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

Code: SW2122A

Title: The Boson Fermion Correspondence Description: Fundamental particles in quantum physics are partitioned into two types, called bosons and fermions, depending on how they behave under being permuted. The wave function of a collection of identical bosonic particles is invariant under any permutation of these particles, while the wave function of a collection of identical fermionic particles is multiplied by the sign of the permutation. Despite the seemingly physics focused introduction to this project, no prior knowledge of quantum physics is required. However, it can help provide background motivation and context. The goal of this project is to explore and understand surprising combinatorial identities between the quantum field theories of free bosons and free fermions. This will be done by first learning about the symmetry structures, called vertex operator algebras (essentially a generalisation of a ring), underlying these two field theories, studying their modules and constructing suitable isomorphisms, which imply interesting combinatorial identities.

Code: SW2122B

Type: Double

Supervisor: Dr S. Wood

Prerequisite modules: MA2013 Groups, MA3014 Algebra II: Rings

Maximum number of students: 1

Title of project:
Derived Categories and Algebraic Geometry

Code:
TL2122B

Supervisor: 
Dr Timothy Logvinenko

Project description:
Algebraic geometry studies geometrical objects by attaching invariants to them. For example any Riemann surface has its genus — the number of holes in it. More sophisticated examples include Betti and Hodge numbers, cohomology groups and ultimately – the derived category. The latter became over the last two decades the main technical tool of algebraic geometry, a sophisticated, abstract but very rewarding instrument.

This project would first introduce students to the abstract notion of a derived category of an abelian category. The main example we want to study is the bounded derived category of vector bundles on a projective space. Ultimately, the students should learn to compute the derived categories of P^1 and P^2, one- and two-dimensional projective spaces.

The project will involve guided reading, review of literature and writing up a report. There will be also be a minor computational element.

Students who wish to take this project in Year 4 are strongly recommended to take this supervisor’s Category Theory project in Year 3.

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

Prerequisite 3rd year modules:
MA3013: Algebra II – Rings
MA0322: Algebra III – Fields

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

Title of project:
Random fish schooling – hydrodynamic perspective

Code:
UK2122A

Supervisor:
Dr Usama Kadri

Project description:
One of the most highly debated questions in the field of animal swarming and social behaviour, is the collective random patterns and chaotic behaviour formed by some animal species, in particular fish schooling. Recently, we found that some fish species do not have preferred orientation and they swarm in a random pattern mode. Nevertheless, there is a clear coordinated movement of the school as a whole. Several projects are available that concern the three-dimensional random movement in fish schools.

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

Title of project:
Radiation of acoustic-gravity waves by an impacting object

Code:
UK2122B

Supervisor:
Dr Usama Kadri

Project description:
When solid objects, such as an aircraft or meteorite, impact the water surface in the deep ocean they generate propagating compression-type waves known as acoustic-gravity waves (AGWs). AGWs have unique and measurable bottom pressure signatures that travel at speeds near the speed of sound in water, which turns them into near-real time signals. Employing these signals we want to identify the location and form of impacting objects, with a focus on falling meteorites.

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

Title of project:
Effect of acoustic pulsation on gas-liquid flow in horizontal pipelines

Code:
UK2122C

Supervisor:
Dr. U Kadri

Project description:
Long liquid slugs reaching a length of several hundreds of pipe diameter may appear when transporting gas and liquid in horizontal or nearly horizontal pipelines. These long slugs may cause system vibration, separator flooding, and operational problems for the downstream processing facilities. In this work, we develop a mathematical model for the remote control of slug flow by pulsating low frequency sound waves.

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

Title of project:
Early detection & mitigation tsunami by acoustic-gravity waves

Code:
UK2122D

Supervisor:
Dr Usama Kadri

Project description:
Acoustic-gravity waves (AGWs) are compression-type waves propagating with amplitudes governed by the restoring force of gravity. They are generated, by wind-wave interactions, surface wave interactions, and movements of the tectonic lithosphere plates. They travel at the speed of sound in water carrying information on their source. Employing AWG theory we aim to improve current sea-state warning systems with a particular focus on early detection and mitigation of tsunami.

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

Title of project:
Faraday waves

Code:
UK2122E

Supervisor:
Dr Usama Kadri

Project description:
A surface wave disturbance over a fluid layer that is subject to sinusoidal vertical oscillation at frequency 2ω may excite subharmonic standing field of waves of frequency ω known as Faraday waves. For many applications the driven flow due to the interaction of the disturbance with the oscillating bath is attributed to parametric instability (or parametric resonance). Here, we model the oscillating bath as an infinitely long-crested compression (acoustic) wave that exchanges energy with surface (gravity) waves via resonant triad interactions providing an alternative perspective.

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