Maths and Engineering Trip to Cambridge

Fifteen Lower Sixth students participated in a trip to mathematical sites in Cambridge on Tuesday 20 March.

King’s College
The day started with a visit to King’s College, the alma mater of notable British Mathematicians Alan Turing and Frank Ramsey. Unsurprisingly the highlight was the magnificent gothic chapel, and the budding engineers were able to marvel at the largest self-supporting fan-vaulted ceiling in the world. Our students met with the Head of the Department of Pure Mathematics, Professor Martin Hyland, a Fellow of King’s and an academic grandson of Turing. Professor Hyland was my Director of Studies and he is one of the most inspiring teachers and supervisors of mathematics. Our students were able to ask him both about the Mathematics Tripos, and the quirks and subtle differences between the Colleges. I was unsurprised to hear that King’s still has more of a reputation for political activism than for its sporting prowess.

We had lunch in The Maypole (my vote for the best pub in town) with Old Sennockian Felicity Thompson, who is currently in her final year at St Catharine’s College. She was able to give our students invaluable information both about the transition between IB and University and the practicalities of life as an undergraduate.

Trinity College
After lunch we were given a guided tour of Trinity by Dr Anson Cheung, a Research Fellow. Dr Cheung took us to several lesser-known corners of the college, of which my favourite was the Fellow’s Bowling Green, hidden away behind the Master’s Lodge, where Isaac Newton devised a complicated version of bowls to be played by Trinity fellows. A quiet corner of this court also contained the table where G H Hardy produced most of his Mathematics during the early part of the 20th century.

Dr Cheung talked us through the mathematical history of Trinity, including a reconstruction of the original experiment used by Sir Isaac Newton to obtain the first measurement of the Speed of Sound using the echoes of Neville’s Court. The highlight of this tour was our visit to The Wren Library which includes displays of Newton’s own private papers and his own personal copy of the Principia Mathematica, though some of our students seemed equally impressed by a 1623 First Folio of Shakespeare and A A Milne’s handwritten manuscripts of Winnie the Pooh.

As well as the impressive public buildings our students also had an equally important glimpse at some of Trinity’s accommodation – and there was some excitement to realise that they had the same chairs as the IC. Dr Cheung gave our students a very thorough and extremely honest overview of his experience as an Admissions Tutor for Natural Sciences, and was able to outline exactly what qualities are expected of top Mathematicians and Engineers.

After a packed day our students left laden with prospectuses for both their chosen colleges and departments, and they had plenty to mull over on the train journey back. It was a thoroughly enjoyable day and we are particularly grateful to the kind hospitality shown by Professor Hyland, Felicity Thompson and Dr Cheung.

David Vaccaro

Year 8 Code Breaking day

One of the most intriguing stories from the Second World War is how an elite group of British code breakers, led by Cambridge Mathematician Alan Turing, successfully deciphered the supposedly unbreakable German Military Code, known as Enigma. Their achievements helped safeguard Allied shipping routes across the Atlantic, and helped ensure the success of the D-Day Landings.

As part of our celebration of the Alan Turing Turing Centenary, all Year 8 students were involved in a day of code breaking activities organised by Dr James Grime, from Cambridge University Millennium Maths Project. As well as a hugely entertaining presentation on the history of encryption and secret communication, the students also had the exciting opportunity to see a genuine Enigma Machine which had been used by the German army in occupied France. 

The day started with a hugely entertaining presentation tracing the history of code breaking from ancient times up to internet encryption. Throughout his talk, Dr Grime emphasised two points: firstly that the ability to crack codes changed history altering the outcome of wars and saving lives, and secondly that the ingenuity and problem solving skills required by code breakers was at the heart of Mathematics.

The students then had the chance to put their own code breaking skills to the test, as they worked through a selection of classical and modern ciphers. It was hugely fun, and many groups pitted their wits against some of the hardest codes. During these sessions they also had the chance to use the Enigma machine and to look closely at its inner workings, including the complicated system of rotating scrambling wheels and plug board.

Visiting Fellow: Professor Imre Leader

On 12 January 2012 the Mathematics department was delighted to welcome Professor Imre Leader, Trinity College, Cambridge, for a two-day visit. Professor Leader is an expert in the field of Combinatorics, and as well as being an active researcher he also is a popular lecturer at the University. Professor Leader represented the UK in the 1981 International Mathematics Olympiad, and he later managed the team between 1999 and 2001. He has published widely and is a co-author of The Princeton Companion to Mathematics.

Away from Mathematics he is one of the country’s top players of Othello (Reversi), and he has won the British championships an unprecedented ten times, as well as being a runner-up in the world championships. Professor Leader was invited to visit the school not only for his eminence within Mathematics, but also his ability to communicate subtle and beautiful ideas.

