## Applied Projects

## Nils Andersson: Gravitational wave sources

With several hypersensitive kilometer-long interferometric gravitational wave detectors due to come on line within the next year, we are standing on the doorstep of a revolution in astronomy. Once Einstein's elusive gravitational waves are caught, we can hope to learn much about the very extremes of physics. This project provides the student with an insight into this fascinating area of applied general relativity. Following an introduction to the nature of gravitational waves, the main focus will be on the various sources that are likely to produce detectable waves, such as colliding black holes or rapidly spinning neutron stars. The aim of the project is to produce an up-to-date survey of the many exciting possibilities. Although it is not an absolute requirement for this project, the student would benefit from the first half unit MATH 3006 Relativity, Black Holes and Cosmology , which includes a brief consideration of gravitational waves. [Level 3]

Background reading:

R. A. d'Inverno, Introducing Einstein's Relativity , Clarendon Press

B. F. Schutz, A First Course in General Relativity , CUP

K. S. Thorne, in Three Hundred Years of Gravitation (ed.S.Hawking, W.Israel), CUP

## Nils Andersson: Gravitational wave asterology

Recent evidence supports the notion that most stars pulsate. Such stellar oscillations share many properties with waves in the Earth's oceans and atmosphere. This project is aimed at providing the student with an introduction to the stellar pulsation, with particular focus on those modes of oscillation that may prove relevant for gravitational-wave astronomy. Starting from the standard Euler equations from hydrodynamics, we will study the nature of waves in spherical bodies, the possible relevance of various pieces of physics (equation of state of matter, viscosity) etc. One possible extension of this work would make contact with current research on oscillations in rapidly rotating (and therefore no longer spherical) stars. Although it is not an absolute requirement for this project, the student would benefit from the first half unit MATH 3006 Relativity, Black Holes and Cosmology , which includes a brief consideration of gravitational waves. [Level 3]

Background reading:

R. A. d'Inverno, Introducing Einstein's Relativity , Clarendon Press

B. F. Schutz, A First Course in General Relativity , CUP

J. P. Cox, Theory of Stellar Pulsation , Princeton

## Leor Barack: The causal structure of black holes (not available 2014-15)

The aim of this project is to expose you to some of the most fascinating consequences of Einstein's General Relativity. You will first familiarise yourself with the idea of *conformal diagrams*---one of the most elegant and useful methods of relativistic theory. With the help of these, you will define notions such as those of "null infinity", black hole and white hole, event horizon and Cauchy horizon; and will then be able to explore the causal structure of spacetime outside and inside different types of black holes. Looking first at a non-rotating black hole, you will learn about the maximal extension of the Schwarzschild geometry, and explore the fate of an observer falling through the event horizon. You will then investigate into the much richer structure of charged and rotating black holes, strolling through a wonderland of ring singularities, closed timelike curves, wormholes and multiverses. [Level 3 or 4]

Prerequisite course: General Relativity.

Background reading:

Gravitation (Misner, Thorne and Wheeler)

An Introduction to Einstein's General Relativity (Hartle)

A Relativist's Toolkit (Poisson)

## Leor Barack: Black hole orbital mechanics (not available 2014-15)

The project will explore the behavior of objects moving in the vicinity of a black hole, looking at orbits of test-mass satellites as well as the trajectories of light particles. Using the mathematical tools of General Relativity, you will try to describe and understand curious phenomena such as the gravitational deflection of light; periastron precession; the existence of an innermost stable orbit; the bizarre "zoom--whirl" behavior; and the possibility of light rays moving in a closed circular orbit. The project will begin with a review of Keplerian celestial mechanics, and, through analogy, proceed to investigate general-relativistic orbits around a non-rotating (Schwarzschild) black hole. [Level 3 or 4]

Prerequisite course: General Relativity.

