My research lies at the junction of mathematics, physics, engineering, and computation.  In my thesis and early postdoctoral work, my research focussed on adaptive mesh discretisations: changing the computational mesh in some way to make orders of magnitude efficiency gains in solving complex models. In this work, I solved a problem of computational geometry that had been open in the literature for over twenty years. In my later postdoctoral work, I have focussed on the problem of automatically deriving adjoint models. Adjoints are absolutely essential for many important problems, such as weather prediction, optimising the shape of wings, and quantifying the accuracy of nuclear simulations. I have developed an entirely new approach to deriving adjoint models, which yields dramatic gains in automation, robustness and efficiency -- in some cases, from years to days. With these methods in hand, it is now possible to contemplate the automated solution of optimisation problems constrained by the laws of physics. Such applications are of huge interest and importance across all of engineering and the quantitative sciences.