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

Title of project:
Blebbing on Oscillating Surfaces

Code:
TW2122B

Supervisor:
Dr. Thomas E. Woolley

Project description:
We propose to construct a mechanical model of muscle stem cell motion combining solid mechanics, stochastic theory and numerical simulations. During healing muscle stem cells adopt an amoeboid form of translocation, known as blebbing, in which cells are round and display highly dynamic plasma membrane extensions and retractions. The model will be a novel testing bed for biological hypotheses of cellular motion and illuminate connections between the multiple scales, as well as provide new mathematical insights into the connection between motion and surface geometry.
The aim is to model this mechanism of cellular motion and characterize the relationship between cell movement and the surrounding environmental geometry e.g. size, shape and topology. This relationship is usually discounted in experiments as extracted cells are placed on flat plates, or gels, which do not match the complicated heterogeneous environments that cells would have to naturally contend with. Mathematical modelling provides us with a way to test these disregarded factors and challenge current biological knowledge as to their importance.

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

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

Title of project:
Euler Characteristic and the Classification of Surfaces

Code:
UP2122B

Supervisor: 
Dr Ulrich Pennig

Project description:
In the first lecture of the Algebraic Topology course we encountered the Euler characteristic of polytopes as a topological invariant. Polytopes are special cases of combinatorially defined topological spaces called simplicial complexes and the definition of the Euler characteristic can be adapted to work for these as well. If we triangulate a surface, it gives rise to a simplicial complex and therefore always has an associated Euler characteristic, which is independent of the choice of triangulation as we will see. Another important property of surfaces is their orientability, which tells us whether we can pick a full set of coordinate patches, such that the handedness of the coordinate system does not change going from one patch to the other. The torus from the lecture is orientable, while the Klein bottle is not.

The goal of this project is to understand the Euler characteristic of simplicial complexes in full generality and of surfaces in particular. We will then look at the orientability of surfaces. It turns out that the pair of Euler characteristic and orientability of a surface without boundary provides a complete homeomorphism invariant. This means that two surfaces without boundary are homeomorphic if and only if their Euler characteristics agree and they are both either orientable or not orientable. After having understood the definitions we will focus on proving this theorem.

Prerequisite Modules:
Algebraic Topology (MA3008)
Groups (MA0213) or Rings and Fields (MA3003)

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