Background reading: Gravitation (Misner, Thorne and Wheeler)

An Introduction to Einstein's General Relativity (Hartle)

A Relativist's Toolkit (Poisson)

## Giampaolo D'Alessandro: Using conformal maps to solve potential problems

**Aim:** Analytical and (optionally) numerical study of how to use conformal maps to solve partial differential equations for two dimensional potentials. [Level 3 or 4]**Description:** A conformal map is an analytic function with non-zero derivative. This rather abstract definition hides the fact that these functions can be used in one of the most intriguing applications of complex analysis, namely solving potential problems. A potential, in this respect, is a function of two variables, defined in a region of the plane, that satisfies Laplace's equation. Examples of potentials are the electrostatic potential and the fluid velocity potential.

It is normally quite easy to solve Laplace's equation in rectangular or circular domains. It becomes much more complicated to do so when the domain does not have such a simple shape. It is in this case that conformal maps come to the rescue: we can solve the problem in a simple region and transform it, using the map, in the solution for the same problem in a much more complicated region.

In this project the student will learn the main theory of conformal maps and apply it to some simple problems. If time allows we will also explore how conformal maps are implemented numerically.

## Giampaolo D’Alessandro: Solving partial differential equations numerically without using a mesh

**Aim:** It is possible to solve partial differential equations numerically by scattering a set of points in the domain of definition and solving a linear system? In this project you will learn how to do this.

**Description: **In the numerical methods module you learn the method of finite differences to solve partial differential equations. This method is simple to implement on a rectangular domain. Suppose, though, that you need to solve Laplace’s equation in a three dimensional “blob”, an object that has no “nice” geometrical shape. In this case, it is rather hard to define a mesh and, hence, apply a finite difference method.

Many other methods have been developed to solve partial differential equations in arbitrary geometries. In this project we will study Radial Basis Function methods: these consist in scattering a random set of points (called collocation points) in the domain of definition of the solution (the “blob”) and imposing that the function satisfies the equation at these points. It is a very simple method to implement, but requires us to do some mathematics first: in particular, we need to define appropriately what it means to “satisfy the equation at the collocation points”. We will study the mathematics behind these methods and also, if time allows, code them in Matlab. In a one semester project we will consider only one type of Radial Basis Function methods. In the two semester project will compare different methods and/or, depending on the interests of the students, apply them to solve some specific mathematical model. Prerequisites: MATH3018 - Numerical methods. [Level 3 or 4]

Further reading:

M. D. Buhmann, Radial Basis Functions : Theory and Implementations, Cambridge University

Press, West Nyack, NY, USA, 2003.

## Carsten Gundlach: Making small black holes

A massive object will collapse under its own weight, and turn into a black hole, if it is sufficiently dense. Interesting things happen at the black hole threshold: when the object is only just dense enough, or only just fails to be dense enough. The black hole threshold shows many mathematical similarities to a critical phase transition, for example in a fluid: a small change in temperature and pressure turns a liquid into a gas, or vice versa. Understanding these phenomena requires a little general relativity and a little knowledge of dynamical systems, and brings together two widely separated areas of physics: gravitational collapse and statistical mechanics. Some of the basic mechanisms can also be explored using a nonlinear wave equation as a toy model. A one semester project [Level 3] could be centered on reading, and a two semester project [Level 4] on writing a numerical code to simulate gravitational collapse.

(MATH3006 Relativity, Black Holes and Cosmology is helpful but not required.)

## Carsten Gundlach: Detecting gravitational waves

General relativity predicts that whenever masses are accelerated, they produce gravitational waves: distortions in space itself that travel at the speed of light. In principle, these waves can be detected because they deform objects they pass through. In practice, the waves are very weak: even violent astrophysical events such as the collision of two stars, once the waves reach us, produce a deformation of objects of only one part in 10^{24} or so. Nevertheless, if current mathematical models of the sources are correct, and if the instruments work as planned, it seems likely that gravitational waves will be detected in the next few years by instruments already under construction. These instruments use laser interferometers to measure tiny changes (a fraction of the size of a nucleus) in the length of a pair of arms 4 kilometers long. The project could focus on the theory of why gravitational waves can be measured at all, or on how specific detectors work, and on the weird and interesting sources of experimental noise one has to consider when trying to measure something to a precision of one part in 10^{24} [Level 3 or Level 4]

(MATH3006 Relativity, Black Holes and Cosmology is helpful but not required.)

