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Contents 2 The Former President of the Maths Society. 2 A Few words from the Head of the School of Mathematics

and Statistics. 3 International Congress of Mathematicians ICM2002 Beijing

with Professor Cheryl Praeger 4 Dr. Patrick Hew: Maths Got Me A Job 5 An Interview with Dr. Ron List 7 Top Five Reasons Not to do A PhD! 9 Modern Security, by Jon Cohen 13 Research In Probability And Statistics 14 What’s happening in Pure Mathematics? 14 What’s happening in Applied Mathematics? 15 Maths Courses at UWA, and why they are useful! 16 Degrees Beyond Graduation 17 Perfect Numbers and the Order of Pythagoras 19 World News in Mathematics

Editor’s column Hi everyone! This is the first issue of AfterMath”, and we do not know yet whether or not this will carry on into a sequence of seasonal editions. From 1963-1965, the Weatherburn Mathematics Society (of UWA) published a series of magazines like this one, called “WAMMS”. It was resurrected as “Wammette” in 1973, which was renamed to “Aftermath” in 1974. This magazine eventually ceased in 1991, and so did the Weatherburn Mathematics Society shortly after. Last year, a new maths society was borne, this time called the “Maths Society”, and we hope to keep it running as long as we can! In this first issue, we’ve tried to include as much stuff as possible, including feature articles, jokes and puzzles, and serious information. We strongly encourage our readers to contribute articles for the next edition (send them to me) and we welcome any suggestions you might have for possible improvements. Finally, I would like to thank the editorial team (see below), as well as Dr. Patrick Hew, Dr. Ron List, Prof. Cheryl Praeger, Jon Cohen, Dr Michael Giudici, Scott Brown, A/Prof Tony Pakes, Dr Martin Hazelton, Dr Oreste Panaia, A/Prof Les Jennings & Geoff Pearce.

-John Bamberg

EDITOR: John Bamberg SUB-EDITORS: Sophie Ambrose, Robin Milne, Cai Heng Li, Des Hill LAYOUT AND PRODUCTION: Belinda Dodd & Val Moore

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The Former President of the Maths Society Hi all and welcome to the first ever edition of AfterMath! For those of you who don’t already know, the Maths Society is a student club, which was formed to provide a social environment for mathematics and statistics students. We have a common room on the ground floor of the maths building, first door on the left (by the way, any couch donations would be very much appreciated). We also have a website at http://www.maths.uwa.edu.au/~MS. Second semester will include the ramping up of the Weatherburn Lectures to provide interesting talks at the undergraduate level. They will be held on Fridays in the Blakers Lecture Theatre and anyone interested in learning some cool things should come along! In addition, if you have an interesting topic which you would like to share, you are welcome to present a talk. Also in the pipelines is a T-Shirt and regular publications. Given this large increase of activity, we are desperately short of enthusiastic people who want to help out with organizing functions, designing T-shirts and editing the magazine. Anyone who is keen to help out is encouraged to come along. Jon Cohen Maths Society President Ed: Stacey Cole is the new President of The Maths Society.

A Few words From the Head of The School of Mathematics and Statistics

Communication has a place at the centre of Science. It allows people to keep in touch with what others are doing, thinking, creating. In order to communicate science in an orderly fashion, we have some formally defined ways of doing it. In fact much of the learning of science is spent in learning the rules of the formal languages of science which describe what is there, how the various bits relate to each other, and trying to work out the rules of causality using logic. But the formal languages can be rather dry. Spoken word is much less dry usually, and visual clues between speaker and listener enhance the communication. There are problems, mostly financial, in recording this communication experience so others can see and hear the science being communicated. The written word, either on paper or now on the web, is the usually acceptable compromise.

