Contents 
News from the CMS 

Professor Thanasis Fokas is the first applied mathematician to be elected as a full member of the Academy of Athens. Professor Fokas is the youngest of the 45 members of the Academy, and only the sixth mathematician to be elected. In January 2005, as one of only 15 people honoured by the President of the Hellenic Republic, he was created a Commander of the Order of Phoenix, which is the second highest honour bestowed by the President. In addition, in December 2004 he was awarded Honorary Degrees from the Technical University of Athens (Polytechneion), the University of Ptras and the Technical University of Crete. 

In January 2005, Professor Artur Ekert was one of the recipients of the Descartes Prize, which covers all areas of scientific endeavour where outstanding results have arisen from European collaborative research. Professor Ekert is head of the Centre for Quantum Computation within the Department of Applied Mathematics and Theoretical Physics (DAMTP), and one of the inventors of quantum cryptography, the subject of the prizewinning QuComm collaboration. 
Professor Artur Ekert 
Professor Sir Martin Rees, Professor of Cosmology and Astrophysics and Master of Trinity College, has been awarded the Crafoord Prize 2005 for "contributions towards understanding the largescale structure of the Universe". The Crafoord Prize is awarded by the Royal Swedish Academy of Sciences in recognition of research in areas not covered by the Nobel Prizes. In the Department of Pure Mathematics and Mathematical Statistics (DPMMS), Dr Steven Brooks has been awarded a Leverhulme Prize for his work in statistics., and Dr Ben Green received the 2004 Clay Research Award in recognition of his joint work with Dr Terry Tao on arithmetic progressions of prime numbers. In January 2005, DAMTP was delighted to welcome Professor John Papaloizou , who joins the Astrophysical Fluid Dynamics group. Elected as a Fellow of the Royal Society in 2003, Professor Papaloizou's special areas of interest include accretion discs, planet formation and the dynamics of young planetary systems. 

Rising DAMTP: Neil Turok, Sam Webster, Guido Carlett, Claire Lambert, Fabio Riccioni, Ben Allanach, Nick Dorey  
The start of the Michaelmas Term saw the arrival of four new University Lecturers in DAMTP and three in DPMMS. Dr Johann Paulsson joined the growing group of researchers in the field of mathematical biology, while Dr Ben Allanach, Dr Nick Dorey and Dr David Tong joined DAMTP's High Energy Physics research group. Drs Dorey and Allanach were swiftly enlisted as guitarists in "Rising DAMTP", the Department's recently formed band which performed a series of Sixties' hits to a stunned audience at the DAMTP Christmas Party. In DPMMS, Dr Michael Mandell joined the established Geometry/Topology Group, while Dr Mihailis Dafermos' interests in partial differential equations and general relativity will contribute to the further development of collaboration between the two Departments. Dr Peter Friz joined the Statistical Laboratory, specialising in the field of mathematical finance. 

Back to top 
Game Theory and "Who Wants to be a Millionnaire?"Richard Weber 

In September 2003 I received an email that I almost deleted as spam. The sender introduced herself as a television producer and invited me to appear on an unspecified television game show. Following an interview, she revealed that the show was actually a special edition of "Who Wants to be a Millionaire?" in which professors and freshers would play in teams. Apparently, the producers had been browsing web sites and had invited me to participate because of the recreational material on my site and my interest in game theory. Coincidentally, I had recently set a Part II exam question on determining the optimal play in a version of "Who Wants to be a Millionaire?" using the technique of dynamic programming. This is a technique, popularized by Richard Bellman in the 1950s, which is useful in solving optimization problems that are posed in discrete stages over time. It is widely used to solve problems of scheduling, vehicle routing, pricing financial derivatives and so on. The key idea is to work backwards from the problem's last stage  an idea that is familiar to anyone who has noticed that it is sometimes easier to find the path through a maze by working backward from the exit rather than forward from the entrance. As Kierkegaard once said "Life can only be understood backwards; though it must be lived forwards". When deciding how to best tackle questions in Millionaire, one should first figure out tactics for answering the £1,000,000 question, and then figure out tactics for the £500,000 question. Only after figuring out tactics for all higher money questions can one really say how optimally to play for the initial £100. 

