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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 1 June. You should know who your supervisor is by 8 June and should then contact them. You should also attend the LaTex training course in late May/early June 2018.

## 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 **1 June 2018**. 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 31 May/1 June.

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 7^{th} March 2019. 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 13th March, or on Wednesday 20th March 2019. 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.

### 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 and on mathematical writing on 31 May/1 June 2018.

### 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 2018-19**

**Dr Stephen Baigent**

• Discrete dynamical systems in ecology:

A survey of some models from ecology that involve the iteration of maps, such as the Logistic map in one dimension, but also looking at models in higher dimensions.

Pre-requisites: Some knowledge of Mathematica would be useful.

• Dynamics of Lotka-Volterra population models:

The project will first review some planar Lotka-Volterra models and look at some new theory such as existence or non-existence of periodic orbits. Next we will look at some Lotka-Volterra models in higher dimensions, with a particular focus on 3 dimensions where we will study phase portraits using the Mathematica package `CurvesGraphics6’.

Prerequisites: Mathematical Ecology (MATH3506). Some knowledge of Mathematica would be useful

**Dr Costante Bellettini **

• Minimality of the Simons cone:

Discovered to be a stable minimal hypersurface by J. Simons, this 7-dimensional cone in R^8 was proved to be a minimizer of the area by Bombieri, De Giorgi and Giusti (about 50 years ago). The discovery of this area-minimizer with a singular point indicates that the minimizing problem has to be posed in a class of non necessarily smooth hypersurfaces (sets of finite perimeter in geometric measure theory, a field where measure theory and differential geometry merge). A singularity formation of this type does not arise in dimensions up to 6 and it is still mysterious nowadays what makes dimension 7 so special.

• Other topics in geometric measure theory and in elliptic partial differential equations (Monge-Ampere equation, harmonic maps, etc)

Pre-requisites for both projects: Analysis 4 (MATH7102). Any of Measure Theory (MATH3101), Linear Partial Differential Equations (MATHM110) and Differential Geometry (MATH3113) would be helpful.

**Dr Christopher Birkbeck**

• Modular forms:

Modular forms are some of the key objects in number theory and have very deep connections with other parts of mathematics, which makes them widely studied. The first part of the project would be to learn what modular forms are (if not already known), after this there are several possible directions one could look at, including some possible original work.

Pre-requisites: Number Theory (MATH7701) and a little Complex Analysis (MATH2101).

**Prof Steven Bishop**

• Anomalous behaviour caused by rapid oscillations:

If a simple pendulum is vertically driven with a periodic force then, for some frequencies, the inverted position stabilises. This project will explore this behaviour and other similar unusual outcomes of periodically driven systems. Requires the numerical solution of systems of ODEs.

Pre-requisites: unspecified

**Dr Christian Boehmer **

• 1. Continuum mechanics with microrotations

• 2. Azimuthal geodesics in Cosmology

• 3. Vaidya type spacetimes in General Relativity

Pre-requisites: 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. In addition,

Project 1: No particular prerequisites.

Project 2: This project can be done without any cosmology knowledge, it is about understanding the properties of a certain ODEs which needs to be solved numerically.

Project 3: Mathematics for General Relativity (MATH3305) essential.

**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.

Pre-requisites for both projects: Fluid Mechanics (MATH2301)

**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.

Pre-requisites: Numerical Methods (MATH3603)

**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!

Pre-requisites: Fluid Mechanics (MATH2301) and Mathematical Methods 3 (MATH2401)

**Dr Jonny Evans **

• Hypersurface singularities

• Convexity and commuting Hamiltonians

See my webpage for more details: http://www.homepages.ucl.ac.uk/~ucahjde/projects.htm

Prerequisites: Multivariable analysis (MATH3109)

**Prof Rod Halburd**

• Geometry of the Painlevé equations:

This project will involve the use of methods from algebraic geometry to de-singularise some important differential and discrete equations. The geometry of the resulting space of initial conditions reveals very deep structure underpinning the symmetries and classification of these equations.

Pre-requisites: There are no formal pre-requisites beyond core modules but a good degree of mathematical sophistication will be necessary. I recommend that you take Lie Groups and Lie Algebras (MATHM206) and Algebraic Geometry (MATHM211) while doing the project.

• Topics in general relativity:

Possible directions include (1) the role played by different choices of coordinates and gauge, (2) approximations methods and (3) exact solutions arising from the theory of Riemann surfaces.