There is also the human side of science to consider. Scientists, despite the movie stereotypes, are social animals, and like to feel part of something larger than themselves. This magazine will communicate some different aspects of the world of Mathematics and Statistics using an informal approach. AfterMath has a variety of aims, but entertainment and information would be foremost in the Editor's thinking. Defacto, it will be part of the fabric that helps to bind people into the social and scientific groups which have a need for Mathematics and Statistics. Despite the computer, these groups are becoming larger, that is, more and more scientists and engineers need to have some familiarity with parts of Mathematics and/or Statistics. The readers being aimed at are the students of science and engineering in the broadest sense. This includes high school students, university students and the teachers of these students. I would like to congratulate all involved in producing this magazine. Energy has to be used to produce order from chaos and I hope that others in years to come will find the energy to continue producing the magazine at opportune times. The ideas displayed are coming from the younger members of the mathematics community at UWA, so in a sense reflect the health of the community. Enjoy.

- Associate Professor Les Jennings (HOS)

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International Congress of Mathematicians ICM2002 Beijing.

By Prof. Cheryl E. Praeger

Approximately every four years mathematicians from around the world meet together at an International Congress of Mathematicians. These congresses attract the attention of all mathematicians, not only those who are able to attend, since at the opening ceremony of the Congress the winners of the Fields Medals are announced. These medals are the equivalent for mathematicians of the Nobel Prize. They are very special, since no Nobel Prizes are awarded in mathematics; and they are different from the Nobel Prize as the recipient of a Fields Medal must be less than 40 years of age. Thus when Andrew Wiles proved Fermat’s Last Theorem (Mathematicians had been trying for 350 years to find a proof) he was unable to receive a Fields Medal because he was over 40, and so he was awarded a special one-off Medal at the ICM in Berlin in 1998. As well as the Fields Medals the ICM includes a carefully designed programme of plenary and invited lectures covering the most important advances in Mathematics over the past four years. It is a great privilege to be invited to give one of these lectures. In 2002 an International Congress of Mathematicians ICM2002 was held in Beijing. This was the first ICM in the new millennium, and the first in a developing country. The opening ceremony took place in the Great Hall of the People adjacent to Tienanmen Square, right in the centre of Beijing. Present were over 4000 delegates from more than 100 countries, and many thousands of others, including hundreds of volunteers (mostly Chinese mathematics undergraduate students) who helped participants throughout the congress. Also present were Jiang Zemin, President of the People’s Republic of China, who granted the Fields Medals to the two Fields Medalists: Laurent Lafforgue from Paris for his proof of the Langlands correspondence over function fields, a thirty-five year old conjecture in Number Theory, and Vladimir Voevodsky from Princeton for developing motivic cohomology theory. I will not say any more about there achievements. However there is a UWA connection with Lafforgue: Ngo Dac Tuan spent a research semester at UWA under my supervision in connection with his undergraduate degree at the Ecole Polytechnique in Paris. He is now

studying with Lafforgue for his PhD degree at the Universite de Paris-Sud. I gave one of the invited lectures in Algebra at the ICM2002, to hundreds of mathematicians in the largest auditorium used during the Congress, the one used also for the plenary lectures. I spoke about the theory I had developed of quasiprimitive permutation groups, and the way this could be applied to study graphs and designs. A couple of years before the ICM I had been asked to send my CV to an eminent colleague who told me that he wished to put my name forward for the Algebra section. Eventually, around a year before the ICM an invitation arrived. There were strict rules about the preparation of the full paper, and it had to be ready many months before the congress so that the volumes of proceedings could be distributed to participants in Beijing.

The auditorium where I gave my lecture was also used for a special evening lecture by John Nash, the mathematician who was awarded the Nobel Prize in Economics for his seminal work in Game Theory. My

husband John and I were seated near John Nash and his wife at a special dinner at Peking University. When Alicia Nash discovered that we were Australians she asked what we thought of Russel Crowe. They had met Crowe on several occasions and had long discussions with him in connection with filming “A Beautiful Mind”. There were numerous specialist “satellite conferences” organised both before and after the ICM2002, and these gave the opportunity for exchanging more detailed information about new mathematical breakthroughs than in the “big picture” lectures at the ICM itself. Some of my colleagues had attended one of these in Tibet. I spoke at one on Combinatorics in Shijiahuang before the ICM, and at a second satellite conference on Algebra in Suchou after the congress. Altogether it was a very exciting time. I was especially pleased that the Congress to which I was invited was held in Beijing, both because it is in our region and also because of my many mathematical Chinese friends and colleagues.