## Ian Hawke: Dam breaking and shocks

Some physical phenomena involve large changes on very short scales, which are often modelled using discontinuous functions. Examples include tsunamis and shock waves in gas and fluid dynamics. This project will look at the structure and evolution of simple systems containing shocks, particularly the dam break problem for the shallow water equations.**Pre-requisites:** A good knowledge of PDEs. At level 3 some use of Maple will be expected. At level 4 some more detailed knowledge of numerical methods will be required.**Background reading:**

E. Toro: Riemann Solvers and Numerical Methods for Fluid Dynamics (Springer)

R. Leveque: Finite Volume Methods for Hyperbolic Problems (CUP)

## Wynn Ho: Neutron star cooling

Neutron stars are created in the collapse and supernova explosion of massive stars, and they begin their lives very hot but cool rapidly. The processes that govern this cooling depend on the detailed but uncertain properties of matter at the extremely high densities that exist inside neutron stars. By comparing astronomical observations to predictions from theoretical models, we can learn about not just the neutron star interior but also fundamental physics. The aim of this project is to understand the energy and heat flow equations that govern neutron star cooling and calculate (including computational work) some simple evolutions.

Background reading: Page, Geppert, and Weber, "The cooling of compact stars", Nuclear Physics A, vol 777, pg 497-530 (2006)

## Wynn Ho: Modelling stellar and planetary spectra

Measurements of the brightness as a function of photon energy (or spectrum) of stars and planets can reveal important information about their physical properties, for example, the temperature and composition of the star or planet. The aim of this project is to study the radiative transfer equation which determines the spectrum of stars and planets and to use this equation to calculate (including computational work) spectra for simple situations.

Background reading: Rybicki and Lightman, Radiative Processes in Astrophysics, Wiley

## Chris Howls: Lenses with Negative Refractive Index (Not available 2014-15)

The art of making lenses involves geometry and Snell's law sin(*i*)/sin(*r*)=*n* where *i* is the angle of the incident ray to the surface normal, *r* is the angle of the refractive ray and *n* is the refractive index of the lens. In naturally occurring material *n* >1 (*n* = 1 for a vacuum).

Lens design effectively involves fixing the lens shape in order to focus the rays, etc. Recent developments in photonic materials have led to the possibility of man-made substances with negative refractive index (*n* < -1), which would have remarkable focussing properties.

The purpose of this project is to investigate how the geometrical optics of lens design would be affected by using materials with negative refractive index, using geometry and ray-tracing methods. This project requires no prior knowledge of physics. [Level 3]

## Chris Howls: Conflict modelling (Not available 2014-2015)

You may already be familiar with predator-prey models of populations. Similar coupled differential equation models have been applied to attrition rates in military combats, most notably by Lanchester. Different types of warfare require different consideration. For example, conventional warfare on an island with regiment-sized deployments should be treated differently from a model of an asymmetrical guerilla war where the opportunities for each side to kill one-other are much smaller. Effects such as reinforcements and resupply can also be included. Such models have been used retrospectively to exam the causes of defeat of large powers in certain conflicts in the post World War 2 era.

This project will consider a variety of warfare scenarios and apply Lanchester-type models to them. One aim is to carry out a time-dependent model of a recent guerilla conflict based on available public data. If time allowed, a stochastic element might also be considered. Skills involved: interest in mathematical modelling, ability to solve differential equations, phase plane analysis, use of software packages, ability to research data, ability to search out papers, ability to write reports.

Background reading:

Lanchester FW, 1956,Mathematics in warfare, in The world of mathematics vol 4, Newman JR, ed. (Simon and Shuster: New York)

Tung, KK, 2007 Topics in Mathematical Modelling, (Princeton University Press).