Pre-requisites: Mathematics for General Relativity (MATH3305) (essential)

• Topics in complex analysis:

Possible directions include (1) Riemann-Hilbert problems (which have many applications in the theory of differential and integral equations), (2) the value distribution of meromorphic functions and (3) singularities of solutions of differential equations in the complex domain.

Pre-requisites: None beyond core modules.

**Dr David Hewett **

• Integral equation methods in wave scattering:

Wave scattering problems arise in many areas of science and technology, including e.g. acoustics and electromagnetism. Mathematically they are typically modelled as boundary value problems involving a PDE 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, and are also of significant mathematical interest in their own right. Depending on the interest of the student, the project could focus on

- functional analysis (Banach and Hilbert spaces, bounded linear operators, compact operators, variational formulations, distribution theory and function spaces etc.), building on e.g. Measure Theory (MATH3101), Functional Analysis (MATH3103), Linear PDEs (MATHM110); - applications to PDE theory (e.g. in the study of wave scattering problems in acoustics or electromagnetism), building on e.g. Methods 4 (MATH7402), Linear PDEs (MATHM110), Waves and Wave Scattering (MATHM402);

- numerical analysis (approximation theory, numerical quadrature

etc.), building on e.g. Computational Methods (MATH7601), Numerical Methods (MATH3603).

Prerequisites: no formal prerequisites but some guidance as to relevant courses (depending on the intended focus of the project) is given above.

**Dr Richard Hill **

• Topics in Number Theory

Prerequisites: the exact prerequisites will depend on which topic we choose, but you should have taken at least three of the modules Number Theory (MATH7701), Algebraic Number Theory (MATH3704), Elliptic Curves (MATH3705), Prime Numbers and their Distribution (MATHM703) by the end of the third year.

**Dr Peter Humphries**

• Weighted Prime Number Races:

A famous observation of Chebyshev is that most of the time, there are more primes of the form 4k + 3 that are less than x than those of the form 4k + 3. This project would be investigating how this phenomenon changes when the primes are weighted to either favour small primes or large primes.

Prerequisite: Prime Numbers and their Distribution (MATHM703).

• The Mertens Conjecture for Quadratic Number Fields:

Mertens conjecture in 1897 that partial sums of the Möbius function have square-root cancellation; such a bound would imply the Riemann hypothesis, but not conversely. This conjecture was subsequently disproved via numerical calculations by Odlyzko and te Riele. This project would investigate number field analogues of this conjecture.

Prerequisites: Prime Numbers and their Distribution (MATHM703). Algebraic Number Theory (MATH3704) and some experience in a numerical programming language are recommended.

**Prof Ted Johnson **

• Inertial waves in a half-cone:

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.

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.

• Instability of a draining vortex.

When fluid with vorticity flow down a drain it forms a draining vortex. Under certain conditions large spiral waves are seen circling the drain. This project will attempt a description of such waves starting by examining small amplitude waves of the vortex surface.

• The instability of Hill's vortex

Hill vortex consists of a sphere of vortical fluid. It is an exact solution of the Euler equations of motion and propagates at uniform speed without change of form. In fact it behaves like a spherical vortex ring or smoke ring. However it is known to be unstable in that it leaves in its wake a thin stream of vortical fluid. This project will survey the existing literature and then exploit some hitherto unused geometric results to simplify existing results and explain them more clearly.

Prerequisites: Fluid Mechanics (MATH2301) or equivalent (Inviscid fluid dynamics) For project 1 Geophysical Fluid Dynamics (MATH3304) would also help.

**Prof Francis EA Johnson**

• Cohomology groups of finite groups

Prerequisite: MATH7202, Algebra 4, Algebraic Topology (MATH3203)

Algebraic Topology and Representation Theory (MATHM204).

**Dr Ilia Kamotski **

• Topics in homogenisation theory

Prerequisite: Linear Partial Differential Equations (MATHM110)

**Dr Jack Lamplugh**

• Congruent Numbers and Heegner Points

Pre-requisites: Elliptic Curves (MATH3705) and Algebraic Number Theory (MATH3704).

**Dr Jason Lotay**

• De Rham Cohomology:

One of the greatest challenges in geometry is: how do we know when two spaces are different? An important way to distinguish spaces is using invariants. Given any manifold, one can define a collection of vector spaces using the differential forms on the manifold called the de Rham cohomology. De Rham cohomology is an invariant of the manifold which is in fact dual to singular homology, and classes in de Rham cohomology have canonical representatives which have "least energy" known as harmonic forms (in the case of functions they are just the solutions to Laplace's equation). De Rham cohomology is a fundamental tool in differential topology which has many applications throughout geometry and topology.