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Dr. Patrick Hew: Maths Got Me A Job BSc (Hons) Pure Mathematics and Applied Mathematics 1995

PhD Mathematics and Intelligent Information Processing Systems 1999 It was when I was soaked to my chest in mud, wearing Disruptive Pattern Camouflage Units and being addressed as “Dr Hew” by an Army Warrant Officer 2 that I appreciated the value of my university education. I entered The University of Western Australia as a Science/Engineering student in 1992. In 1995, I continued my Science major in Pure Mathematics and Applied Mathematics into Honours, having just finished 11 weeks of Engineering work experience with the Shell Company of Australia (they had an excellent work experience programme). The combination of this experience, an Honours project that was going pretty good, and browsing the Engineering course book, made me decide that Engineering wasn't for me, so I decided to “go postgrad”. I started my postgraduate study in Mathematics and Intelligent Information Processing Systems in 1996 and submitted my thesis in 1999. 1999 was a very good year to be interviewing for work, with the .com boom going great guns and technical skills in high demand. My path was quite conservative - applied to Telstra Research Laboratories and the Defence Science Technology Organisation, got interviews and offers from both, and went with DSTO over TRL because the work sounded more interesting. To me, that is – I’d spent high-school trying to be a fighter pilot and undergraduate hoping to be an astronaut, and so in my (copious) spare time I was reading and viewing everything I could get my hands on about the military and space, and technology in general. (I no longer call it a “mis-spent youth” because I couldn’t have tailored my education to better ready me for DSTO ... square pegs and round holes and all that.) So 2000 saw me in Adelaide, joining a DSTO team that provided advice on the conduct of operations, and on concepts for command support tools. This kind of work falls under the blanket term of “operations research” (in military circles - the civilian equivalent is “management science”). In 2002 I took an internal transfer to Canberra, joining a team providing advice on capability development issues with specific attention to the implications of future technology. Capability development is the process by which Defence acquires new equipment – it’s important that this is done not just “platform for platform”, but thinking “systems” encompassing equipment, doctrine and people. Throw future technology into the mix, with its potential for disruptive change, and you get a lot of interesting questions that aren’t easily answered. This is the primary use of my mathematics training, particularly my pure mathematics background. My PhD supervisor once suggested that, “the best place for doing

(Photo: Patrick Hew rides the gun position on an Australian Army ASLAV-25 Light Armoured Vehicle (Reconnaissance), on the Defence Civilian Army Familiarisation Course 1/2000.)

new, original mathematics is where there currently isn't any” and I’m tending to agree. In these kind of high-level “strategic” areas, the problems are generally ill-posed and not well understood – if they were easy to answer, they would have been handled at a lower level in the bureaucracy. As a mathematician, I (try to) bring clarity, precision and order to problems, helping our Defence clients to understand the issues. Sometimes the problems are amenable to mathematical solution or insight, which is where I bring my applied mathematics and software skills to bear; modern computing and software has given operations research a booster shot in the arm (and a booted kick in the pants). DSTO is a civilian organisation in the Department of

