## Oliver Club

Decoupling theory is a modern branch of Fourier analysis which has had surprising applications to problems from number theory and PDE. In Fourier analysis, we decompose functions as linear combinations of waves which oscillate at many frequencies. In decoupling theory, we restrict our attention to functions \(f\) whose Fourier expansion only has frequencies which lie in special sets like the parabola or the cone. These functions arise naturally as solutions to the Schrödinger and wave equations, respectively. By measuring the size (using even exponent \(L^p\) norms) of functions with special Fourier support, we obtain additive combinatorial estimates like those that appear in the Vinogradov mean-value theorem. Using the example of the parabola, the idea of decoupling is to break the Fourier support (frequencies) of \(f\) into small pieces which are localized to almost linear arcs of the parabola. A decoupling inequality measures the size of \(f\) in terms of the simpler-to-understand component pieces coming from each of these arcs. I will present some known, in-progress, and open questions related to decoupling theory and give some basic idea of the tools and intuitions we use to approach these problems.