My main PhD project was on developing (Weyl) pseudodifferential calculus in new settings in which most of the classical tools of pseudodifferential operator theory cannot apply. For example, in settings in which symbol classes which give rise to bounded operators have necessary holomorphic requirements, so that the symbol calculus of classical pseudodifferential calculus theory cannot be applied. The techniques involved thus have a stronger focus on algebra than the classical theory, such as exploiting representation theory of the Heisenberg group and related algebras behind the Weyl quantisation.
Towards the end of my PhD I also start developing a theory of Haar bases on a class of locally compact groups and associated homogeneous spaces. I am exploring applications in the L^p analysis of Fourier multipliers on such spaces, a setting in which the Fourier transform is unwieldy.
Currently I am working on developing a theory of self-similar "sets" and decompositions in non-commutative geometry, based in the theory of quantum metric spaces and with potential applications to the study of Fourier multipliers/harmonic analysis on non-Abelian locally compact (quantum) groups.
Self-similar states and projections in noncommutative metric spaces, ArXiv preprint, April 2023.
ArXiv version at https://arxiv.org/abs/2304.13340
Weyl Pseudodifferential Calculus and the Heisenberg Group in New Settings, PhD Thesis (supervised by Dr. Pierre Portal), 2020, available here.
A Weyl Pseudodifferential calculus associated with exponential weights on R^d, Illinois Journal of Mathematics 65(1), 121-152, April 2021.
ArXiv version at https://arxiv.org/abs/2001.04572
Optimal angle of the holomorphic functional calculus for the Ornstein-Uhlenbeck operator, Indagationes Mathematicae 30(5), 854-861, 2019. MR 3996768.
ArXiv version at https://arxiv.org/abs/1812.08300
Scattering Theory, Honours Thesis (supervised by Professor Andrew Hassell), 2016, available here.