## Ian Jones: Testing General Relativity using binary pulsars

Ever since its conception in the early 20th Century, experimenters have attempted to put general relativity to the test. The most impressive tests so far have involved making use of laboratories supplied by Nature herself - the so-called binary pulsar systems. These are distant neutron star-neutron star binaries, where (at least) one star is seem as a pulsar, i.e. a rotating lighthouse illuminating the Earth with radio waves once per rotation. This rotation is remarkably steady. This in effect means that Nature has supplied us with a very accurate clock moving in a relativistic spacetime. By analysing the received time of arrival of these pulses, astronomers have carried out remarkably precise tests of many predictions of General Relativity, including energy loss to gravitational waves, which won Hulse & Taylor the 1993 Nobel Prize. The aim of this project is to understand the theory behind these tests and what they tell us about relativistic gravity. The semester 2 course MATH3006 Relativity, Black Holes and Cosmology is very useful but not essential for this project. [Level 3/4]

Background reading:

R. A. d'Inverno, Introducing Einstein's Relativity , Clarendon Press

B. F. Schutz, A First Course in General Relativity , CUP

A. Lyne and F. Graham-Smith, Pulsar Astronomy, CUP

## Ramin Okhrati: Generalized functions: Theory and Application

The theory of generalized functions (also known as theory of distributions) extends the concept of ordinary functions. This extension will have lots of advantages both theoretically and form an applied point of view. For instance, despite ordinary functions, all generalized functions are differentiable. The purpose of this project is to develop the theory of generalized functions and then study their applications. While this theory is mostly applied in physics and engineering, this project focuses on its financial applications. Using this theory, the sensitivity of an expectation (or simulation) with respect to its underlying parameters can be measured. This is especially useful when there is no simple formula for this expectation. In particular, one can then estimate the sensitivity of an option price with respect to its underlying parameters. [Level 4]

J. Ian Richards, Heekyung K. Youn, (1995). The Theory of Distributions: A Nontechnical Introduction. Cambridge University Press.

## Giles Richardson: Modelling of Magnetically targetted drug delivery

In conventional (systemic) drug delivery the drug is administered by intravenous injection; it then travels to the heart from where it is pumped to all regions of the body. Where the drug is aimed at a small target region this method is extremely inefficient and leads to much larger doses (often of toxic drugs) being used than necessary. In order to overcome this problem a number of targeted drug delivery methods have been developed. One of these, magnetically targeted drug delivery, involves binding a drug to micron-sized biocompatible magnetic particles, injecting these into the blood stream and using a high gradient magnetic field to pull them out of suspension in the target region. Once on the vessel wall the drug can either be released directly into the blood stream or a biological technique can be used to ensure uptake of the particles into the tissue.

The aim of this project is to formulate a simple model for the transport of drug carrying magnetic particles in the blood stream, use it to track the distribution of particles within the bloodstream and to assess where the particles are pulled out of suspension. [Level 3 or level 4]

## Giles Richardson: The mathematics of musical instruments

In this project we will look at solutions to the wave equation describing the vibrations in a musical instrument. These solutions can be characterised by an eigenvalue problem whose eigenvalues give the frequencies with which the instrument vibrates (and hence the frequencies of the notes produced by the instrument). The lowest eigenvalue to this problem gives the fundamental frequency of the instrument (e.g. the G generated by the G-string on a guitar) while the higher eigenvalues give the harmonics associated with the tone of that particular instrument. It is thus possible to characterise the tone of an instrument by the eigenspectrum of its characteristic eigenvalue problem. Indeed a perfect 5th and a major 3rd are prominent harmonics of a stringed instrument (or of a wind instrument) and this is probably the reason for their importance in Western music.

## Giles Richardson: Organic Solar Cells

In semiconductors electric current is transported both by negatively charged excited electrons lying in the conduction band of the material and by the positively charged ‘holes’ that they leave behind in the valence band of the material. In an organic solar cell two organic semiconducting materials, with very different properties, are joined together to form a junction. These materials are chosen so that one shows a strong affinity for holes and the other a strong affinity for excited electrons and are termed hole and electron carriers, respectively. The junction between the two materials is used to separate hole-electon pairs created by incident sunlight so that positively charged holes end up in the hole carrier while negatively charged electrons end up in the electron carrier. This charge separation between the two halves of the solar cell can be used to drive an electric current around a circuit and is thus a means of converting solar power to electric power. The project will focus on the modelling of the charge transport in the device and the generation of charged pairs via solar radiation. Charge transport within semiconductors can be modelled by straight-forward advection diffusion equations for the motion of the electrons, and the holes, that account for the diffusive motion of these particles and their motion due to the forces they experience from the electric field in the device. In turn the electric field in the device is coupled to the densities of the charged particles. [Level 3 or 4].

