Modular forms, which can be seen as functions on $\mathrm{SL}_2(\mathbb{R})/\mathrm{SO}_2(\mathbb{R})$ with certain discrete symmetries, are connected to many areas of both mathematics and physics, and their Fourier coefficients have long been studied for their ability to count various arithmetic objects. After briefly giving an example of this phenomenon I will discuss automorphic forms, which can be seen as their generalizations to groups of any rank. I will give an overview of how to Fourier expand periodic functions on real Lie groups, and of different kinds of Fourier coefficients together with what is known about them.

Much is known about Fourier coefficients with respect to maximal unipotent subgroups, but less so for other unipotent subgroups. I will summarize recent work on how to express the latter in terms of the former and when this is possible (e.g. for small automorphic representations of simply laced groups) which is joint work with Gourevitch, Kleinschmidt, Persson and Sahi. I will also briefly describe how this work has applications to string theory where the above Fourier coefficients are interpreted as contributions to graviton scattering amplitudes from non-perturbative effects such as instantons and black holes.