Yi Sheng graduated from the University of Warwick in 2017 under the MMORSE programme. His final year project was in the area of stochastic analysis, where the goal was to state and prove a change of variables formula, or “Ito’s lemma”, for the process (B_{.-1}, B.), B being the standard Brownian motion. The key point is that this 2 dimensional process is constructed from a single Brownian motion. Outside of maths, Yi Sheng enjoys jogging, hiking, and hokkien mee with sugarcane juice (iced, no lemon).

Research project title:
Localization of waves in random media with resonant inclusions

Supervisor(s):
Kirill Cherednichenko, Hendrik Weber

Project description:
Consider waves travelling through a domain with two key features: (1) that it has numerous tiny regions of high contrast (“resonant inclusions”), and (2) a random landscape outside these microscopic inclusions. Physically, we may think of this media as a composite material with disorder; mathematically, a wave equation where the coefficients are very large at (1) and random at (2). The goal of Yi Sheng’s project is then to show that waves are “caught” by the environment, with probability one. This is in contrast with the classical picture of waves travelling through a non-resonant and ordered environment (e.g. sound waves propagate through air over large distances). The fact that this is possible in a random environment is one of the most important discoveries of physics in the late fifties. However, a mathematically rigorous study of this “Anderson localisation” phenomena is only available for a few specific models, so work is far from complete. Considering the inclusions (1) and the disorder (2) separately, we also know that (1) has strikingly similar spectral properties to (2). Since the goal of proving that waves are localised may be formulated in terms of the spectrum, Yi Sheng asks if the resonant inclusions may “assist” in trapping waves. This will be made rigorous using multiscale analysis as the primary tool, accompanied by ideas from the theory of random operators.

Fun fact(s):
I like Hokkien mee.