Third year projects 2005/06

LIST OF THIRD YEAR PROJECTS FOR THE ACADEMIC YEAR 2005/06

Project 1:  How can we generalise the rational numbers?

We begin our mathematical lives by learning about the natural numbers 0,1,2,... These quickly turn out to be insufficient for our needs, so we introduce negative numbers and fractions to allow us to do subtraction and division. At University we find that these collections of numbers are still not "large enough", and we begin to study real and even complex numbers. Could we possibly imagine any other types of numbers?

In fact, there are many different types of numbers used in Mathematics. Examples include finite fields (which are used in crpytography and computer science), p-adic numbers, and even a strange variant of the reals called surreal numbers. The aim of this project is to write a clear introduction to this area and its applications.

Project 2:  Option Value Theory with Time-Dependent Volatility

The main objectives of this project are to obtain solutions of the Black-Scholes equation in cases where the volatility is time-dependent and to investigate how the time variation of the option value is related to that of the volatility. It is expected that the investigation will involve both numerical and analytical techniques and that the results will be interpreted in the context of financial applications.   

Project 3:  The Science of the Ethics of Finance:
Islamic Finance and other Ethical issues treated Mathematically

Most financial instruments arising in 'western-style' finance may be understood mathematically. Many such instruments are  designed to work without ethical constraints, or consideration of their social implications. On the other hand some financiers choose to apply ethical and related constraints to their financial instruments. A prime example is the instruments of Islamic finance (although there are plenty of others). These instruments still operate  in a framework which is prescribed by Mathematical considerations, but they are also constrained by (say) consideration of Sharia law. The purpose of this project is to provide a mathematical, logical, semantic analysis of such instruments. The first aim is simply to explain the instruments mathematically,  and analyse their relationship with the ethical constraints which motivated them. The second aim is to use this machinery to  discuss their fitness for purpose, and perhaps to consider alternatives (and the reasons why conventional instruments may not be acceptable). This is a rigorous project - following a strict definition-based construction of components, with logic based arguments. It is anticipated that some of these arguments will use mathematical language, and others logical argument in English. As such a good mastery of English is (unfortunately) a prerequisite. The project is suitable for individual student study, or possibly for a team of two students developing arguments in dialogue (but then writing independent reports).

Project 4:  The mathematics of juggling

What kind of mathematics can be used to describe juggling? And what kinds of questions can it answer? This project will be an introduction to Combinatorics -- a branch of mathematics used in many areas in Computers Science, Statistics, and general Mathematics -- concentrating on juggling as a source of particularly simple and easy to understand examples.  

Project 5:  Real numbers

Zorg has been rotated into our universe from the Zeta quadrant. She has a good grasp of the system of natural numbers (the whole numbers), and has learned English, but has never seen a real number (such as an irrational number) before. It is your job to explain the real number system for her (perhaps using Cauchy sequences?).

Project 6: Nonlinear stock market price dynamics

Stock market prices are strongly influenced by two classes of investors, so-called fundamental value investors and so-called trend followers. Recently (2001) a simple two-dimensional model has been proposed which simulates the influence of such type of investors on prices in the stockmarket in a nonlinear fashion. One of the two dynamical variables is taken to be the logarithm of the ratio of the observed price over the fundamental price. The other variable is the market trend. The system is known to possess an unstable fixed point when the observed price equals the fundamental one and the market trend is zero, that is when the stock does not know  in which direction to develop. The purpose of the project is to study this model in some detail depending on various market parameters. This project can accommodate 1-3 students.

Project 7: Graphs and maps

Given a map of an area divided into regions, how many colours are needed so that no two adjacent regions have the same colour? Can you redraw the map so that each region is replaced by a square, without altering which regions are adjacent? Can you visit every region exactly once without retracing your steps?

These are examples of problems that can be solved using Graph Theory, an area of mathematics with many practical applications (for example in computing and telecommunications). This project will provide an introduction to the methods used to solve such problems.

