Unusual applications of analytic number theory to algebraic geometry

The circle method is a classical technique in analytic number theory, originally developed to count solutions to polynomial equations. In this talk, I will sketch how this method can be adapted to study moduli spaces of curves on hypersurfaces — objects that have traditionally been approached using purely geometric techniques. Part of this work is joint with Jakob Glas. No prior familiarity with either the circle method or moduli spaces is required.