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## Year 4 Project

Here is information about the project for fourth year MSci students on Mathematics degrees (single and combined honours). Please note that you must submit your list of preferred supervisors on-line by 2 June. You should know who your supervisor is by 9 June and should then contact them. You should also attend the LaTex training course in late May/early June 2017.

## Choice of Project

See the list of project titles offered by members of staff. When you have found some that interest you, please then go to the relevant members of staff to discuss the project with them. If you particularly wish to do a project on a topic that is not offered, this may be possible; please discuss with a suitable member of staff.

Once you have found some suitable projects, please complete this form by entering your list of possible supervisors, in order of preference. This should be done by **2 June 2017**. Please note that you may not necessarily be able to do the project with the supervisor you want; members of staff cannot supervise too many projects, and will need to believe that the project they are offering is suitable for you. For this reason, you should include at least three possible supervisors in your list.

Students will be assigned to supervisors on 6 June and you will be told who your supervisor is shortly after this. You should then contact your supervisor (in person or by e-mail) to discuss preparatory work to be done over the vacation.

Please also register for and attend the LaTex course on 28/29 May.

If you have any queries or cannot find a suitable project, please discuss this with Dr Roberts in the first place.

##### Combined degrees

If you are doing one of the combined degrees Maths and Physics or Maths and Statistics, you will probably be able to do a suitable 1-unit or 1.5-unit project course in the other department rather than MATHM901 if you wish. In this case, you follow the guidelines, timetable, etc, of that project, and the rest of this document is not relevant to you. In this case, please discuss with Dr. Roberts in person or by e-mail.

## Structure of the Project

The project is a one-unit course. Thus it accounts for 25% of your fourth year courses, and can be expected to take up a considerable amount of your time (nominally about 200 - 250 hours). The project has two parts, a written project and an oral presentation.

### The written project.

This will have to be completed by the 8^{th} March 2018. It will be marked by the first examiner (who will be the supervisor) and the second examiner, and will be assigned 70% of the marks. The written project should normally be 5,000 - 10,000 words. Its form is likely to depend on the exact project, but it might typically include:

(a) an introduction;

(b) an exposition of the general area of the project; this should be at an appropriate level, generally assuming knowledge of the first 3 years of a Maths degree,

(c) any problems solved;

(d) a clear list of references.

In a project where the main emphasis was on researching
a given area and understanding and collecting together the results, (b)
might dominate; in one where there was more emphasis on solving problem(s),
(c) might take up most of the space. Please
note that a project consisting entirely of copied-out book-work would not be acceptable. You might also
do a project with some computer work involved; in this case you would
also have to submit the program in written form and on disc, and give
some explanation of it in your written project.

You will get advice and help on your project from your supervisor. Your project must be word-processed.

### An oral presentation

This will take the form of a talk to the examiners and other students on either Wednesday 14th March, or on Wednesday 21st March 2018. This will be assigned 30% of the marks, and will be assessed principally by your supervisor and the second examiner, with some input from the other examiners. The talk will last approximately 18 minutes, with a few minutes for questions at the end. You will be given advice on how to structure and give your talk during the presentations skills training, and will give at least one practice talk.

## Help with the Project

### Supervision

You will meet with your supervisor on a regular basis (probably once every week or two weeks) to discuss your progress and to get guidance on what to read, what to try to do next, etc. You will be asked to write a short progress report at the end of the first term.

### Mathematical Writing and LaTeX

Projects are normally expected to be typeset using the mathematical word-processing package LaTeX. There will be a short course on this on 28/29 May 2017. It is also expected that a course on mathematical writing will be offered during the Autumn term (TBC).

### Presentation skills

During the middle of the autumn term you will get a short course on giving talks which will include the chance to see yourself on video. This can be very illuminating, and hopefully will make it easier to talk and present material effectively - an ability that is likely to be useful not only in the presentation of your project, but also after you have finished your degree, whether you are in teaching or other jobs.

## Project Timetable

The project is quite a large-scale task and will take up quite a lot of time. It is therefore important that you start to work on it quickly, and follow the timetable given below.

1. **Now **(Spring term 2017): start thinking about which project you want to do, talk to supervisors, etc. Register for post-exam course on mathematical writing.

2. **By 2 June 2017: ** submit on-line your list of choice of supervisor.

3. **28/29 May 2017: **Attend the LaTex course.

4. **After 8 June: **when you know your supervisor, contact him/her in person by e-mail to discuss the project.

5. **Summer vacation: **do some background reading/preliminary work as agreed with your supervisor.

6. **Beginning of Autumn Term 2017: **meet supervisor and discuss your progress

so far.

7. **Mid-term autumn 2017: **training on presentation skills.

8. **End of autumn term 2017:** brief report on progress completed by supervisor

and student and returned. Substantial progress should have been made on

the project by this stage.

9. **Early spring term 2018:** You should start giving practice talk to fellow students. Also draft copy of written project submitted to supervisor.

10. **Thursday 8th March 2018, 4.00pm: **submission of three copies of your written project to Departmental Office, Room 610, 25 Gordon Street.

11. **14-21 March 2018: **during the last two weeks of spring term project presentations (talks).

