I am currently a second year PhD Mathematics student at the University of Manchester, working in the Analysis, Geometry and Dynamics Group and within the Sahlsten Research Group. I am jointly supervised by Dr. Tuomas Sahlsten (Manchester) and Dr. Etienne Le Masson (Cergy).
My main research interest is broadly categorised by the field Quantum Chaos. This area concerns itself with a mathematical formalism of quantum mechanics in settings where the underlying dynamics are chaotic. In particular, I am interested in the Laplacian operator and its eigenfunctions. When normalised, these eigenfunctions can be interpreted as probability measures determining the probability that a quantum particle is in a certain position at a certain time on a compact manifold. Some good overviews of this area are:
- What is Quantum Unique Ergodicity? by Andrew Hassell. Austral. Math. Soc. Gaz 38, no. 3 (2011): 158-167.
- Mathematics of Quantum Chaos in 2019 by Steve Zelditch. Not. Amer. Math. Soc. 66, no. 9 (2019): 1412-1422.
Currently, I have been investigating how norms of Laplacian eigenfunctions depend upon the geometry of the manifold. More specifically, I have been looking at their behaviour in the large genus limit on compact hyperbolic surfaces.
Shadowing in Topological Dynamics
I am also interested in the investigation of variations of shadowing in topological dynamical systems. Shadowing is concerned with the ability to approximate sequences of points in a metric space that look like the orbit of a point under some continuous self-map by a ‘true’ orbit of a point.
This has an easy real life interpretation. Suppose that you have a computational process that takes some numerical input and returns a numerical output. The output that you receive potentially is only given up to some rounding error, and so continued iteration of this point under the same numerical computation will lead to a sequence of points that are close to the true values that should have been obtained through the computation. Since this error may grow quite large under many iterations, a reasonable question to ask would be: does this sequence represent (up to a small error) the actual numerical outputs of continued iteration of a single point under the computation? If the system possess the shadowing property, then we are able to answer this question affirmatively. Thus, shadowing is a question of the stability and potential chaotic nature of a dynamical system.
My particular study in this area has been to look at characterisations of various types of shadowing in systems in terms of various topological properties of the underlying space. In addition, I have studied characterisations of semiconjugacies between dynamical systems that preserve various notions of shadowing.