Research
Overview
"同一物理定律的不同的数学表述，尽管在物理上是等价的，但实际并不等效。"- 冯康
"Different mathematical representations of the same physical law, while equivalent in physics, are not necessarily equivalent in practice。" - Kang Feng
The numerical analyst is the one who works between the infinite-dimensional space and the finite-dimensional space. When I work in the infinite-dimensional space, the beauty of abstraction attracts me. While I am in the finite-dimensional space, the practical motivates me. Therefore, I am interested in the finite element exterior calculus, structure-preserving scheme and preconditioning of solving partial differential equations. The tension between the pure and the applied is the core of my work.
Current Research
Past Research
The University of Nottingham 2020-2022
Summer Research Intern: Transfer Operator Methods - Transport 2D Electron Gases
Funded by The School of the Mathematical Sciences, University of Nottingham
Advisor: Martin Richter
Project Description: The Aim of this project is to derive a formula for the electronic transport in a so-called two-dimensional electron gas (2DEG). These are specifically designed interfaces between semiconductors in which electrons are restricted to two dimensions. From a dynamical systems point of view, interesting phenomena can arise when the geometry or other parameters, like a perpendicular magnetic field, are changed. They lead to bifurcations and a generic KAM route to chaos thereby highly influencing the transport properties.
Results: The transfer operator method successfully reproduced the complex dynamics of the electrons in the small cavity. The bifurcation corresponding to the magnetic field can be seen via the Poincaré section. Take a look at my poster here.
Bachelor thesis: When does the Repressilator Oscillate? Hopf Bifurcation and Stability of the Differential System. Here
Advisor: Etienne Farcot
Project description: The repressilator is a genetic network in which three genes code for proteins that, upon their presence, induce cellular repression of the next gene, which forms a negative feedback loop with nonlinear interactions. Elowitz and Leibler describe this network based on six nonlinear first-order differential equations. In such a system, oscillations may occur, which is valuable for studying since this model can be tested experimentally via artificial regulatory networks.
Results: This project investigates the mathematical conditions of exhibiting oscillatory behaviours in repressilators. We study the differential stability using both mathematical and computational approaches and find that significant cooperative binding effects, efficient repressions, and high transcription rates promote oscillations. We confirm the existence of Hopf bifurcation by demonstrating the bifurcation diagram and the formation of the limit cycle numerically. We also present the numerical simulations of the dynamical behaviour of the repressilator with respect to different parameters. Finally, we find more complex behaviours of the genetic network by extending the repressilator in two ways:
adding spatial effects, leading to synchronization property.
adding more genes in the loop, leading to the change of frequency and amplitude of the oscillation associated with different parameters.