Defence, and I am employed in the Australian Public Service. My salary is less than (what I regard to be) comparable jobs in private management consulting firms, but not unreasonably less, and in terms of an overall package of work and life, I have a good balance. The integration of civilian

professionals (scientists in particular) into Defence planning has quite a long history, with civilians operating with the team but outside the chain of command. This gives us the ability to talk to people at all levels in the organisation (one researcher I know has interviewed people ranging from troops in the field right up to Chief Defence Force), and we (should) have the independence, integrity and intellectual capacity to provide a different opinion if needed. This was my epiphany in the mud, my introduction to the military through the Defence Civilians Army Familiarisation Course (I broke the Golden Rule of “Never Volunteer For Anything”; believe it or not, DCAFC has a moderately competitive entry system, and it’s often oversubscribed...). The Army trainer was a professional soldier, a leader of men and women, and an expert in his domain. I was at least 15 years younger, yet his use of the title “Dr” was due courtesy for what I had achieved, as a developing expert in my domain. This feeling of respect is both a tremendous boost to my ability to do what I do, and at the same time an anormous promise to live up to. But in the final analysis, it’s what the PhD really means to me ... and I wouldn’t have it any less. (The views in this document are those of the author, and not necessarily those of the Department of Defence or The University of Western Australia. Contact the author at Patrick.Hew@dsto.defence.gov.au .)

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An Interview with Dr. Ron List Ron, has been a member of this school for just over forty years, and has just recently retired. He has been responsible for creating ties with the geophysicists, and has recently supervised two graduate students who have been awarded PhD’s with distinctions. John Bamberg: You did your undergrad here, then afterwards you did your PhD at Sheffield. After that, what did you do? Ron List: After I left here, I did a masters degree at the University of New England and then I came back on the staff here. Then I accumulated some lot of leave, and then with my leave, I spent time at the University of Sheffield. Back in the 60’s, Australian universities hadn’t got the confidence up to go for higher degrees, that’s only something of the more recent past - you normally packed up and went overseas. Most of the staff were involved in doing a lot of teaching actually. The emphasis very much was on how many hours of teaching you did. Larry Blakers made a comment “I expect you to do forty hours of teaching”, and we raised our eyebrows and said “well, the eastern state’s universities are currently saying 25 hours for the junior appointments”. Or something of the line of “you should be able to complete higher degrees in twice the normal length of time for a student”. What was happening in the 60’s and 70’s was that you were taking students on as tutors and senior tutors and things like that and this was the way at which you could fund your way through for a scholarship. If you look at the history you’ll find that Jorg Imberger spent sometime here, and Nev (Fowkes) spent some time here while he was finishing his PhD, and that’s how things were funded. JB: How many staff were here when you started? RL: I started in 1962, and you can see the gradual build up after that. There was money from the Murray report and they were building up all the junior staff. The early 60’s are a time of a lot of people coming. We’ve only basically got up to 1960 one professor and then Levey gets promoted and you can start to see the school getting larger. Things were quite different back then - it was a period of expansion. JB: That kept on happening until the 1970’s? RL: Yes, you’ll find that they were expanding quite happily into the 70’s then clamped down. That’s why so many of us in the school are over 50. Back then, you couldn’t move very freely between universities unless you were a real high flyer and there weren’t places for new staff to come in. The good new graduates were being moved from one campus to the other...two/three year appointments here and there. It wasn’t very good for the student or the new staff member. So you should see a real opening up at any rate, as all the retirements start coming! JB: When you began your research in your early years, what were your interests? RL: Those days my interests were in fracture mechanics. Fracture mechanics wasn’t going very far in those days…it