During the visit Professor Leader sat in on various lessons, including Dr Fugard’s Mathematics Extension set in the Lower Sixth, Dr Williams’ top set Year 10 and Mr Vaccaro’s Year 8 set. He commented on both the level of achievement and the warm and friendly atmosphere of these classes, with students unafraid to push themselves and share ideas when faced with non-routine, challenging problems. He also visited Mark Beverley’s Lower Sixth TOK class, and was greatly impressed with the role of this synoptic course in encouraging students to think freely and develop critical sensibilities. He thought the material on logical fallacies was especially worthwhile, as forming rigorous proofs and arguments is a vital skill for higher Mathematics.

Lucky students from Years 7, 8 and the Lower Sixth also had the opportunity to work in small groups with Professor Leader, looking at Olympiad style questions. The students were chosen either because of their mathematical ability or their attendance at Mathematics Society events and lunchtime Puzzle Clubs. The younger students looked at tiling problems, for example whether it is possible to tile a 7x4 rectangle using the five different tiles from the game Tetris, and the Lower Sixth group looked at challenging problems about the properties of primes. The Maths department feels that the sessions with the younger students were particularly important, as it is at this age where a lifelong passion for Mathematics can develop. Thea Beadle from Year 7 and Miles Caven and Hethvi Gada from Year 8 were notable for their enthusiasm and problem solving skills.

The highlight of the visit was Professor Leader’s own lecture delivered to the Lower Sixth, where he presented a recent and surprisingly difficult result from graph theory. At the start of the lecture he posed the innocuous looking question: ‘Does a long string of red and blue beads always contain four equally spaced beads of the same colour?’ The answer turned out to be yes, but the numbers involved in the proof soon became astronomically, almost inconceivably large. It was a masterly presentation of a difficult result, with several neat tricks and shortcuts, and without a single equation in sight. What a wonderful advert for Pure Mathematics, showing how much more rich the subject is than the all too prevalent fare of rote learned techniques and standard methods.

It was wonderful to see such a successful academic take a keen interest in all areas of school life; throughout his visit he was full of questions for both staff and pupils, and he seemed extremely keen to find out more about the wider aspects of school life and the IB, even attending a Sixth Form student's presentation on his Extended Essay. He was also happy to discuss all areas of Mathematics with the pupils, and led a very productive Beginners Guide to Mathematics at University in which he talked students through the various options and debunked several myths about Oxbridge admissions and interviews. (Disappointingly, all the media reports of psychological warfare and trick questions are completely made up!)

It was a pleasure and a privilege to welcome Professor Leader to the school, and we very much hope that he would like to return in the future. However most of all we hope that his manifest passion for the subject will enthuse and inspire some of our current pupils about the joys of mathematical ideas and problem solving.

The man who stole $65 billion dollars…

On Thursday 12 January, Ilya Muzykantskiy (Upper Sixth) gave a fun-packed and informative lecture on the Mathematics underlying Ponzi Schemes, a financial scam involving dubious investment opportunities which promise more than they can deliver. The talk started with a brief history of these schemes tracing their origin back to Boston fraudster Charles Ponzi, before giving more detailed information about the scandal which shocked Wall Street in 2009, the collapse of Bernard Madoff’s hedge fund. This was the largest financial crime in history and resulted in a staggering loss of $65 billion, and an unprecedented 150 year jail sentence for Madoff, who personally embezzled an estimated $18 billion. Ilya outlined how both the naivety and greed of Madoff’s clients led to vital clues to the funds unviability being missed.

The main body of the talk however was a technical, Mathematical analysis of the evolution of Ponzi schemes which Ilya himself had developed. His model used standard assumptions taken from the insurance industry (maximization of log-utility) to predict the behaviour of individual investors, and then built differential equations to investigate the growth of the whole scheme, as both the underlying economy grew and as more people heard about the scheme.

In the introduction to the talk, his Extended Essay supervisor, Mr Vaccaro, explained how Ilya was determined that his Extended Essay should contain original research, tackling an interesting problem which had never been studied before. Mr Vaccaro described his frustration as Ilya rejected suggestion after suggestion for a possible topic, until discovering there is virtually no published literature on Ponzi Schemes. Ilya’s inspiration for choosing this subject stemmed from his an experience as a 15-year-old, when he lost all his savings on an internet-based fraud known as 12 DailyPro.

In the course of the lecture, Ilya outlined the central role computers had played in allowing him to solve his system of Differential Equations, which were harder than those encountered on the Higher Level syllabus. All Sevenoaks Maths department computers are installed with Wolfram Mathematica, which is probably the most powerful and sophisticated industrial Maths package. The use of Mathematica was central to Ilya’s research and enabled him to finding significant non-trivial results. Ilya’s experience with Mathematica programming led to him choosing to apply for Computer Science at Cambridge, and he thinks that a significant part in his success at receiving an offer was down to the skills he learnt while working on his Extended Essay.