10. **After that: **preparing for your exams!

## Assessment of MATHM901 Project

### The presentation

This is assessed by all examiners present using the presentation assessment form. The first examiner (the supervisor) and the second examiner then decide the final mark for the presentation using this information.

### The written project

This is assessed by the first and second examiner independently and then a mark agreed. Rough criteria for the assessment are given in the guidelines for marking M901 project. Please click here to view the assessment guidelines.

### The final mark

This produces an overall mark for the project (30% presentation, 70% written project). These marks and the projects are considered (in rough subject groups) by all the examiners to see if they are all in agreement on the marks assigned, and some changes may be made if necessary. Each project is then seen by an external examiner, who may also recommend changes to the mark.

The final mark will be made available to you at the same time as the other examination marks.

## Available Project Titles

**M901 Project Titles 2017-18**

**Dr Stephen Baigent**

• Dynamics of 3-dimensional discrete-time population models.

• Dynamics of competition models.

• Optimisation problems associated with maximising fitness.

**Dr Costante Bellettini **

• Minimality of the Simons cone

Discovered to be a stable minimal hypersurface by Simons, this cone was proved to be a minimizer of the area by Bombieri, De Giorgi and Gusti. The cone has an isolated singularity and, in view of this, the minimizing problem must be phrased allowing also for non-smooth competitors (sets of finite perimeter in geometric measure theory).

** **

**Dr Timo Betcke **

• Fast boundary integral equation methods and their applications

• Spectral properties of boundary integral operators

Please note that both projects require a good deal of programming in Python.

It is therefore essential that candidates have some programming background and are willing to invest effort into learning Python development.

**Dr Christian Boehmer **

• Continuum mechanics with microrotations

• Geometric identities in General Relativity

• Rotating spacetimes in General Relativity

Please note that these projects require a good deal of programming in Mathematica. It is therefore essential that candidates have some programming background and are willing to invest effort into learning Mathematica development.

**Dr Robert Bowles **

• Inviscid flow passing the edge of a horizontal plate

Complex variable methods are very powerful in description of irrotational inviscid free-surface flows. One such flow that seems to have received relatively limited attention is the simple acceleration an oncoming fluid layer off of the edge of an horizontal plate. This, mainly numerical, project investigates the application of conformal mapping and Newton iteration (using a NAG package, or similar) to approach the problem.

• Time-evolution of a free-surface flow

The project concentrates on the numerical solution of either an extended Korteweg de-Vries equation, or the full boundary-layer equations, describing the response of the flow to the introduction of an obstacle into a free-surface flow upstream of the point at which the flow falls off the edge of a horizontal plate. The equation can be solved using available NAG packages.

**Prof Erik Burman **

• Computational methods for data assimilation

In many applications in computational medicine and meteorology one wishes to compute approximations of the solutions of a partial differential equation for which the available data are not such that the problem is well posed. Recall that the mathematical theory for partial differential equations requires certain quantities to be known to guarantee that a unique solution exists and is stable under perturbations. In this project we will consider recently designed computational approaches for the heat equation that can be shown to produce approximations that are optimally accurate with respect to the approximation order of the scheme and the stability of the (ill-posed) data assimilation problem. The objective is to study how these methods perform on the convection—diffusion equation for high or low Peclet number. The computational package FreeFEM++ will be used for the computations.

• Finite element methods for contact problems

In this project we will consider the contact problem, i.e. the problem in elasticity where one elastic body under deformation comes into contact with another (or a rigid wall). The aim is to study the basic model for frictionless contact and compare some different finite element approaches for the numerical approximation of this type of problem.

Either a one dimensional situation can be considered in which case the design of a Python program is required, or computations on more advanced problems in FreeFEM++ could be carried out.

**Prof Gavin Esler **

• Flooding in a Manhattan-like network of streets: Solve the shallow water equations on a grid to uncover the physical processes occurring when a city or town is flooded.

• Stochastic differential equation methods in advection-diffusion problems.

• Phase transitions in point vortex dynamics: explain why the mean circulation of a set of chaotically evolving vortices can change spontaneously as parameters are varied!

**Dr Jonny Evans **

• Hypersurface singularities

• Convexity and commuting Hamiltonians

• Small exotic 4-manifolds

There will be more extensive descriptions given on my webpage in due course: http://www.homepages.ucl.ac.uk/~ucahjde/projects.htm

**Prof Andrew Granville **

• Primality testing and factoring algorithms

• Elementary proofs of the prime number theorem

• Large gaps between primes

**Prof Rod Halburd**

• Integrable systems

• Topics in complex analysis

**Dr David Hewett **

• Integral equation methods in wave scattering

Wave scattering problems arise in many important areas of science and technology, e.g. acoustics, electromagnetism and elasticity.

Mathematically they are typically modelled as boundary value problems involving a partial differential equation such as the wave equation, Helmholtz equation or Maxwell's equations. Integral equations provide a natural framework for the analytical and computational study of such problems. The focus of the project could be on analysis or computation, depending on the student's interests.