Project 8:  Applications of physics to economics and finance:
Money, wealth and the stock market

Many problems arising in the context of economics and finance can be analyzed by using methods and concepts from physics. For instance, by analogy with energy, the equilibrium probability distribution of money must follow the exponential Boltzmann-Gibbs law characterized by a temperature equal to the average amount of money per economic agent. Based on this observation one can construct a  thermal machine which extracts a monetary profit between two economic systems with different temperatures. With such analogies one can investigate stock market fluctuations, probability distributions of income and many other problems in economics and finance. The purpose of the project is to test such ideas which were presented in a recent (2003) essay. This project can accommodate 1-3 students.

Project 9: Perspective and Projective Geometry

We are all familiar with the idea of perspective : train tracks seem to converge in the distance, objects look smaller as they move further away from us, etc. These phenomena occur when our 3-dimensional space is projected onto a 2-dimensional plane (our retina, a photograph, a painting...). Projective Geometry can be seen as the mathematics behind perspective. Since the Renaissance, painters have used it to represent accurately our 3-dimensional world onto their canvas. More recently, scientists have used Projective Geometry to reconstruct a scene from photographs or paintings.

The aim of these projects is to give a rigourous introduction to Projective Geometry, focussing on the projective plane and its properties, and to study some applications.

Project 10: Quantum finance

Most financial models which simulate market behaviour are of stochastic type. In such models coherent effects, such as the influence of the trade of securities on the price distribution of the next trade are not captured. Recently (2003) it has been proposed to use a quantum theoretical description, which can take such effects naturally into account. The idea of the project is to describe such a quantum theoretical model. Key ideas are for instance that all possible realizations of investors holding securities are taken as a basis for a Hilbert space. Linear operators acting in this space describe basic financial transactions, such as cash transfer and buying and selling of securities. Simple Hamiltonians can be used to describe the temporal evolution of the market. This project can accommodate 1-3 students.

Project 11: The geometry of animation 

Shrek is an oscar winning animated film which was `shot' by creating a virtual world in a computer and then rendering scenes from this world entirely in software. The rules for this world were: 1. obey the director; 2. obey a few rules of ordinary physics, and 3. obey the rules of 3D geometry (such as those studied in the Geometry and Vectors course). The idea of this project is to exercise or reinforce our understanding of this applicable geometry by making a `mini--Shrek' (perhaps 5 seconds of animation). Step 1 is to create a virtual geometrical world consisting, say, of a single human hand with water (perhaps rain) running down it and collecting in a droplet at one fingertip. Here `create' means to give a geometrical description of the shapes and positions of objects at each of a number of moments in time. This might be done using one of the standard computer languages for describing geometry (these languages are easy to learn, and no prior knowledge would be required). Step 2 is to `film' this action (i.e. to choose where to view it from, how to move the `camera' etc.). Step 3 is to render this scene into a format suitable for viewing on your online CV/web page.

THIS PROJECT IS SUITABLE FOR UP TO 4 PEOPLE WHO WOULD PRODUCE SEPARATE SOLUTIONS TO STEPS 1 AND 2, BUT COULD COOPERATE ON THE UNDERLYING THEORY AND ON STEP 3.

Project 12:  Quantum computing

Computer chips are made smaller and smaller and one naturally imagines that eventually a limit is reached in this process once the atomic scale is reached and quantum mechanical effects become relevant. However, at this scale new possibilities emerge due to the occurrence of a phenomenon called entanglement, which is specific to quantum mechanics and shall be explored in this project. In fact P. Shor showed in 1995 that one can exploit this characteristic of quantum mechanics and design computers which are far more efficient than classical computers. The idea of this project is to give an account of the general principles and concepts of quantum computing, in particular on Shor's algorithm. By comparing the amount of additions and multiplications needed in a classical computer with the number of quantum gates needed, the potential power of a quantum computer will become apparent. This project can accommodate 1-4 students.

Project 13:  Solving unsolvable problems

For most problems in theoretical physics or applied mathematics analytical solutions are not known. Often this is only a matter of consistent investigations, but many times the situation is hopeless as a proper mathematical framework does not exist. Non-linear integral equations are examples for this, as there is no systematic solution procedure developed for them. With the advance of computer technology more and more problems allow for a numerical treatment. The idea of this project is to write ashort computing program to solve a set of coupled non-linear integral equations which play an important role in the theory of quantum integrable models. The problem can be treated without reference to its background in physics, but according to preferences one can also learn about the contextin physics where they occur. This project can accommodate 1-3 students.