When I entered Cambridge as a fresher there was a matriculation requirement for a foreign language. That helped me win the 'fastest finger first' competition by putting the Latin names of bones in the arm in order from fingertips to elbow. I chose as my partner, Jack Steege, from Newcastle University.  Jack Steege and Richard Weber 
One thing that makes Millionaire interesting is that players have different attitudes to money and risk. It is particularly interesting when such players are a team and must agree on their tactics. When we reached the £16k question, Jack was, quite reasonably, reluctant to risk his share, which would pay the costs of his entire university course, whereas for my utility function a gamble between £32k and £1k (with a further possibility to play for £64k) was the mathematically more rational choice. Of course people do not always gamble as rational mathematics might instruct them to do. Suppose you must choose between A: £1m guaranteed, or B: 89% chance of £1m, 10% chance of £2.5m and 1% chance of nothing. Most people prefer A to B. Suppose you must choose between C: 89% chance of nothing and 11% chance of £1m, or D: 90% chance of nothing and 10% chance of £2.5m. Most people prefer D to C. Yet these preferences are irrational, since to prefer A to B is to say that you think that u(1) > .89u(1)+.10u(2.5)+0.01u(0); whereas to prefer D to C implies exactly the opposite. I have been told that in Saudi Arabian version of Millionaire the contestants do not like to quit, and continue answering questions until they give a wrong answer. In Russia it is usual that when a player 'asks the audience', the audience deliberately tries to steer the contestant to the wrong answer. Fortunately, my audience were British, and many wore football shirts. When I asked 'Who beat Ireland in the second round of the World Cup 2002?', I gave instruction that they should vote only if they were sure they knew the answer. I was pleased that the audience understood my statistical reasoning and gave a clear signal for Spain. Several people came to tell me later that they had understood why they ought not to vote. Since appearing on Millionaire I have given talks on related mathematics, both to 13 year old Cambridgeshire school children and to retired members of Queens' College. I have described how one can use a spreadsheet and dynamic programming to calculate the optimal play. Of course this depends on the prior probabilities with which you think you can answer the questions and how useful your lifelines will be. However, for some reasonable values of these parameters, I estimate that at the start the game one's expected winnings are in the region of £14k. Jack and I were very happy with our £16k win and enjoyed a very interesting chance to put game theory into practice. 

Back to top 
Student Representatives on the Faculty Board of MathematicsIain Mathieson 

A few weeks before Christmas 2003, Owen Jones and I became student representatives on the Faculty Board of Mathematics. Our motivation was a mixture of curiosity about the inner workings of the Maths Faculty and a desire to improve the lot of the undergraduate mathematician. Generally, the reps are elected by all the undergraduate mathematicians (the graduates elect their own rep), but perhaps fortunately for us there were no other candidates. A month or two after this striking display of student apathy found us in the CMS café waiting for Dr. Stephen Siklos to brief us before the first meeting of the Faculty Board. As Dr. Siklos explained, the Board meets twice a term and is composed of a slowly varying combination of Fellows, as well as the three annually elected student reps. It has general responsibility for the running of the Faculty, which comprises two Departments: Applied Mathematics and Theoretical Physics (DAMTP) and Pure Mathematics and Mathematical Statistics (DPMMS). Specific matters can also be referred to other committees such as the Curriculum Committee and the Teaching Committee. Sitting on the Board, especially for the first time, can be quite a daunting experience. It is not easy to make sensible points when surrounded by various professors, lecturers and directors of studies who can, as I'm sure you can imagine, be extremely intimidating, even when they're not trying. Luckily for us, the Chair of the Board, Professor Anne Davis, was extremely helpful at making us feel comfortable. Throughout the year, various issues came before the Board. Perhaps the most important was the introduction of a new, unified Part II Tripos, to replace the old division into A (general) and B (specialised) courses. The Board spent hours debating various minutiae of the new system, some, I think it is fair to say, more important than others. That said, one of the strengths of the Board is that with the number of different views represented even the smallest issue is thoroughly considered. As well as putting forward students' views on major issues like the Part II reform, we also raised smaller problems. These ranged from the availability of example sheets on the internet to the temperature of lecture rooms in the CMS (too cold). 

