Explore new avenues for the circle method in this lecture by Jayce Getz. Delve into an adelic version of the delta-method developed by Duke, Friedlander, Iwaniec, and Heath-Brown, and examine a proposed nonabelian analogue that merits further investigation. Discover Poisson summation formulae and Fourier transforms for special families of varieties, including the zero locus of a quadratic form. Learn how these techniques may allow for broad generalizations of standard analytic number theory methods that rely on Fourier theory in vector spaces. This 59-minute lecture, presented at the Hausdorff Center for Mathematics, is motivated by research arising from automorphic representation theory and aims to open up new research possibilities in the field.