## Janne Ruostekoski: Numerical integration of nonlinear propagation equation

Nonlinear propagation equations play an important role in numerous applications and frequently require numerical solutions. In this project you will learn how to integrate numerically the Gross-Pitaevskii equation (also known as the nonlinear Schroedinger equation),

using the Crank-Nicholson algorithm. The Gross-Pitaevskii equation is an important PDE that describes the dynamics of coherent matter waves, superfluidity and light propagation in nonlinear optics. [Level 3 or Level 4]

Prerequisites : Differential equations. Some knowledge of a numerical software (e.g., Matlab or Maple) or a programming language (e.g. Fortran).

## Ruben Sanchez-Garcia: Community detection in networks

Community structure is one of the most relevant features of graphs representing real-world systems (networks). It consists on identifying groups of vertices (clusters or communities) highly connected among themselves while poorly connected to other vertices.

This project consists firstly on a description of the basic terminology on network theory and community detection, and an overview of the most popular methods, with an emphasis on spectral and modularity-based methods.

Then the student will choose one of the community detection methods to be studied in depth and implemented as a computer algorithm. The algorithm will then be validated on a small set of real-world network examples.

[Level 3 or 4]

Reference: Fortunato, Santo. Community detection in graphs. Physics Reports (2010).

Pre-requisites: None, but programming experience (e.g. Matlab in MATH3018) desirable.

## Kostas Skenderis: Spacetimes of constant curvature

Spacetimes of constant curvature are solutions of Einstein's equations with a cosmological constant. The solution with positive cosmological constant, the de Sitter spacetime, was found by de Sitter in 1917 and describes a Universe undergoing a rapid, exponential expansion. The dynamics of such a University is dominated by the cosmological constant which corresponds to dark energy in our Universe or the inflaton field in the very early Universe. When the cosmological constant is negative, the solution, the Anti-de Sitter solution, describes a negatively curved Universe. This solution has played a prominent role in many recent developments in theoretical physics. The aim of this project is to understand the main properties of these spacetimes. (Level 3 or 4)

Prerequisite course: General Relativity.

Background reading:

Gravitation (Misner, Thorne and Wheeler)

An Introduction to Einstein's General Relativity (Hartle)

A First Course in General Relativity ( Schutz)

## Tim Sluckin: Phylogenetics

The idea of a “tree of life” goes back to biblical times, and the idea that similarities between organisms is due to recent or less recent common descent goes back to Charles Darwin. The mathematical discipline which builds up family trees of species, subspecies or organisms from comparing common phenotypic or genotypic properties is called *phylogenetics*, and the resulting family tree is known as a *phylogeny*. In earlier years of the last century the only traits available for comparison were phenotypic (i.e. more or less, what the animal looks like). As progress is genetics improved, it became possible to compare biochemical properties in the organism. More recently still it has been possible to compare directly DNA sequences. [Level 3 or 4]

**What the project involves:**

- Learning and explaining mathematical approaches to the construction of phylogenies in general
- Learning how to use one of the standard phylogeny engines available on the internet, and using it to construct a phylogeny from data available on the internet.
- My particular interest is in human population history, and there is much current interest in the evolutionary history of human groups.

**References:**

M. Nei and S. Kumar, *Molecular Evolution and phylogenetics *(OUP 2000)

J. Felsenstein, (2009) ‘*Theoretical evolutionary genetics’*, Seattle. http://evolution.gs.washington.edu/pgbook/pgbook.pdf

See also Felsenstein’s web page: http://evolution.gs.washington.edu/felsenstein.html

## Tim Sluckin: the population singularity

Some demographers, usually with a background in mathematics, physics or econometrics, claim that population data over the last several thousand years, as well as more recent indices of economic growth such as the Dow Jones stock market index, all exhibit *hyperexponential* behaviour. Hyperexponential means growing faster than exponential, and in principle may imply a divergence in the relevant property at some time in the future, although obviously in the case of populations this is impossible. Nevertheless, the imputed date of the divergence may represent something interesting which will happen in the future. Hyperexponential behaviour is not predicted by standard logistic models, which all include some population saturation. In the case of humans, the implicit idea is that many people eat more food, but also do more work, thus changing the environment, and increasing the effective carrying capacity. This may lead to runaway effects. [Level 3 or 4]