never went very far. Mathematicians came along and applied some beautiful mathematics but it didn’t have much physical reality. It just faded out, and from a mathematical point of view, a lot of the problems are still unsolved. The approach being used was beautiful but it wasn’t really solving the problems. Barber (the former Pro-Vice Chancellor for Research)…that was one of his interests, fracture mechanics. He reckoned that finally…I was talking to him a few years ago…he said, “I can see how we can crack into this, we just need a few PhD students etc”. But you really need to get your hands dirty and see exactly what’s happening with the fracture process. It was a real fudge in those days. The mathematical theory was that fractures happen in straight lines because that fitted the mathematics, but cracks don’t go in straight lines – they go in angles. So I was actually talking to an experimentalist, and he said “oh yes, we’ve had a lot of trouble with this”. So you can see that the physics wasn’t matching the maths. JB: And then you changed to geophysics… RL: It was a linear progression really. Geology was the only other interest I had. My old math’s teacher told me when I finished high school “you ought to do something different”, so I did geology. I wasn’t all that enthused about geology. Geology of that time was just getting into the story of plate tectonics. If the lecturers were prepared to tell you all about the new plate tectonics in the literature, it would have been a fascinating subject to listen to. They just wanted to tell you about the 13 crystal classes, mineralogy which wasn’t very fascinating - fossils were fascinating. But that was one of my areas of interest which later came on as geophysics. There was only one senior lecturer in the Geology Department, and I was friends with him the last 10 years he was there, then they made Mike Dentieth the new appointee in 1990. So the university decided they were going to try and get something going. Mike and I hit it off, and there was a lot of cooperation. Mike Partis, who was head of the school at the stage, was very supportive, then Cheryl Praeger was very supportive after that. They helped get things moving along. JB: You’ve had some successful graduate students of late. RL: In the last few years we’ve had 7 or 8 honours students, 3 PhDs, and well into the double figures of pass graduates. In terms of a minor program, it was quite successful. JB: Who are your recent graduates? RL: Amanda Buckingham, Tom Ridsdell-Smith, and Fabio Boschetti.

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JB: You’ve been around a while. Do you have any wisdom or items from your experience that you could share with a young person such as myself? RL: The point I was trying to make with you on the stairs the other day, is not to look back but to look forward. The school really is going to be invigorated. Hopefully we’ll get a new young dynamic applied professor. And then you’ve got 8 or 9 retirements due, and a large number of the school will be under 40. Under 40’s stir the place up. In the old days there used to be a mix, so at least half the staff were under 45. The thing I discovered with the geophysics is that there is this wonderful opportunity that many of our graduates never see. There are people out there who are just dying for some help, and they don’t know how to ask the help, and you don’t know enough about them or know enough about there subject to give them help. That’s where the success of the geophysics program is. You’ve had two people – a geophysicist who can see the point of the mathematics and a mathematician who can see part of the geophysics idea. So we’re complementary, and we’ve created a bridge between the two subjects. And we’ve been able to train students up in both subjects so that they have a strong enough background. If you do geophysics, you need a pass degree in applied mathematics and yet you have to have a strong geology background too. And so you were very well equipped to understand the problem and at the same time, you had a strong mathematical background to do something with it. With applied mathematics it’s not a matter of just, here’s a problem, here’s the mathematics, let’s solve the problem – like fracture mechanics, “a crack obviously moves in a straight line”. JB: How should we teach students in applied mathematics? Well that’s not for me, there’s a new order coming in, isn’t there?! You can see what mathematics you’re going to need and certainly the electrical engineers are keen to have their students trained reasonably well. They’re the main group that comes through in first, second, and third year. It does make a difference if it’s steering towards the applications. In

case the of the electrical engineers, most of the 217 students are studying a serious electromagnetics course later on, there’s Maxwell’s equations, so you need divs, grads, curls, Stoke’s Theorem, and Gauss’ Theorem. They all start to have a physical meaning. JB: Do you think it’s important for engineering students to understand some of the philosophy of applied mathematics? RL: I think a lot of the engineering students can see the modeling, but they miss out on getting enough mathematical background, they often pull out after second year mathematics, and so they don’t go on to see the loads of interesting problems. There’s got to be some way to get back to them later down the track to getting them to build their mathematical tool kit up. There are new tools coming out all the time. Tom Ridsdell-Smith was able to show with his

wavelet theory, just how applicable it was. It’s not the answer in every case, and there are other tools that will do something similar to wavelets. But it does show a lot of different insights, and that’s what I mean. A lot of these other students don’t see a lot of these tools. But that’s something that can come from the

math’s school interacting with the engineering department. Certainly, pure math’s, the thing you’re interested in, is not without applications over there. But it takes a lot of effort for the contacts to be made and kept. That’s the real difficulty. We found it very helpful to send out students at the end of their honours year. Most of them have gone out and understood the practical problems, and got a clear idea of the different problems there are to solve. JB: What has been the favourite component of your research? RL: The geophysics. JB: What has been the favourite part of your teaching? RL: Teaching engineers.