The student reps also have a role outside the Faculty Board. We have a website where we host some information about the Faculty and lecture notes for a selection of courses. The website was excellently maintained by John Rudge, the graduate rep. We answer emails from students and non students, and we liase with CUSU, the University Student Union on universitywide education matters. One of our last and most difficult tasks was to sit on the student committee which recommended Dr Natalia Berloff as the Faculty's candidate for a Pilkington Teaching Prize.  Iain Mathieson, Anne Davies and Owen Jones 
All in all, being student rep was an illuminating experience. It was extremely interesting to see how the Faculty deals with the important issues that affect everyone here in the CMS. A year after we began the job, Owen and I carried out our final duty: supervising the elections for the new batch of student reps. In contrast to our 'election', there were seven candidates standing. Whether this is a reflection on our term, I wouldn't like to say. But if our only contributions were to reduce student apathy and warm up the lecture rooms then I feel we have achieved something worthwhile. Finally I would like to thank the entire Faculty Board, but particularly Professor Davis and Dr. Siklos for being so helpful and accommodating all year, and also to say good luck to the new student reps, Ed Saperia and Peter PembertonRoss. 

Back to top 
CATAM  ComputerAided Teaching of All MathematicsNikos Nikiforakis (Course Director), Tim Pedley and Robert Hunt 

Computational methods have a fundamental place in the study and application of mathematics. In research laboratories throughout the world, the combination of mathematical modelling, numerical methods and high performance computing facilities are making many formerly intractable projects possible. The Department of Applied Mathematics and Theoretical Physics was the first UK university mathematics department to introduce computeraided learning into its syllabus. Since the introduction of CATAM in 1969, many other mathematics departments now have similar projects. 

Students working on a statistical modelling project. 
CATAM is introduced through small practical projects providing students with the opportunity to engage in independent study, typically writing programs to explore the properties and structure of the mathematics, which helps develop their ability to solve mathematical problems. The student will then present a written report detailing his or her findings and conclusions, relating these to the physical and/or mathematical system being studied and extending the theory beyond what is found in the associated lecture courses. 
The undergraduates use the C language with the assistance of mathematical and graphical routines from a locallywritten library. This library has been carefully designed to facilitate the programming of projects in C, removing much of the tedious work in equationsolving, graphics and screen layout. New users can begin with the highlevel graphics routines where minimum knowledge is required, and gradually develop their expertise towards the lower levels which require a more detailed understanding. There are varied projects in the second year covering key topics from applied, pure and applicable mathematics. In the third year, the choice is very wide and the projects relate directly to specific lecture courses in the Mathematical Tripos. These include fluid and solid mechanics, quantum theory, general relativity, astrophysics, optimisation, dynamical systems, number and group theory, algebra, analysis, statistics and probability. CATAM provides students with the opportunity to acquire the valuable skill of programming whilst also helping them to understand the mathematics involved. Successful completion of CATAM projects implies the development of important programming and investigative skills of widespread value in industrial and commercial work, as well as in scientific research. A recent development is the establishment of a link between Department of Applied Mathematics and Theoretical Physics and the Mathematics Faculty at Massachusetts Institute of Technology (MIT), generously funded by the CambridgeMIT Institute. The mathematicians at MIT are interested in creating a "math lab" course of their own. MIT will benefit greatly from our expertise in this area, while we will benefit from new projects and material written by the MIT faculty in their own areas of expertise. It is hoped that this link will provide educational advantages on both sides of the Atlantic and bring the best of each curriculum to bear. 

One of the CATAM projects available to third year students gives an introduction to protein comparison in bioinformatics. The comparison and alignment of biomolecular sequences (both DNA and protein), combined with the systematic collection of databases of such sequences, has become an essential part of modern molecular biology. Molecular sequence data is important because two proteins with similar sequences often have similar functions or structure: so we can often learn about the function of a protein in humans by studying the functions of proteins with similar sequences in simpler organisms such as yeast, fruit flies or frogs. Since proteins are essentially long sequences of amino acids, each of which can be represented by a single letter, the problem is mathematically equivalent to asking how similar are two (very long) "words". The project starts with a simple example (how similar are "concentrate" and "comfortable"?) before moving on to comparing four real proteins (a human cancer cell protein, for instance, and a cellular transport protein in toads). At this stage it is necessary to use scoring methods (from the work of biologists) to rate the relative likelihood of various kinds of mutation (e.g. letter deletion, letter replacement, etc.). 

Back to top  
To discuss any aspect of making a donation in support of mathematics at Cambridge, please contact Professor Anne Davis (A.C.Davis@damtp.cam.ac.uk) or Mr Christopher Hesketh (ch10002@cam.ac.uk). 