Project 14: Dimensions and Fractals: measuring complexity

The notion of dimension we are mostly familiar with goes back to Euclid's definitions of point, line, surface and solid and to Descartes' coordinate geometry. This is the notion of topological dimension, which tells us that points have dimension 0, lines have dimension 1, the interior of a square has dimension 2 and that of a sphere 3 etc. However this concept of dimension can not always be used, for instance if we imagine a curve such that it covers all the 2-dimensional plane (as for example the famous Peano curves) and try to answer the question of what its dimension is we would not know whether to say 1 or 2. Precisely fractals are objects for which a similar problem arises and a more suitable definition of dimension (fractal dimension) need to be introduced. Examples of those are the so-called Hausdorff dimension (1919) or the box-counting dimension, which for most examples coincide. It turns out that we can find many systems for which the fractal dimension and topological dimension do not coincide. In fact, a definition of fractal introduced by B. Mandelbrot is that of a set whose dimension is strictly larger than its topological dimension. Nonetheless, the essential property of a fractal is that it has and infinitely complex structure which extends over all length scales. In other words, if we were able to magnify a picture of a fractal as much as we wish, we will see that no matter how small the part we look at is, it always possesses a complex structure, which when subsequently magnified is showing new details. Often these structures are self-similar, which means that, as we magnify a fractal we find structures which look very similar to the ones we had at  larger scales. It is also a mathematical feature of fractals that they are often generated by recursive equations. That is the case of two of the most widely studied groups of fractals known: the Julia sets and the Mandelbrot sets.

Fractals are very interesting objects from a mathematical point of view but they also resemble very often real structures or patterns we find in nature (like tree leaves, snow flakes, shells or the color patterns of certain animals). That makes them all the more fascinating!

In this project I would like the students to write an essay about fractals: how did they first appeared?, how has their study developed?, what are their main characteristics from the mathematical point of view? etc. Two or more students could do this project, in which case each of them should concentrate (after a general introduction) on a different kind of fractal (e.g. Mandelbrot sets, Julia sets etc) and describe the fractal of his/her choice in more detail. It would be very interesting if the students could actually generate some fractals numerically and therefore understand better how they come about. There are many "fractal-generating" programs which can be found on line for free or can be used on-line, and also many web pages with beautiful pictures of fractals and more or less rigorous explanations about them.

Project 15:  Einstein's miracle year

In recognition of the 100-th anniversary of Albert Einstein's "Miracle Year", the year 2005 has been announced as the World Year of Physics. In this year Einstein published four landmark papers, nowadays referred to as the Annus Mirabilis Papers, which have changed significantly many viewpoints in physics over the last 100 years. The papers deal with the introduction of the theory of special relativity, the explanation of Brownian motion, the description of the photoelectric effect and on matter energy equivalence. These papers explained experimental findings, which had up to 1905 no proper underlying theory and it is widely agreed that at least three of them were worthy of the Nobel prize. The purpose of the project is to characterize these papers (possibly not all of them) and discuss their significance by focussing on their mathematical content. Depending on preference one may include data about Einstein's life into this project and turn the project into a conference type poster. A satisfactory result can be displayed in the Centre of Mathematical Science. This project can accommodate 1-4 students.

Project 16: Neural networks

The description of nature is usually carried out by means of some model or theory based on various mathematical equations. However, often such model is not at hand or the situation one would like to describe is simply too complex. Neural networks are designed to overcome this deficiency and have the ability to derive meaning from very complicated or even imprecise data. They can be used to extract patterns and detect trends that are too complex to be noticed by either humans or other computer techniques. There are different kinds of neural networks, such as biological neural networks, as for instance the human brain or parts of it and also artificial neural networks originally referred to electrical, mechanical or computational simulations or models of biological neural networks. Meanwhile the field has expanded so much that some applications do not clearly resemble any longer an existing biological counterpart. The adaptability of neural networks is so large that their applications nowadays range from fundamental subjects as biology and physics even to banking, finance, insurance, marketing, manufacturing, etc. The purpose of the project is to provide an introduction to neural networks, give their brief history, investigate simple network architecture and consider some realistic applications. Students are encouraged to write a simple program for some neural network, such as for instance for pattern recognition or similar. The programs could written in MATLAB, Mathematica or Maple or similar languages. This project can accommodate 1-4 students.

