LIST OF THIRD YEAR PROJECTS FOR THE ACADEMIC YEAR 2007/08

Project 1:  Can we distinguish between a pair of knots? [up to 3 students]

Take two pieces of string and tie a knot in each. Although they may look very different, is there any way to determine whether the two knots are in fact the same? This is one of the fundamental problems in Knot Theory, and this pair of 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 the two projects offered 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 2:  Hittings and Crossings: ruin, wear, and degradation [up to 7 students]

In many areas of applied mathematics we are concerned with the first time to hit or cross a boundary.

• Bridges fail and buildings collapse when the loads on them exceed their capacity

Seven Die In Rush-Hour Bridge Collapse: About 20 people are still thought to be missing after a motorway bridge collapsed in Minneapolis, sending cars, lorries and pedestrians tumbling into the River Mississippi and leaving at least seven people dead.

Hurricane Katrina: Hurricane Katrina was the costliest and one of the deadliest hurricanes in the history of the United States.  It was the sixth-strongest Atlantic hurricane ever recorded and the third-strongest hurricane on record that made landfall in the United States.... The most severe loss of life and property damage occurred in New Orleans, Louisiana, which flooded as the levee system catastrophically failed.

•  Financial

Rush on Northern Rock continues: Customers have been withdrawing money despite appeals for calm. The rush of customers taking money out of Northern Rock continued for a second day on Saturday, amid concerns over its emergency Bank of England loan.

Cocoa futures: Traders who have sold short close their contracts when the price falls below far enough below the price they sold at.

Nuclear future: Will the price of nuclear electricity ever be low enough to justify investing in nuclear power in the future?

Insurance: Insurance premiums have to be set to ensure that the insurer always has enough funds to meet the claims on the fund.

•  When should plant and machinery be serviced?

Plant and machinery deteriorate as they are used.  Repair and maintenance have to be scheduled in an optimum manner to avoid breakdowns while not spending excessive amounts on maintenance. 

When should the local authority replace a road rather than just patching the surface?

Stochastic Processes and Boundary Crossings

I am interested in formulating many of the above problems in terms of stochastic processes. For the projects this would entail looking at the properties and applications of Markov processes, or renewal-reward processes, or Brownian motions and Wiener processes, or extreme value distributions.  There are various excursions you can make: financial mathematics; the fractal nature of Brownian motion; simulation; optimal decisions under uncertainty; extreme value statistics; finding friendly Markov chains in more complex less friendly processes; even some analysis.  The elements are indicated in the innocent looking Figure below.

Project 3:  Vedic Mathematics [the combined number of students on projects 3,5,12,16 and 18 should be 12]

Vedic Mathematics is comprised of a system rules, which have been discovered or redicoved by Jagadguru Swami Sri Bharati Krishna Tirthaji Maharaja between 1911 and 1918 from the ancient sacred Hindu text, the Vedas. The mathematical system developed in this way is based on sixteen Sutras in word form. It is still debated whether Vedic Mathematics is actually vedic in its origin or whether it is Mathematics at all. The aim of the project is not to answer the first question, which is left to Hindu scholars, but to address the second question concerning the mathematical content of the discovery. Certainly Vedic Mathematics provides some rules and techniques, which are of use for concrete computations and applications. The idea of the first part of the project is to explain these rules in conventional mathematical terminology and thereafter to provide proofs for these statements. Some elementary number theoretical proofs exist in the literature. The project should enable to answer the grant claim of whether Vedic Mathematics can encompass all of Mathematics.

Project 4:  The mathematics of juggling [up to 3 students]

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: Solitons [the combined number of students on projects 3,5,12,16 and 18 should be 12]

The first reported observation of a soliton was made in1834 by the Scottish naval engineer John Scott Russell. In the union canal near Edinburgh he observed a wave, which was moving at a constant speed whithout changing itsshape. He was even able to follow the wave on his horse for several kilometers. Despite the fact that John Scott Russell was convinced of the importance of solitons and could reproduce them experimentally in a tank, which he built in his garden, the subject was not studied until sixty years later, when Korteweg and deVries found some equation, which admitted a solitary wave as solution. Thereafter many more equations have been found, which exhibit such solutions and various applications have been developed ranging from fibre optics to taking solitons as a model for elementary particles and even some financial market systems have solitary wave solutions. The aim of the project is to understand these phenomena and to explain the Mathematics behind the phenomenon. In particular, the Hirota method should be explained and applied. Possibly one could touch on Sato's theory which is the foundation of these techniques.

Project 6: Option Value Theory [up to 14 students distributed in groups of 4 or 5 for topics (a), (b) and (c)]

The main objectives of this project are to obtain solutions of the Black-Scholes equation focusing on one of the following areas of investigation:
(a) barrier options,
(b) time-dependent volatility,
(c) numerical methods,
and to examine the dependence of the solutions on the various financial parameters involved. Further information on each of the above topics may be found, for example, in chapters 12, 6 and 8 respectively of `The Mathematics of Financial Derivatives: a Student Introduction' by P Wilmott, S Howison and J Dewynne, published by Cambridge University Press.

