Why number-theorists need physics

Applications of physics to geometry have deep historical roots going back at least to Archimedes, while recent decades have seen structures of classical and quantum gauge theory lead to enormous advances in the theory of three and four manifolds. Meanwhile, geometry provides conceptual tools for physics as foundational  as variational principles, the kinematics of spacetime, and the topological  classification of matter. This talk will describe some possibilities for bringing number theory, more specifically, *arithmetic geometry* into this interaction. After giving some examples of the suggestive interplay of  number theory and physics, I will speculate somewhat on how ideas of quantum field theory may elucidate some of the deepest and hardest problems of number theory, especially the effective Mordell conjecture and the Hasse-Weil conjecture.