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Friday, March 5, 2021

**Abstract:** Suppose one wants to decompose a function on into parts that have disjoint Fourier supports. Due to linearity of the Fourier transform, the sum of these parts must be the original function. Suppose, however, that you wish to understand how the norms of these parts relates to the norm of the original function. While this task isn’t so straight forward, Bourgain and Demeter (2015) were able to resolve a conjecture related to this: the $L^{2}$ Decoupling Conjecture.In this talk we will introduce the conjecture, provide applications of it to both PDEs and Number Theory, prove an earlier result of Bourgain for a smaller range of admissible exponents, and (time permitting) comment on how to adapt the ideas presented to prove the full result.