Members of the School of Mathematics and Statistics in 1962: Professors: (A.L. Blakers & H.C. Levey) Readers (J.P.O. Silberstein, D.G. Hurley, N.U. Prabhu) Senior Lecturers: (F. Gamblin, R.J. Storer, E.W. Bowen, P.E. Wynter) Lecturers: (B. Briner, B. Hume, M. Hood, W.S. Falk, B.S. Niven) Temp Lecturers (J. Billings, R.L. Duty) Senior Tutor: (R. List) Tutor: (R.M. Coghlan) Temp Tutors: (E.J. Giles, K.G. McNaughton, R.A.H. Jackson, C.L. Jarvis, P.N.Kennedy ) Graduate Assistants: (U.N. Blat, K. Freeman)

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Top Five Reasons Not to do A PhD

Hello. I’m Sophie Ambrose, a mild-mannered Pure Mathematics Phd student, and I’ve been asked to tell you all about the joys of postgraduate-ness here at the UWA School of Maths and Stats. Now first off, I’m not going to

pretend that a postgraduate degree is for everyone. It’s probably a good idea that you like Maths and/or Statistics a fair bit, and don’t mind the odd bit of research. (Although then again there’s always Masters by Coursework.) But one thing I know from my own experience and talking to other people, is that there are some misconceptions as to what being a postgrad is like and who should do it. I’m basing this on my own experience as a PhD student, but this applies to the other postgraduate degrees too. So, here are some reasons not to do a PhD, and why they’re well, wrong. 1. Honours was bad enough, I don’t want more of it! 2. I’m sick of uni, it's time I grew up and a got a real

job. 3. Who wants to spend the rest of their life hanging

around mathematicians? (Or statisticians) 4. I’m not PhD material. With all these good reasons how on earth did I end up as a PhD student? Well, to be honest, one major reason is that I realised I didn’t like the look of any of the jobs I could get with my degree. I’m one of those people who, loves theory and hates practice, and if I didn’t do postgrad I’d probably end up being stuck doing practical stuff for the rest of my life. But what about my objections? Well I’ve been doing this for over two years, and I’ve discovered: 1. Postgrad is nowhere near as bad as honours. Yes, you are doing the same sort of thing, and the beginning of my PhD had a lot of the same feeling of overwhelmed confusion as most of honours (there’ll probably be the same sense of panic at the end too) but it’s all spread out, and in the middle you have time to understand what you're doing and do it properly. That and hopefully you’ve already got the hang of the basics. If honours was one long living hell, you probably shouldn’t do postgrad. If it was stressful but rewarding, you possibly should. (To any potential honours students – don’t let this description put you off!) 2. Postgrad students are grown-ups. Ok, one of the main points I took a while to grasp - they PAY YOU to do a postgrad degree! (This is provided you get a scholarship, which isn’t as hard as getting an undergraduate one.) There is also all the tutoring, which is rewarding in and of itself. There are no exams. No one is telling you what to do. While in theory I am still a student I really feel like a grown-up in a

way I didn’t as an undergraduate. Even if, like me, you did your undergrad at UWA, it’s a whole new place. You get an office and access to the common room, which has useful things like cheap tea and a microwave. All those scary lecturers suddenly turn out to be nice people with whom you can have an interesting conversation. I’m on a committee that lets me be involved in school decisions and organising stuff like staff/student barbeques and this magazine. At the same time you can hang out with all the other postgrads, a ready-made social group who won’t think you’re a freak for liking maths/stats. There’s fun seminars (really, they are),

get-togethers, and of course excursions to the pub. And if you want, you can still do all the old student things you did before.

