Moduli spaces of curves on hypersurfaces via the circle method

In this talk, I will discuss analogies between rational points and rational curves on varieties and how tools from analytic number theory can be used to study the geometry of moduli spaces of curves on hypersurfaces. In particular, I will explain how the circle method can shed light on geometric properties such as dimension, irreducibility and singularities of these moduli spaces. This is based on joint work with Matthew Hase-Liu.