Title of project:
Embedded Eigenvalues of Schrödinger Operators

Code:
KMS2122B

Supervisor: 
Prof Karl Schmidt

Project description:
The spectrum of Schrödinger operators, corresponding to the set of admissible energies of a quantum mechanical system, consists of a half-line of continuous spectrum and additional discrete eigenvalues in many physically relevant situations. It is a rarer and more unstable phenomenon to have eigenvalues embedded inside the continuous spectrum. A first example of such a Schrödinger operator was first constructed by John von Neumann and Eugene Wigner in 1929.

The project will be focussed on understanding a recently published construction which allows to ensure the presence of any finite number of predefined embedded eigenvalues. This challenging project gives the opportunity to learn and apply some techniques of the spectral analysis of ordinary differential operators.

Prerequisite 2nd/3rd year modules:
MA2006 Real Analysis
MA3012 Ordinary Differential Equations
MA3005 Introduction to Functional and Fourier Analysis

Number of students who could be supervised for this project:
1

Title of Project:
Inverse Eigenvalue Problems in One Dimension
Code: MM2122B
Supervisor: Professor Marco Marletta
Project description:

Number of students who can be supervised on this project:
1

Title of Project: Inverse Problems for Elliptic PDE’s
Code:
MM2122C
Supervisor:
Prof. Marco Marletta
Project Description:

Project offered as double module, single module, or both:
Double
Prerequisite 2nd year modules:
None
Number of students who can be supervised on this project:
1

Title of project:
Representation Theory Of Finite Groups

Code:
MP2122B

Supervisor: 
Dr. M. Pugh

Project description:
In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces. In particular, group elements can be represented as matrices so that the group operation can be represented by matrix multiplication. Representations of groups are important because they allow many group-theoretic problems to be reduced to problems in linear algebra.

The initial task of this project will be to classify (real and complex) representations of the dihedral and cyclic groups. For the second half of the project there are a number of directions the project could take, including (but not limited to): extending the representation theory of dihedral and cyclic groups to fields of positive characteristic; or, considering different approaches to classifying irreducible representations of the symmetric group.

Project offered as double module, single module, or both:
Double

Number of students who could be supervised for this project:
1

Title of project:
Qualitative Theory of Partial Differential Equations

Code:
ND2021B

Supervisor: Prof. N. Dirr

Project description:

Certain partial differential equations of the form

∂tu(x,t)=∆u(x,t)+f(x,u(x,t))
can be transformed in an integral equation and solved by a Peano-Iteration in a similar way as it is done for ODEs. The difference is however, that we have to work in infinite dimensional vector spaces instead of Rn. Combining ideas from ODEs and Functional Analysis, a lot can be said about the qualitative behaviour of such equations (stability, long-time behaviour etc.) similar to the ODE case.
These equations, called reaction-diffusion equations, have applications in chemistry, biology and physics.
A project should always contain some proofs, but, depending on the interest of the student, could then focus on numerics or on analysis.
Background Reading:
D. Henry, Geometric Theory of Semilineat Parabolic Differential Equations Springer Lecture Notes in Mathematics 840
L.C. Evans, Partial Differential Equations, AMS Grad. Studies in Math. 19

Project offered as double module, single module, or both:
Double

Prerequisite 3rd year modules:
Measure Theory, Ordinary Differential

Title of project: Index for quantum cellular automata

Code: PN2122A

Supervisor: Dr P. Naaijkens

Project description: Quantum cellular automata (QCA) can be seen as discrete time quantum systems with strict locality properties. They are a quantum analogue of classical cellular automata, such as the ‘game of life’. Such systems are local in the sense that the rules to ‘update’ them after a time step are local. That is, they only depend on neighbouring sites. Quantum cellular automata are of interest because of their applications to quantum information theory and quantum computing. They are also relevant as toy models for more complicated physical systems. An interesting problem is to classify all quantum cellular automata. In this project you will look at an invariant, called the index, for 1D QCAs. This index can be used to distinguish different equivalence classes. Possible directions to explore are stability under perturbations of the QCA, or looking at the classification in the presence of additional symmetries. No prior knowledge of quantum mechanics is required for this project.

Project offered as a double module, single module, or both: Double

Prerequisite modules: MA2008 Linear Algebra II MA3005 Functional and Fourier Analysis

Recommended module for concurrent study in year 4: MA4016 Quantum Information

Number of students who could be supervised on this project: 1

Year: 4

Title of project: Holography and quantum error correction

Code: PN2122B

Supervisor: Dr P. Naaijkens

Project description: Error correction is indispensable in modern computers to ensure a fault-tolerant operation. This is true even more so for quantum computers. Not only are quantum systems more susceptible to noise due to their delicate nature, a single qubit (the quantum analogue of a bit), can undergo a continuum of possible errors. Nevertheless, it is possible to design quantum error correcting codes that tackle this issue. Moreover, if one can make the individual components of a quantum computer (a ‘quantum gate’) good enough, it is possible to build a fault-tolerant computer that can correct all kinds of errors. An interesting class of examples come from ‘holography’, where the bulk of a physical system is described completely by what happens on the boundary. An example is the so-called ‘HaPPY code’, which provide a toy model for the AdS/CFT correspondence in physics. The key point of the latter is a bulk-boundary correspondence, where the behaviour in the bulk of a system is completely determined by what happens at the boundary. In the HaPPY code, this can be understood as a consequence of error correcting properties. In this project you will look at the main features of such holographic quantum codes and study them in a mathematical setting. A non-technical introduction can be found at https://www.quantamagazine.org/how-space-and-timecould-be-a-quantum-error-correcting-code-20190103/ Knowledge of quantum mechanics is not required for this project, but the student is advised to take MA4016 concurrently with this project.

Project offered as a double module, single module, or both: Double

Prerequisite modules: MA3005 Functional and Fourier Analysis, not necessary, but helpful: MA3007 Coding Theory

Recommended module for concurrent study in year 4: MA4016 Quantum Information

Number of students who could be supervised on this project: 1

Year: 4