**Dr Richard Hill **

• Topics in Number Theory

**Prof Ted Johnson **

• Inertial waves

Inertial waves are oscillations in a fluid in a rapidly-rotating fluid where the restoring force can be regarded as a tendency to conserve angular momentum.

• Inertial waves in a half-cone

Consider a closed domain formed by taking a cone of finite height with axis vertical and apex down. Let the cone be cut in half by a vertical plane through containing the axis and take the flow domain to be one of the half-cones so formed. Now let this half cone be rotating rapidly about the cone axis. This system supports oscillatory modes of internal disturbances called inertial waves. The project will consist of analysing these modes by solving a two-dimensional eigenvalue problem (perhaps most simply by MatLab) and then comparing the results with existing three-dimensional finite element simulations and experiments.

• Initial value problems for inertial waves

Suppose a body is set into motion in a rotating fluid. This transient motion sets up an inertial wave field. For some geometries this field can be found semi-analytically. This project would examine one of these.

**Prof Francis EA Johnson**

• Cohomology groups of finite groups

• Algebraic Topology

• Representation Theory

**Dr Ilia Kamotski **

• Topics in homogenisation theory

**Dr Jack Lamplugh**

• Congruent Numbers and Heegner Points (normal pre-requisites MATH3705 Elliptic Curves and MATH3704 Algebraic Number Theory)

**Dr Jason Lotay**

• De Rham Cohomology

• Holonomy

• Calibrated geometry

**Dr Lars Louder**

• Free groups and topology of finite graphs

• Stallings' theorem on groups with infinitely many ends and groups of cohomological dimension 1.

**Prof Robb McDonald**

• Exact solution methods for Laplacian growth

• Loewner's equation: analytical and numerical solutions

**Dr Lauri Oksanen**

• Inverse problems for the wave equation

• Hyperbolic partial differential equations

• Topics in Lorentzian geometry

**Dr Nick Ovenden**

• Biomedical Flows

• Sound transmission and propagation

**Dr Karen Page**

• Topics in Mathematical Biology

**Prof Leonid Parnovski**

• Periodic operators and lattice points counting

**Dr Yiannis Petridis**

• The hyperbolic prime number theorem

• Lattice counting problems in Euclidean and hyperbolic spaces

• Central values of L-functions and modular symbols.

• Other problems in analytic number theory.

Projects normally require that M703 Prime Numbers and their distribution.

**Dr Mark Roberts **

• Non-commutative unique factorisation domains

• Other projects in algebra

**Dr Felix Schulze **

• Isoperimetric inequalities

• Curve shortening flow on surfaces

**Dr Nadia Sidorova**

• Topics in probability

**Prof Michael Singer **

• Hodge Theory

• The Euler – Maclaurin formula and lattice points in convex polytopes

**Dr Iain Smears**

• Numerical Methods and Applications in Stochastic Optimal Control Theory

Please note that this project requires a good deal of programming in a numerical programming language (at least one of Matlab, C++ or Python). It is therefore essential that candidates have some programming background and are willing to invest effort into learning development in a programming language.

**Prof Frank Smith **

• Industrial modelling problems

• Biomedical flows

• Modelling of bioprocessing problems

**Prof Valery Smyshlyaev **

• High frequency scattering: asymptotic methods and analysis

• Multi-scale problems and homogenisation: asymptotics and analysis

**Prof Alex Sobolev**

• Pseudo-differential operators

• Mathematical theory of wavelets

**Dr Isidoros Strouthos**

• Topics in homological algebra / algebraic K-theory / algebraic topology

**Dr John Talbot**

• Probabilistic methods in extremal combinatorics. (Possible directions include random graphs or flag algebra computations.)

• Extremal problems for graphs, hypergraphs and the hypercube.

Both projects are suitable for students who have taken Graph Theory and Combinatorics MATH3503.

**Dr Sergei Timoshin**

• Two-fluid flows

• Flow past a rough wall

• Transonic boundary layer: viscous/inviscid interaction

**Prof Jean-Marc Vanden-Broeck**

• Analytical and numerical studies of waves of large amplitude

**Professor Dmitri Vassiliev**

• Topics in spectral theory of partial differential operators and microlocal analysis

**Dr Helen Wilson**

• Topics in non-Newtonian fluid mechanics.

Possible directions include real, industrial modelling problems or more theoretical work on flow instabilities. A background in fluid mechanics is useful but I do not expect you to know anything about non-Newtonian fluids beforehand.

• The mathematics of rainbows.

**Dr Andrei Yafaev **

• Topics in arithmetic algebraic geometry

**Prof Alexey Zaikin **

• Intelligent cellular decision making

Similar to intelligent multicellular neural networks controlling human brains, even single cells surprisingly are able to make intelligent decisions to classify several external stimuli or to associate them. This happens because of the fact that gene regulatory networks can perform as perceptrons, simple intelligent schemes known from studies on Artificial Intelligence. Recently we have studied a model of genetic perceptron modelled with differential equations using Kaneko's approach. In this project one should construct and investigate similar model with different simpler equations enabling understanding of chemical reactions behind the perceptron functionality. From the methodological point of view the project includes numerical simulations to solve ordinary differential equations.