Teaching / Projects

In addition to the usual teaching, I also supervise a number of undergraduate (BSc) and master's (MSc) projects every year. These involve a student exploring in detail a topic that is not in the usual curriculum and then writing a dissertation.

Control of Differential Equations

Location: Queen Mary University of London
Level: BSc, MSc
Project page (on QMPlus)

Differential equations and their solutions are used to model a wide range of phenomena in the real world. An important general question in engineering is to determine whether one could steer these physical systems to a desired state using a limited number of inputs, or controls. For example, one wishes to bring a car into a specific parking space, but can only steer the car through limited inputs (e.g. pedals, steering wheel). Such questions comprise a field known as control theory, and this project aims to study some of the mathematics in this area.

For this project, we focus mainly on simple dynamical systems, namely, linear systems of differential equations with constant coefficients: \[ x' = Ax + Bu \text{,} \qquad x(0) = x_0 \text{.} \] Here, \(x: \mathbb{R} \rightarrow \mathbb{R}^n\) is a vector-valued function, with \(x (t) \in \mathbb{R}^n\) representing the state of our physical system at time \(t\). The vector \(x_0\) then represents the initial state that we impose on our system. In addition, \(A\) is an \(n \times n\) matrix describing how our physical system evolves in time. Next, \(u: \mathbb{R} \rightarrow \mathbb{R}^m\) is the control that we can impose on our system, with the components of \(u\) representing the \(m\) inputs we have at our disposal. Moreover, \(B\) is an \(n \times m\) matrix describing how the control interacts with our system.

BSc projects in this direction deal with the problem of eigenvalue placement. Here, one imposes a feedback control \(u = -Kx\) depending only on the system state, with \(K\) an appropriate matrix relating the state to the control: \[ x' = (A - BK) x \text{,} \qquad x(0) = x_0 \text{.} \] The key question is then to investigate whether the eigenvalues of \(A - BK\) can be prescribed by an appropriate choice of the relation \(K\). This is closely connected to the problem of stabilization, i.e. whether one can force the solution to decay to an equilibrium state.

MSc projects, which require more extensive analysis background, will instead focus on the full question of exact controllability: whether one can find a control \(u\) that can steer solutions \(x\) of our system to a desired state \(x_T\) at a later time \(T > 0\).

If time permits, the project can also explore some basic control theory for nonlinear differential equations, as well as numerical simulations of controlled systems.

(Below are some short notes I have written on these topics:)

The Eigenvalue Placement Problem: These short notes give a rigorous mathematical treatment on the eigenvalue placement problem for linear systems of ODE with constant coefficients.
- (Original 02/2025)

Fourier Transforms and Applications

Location: Queen Mary University of London
Level: BSc
Project page (on QMPlus)

The Fourier transform, which roughly decomposes a signal into its constituent frequencies, is one of the most important tools in mathematics, with a multitude of applications in science, computing, engineering, and even within mathematics itself. The objective of this project is to learn some basic properties of Fourier transforms, and then study in detail one or more of its applications.

The first part of the project investigates the basic theory of the Fourier transform:

Fourier and inverse Fourier transforms.
The Fourier inversion theorem.
The Plancherel and Parseval theorems.

The aim of the latter part of the project is to explore one application of Fourier transforms:

(a) Partial differential equations: Here, we use Fourier transforms to derive solutions to various partial differential equations, e.g. Laplace, heat, wave, and Schrödinger equations. This gives an alternative way to study many of the above equations, as well as produces new insights on their solutions.

(b) Fast Fourier transform: The Fast Fourier Transform (FFT) is, as you may guess, an algorithm for quickly computing Fourier transforms. It is heralded as one of the most important algorithms of all time. Here, the aim is to study how and why the FFT works, and to then implement the FFT on a computer.

(c) Image compression: Do you ever wonder how computers can take a large image file (e.g. a photo) and compress it down to a much smaller file? The Fourier transform is a key tool for many image compression algorithms, with the JPEG format as a well-known example. The goal is to understand how the Fourier transform is used, and then to try implementing a simple image compression system.

Calculus of Variations

Location: Queen Mary University of London
Level: BSc
Project page (on QMPlus)

Early on in calculus, one learns its fundamental concepts—limits, differentiation, and integration—on both the real line and on finite-dimensional spaces. In more advanced settings, one learns to deal with these concepts more rigorously, and in unusual settings.

In calculus of variations, one extends the notion of differentiation to various "infinite-dimensional spaces". Though this may seem strange at first glance, these ideas have been instrumental in solving many problems in physics, engineering, and even within mathematics. The aim of this project is to gain an understanding of some of the fundamental ideas within the calculus of variations, and to then apply these ideas toward some historically famous problems, for instance:

(a) Shortest curve: It is well known that the shortest curve between any two points \(A\) and \(B\) in space is a straight line segment. Though this is intuitively obvious, one can use calculus of variations to rigorously verify this. Students with some background in differential geometry can also extend this study to finding geodesics, i.e. distance-minimizing paths on curved spaces.

(b) Brachistochrone problem: Suppose you want to roll a ball down a ramp, from a higher point \(A\) to a lower point \(B\). What should the shape of your ramp be so that the ball rolls from \(A\) to \(B\) in the least possible time? One can apply calculus of variations ideas to find the solution. (It's not a flat ramp!)

(c) Dido's problem: Suppose you are given enough material to build a fence, with total perimeter \(L\), to enclose some area of land. What shape should your fence be if you wish to enclose the maximum possible area of land?

Nonlinear Wave and Dispersive Equations

Location: Queen Mary University of London
Level: MSc
Project page (on QMPlus)

The wave equation is one of the most important examples of partial differential equations. Waves can be found within many fundamental equations of physics, such as the Maxwell equations (electromagnetics), Yang-Mills equations (particle physics), Einstein field equations (gravitation), and Euler equations (fluid dynamics). Similarly, the Schrödinger equation is the fundamental partial differential equation in quantum mechanics. One shared feature between these equations is that they are dispersive, i.e. that solutions tend to "spread out" over time.

This project aims at developing a mathematical understanding of such nonlinear dispersive equations and their solutions. In particular, the goal is to develop the mathematical tools needed to solve these equations from initial conditions, and to derive some basic properties of these solutions.

The first part of the project involves revising a number of prerequisite topics:

Revision for the Fourier transform and its properties.
Revision of concepts from analysis and topology (metric spaces, contraction mapping theorem, Lebesgue and Sobolev spaces).
Revision of ideas from differential equations (existence and uniqueness, solving linear systems, Duhamel's principle).

The second part of the project involves studying either the wave or Schrödinger equation in detail:

Solutions of the linear wave/Schrödinger equation: \[ - \partial_t^2 u + \Delta_x u = 0 \text{,} \qquad i \partial_t v + \Delta_x v = 0 \text{.} \]
Existence and uniqueness of solutions to nonlinear wave/Schrödinger equations: \[ - \partial_t^2 u + \Delta_x u = \mathcal{N} ( u, \nabla u ) \text{,} \qquad i \partial_t v + \Delta_x v = \mathcal{N} ( v, \nabla v ) \text{.} \]
Decay and dispersion properties of solutions.