Project 7: Error correcting codes [up to 3 students]
      

In the Internet age, reliable communication between computers isessential. However, it is not always possible to guarantee that thecommunication channels are reliable - particularly with the rise of wireless and mobile technologies. Error correcting codes are a meansof transferring information in such a way that the content can berecovered even if some part of the data is corrupted.

These projects will be concerned with various aspects of the theory behind error correcting codes. There is the possibility of including a computation element (if so desired).

Project 8: Quantum graphs [up to 3 students]

Graphs represent a powerful graphical way of dealing with a wide range of problems related for example to networks. Basically, they consist of a collection of edges and vertices. Motivated by quantum mechanics, it is interesting to define differential operators on the edges while the vertices represent boundary conditions. This gives rise to the notion of quantum graphs which offer many promising perspectives in terms of applications in nanotechnologies for instance.

Mathematically speaking, the question usually dealt with is the contruction and classification of all the self-adjoint extension of a given operator on a given graph. This project could be undertaken by several students, each one focusing on different aspects e.g. the theory of self-adjoint extensions, domain of applications of such graphs, etc.

Project 9: Games and surreal numbers [up to 2 students]

How can mathematics help us to play games? One possibility is to study a particular game, and find ways to determine the best move to make in any given situation.  However, this is rather hard, and the games that can be considered are all very simple.\medskip

Instead, we might want to determine who should win a game (if they play well) and by how much. The aim of this project is to study this problem, and explain how even the simplest looking games can leading to some remarkable mathematics.\medskip

In particular it will be necessary to introduce the surreal numbers. As the name suggests these are much stranger than the ordinary real numbers - for example, we can define numbers like the square root of infinity! The project will discuss some of the basic properties of such numbers, and how they arise in the study of games.

Project 10:  The mathematics of holograms [up to 4 students]

Everybody knows holograms, these fancy 3D pictures created from 2D support. But not everybody knows the mathematics behind them, for instance some of the equation describing the physical behaviour of light. There are many aspects that can be covered in this project and the students who are interested could concentrate on some of the following non-exhaustive list: Historical account of the holograms which led to a Nobel Prize for Dennis Gabor. Why did that deserve a Nobel prize?

Mathematical description of the phenomenon (derivation of some set of equations and their interpretation). Are there different types of holograms? What distinguishes them?

Applications : leisure, security (bank notes, credit cards, official documents, etc), 3D imaging, etc. Why are they so useful and what advantages do they offer?

Project 11: The discrete logistic equation [up to 4 students]

In 1976 R.M. May (now Lord May) published a pivotal paper looking at "Simple Mathematical models with complicated dynamics''. This paper is available at http://organic.usc.edu:8376/reference/01May76.pdf. The idea of this project is to look in more detail at some of the interesting mathematical results quoted and in subsequent works by others on the discrete logistic equation, giving understandable proofs of the many interesting results to be found.

Project 12:  Quantum computing [the combined number of students on projects 3,5,12,16 and 18 should be 12]

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.

Project 13: Breaking Ciphers [up to 5 students]

Over the years many attempts have been made to ensure the secrecy of messages being carried from one person to another. These range from what are now thought of as simple cyphers such as the Caesar shift cypher:

DZXP NZOPD ESLE HPCP ESZFRSE DPNFCP EHZ ESZFDLYO JPLCD LRZ NLY MP MCZVPY MJ NSTWOCPY.

and simple substitutions such as in this example:

``VQN TSVNMMTANSUN INPHTUN UOSUMWENI VQFV TPFX QFI UQNDTUFM FSE LTOMOATUFM
RNFYOSI, VQFV IFEEFD QFI UOSVTSWNE VO YPOEWUN VQND, VQFV QN QFI NZTIVTSA
FSE FUVTHN DTMTVFPC YMFSI BOP VQN WIN OB UQNDTUFM FSE LTOMOATUFM RNFYOSI,
RQTUQ UOWME LN FUVTHFVNE RTVQTS 45 DTSWVNI'' - VOSC LMFTP

More complex codes and their breaking need a more detailed understanding of, for example, permutations, statistics and the properties of prime numbers as well as some guessing!

In these projects students will focus on the breaking of cyphers and how mathematics, statistics and logic are used in their cracking. We will look at some classic codes, including some examples such as the above or looking at the way that Alan Turing and others managed to break the Enigma cypher in World War II.

See Simon Singh's book ``The Code Book''.