**What the project involves:**

- Reading the current literature on the subject
- Constructing and checking mathematical models which lead to hyperexponential population growth
- Comparing different data sets which suggest the existence of a “singularity”, to check the validity of claims in the literature

**References:**

D. Sornette, *Why Stock Markets Crash (Critical Events in Complex Financial Systems) *Princeton University Press, 2003

http://eclectic.ss.uci.edu/~drwhite/pub/WorldPopulations.pdf

Korotayev et al, http://www.academia.edu/3116549/Secular_cycles_and_millennial_trends

Johansen and Sornette, Physica A **294 ** 465-502 (2001)

S. P. Kapitza , *The phenomenology of world population growth* Sov. Physics Uspekhi ** 39**, 57-71 (1996) http://ufn.ru/en/articles/1996/1/c/

## Tim Sluckin: Cliodynamics

History has traditionally been one of the most qualitative subjects. But historians make analyses of current and ancient politics to draw conclusions about cause. One might argue that history is merely the laboratory of anthropology, which itself is the internal population dynamics of human societies, and as such a branch of population biology. Isaac Asimov in his *Foundation * series of science fiction books fancifully postulated the existence of a probabilistic theory of history, which he called *psychohistory*, but until recently no-one has tried to put such a programme into practice. But history poses all sorts of questions which might have a precise answer. Examples of this might be “why is China a unitary state, but Europe, about the same size, seems to be divided into small statelets?”, or “is there some logic to the present division of Europe into states, or is the division completely random?”, not to mention, “why did Spain invade Mexico, and not the Aztecs invade Iberia?”. Recently a discipline, known as *cliodynamics*, has developed which seeks to put some of these problems into a quantitative state. This project will look at the state of the art in this field. [Level 3 or 4]

**What the project involves:**

- Reading and reviewing the current literature on the subject
- Constructing and checking mathematical models which lead to observable historical conclusions
- Perhaps doing some statistical research to find temporal correlations between historical events in recent or less recent history.

**References:**

The leading worker in this field is Professor Peter Turchin of the University of Connectitut. He has a blog on http://cliodynamics.info/, which gives a load of useful references and a summary of the field as it is now. His book *Historical Dynamics: Why States Rise and Fall*, Princeton University Press (2003) is particularly interesting.

## Marika Taylor: Perturbation and stability methods

Many problems of physical interest can be dealt with using asymptotic limits, in which a parameter or coordinate in the problem takes large or small values. Such perturbative, asymptotic, methods can be used to gain insight into the underlying structure and in many cases give a sufficiently accurate solution of the problem at hand. Given a solution, the next question is that of stability, which is very important in most physical problems: if we make small changes to the initial state, will the behaviour at late times be similar or very different? In this project a number of perturbation and stability methods will be explored and then applied to a range of physical problems. For example, the so-called matched asymptotic expansion method for solving differential equations is applicable to many fields, ranging from fluid problems involving regions of rapid variation right through to astrophysical and quantum mechanics problems. The physical applications explored in this project will be decided jointly with the student, following their interests. [Level 3/4]

Prerequisites: Confidence in analysis, and some prior interest and knowledge about fluid dynamics and electromagnetism, such as is covered in MATH 2044 and MATH 3071.