Prerequisites: Multivariable Analysis (MATH3109)

• Holonomy

In Riemannian geometry, so on curved spaces, parallel transport gives a map between the tangent spaces at the start and end point of a curve. In flat space parallel transport is just translation, but in other Riemannian manifolds it can be far more interesting. If your curve happens to be a loop, parallel transport around the loop gives you an isometry of the initial tangent space, and by taking different loops based at the same point you can form a group using the parallel transport maps. This group is called the holonomy group and is an invariant of the Riemannian manifold. For flat space the holonomy group is trivial but for the sphere it is the special orthogonal group. The classification of holonomy groups is very surprising, with connections to the quaternions and octonions as well as Ricci-flat and Einstein metrics, and inspires hot topics in current research.

Prerequisites: Differential Geometry (MATH3113)

• Calibrated geometry

Minimal surfaces have formed a fundamental part of mathematics for more than 250 years, with important contributions from key figures in mathematics such as Euler, Lagrange, Gauss and Weierstrass, and continue to play a major role in current reseach. The minimal surface equation is a second order partial differential equation, so is very difficult to solve and analyse in general. In 1982, Harvey and Lawson introduced the notion of calibrated submanifolds, which are minimal but are defined by a first order equation. Calibrated geometry includes the classical subject of complex geometry in Kaehler manifolds, but also relates to current research in Calabi--Yau manifolds and manifolds with exceptional holonomy, Lagrangian mean curvature flow, gauge theory, and theoretical physics.

Prerequisites: Multivariable Analysis (MATH3109) and Differential Geometry (MATH3113)

**Dr Lars Louder**

• Free groups and topology of finite graphs

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

Pre-requisites: Topology and Groups (MATHM205) is compulsory, and Groups and Geometry (MATH7202) is recommended.

**Dr Djordjo Milovic**

• Modular forms modulo 2 (algebraic number theory):

The goal is to investigate the claim in J.-L. Nicolas and J.-P. Serre (Formes modulaires modulo 2 : structure de l'algebre de Hecke, C. R. Acad. Sci. Paris, Ser. I 350 (2012)) that the coefficients of Hecke operators on a certain space of modular forms modulo 2 are "frobenienne".

Prerequisites: Basics of algebraic number theory are essential. Willingness to read a bit of math in French is necessary. A good understanding of the Chebotarev Density Theorem would be helpful but not necessary.

• Units in real multiquadratic fields (computational number theory)

The goal is to develop a set of conjectures, backed by ample numerical evidence, about the interactions of unit groups in real multiquadratic fields.

Prerequisites: Basics of group theory and willingness to write code in Sage are both essential. Basics of algebraic number theory would be helpful but not necessary.

**Dr Lauri Oksanen**

• Inverse source problem:

The goal is to understand an inverse source problem, arising for example in medical imaging (see e.g. http://www.medphys.ucl.ac.uk/research/mle/). The problem can be studied either from computational or from theoretical point of view.

Pre-requisites: In the former case (computational point of view) a degree of fluency in Python or Matlab programming is a pre-requisite, and recommended but not necessary pre-requisite modules are Computational Methods (MATH7601) and Numerical Methods (MATH3603).

In the latter case (theoretical), Linear Partial Differential Equations (MATHM110) is highly recommended, and Functional Analysis (MATH3103) is helpful.

• Inverse boundary value problem:

The goal is to understand a problem that gives an idealized mathematical model for both geophysical and ultrasound imaging. Also this problem can be studied either from computational or from theoretical point of view.

Pre-requisites: The pre-requisites are the same as for the first project. In the 1+1-dimensional case the problem can also be studied without any pre-requisites.

• Topics in Lorentzian geometry:

Both the below topics are covered in the book Semi-Riemannian geometry by O'Neill. No original contributions are expected and the projects aim for a mark in the upper second bracket.

Robertson-Walker models in cosmology.

Hawking's singularity theorem.

Pre-requisites: Differential Geometry (MATH3113) or Mathematics for General Relativity (MATH3305) is recommended.

**Dr Nick Ovenden**

• Biomedical Flows

Prerequisites: Real Fluids (MATHM301) and Mathematical Methods 4 (MATH7402)

• Sound transmission and propagation

Prerequisites: Mathematical Methods 4 (MATH7402) Waves and Wave Scattering (MATHM402).

**Dr Karen Page**

• Topics in Mathematical Biology

Pre-requisites: not specified

**Prof Leonid Parnovski**

• Periodic operators and lattice points counting

Pre-requisites: Functional Analysis, Multivariable Analysis. Spectral Theory would also be desirable.