Project 14: Dimensions and Fractals: measuring complexity [up to 4 students]

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. 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 appear?, how has their study developed?, what are their main characteristics from the mathematical point of view? etc. Each student on the project should concentrate (after a general introduction) on a different kind of fractal 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. 

Project 15: The theory of special relativity [up to 4 students]

The theory of special relativity was proposed by Albert Einstein in 1905 in his paper entitled "On the Electrodynamics of Moving Bodies". About 300 years earlier, Galileo had already stated that all uniform motion was relative, and that there was no absolute state of rest. In other words whether an object moves or not depends on the relative position of the observer with respect to the observed object. Einstein's theory  is based on two fundamental axioms:

• Galileo's relativity principle: No experiment can mesure in an unambiguous way the velocity of an inertial observer.
• The universality of the speed of light: The speed of light is a universal constant, which is independent of the relative velocity between the source emiting light and the observer.  In addtion, no material object can travel faster than light.
  

As we have said, the first axiom was already quite old, but the second one implied a completely new understanding of the way space and time are tied together and, in particular, contradicted the Newtonian notions of absolute space and time. It had various a priori "surprising" consequences that have since been verified experimentally: the equivalence of matter and energy expressed in the famous equation E=mc², where c is the speed of light, the fact that time and space become relative to the observer and that on the contrary the speed of light becomes an absolute quantity, independent of  the velocity of the observer himself.

The aim of this project is that the student or students write down an introduction to the theory of special relativity, emphasizing the mathematical aspects of the latter and explaining how the two main axioms above imply the relativity of space and time.  The project should also describe the main differences with respect to Galileo's theory and show how the theory of special relativity is consistent with the latter for velocities much smaller than the velocity of light.

Project 16: Neural networks [the combined number of students on projects 3,5,12,16 and 18 should be 12]

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: Cryptology [the combined number of students on projects 3,5,12,16 and 18 should be 12]

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. 

Project 18: The butterfly effect: what is chaos? [up to 4 students]

In mathematics 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. In his original work, Lorenz took a=10 and b=8/3 and varied r. He found chaos for r=28. For these values of the constants, a three dimensional plot of the variables {x(t),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.

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. 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 (or other examples)  and answer 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 or EXCEL) in order to investigate the equations numerically.

Project 19: The millennium problems [up to 4 students]

In May of the year 2000, in a highly publicized meeting in Paris, the Clay Mathematics Institute announced that seven $1 million prizes were being offered for the solution of each of seven unsolved mathematical problems. These problems had been chosen by an international comitee of mathematicians to be the most difficult and most important in the field today. The seven millennium problems are:

1. The Riemann Hypothesis
2. Yang-Mills theory and the Mass Gap Hypothesis
3. The P versus MP Problem
4. The Navier-Stokes equation
5. The Poincaré conjecture (recently solved by G. Perelmann!)
   
6. The Birch and Swinnerton-Dyer conjecture
7. The Hodge Conjecture

You can find more information about them in the Clay Mathematics Institute website www.claymath.org. A very nice and simple introducction to each of the problems is also given in the book: The millennium problems by Keith Devlin. The idea of this project would be that you write a report about each (or some) of this problems. Some of the problems in the list are in fact too involved for a 3rd year project, but problems number 1, 3, 4 and 5 are suitable as dissertation topics.

Project 20: Number theory applied to cryptography [up to 7 students]

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 21: The concept of Entropy and its applications to information and coding [up to 7 students]

Much of the current technology is concerned with the transmission and storage of information. Most of the devices that we use daily either transmit/receive information (telephones, radios, TV sets), store it (MP3 players, CDs, DVDs, memory sticks) or do both (computers, mobile phones). This technology has been made possible by great advances in physics and engineering, but also by the creation by Claude Shannon of a mathematical theory of communication in 1948.

The concept of entropy is central to the mathematical description of information. Given a source that generates a string of symbols (e.g. a text, a numerical code, or the sequence of nucleotides in the human genome), its entropy provides a quantitative measure of how much information it produces and of how much memory is needed to store that information. In most cases it is possible to encode the string of symbols in such a way as to reduce the amount of memory needed, a process known as "data compression'', and again the entropy tells us how much a given piece of data can be compressed without distortion.

In this project we will define the notion of entropy and discuss some of its properties. The theorems relevant to its applications to coding and data compression will be presented and motivated with examples (and, ideally, proved).

Project 22: Wallpaper patterns and crystals [up to 5 students]

When you buy patterned wall paper you will find that the pattern repeats itself, both along the roll and across it. For big patterns the across roll pattern may  only be clear when it is hung. How many repeat patterns are there? For example, in some patterns it will not matter which way up you hang each strip (or even if you hang them horizontally!) while for others it does. Some look the same in mirrors, others don't. In this project you will determine rigorously the number of possible repeat patterns  for wall papers. This can then be extended to look at, say, not repeating patterns in wall paper or the structures of crystals.