Project 17: Knot theory

Take two pieces of strings and tie a knot in each. Although they may look very different, is there a way to determine whether the two knots are in fact the same? This is one of the fundamental problems in Knot Theory, and these projects will investigate the various methods that have been introduced to try to solve it. The study of knots is a relatively new subject, with applications in various areas of Biology, Chemistry and Physics. 

In these projects the students will each explore a different aspect of the subject. The goal is to write a clear but rigorous introduction to the chosen topic, and to apply the results obtained to a variety of examples.

Project 18: Criptology

Cryptology is the science concerned with communications in secure and usually secret form. The term cryptology is derived from the Greek kryptós (hidden) and lógos (word). Security is "guaranteed" to legitimate users, the transmitter and the receiver, being able to transform information into a cipher by virtue of a key, i.e. a piece of information known only to them. Although the cipher is inscrutable and often unforgeable to anyone without this secret key, the authorized receiver can either decrypt the cipher to recover the hidden information or verify that it was sent in all likelihood by someone possessing the key. Until recently this was mainly important to military circles, but in the computer age the applications of cryptology become more and more widespread, as for instance in on-line business transactions. Meanwhile cryptology has developed into an important branch of mathematics and the idea of the project is to write an account on cryptology and explain some of  its main concepts and techniques. This project can accommodate 1-4 students.

Project 19: The Goldbach conjecture

In 1742 Christian Goldbach proposed in a letter to Leonard Euler the following conjecture: "Every integer greater than 2 can be written as the sum of three primes", where he considers one as a prime number. Nowadays the conjecture is usually formulated as: "Every even number greater than 2 can be written as the sum of two primes". Even though this conjecture has been checked by computers up to 2 x 10^17, a rigorous proof is still not known. Between 2000 and 2002 a prize of 1.000.000 $ was offered for the proof, but it remained unclaimed. Goldbach's conjecture remains one of the most challenging unsolved mathematical problems. The idea of this project is to give an account of some of the attempts made so far and understand their limitations and to design a computer program which verifies the conjecture up to X, where X is the challenge. This project can accommodate 1-3 students.

Project 20: The butterfly effect: what is chaos?

In mathematics and physics chaos theory is understood as the study of certain non-linear dynamical systems which under particular conditions exhibit the phenomenon known as chaos. The latter is most famously characterized by the extreme sensitivity of these systems (equations) to the initial conditions. Many examples of such systems appear in the context of physics: the atmosphere, the solar system, plate tectonics, turbulence in fluids but also in other domains such as economics or populations growth. Some of the best known systems which exhibit chaos are:

The Lorenz model: this is one the most interesting and more widely studied systems of equations which exhibit chaos. The equations involved are three coupled first-order non-linear differential equations which describe in a simplified way the convection rolls arising in the atmosphere

               ẋ(t)=a (y(t) - x(t)),      ẏ(t)= x(t)(r-z(t))-y(t)     and     ż(t)= x(t)y(t)-b z(t),

with x(t) representing the intensity of the convection and y(t), z(t) related to the horizontal and  vertical temperature distributions. a, r, b are positive constants. The first two are known as Prandtl number and Reynolds number, whereas b is a constant having to do with the geometry of the system. In his original work, Lorenz took a=10 and b=8/3 and varied r. With the values of a, b chosen by Lorenz, chaos was found for r=28. For these values of the constants, a two dimensional plot of the variables {x(t),y(t)}, {x(t),z(t)} or {y(t),z(t)} for different values of t yields the famous Lorenz strange attractor, characterized by its shape, similar to the wings of a butterfly. In the chaotic regime, the system is extremely sensitive to the initial conditions {x(0),y(0),z(0)}, the main characteristic of chaos. In this context, the famous term "butterfly effect" was coined, to express the fact that the system of equations describing the atmosphere is so sensitive to any tiny change on the initial conditions that even the flapping of a butterfly's wings could have over time far reaching consequences, for instance the development of a tornado!