Background reading: E.J. Hinch, "Perturbation Methods", Cambridge University Press (1991)

## James Vickers: Stellar evolution

This project will consider possible mathematical models for the structure of various types of star. In particular it will consider problems relating to the equilibrium and stability of stars -- including White Dwarfs, Neutron Stars and Supermassive stars -- and to an understanding of how stars evolve. Most of the models will use the Newtonian theory of gravitation, but there will also be an opportunity to study the relativistic stages of the evolution of cosmic objects, including a study of collapse to a Black Hole. [Level 4]

S. Chandrasekhar, An Introduction to the Study of Stellar Structure , Dover.

## James Vickers: Gravitational waves in General Relativity

This project follows on from the first-semester unit MATH3006 Relativity, Black Holes and Cosmology , which includes a brief consideration of gravitational waves. The project will develop this further, first looking at the books of Schutz and Griffiths. Various extensions are possible, depending on the interests developed by the student. [Level 3]

R. A. d'Inverno, Introducing Einstein's Relativity , Clarendon Press B. F. Schutz, A First Course in General Relativity , CUP J. B. Griffiths, Colliding Plane Waves in General Relativity , Clarendon Press.

## James Vickers: Dynamical systems and cosmology

Even if one makes assumptions about the symmetries of our universe it can be very difficult to find exact solutions of Einstein's equations with those symmetries. However in the study of certain types of cosmology it is possible to reduce the Einstein's equations to a non-linear system of ordinary differential equations. Although it is usually still not possible to solve the resulting equations exactly, one can use techniques from the study of dynamical systems to understand the qualitative behaviour of the solutions. This project will look at how one can make such a reduction to a finite dimensional dynamical system and use these methods to study models of both stable and chaotic cosmologies. This is a Level 3 project for a student who has taken MATH3006 Relativity, Black Holes and Cosmology.

Kostas Zygalakis: Numerical solution of Stochastic Differential Equations

Many physical phenomena cannot be described realistically in deterministic terms and we need to allow for the effect of randomness. This randomness might be inherent in the system (quantum mechanics), coming from incomplete information (weather prediction), or coming from dimensional reduction, i.e. describing a complicated deterministic system with many degrees of freedom, using a lower dimensional one (a heavy particle colliding with lighter ones). In all the aforementioned cases the correct mathematical framework for describing our variables is stochastic.

The aim of this project (which can be offered both as a third or fourth year project) is to expose the student to the ideas of stochastic modelling and in particular modelling using stochastic differential equations (SDEs). The project will then focus on some different numerical methods for solving SDEs. [Levels 3 or 4]

Pre-requisites: Some understanding of basic probability and numerical analysis and some familiarity with a numerical software (most preferably Matlab)

Background reading:

[1] Peter E. Kloeden and Eckhard Platen. Numerical solution of stochastic differential equations. Springer, 1992.

[2] Peter E. Kloeden, Eckhard Platen and Henri Schurz. Numerical solution of SDE through computer experiments. Springer, 1994.

[3] Desmond J. Higham. An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM J. Numer. Anal., 43(3): 525–546, 2001

## Kostas Zygalakis: Data Assimilation

Data assimilation is the problem of estimating the state variables of a dynamical system, given observations of the output variables. It is a challenging and fundamental problem area, of importance in a wide range of applications (weather prediction, control theory to name a few). If one considers the class of linear systems, then it is possible to estimate in an exact (probabilistic way) the state variable of the linear dynamical system in question. However, the situation is more complicated when one deals with a non-linear dynamical system since then performing such an exact estimation becomes much more difficult.

The aim of this project is to introduce the student to some of the fundamental concepts of data assimilation. In addition, the student will become familiar with some basic numerical algorithms used in data assimilation and get to apply them in some interesting but simple dynamical systems. [Levels 3 or 4]

Pre-requisites: Some understanding of basic probability, especially Bayes rule and some familiarity with a numerical software (most preferably Matlab)

Background reading:

[1] Tzyh-Jong Tarn; Rasis, Y.; , Observers for nonlinear stochastic systems, Automatic Control, IEEE Transactions on , vol.21, no.4, pp. 441- 448, Aug 1976

[2] Duan, L. and Farmer, C. L. and Moroz, I. Sequential Inverse Problems Bayesian Principles and the Logistic Map Example. 8th International Conference of Numerical Analysis and Applied Mathematics, AIP Proceedings (2010)

[3] E. Kalnay. Atmospheric modeling, data assimilation, and predictability.

Cambridge Univ Pr, 2003.

[4] K. Law and A.M. Stuart: Evaluating Data Assimilation Algorithms. Mon. Wea. Rev. doi:10.1175/MWR-D-11-00257.1, in press