Underdetermined differential operators arise naturally in diverse areas of physics and geometry, including the divergence-free condition for incompressible fluids, the linearized scalar curvature operator in Riemannian geometry, and the constraint equations in general relativity. The duals of underdetermined operators, which are overdetermined, also play a significant role. In this talk, I will present recent joint work with Philip Isett (Caltech), Yuchen Mao (UC Berkeley), and Zhongkai Tao (UC Berkeley) that introduces a novel approach to constructing integral solution/representation formulas (i.e., right-/left-inverses) for a broad class of under/overdetermined operators. They are optimally regularizing and have prescribed support properties (e.g., produce compactly supported solutions for compactly supported forcing terms). A key feature of our approach is a simple algebraic condition on the principal symbol that implies the applicability of our method. This condition simplifies and unifies various treatments of related problems in the literature. If time permits, I will discuss applications to the study of the flexibility of initial data sets in general relativity.