The logistic map: Another system of equations which sometimes exhibits chaotic behaviour is the so-called logistic equation. The latter describes a discrete one-dimensional dynamical system characterized by the equation:
                                                           x(n+1)= a x(n) (1-x(n)),

where n=0,1,2 ..., a is a constant parameter and x(n) takes values between 0 and 1. This equation was originally used  as a simple demographic model, with x(n) representing the amount of individuals of a certain population at year n. One of the most important features shown by this system is its universality, that means that although the behaviour of the system changes dramatically depending on the value of a, that behaviour is characteristic of many different such one-dimensional systems. For values of a>3.57 the system exhibits chaotic behaviour, which in particular means the values of x(n) for n very large are extremely sensitive to the initial value x(0). Before reaching the chaos regime there is a region where another peculiar phenomenon occurs namely that of bifurcation.

The aim of this project will be that the students familiarize themselves with the most basic aspects of the theory of chaos and write an introduction about the subject. Then each student should concentrate on one of the models described above (whose properties can be easily found in the chaos literature) and write an essay about the characteristics of the solutions for every case, answering if possible questions such as: for which values of the parameters does chaos emerge? What are the characteristic of the solutions which lead us to conclude the existence of chaos? Are there regions were stable solutions exist (fixed points)? How can be determine those? how do the solutions of our equations approach these fixed points? What is an attractor? What is a bifurcation? etc. It would be very instructive if the students could use some computer program (such as MATHEMATICA) in order to investigate the equations also numerically. The project is suitable for 2 or 3 students.

Project 21: Criptography

Cryptography is now part of our everyday lives, we use it whenever we send an email or shop online. The aim of these projects is to write a clear introduction to some of the mathematics behind it and to apply the results to a variety of codemaking and codebreaking problems.

Project 22: The isosprectral deformation method

Integrable models play an important part in modern Mathematics and Physics. Especially in Physics they have attracted an enormous amount of attention as they allow to solve various problems exactly as opposed to perturbatively in some small quantities. The starting point for many considerations are usually classical Hamiltonian systems. The purpose of the project is to provide an introduction to the method of isospectral deformation, which has been developed in that context and to apply it to some original problem. The key idea of this method is to obtain integrals of motion of some Hamiltonian system as eigenvalues of a matrix which depends on the dynamical variables. The essence of the method is that the spectrum of this matrix remains unchanged when the variables evolve subject to the equations of motion. This project accommodates 1-2 students with the intention to carry further study in form of a PhD.

Project 23: The mathematical theory of the fight for life:
The Lotka-Volterra equations

The original Lotka-Volterra equations, proposed independently by A.J.Lotka and V.Volterra in the twenties, consist of two coupled non-linear differential equations which describe the evolution in time of a biological system in which two different animal species (a prey and a predator) interact. Denoting by x(t) and y(t) the amount of preys and predators at time t, respectively the equations take the simple form

ẋ(t)= a x(t)-b x(t) y(t) for the prey,
ẏ(t)=-cy(t)+d x(t) y(t) for th predator.

where a,c denote the reproduction rates of preys and predators and b,c are the aggression intensities of the species to one another (in this simple model the predation rate is proportional to the rate at which pray and predator meet). The Lotka-Volterra equations have been later on generalized to capture more realistic features of ecological systems and are still nowadays object of research, not only in the fields of population dynamics, ecology and biology, but also in other very different contexts such as game theory or even finances where the variables x(t), y(t) can be interpreted as players or the wealth of individual investors, respectively. The aim of this project is that by studying a particular generalization of the Lotka-Volterra equations, each student familiarizes himself/herself with the study non-linear dynamical systems. As part of the project, he/she is expected to write an introduction to the subject which will explain notions such as fixed point, types of stability of fixed points and the technique of linearization of the equations around a fixed point. Then he/she should apply those concepts to the analysis of his/her particular system of equations. It should be very instructive if the students could use some computer program in order to solve numerically the equations for different values of the free parameters and therefore illustrate their analytical results with some plots. For example the program MATHEMATICA would be a good one, but if not available, other  programs might also work. This project is suitable for 3 or 4 students.