**Dr Ruben Perez Carrasco**

• Bacterial Flagellar Molecular Motor:

Bacteria can swim by propelling themselves with flagella. Bacteria rotate flagella by using macromolecules that work as microscopic turbines. In this interface between mathematics, physics, and biology; recent advances allow to capture individual motors in the lab and manipulate them mechanically. These experiments show that the force generated by these biological motors is responsive to external conditions, allowing the bacteria to swim efficiently in different situations. Nevertheless, there is still no model able to fully reproduce this experimental data. During the project we will make use of stochastic models (Markov processes, Stochastic differential equations ...) that encode different mechanisms that will be tested against experiments.

Pre-requisites: It is not required any prior knowledge on molecular biology to take the project, but the student is expected to have basic programming skills (Python, C++ or Julia).

• Patterning of embryonic tissues:

Embryo development is the process of formation of living organisms. From an original cell (the zygote), cells multiply an arrange forming tissues that will give raise to all the different cellular types (muscle, neurones, bones, ...). This process is orchestrated in time and space, creating patterns of different cell types that follow a program encoded in the DNA. Strikingly, the precision of these patterns is higher than expected by traditional models, challenging current knowledge. In this project, we will focus on a stripe pattern and study how different mechanisms affect the straightness and precision of the boundaries between cell domains. In order to do so, we will make use of spatial stochastic models (analytically and computationally). It is not required any prior knowledge on developmental or molecular biology to take the project, but the student is expected to have basic programming skills (Python, C++ or Julia).

Pre-requisites: The recommended modules (but none of them are compulsory) for the project would be: Computational Methods (MATH7601),

Mathematical Ecology (MATH3506), Biomathematics (MATH3307)

**Dr Yiannis Petridis**

• Lattice counting problems in Euclidean and hyperbolic spaces

• L-functions of elliptic curves and effective bounds on class numbers of quadratic fields.

• Other problems in analytic number theory.

Pre-requisites: Projects normally require that Prime Numbers and their Distribution (MATHM703). Depending on the project Elliptic Curves (MATH3705) or Geometry and Groups (MATH7112) may be useful.

**Dr Mark Roberts **

• Non-commutative unique factorisation domains

• Other projects in algebra

Pre-requisites: Algebra 4 (MATH7202) must have been taken, and also at least two from:

Commutative Algebra (MATH3201)

Galois Theory (MATH3202)

Algebraic Topology (MATH3203)

Algebraic Number Theory (MATH3704)

Representation Theory (MATHM204)

**Dr Felix Schulze**

• Isoperimetric inequalities

• Curve shortening flow on surfaces

Prerequisites: Multivariable Analysis (MATH3109). Differential Geometry (MATH3113) would be helpful.

**Dr Edward Segal**

• Quiver algebras

Pre-requisites: Commutative Algebra (MATH3201)

• Principal bundles

Pre-requisites: Differential Geometry (MATH3113), and Algebraic Topology (MATH3203).

• Other topics in algebraic geometry or algebraic topology.

**Dr Nadia Sidorova**

• Reinforced random walks:

Reinforced random walk is a random process on the integer lattice (or on a more general graph) that is more likely to cross edges or visit vertices which it has visited before. If the reinforcement is strong the reinforced random walk tends to get stuck in a certain part of the lattice. The aim of the project is to study this effect for various reinforcements.

Prerequisites: Probability (MATHM105) recommended.

• Brownian Motion:

The aim of the project is to introduce Brownian motion as central object of Probability Theory and discuss its properties, putting particular emphasis on sample path properties.

Prerequisites: Probability (MATHM105) recommended.

• Other project in the area of Probability.

Prerequisites: Probability (MATHM105) recommended.

**Prof Michael Singer **

• Hodge Theory

Pre-requisite: Multivariable Analysis (MATH3109) and at least one of the Linear PDE modules and Riemannian Geometry highly desirable.

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

Pre-requisites: Maclaurin, Multivariable and Linear PDE modules highly desirable.

**Dr Iain Smears**

• Topics in finite element methods for partial differential equations

The development of more accurate, efficient and robust numerical methods for partial differential equations is a major are of current research in applied mathematics. The aim of this project is to study some of the most recent and current advances in modern finite element methods (FEM). Possible specific topics for the project include the analysis of time-parallel algorithms for evolution equations, adaptive FEM for solving problems with singularities, or the analysis of high-order methods such as hp-FEM. Depending on the particular details, this could involve some programming of the methods, possibly in C++, Python or Matlab.

