Lattice Weak Gravity Conjecture, whichfurther requires the existence of an infinite tower of particles of allpossible charges under both abelian and nonabelian gauge groups and directlyimplies a cutoff for quantum field theory. It holds in a wide variety of stringtheory examples and has testable consequences for the real world and for puremathematics. We sketch some implications of these ideas for models ofinflation, for the QCD axion (and LIGO), for conformal field theory, and foralgebraic geometry.

Axion Experiments to Algebraic Geometry: Testing Quantum Gravity via the Weak Gravity Conjecture