I will explain how work of Gupta, Kamath, Kayal and Saptharishi on shallow circuits leads to exciting questions in algebraic geometry. While their work was focused on separating complexity classes via shallow circuits, I will address the more modest goal of using the methods they introduce to improve the state of the art in separating the determinant from the permanent. This leads to the study of several longstanding conjectures in representation theory (Hadamard-Foulkes-Howe conjecture) combinatorics (Alon-Tarsi) conjecture, and commutative algebra. This is joint work with several co-authors, I will focus on the most recent work with Hal Schenck regarding Hilbert functions of the ideal generated by sub-permanents.