Academic history
- 2023–: CMI PhD student in Applied Functional and Harmonic Analysis Group, DAMTP. Supervised by Matthew Colbrook and Anders Hansen.
- 2022–2023: Part III Maths, Cambridge
- 2019–2022: BSc Mathematics, Warwick
Research interests
My research is in computational spectral theory. Most of my results are phrased in terms of the Solvability Complexity Index Hierarchy, which measures the difficulty of computational problems. I try to determine when we can solve spectral problems with error control. Error control enables computer-verified proof: with a mathematically verified error bound (and floating-point errors considered), a printout from a spectral algorithm would constitute a proof that a certain operator has (or lacks) spectra in a particular region. Further, error control may be used as a loss function for operator learning, where we use spectral error as a gauge for accuracy of finite-dimensional approximations.
Proofs that certain problems lack error control can help precisely identify why known algorithms fail in pathological cases, and help identify additional assumptions or additional pieces of information that can help reclaim error control. For example, a certain algorithm may fail for general continuous functions, but a modification may work if we also supply a modulus of continuity. Imagine we are talking about a system whose spectral properties are a consequence of real-world physical phenomena, so that there is a "true" spectrum independent of mathematical formalism. A lack of error control indicates that the spectrum is not fully determined in ZFC, indicating limitations to using mathematical formalism to investigate the real world,
I created a WordPress blog here. I do plan to eventually populate it with expositions on my research and any machinery that I found difficult to find, but since all my research is unpublished at the moment I am holding back on this.