I am a Lecturer (Assistant Professor) in Applied Mathematics at the University of Cambridge, with a broad interest in deep learning foundations, especially in verified machine learning and the provable limits of its capabilities. My work encompasses approximation theory, focusing on spectral properties of operators in infinite-dimensional spaces and neural network approximation and trainability. I have developed algorithms with explicit approximation guarantees, addressing challenges in computing spectral approximations in infinite dimensions, a significant mathematical hurdle since the 1950s. Additionally, I have devised techniques for optimal approximation of various spectral properties, contributing to the Solvability Complexity Index Hierarchy and tackling problems like approximating spectral measures and Koopman operators in data-driven dynamical systems. Parallel to this, my research in neural networks has revealed problems where stable and accurate neural networks exist, yet no training algorithm can produce such a network.