3. There’s no rule that says that just because you do a postgraduate degree, you’re doomed to academia. I’m always hearing about people going off after they finish postgrad to work for some company or the government, doing practical applications of their research. And academia isn’t as bad as it’s made out to be. Its pretty exciting to be part of a worldwide effort to discover whatever it is you’re discovering, and contrary to popular opinion, some academics are actually young and, I kid you not, cool. (Ok, not all of them. But more than I expected.) 4. Working on a PhD isn't as insane as you might think. Well, yes, they ARE hard. I wouldn't recommend it to anyone who wasn't getting reasonably high marks in their chosen field, or who isn’t willing to put in a fair bit of work. But that doesn’t mean you had to be the absolute top of the class, or that you’ll have to work 24 hours a day. I’ve met

several students with second class honours doing really well. (Better than me anyway.) But what if you really don‘t want to do a PhD? This brings me to the missing fifth reason:

5. You’re doing masters instead. (You thought I couldn’t count, didn’t you?) It’s shorter and can be consist of a fair bit of course-work if you don’t want to do just research. You can swap from Masters by Research to a PhD and vice-versa if you change your mind. One major, often unmentioned reason for doing a postgraduate degree is the perks. Not only do you get to do the best subject ever and meet all the interesting visitors to the school, you get free stuff. There’s lots of nice lunches and so on, and the cushy scholarship, but my absolute favorite so far has been flying free to America via Japan for a conference. Free buffet breakfast and lunch, dinner with other conference members, and a week discussing my research with the leaders in the field. And I had a two day stopover in Chicago. Total cost to me: under $300. Going to Sydney was pretty cool too, and I might go to New Zealand

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next year. To do maths, of course. I feel obliged to briefly say that if you are going to do a postgraduate degree then this is a pretty good place to do it. UWA is a great uni, with a very active social scene, and this school has a lot of top researchers. If you’re interested, there’s heaps of info all over the place. (Ed: There’s a bit of info in this magazine.)

So, have I persuaded you to come join me here? No? Well, that’s OK. The main point of this article is to “raise awareness”, as they say, and to maybe get a few of you to find out just how rewarding a postgraduate degree in mathematics or statistics can be. Though I must mention one downside I’ve discovered in doing a PhD on algorithms no space: I keep writing everything in lists...

- By Sophie Ambrose

Divides Crossword

1 2

3 4

5 6

7 8

9 10

11

12 13

14

15 16

17

18

19

Across 1. A real number that sounds like a climber. (6) 3. A chaotic monarch and swiss universalist. (5) 5. The image is odd, forgetting the way. (5) 8. Fall behind and 5 across, for a brilliant frenchman. (8) 9. Not many divisors for this popular time in television. (5) 11. Bury an article, including good antiderivative. (9) 13. French genius, at home next to the river – with interest. (8) 15. This shape is told to fish. (7) 18. The result of multiplying, for a drain. (6) 19. English mathematician was surprised by two units. (7)

Down 2. It is in 15 across, three times! (5) 4. The symbols on a map about the Italian mathematician. (8) 6. The study of 15 across will make yogi tormentor forsake nothing (12) 7. To generate a subspace, and measure. (4) 10. One seizure mixed up with unknown pub, for boundlessness. (8) 12. A fraction of a portion, next to metallic element. (8) 13. The french dropped from pile of circular constant. (2) 14. Covert trouble of great magnitude and direction (6) 16. Fuel has america inside, a german genius. (5) 17. He won a Nobel prize, in the northern white. (4)

Did you know that recent studies have shown that crossworders are three times less likely to develop Alzheimer’s Disease in old age?