The main findings of his talk were precise conditions to guarantee the long-term survival of a Ponzi Scheme, and the talk ended with an interesting analysis of whether current financial instruments are in fact giant Ponzi Schemes, most notably Pension Funds and Social Security. There were also ideas for future research, and suggestions about how his results could be used to shed light on the collapse of several banks during the financial crisis.

The audience included Professor Imre Leader, the Mathematics department’s visiting fellow. He praised Ilya’s ability to communicate the big ideas without getting bogged down in the detail, and the maturity and confidence of his delivery. Most of all he was able to see what a thoroughly worthwhile task the IB Extended Essay is, since there are very few other opportunities for school children to produce genuinely original and significant pieces of academic research.

The lecture was thoroughly enjoyed by a large Sixth Form audience, and it resulted in a long question-and-answer session. We hope that students will have been inspired to think of relevant, original and important questions which they can investigate mathematically.

Simon Donaldson Knighted in New Year’s Honours List

Among the 2012 New Year’s Honours was a richly deserved Knighthood for Mathematician and Old Sennockian, Professor Simon Donaldson. Since 1999 Professor Donaldson has held a Royal Society Research Professorship at Imperial College, London, and he is widely regarded as one of the most influential Mathematicians in the world. Professor Donaldson’s Knighthood joins a long list of awards, honours and prizes, including a Fellowship of the Royal Society, The King Faisal International Prize for Science, the Shaw Prize in Mathematical Science and a Fields Institute Medal.

Simon Donaldson shot to fame in 1982, when as a second year DPhil Student he proved a result which, in the words of his supervisor Professor Sir Michael Atiyah, 'stunned the Mathematical world'. Using techniques which originated in Quantum Field Theory, he established the existence of certain 'exotic' four dimensional spaces, and much of his subsequent work has focussed on attempts to classify 4-dimensional manifolds.

In 1985, just ten years after leaving Sevenoaks, Donaldson was awarded a Professorship at All Souls College, Oxford, and one year later he was received the prestigious Fields Medal (often described as being the equivalent of a Nobel Prize for Mathematicians) for his research. His subsequent research has had far reaching consequences in both Pure Mathematics and Theoretical Physics, and it has helped reform our geometric understanding of space and time, as R Stern describes:

In 1982 Simon Donaldson began a rich geometrical journey that is leading us to an exciting conclusion to this century. He has created an entirely new and exciting area of research through which much of mathematics passes and which continues to yield mysterious and unexpected phenomena about the topology and geometry of smooth 4-manifolds.

Sevenoaks School is incredibly proud of the achievements of our former pupil and we hope that this public recognition will serve as an inspiration for the next generation of Mathematicians and Physicists.

David Vaccaro

Year 10 Trip to Bletchley Park

With the upcoming Alan Turing centenary year in 2012 there has been a renewed interest in the story of the British WW2 code-breakers, with major documentaries appearing recently on BBC and Channel 4. It is perhaps for this reason that so many Year 10 students signed up for the Maths department outing to Bletchley Park, and the 73 students who braved the two hour journey around the M25 were rewarded with an informative and fun programme of code-breaking activities and lectures.

Our knowledgeable guides talked us through the history of the site, and there was something extremely atmospheric about walking through the ramshackle huts and portacabins where so much of the important work was done. The guides stressed the huge contribution made by both professional Mathematicians and the large number of machine operators, telephonists and dispatch riders mostly drawn from the Wrens.

The highlight of the trip was undoubtedly the opportunity to use a genuine Second World War Enigma machine captured during the liberation of Europe. I shared the excitement of the pupils as they were able to press the keys, watch the rotors progress, and to see, as if by magic, the scrambled text appear on the lamp board. Our students were also set code-breaking tasks to complete, and I was proud to see one group break the Bletchley Park record by deciphering a mono-alphabetic substitution cipher in under two minutes.

It was a hugely worthwhile trip, which richly conveyed the historical importance of the code-breaking operation at Bletchley, not only did it shorten the war by as much as two years, saving tens of millions of lives, but it also heralded the start of the computer age. During our visit we saw a working replica of the first ever programmable computer, the aptly named Colossus, and it was hard not to be overawed by the complexity of this machine, and the triumph of human ingenuity which it represents. My main hope for the trip was that, as the students listened to their iPods and played computer games on the journey back, they remembered the flashing valves and whirling tickertape of the 1940s Bletchley computers and thought about the hidden Mathematics which underpins their digital age.