Project 24: Number theory applied to cryptography

Number theory is that branch of mathematics concerned with the properties of  integers. It contains many results and open problems that are easily  understood, even by non-mathematicians. More recently, the field has found  applications to practical problems, the most popular of which is  cryptography.

In this project we review the basic ideas of number theory that underpin several cryptographic protocols, with an emphasis on the Diffie-Hellman and  RSA methods.

Project 25: Brownian motion in Physics, Mathematics and Finance

Brownian motion is named after the biologist Rober Brown (1773–1858). While  examining pollen grains and the spores of mosses suspended in water under a  microscope, Brown observed minute particles in the pollen grains executing a jittery motion. He then observed the same motion in particles of dust, and in  other small particles suspended in liquids. Although he did not himself  provide a theory to explain the motion, the phenomenon is now known as Brownian motion in his honour.  Albert Einstein gave in 1905 a physical interpretation of Brownian motion that  helped establish the reality of atoms, which until then had been regarded as  theoretical speculation. More detailed theories followed Einstein's work, and  by now Brownian motion is an important branch of the mathematical theory of stochastic processes. In recent years, it has been found that Brownian motion  provides a rather good description of the fluctuations of financial markets, and has now become a fundamental object in all modern applications of mathematics to Finance. The goal of this project is to present a reasonably complete review of  Brownian motion as a stochastic process and of its various manifestations in the real world.

Project 26: From convection to chaos

The early investigations into chaos are intimately tied into the numerical study of problems in thermal convection. The equations of motion of a convecting fluid can be reduced to three simultaneous simultaneous differential equations - the classic Lorenz equations which display chaotic behaviour. Other related problems such as convection with the addition of a salinity gradient or the imposition of a magnetic field lead to slightly more complex models with five simultaneous simultaneous differential equations. This project will look at the derivation of these equations and see whether they are a good model for convection -- do these equations predict chaos in convection?

Project 27: Wavelets

Wavelets are a method of representing data by the use of orthogonal functions. Unlike Fourier series the individual functions not only convey information about frequency, but also information about position. Wavelets can be used for various applications such as data compression, solving matrix equations and solving differential equations. This project will look at the use of wavelets in the solving one of these three problems.

Project 28:  The use of Laplace transform in ordinary differential equations

Many linear ordinary differential equations do not have explicit solutions. One way that we can examine the properties of the solutions is by using the method of Laplace transforms. This technique will be covered in X3 Mathematical Methods. It is a method that can sometimes reveal how a solution y(x) behaves, for example, for small values of x or as x goes to infinity.  

Project 29:  Copulas and Their Applications

A copula is a function that allows the joint distribution of a collection of random variables to be expressed in terms of their marginal distributions. The copula C is such that the joint distribution can be expressed as

    FX(x)=C[FX1(x₁),FX₂(x₂),⋯,FXn(xn)]

The value of the copula is that it is frequently relatively easy to obtain marginal distributions and difficult to find joint distributions. The copula allows an insight into the joint distribution. Copulas are now widely used in financial mathematics and in many other fields. Students will explore one of their applications.

Project 30:  Stochastic Processes in Reliability and Maintenance

When planning inspection, maintenance and replacement for plant and equipment we need to construct models which describe the behaviour of the system through time. Using these models, optimum decisions can be made concerning when to inspect, repair or replace. Because the behaviour of systems is not deterministic, the models are based on stochastic processes. For example

  ∙  Why are cars generally recommended to have an annual service or MOT test?
  ∙  Is it worth buying a warranty when you buy a new product

This project will require the student to develop some basic knowledge of renewal processes and Poisson processes and describe their applications.   

Project 31:  Extreme Value Modelling

Extreme value modelling deals with the extremes of random variables and stochastic processes. It is typically used in planning against the effects of natural events such as tides and storms (hurricane Katrina, for example). In finance it is used to describe the extreme events which can disturb insurance funds or markets. This project provides an opportunity for the student to become familiar with some basic approaches to extreme value theory and its applications.

Project 32:  Survival Analysis

Survival analysis is at the basis of several areas of applied statistics, for example

 ∙  reliability modelling deals with the time in service of plant and equipment;
 ∙  medical statistics looks at the effects of treatments on the time to recover or the time to death of patients.

    The project is intended to introduce the student to some of the techniques and their applications.