Pre-requisites: Measure Theory (MATHM301), Functional Analysis (MATH3203), Numerical Methods MATH3603)

**Prof Frank Smith **

• Industrial modelling problems

• Biomedical flows

• Modelling of bioprocessing problems

Pre-requisites: The projects above are suitable for students who have taken a full range of methods courses, have experience with theory of fluids and are interested in applying mathematics

**Prof Valery Smyshlyaev **

• High frequency scattering: asymptotic methods and analysis

• Multi-scale problems and homogenisation: asymptotics and analysis

Pre-requisites: Analysis 4: Real Analysis (MATH7102) and Mathematical Methods 4 (MATH7402).

**Prof Alex Sobolev**

• Pseudo-differential operators

• Mathematical theory of wavelets

Prerequisites: Analysis 4: Real Analysis (MATH7102), Functional Analysis (MATH3103).

**Dr Isidoros Strouthos**

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

Pre-requisites: Such projects are due to involve material covered in modules such as Algebra 3: Further Linear Algebra (MATH2201) and Algebra 4: Groups and Rings (MATH7202), as well as material covered in at least one of the modules Commutative Algebra (MATH3201), Algebraic Topology (MATH3203), Representation Theory (MATHM204).

**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.

Pre-requisites: Both projects are suitable for students who have taken Graph Theory and Combinatorics (MATH3503).

**Dr Sergei Timoshin**

• Two-fluid flows

Two-fluid flows can be studied in various approximations which reflect the specifics of the flow (e.g. thin layers), in two and three dimensions, with or without explicit time dependence. There are many interesting and unsolved problems related, for example, to flow separation and instability.

Prerequisites: Knowledge of fluid dynamics at the level of Real Fluids (MATH3301) is essential.

**Prof Jean-Marc Vanden-Broeck**

• Analytical and numerical studies of waves of large amplitude

Pre-requisites: Fluid Mechanics (MATH2301) or equivalent.

**Prof Dmitri Vassiliev**

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

Prerequisites:Functional Analysis (MATH3103) and Mathematics for General Relativity (MATH3305)

**Prof Helen Wilson**

• MaPPeD: The Mathematics of Populations for Persistence-Decay

This project is concerned with "evidence dynamics": the process by which trace evidence (such as gunshot residue or traces of plastic explosive) transfers from one material to another - typically from a finger to a surface. It's of direct interest to DSTL (for airline security) and UCL Forensic Science.

A key experimental measure is the "persistence-decay curve", which is generated by pressing a contaminated thumb on a succession of clean glass slides and measuring the volume of contaminant deposited on each.

The project consists of mathematical modelling of this real problem, calibrated with real experimental data. As such, the mathematical prerequisites (being comfortable with dynamical systems, some programming experience such as the summer python course) are much less critical than your attitude to the project. You need to be prepared to deal with uncertainty, and to communicate with non-mathematicians. Because of its wide applicability, there is a group supporting the student who takes on this project: Helen Wilson as supervisor with PhD students Eleanor Doman and Liam Escott as advisors.

Pre-requisites: see above.

**Prof Andrei Yafaev **

• Topics in arithmetic algebraic geometry

Pre-requisites: Algebraic number theory (MATH3704) and Elliptic curves (MATH3705)

**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.

Pre-requisites: Biomathematics (MATH3307)

**Dr Ewelina Zatorska**

• Hydrodynamic models of collective behaviour:

Hydrodynamic models for collective behaviour are very useful macroscopic models in mathematical biology to efficiently model the behaviour of large populations of agents – particles, cells, or animals – moving due to interactions produced by adhesion, chemical cues, or via the visual or sensory interactions.

This project is mostly analytical (analysis of systems of partial differential equations) but could also have elements of mathematical modelling or scientific computing, depending on student's expertise and preference.

• A new model for pedestrians including congestion:

The project is devoted to analysis of pedestrian flow by means of macroscopic equations (i.e., continuous models). Such models consist of partial differential systems for the mean density and/or the mean velocity of the pedestrians mainly in two space dimensions and occasionally also in one-dimensional settings.

This project requires knowledge of analysis of PDEs and basics of scientific computing. It has three different aspects that can be developed according to the student's taste:

The mathematical analysis of the model (existence and uniqueness of solutions in a one dimensional setting).

The realisation of numerical simulations in one and two dimensions to compare the behaviour of this model with that of the AR model.

The study of the transition between free and congested traffic.

Prerequisites: Linear Partial Differential Equations (MATHM110).