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Modern Security, by Jon CohenMeet Alice. She is a spy who has infiltrated the highest ranks of that really big bad company. You know the one. Alice wants to regularly send reports back to her employer, Bob, who is the vice president of the major competitor. The problem is that Alice and Bob have never met before. How can they engage in secure communications without ever meeting each other? Furthermore, how can Bob be sure that the messages he receives from Alice were really sent by Alice and not by an imposter? The full answer to these questions requires some pretty heavy mathematical machinery. But that’s OK, because by the end of this article you are going to know how it all works. No, you are not going to understand all the advanced mathematical tools and tricks involved, however, you will have a conceptual overview of how the problem is resolved. Setting the scene Let’s be a bit more specific about just what is going on between Alice and Bob. Alice is sending messages from her computer to Bob’s computer along some communications channel. To simplify the scenario, we will assume that the communication channel is a direct link between the two computers and that it is error free (that is, the channel itself does not create any errors in the transmitted messages). So, what’s the problem you say? Alice and Bob have a direct communications channel between each other and the channel is error free. How can that possibly be insecure? The problem is that we cannot assume that there are no eavesdroppers reading messages that are passed across the channel. In fact, our fears are well founded because there is indeed an eavesdropper, Eve. Eve reads each and every message which is passed between Alice and Bob! How do we stop her from knowing what is going on? Enter cryptography, Stage left. Cryptography, coming from the Greek Cryptos, meaning secret, and Graphos, meaning writing, is the study of securing communications which dates back thousands of years. There is a certain amount of jargon attached and I will get that out of the way first. The original message is called plaintext. The process of transforming it so as to hide its meaning is known as encryption. The result of encryption is ciphertext and the process of transforming ciphertext back into the original plaintext is known as decryption. This is summarized by the following figure: Julius Caesar was one of the first notable figures to utilize cryptography in a significant way. When sending a message to his generals, he did not want the messenger to be able to read it so he devised a very simple encryption scheme. The first step is to convert your message from letters into numbers as shown below: A B C D E F G H I J K L M N O P Q R S T U V W X Y Z 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 Next, you add 13 to every number. If you get to a number greater than 26, then you just subtract 26 from that number. Finally, convert these numbers back into letters. For Example, to Encrypt the letter P you would do: P = 16. 16+13 = 29. 29 is more than 26 so we do 29 - 26 = 3. Therefore, P encrypts as C. So, “WE ATTACK AT DAWN” would be encrypted as “JR NGGNPX NG QNJA”. In order to decrypt the received message, the generals would subtract 13, adding 26 if the resulting number is less than zero. This encryption scheme is known as ROT13 in the modern literature and is extremely insecure. Why is that? Well, in cryptography you have to assume that your opponent knows exactly how your encryption scheme works. If anyone knew that the message was encrypted using ROT13, it would be easy for them to decrypt it. As simple as ROT13 is though, there are people that believe it offers a decent level of security. A couple of years ago, a Russian named Dmitri Sklyarov caused a bit of a fuss when he broke Adobe’s e-book encryption scheme – they were using ROT13!

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Modern Security, by Jon Cohen cont. So, if your opponent knows exactly how your system works, how can you make it secure? The trick is to use something known as a “key”. This parallels how security works in the physical world – everyone knows how to unlock a door but only the person with the correct key can do the unlocking. To change ROT13 into a keyed encryption system, we would use a number between 2 and 25 instead of automatically using 13 and this number would be the key. (This system is called ROTN where N represents the key number). So now anyone intercepting a message knows that it was encrypted using ROTN but they do not know what N was used. In order to break the encryption scheme (i.e. for someone other than the intended recipient to recover the plaintext from the ciphertext), they would first need to find out the value of the key. Even this is not a very secure system because it is easy to figure out the key from a piece of ciphertext. (Challenge: What key was used to encrypt the following message: GOVV NYXO). For many years after this, the way in which cryptography schemes were designed was to use an increasingly elaborate system of substitutions (swapping one letter for another) and transpositions (changing letters like in ROTN), so that guessing the key from a piece of ciphertext would be very difficult. In the Second World War, the German U-Boats used a system known as ENIGMA, which uses many transpositions and substitutions on each letter to produce the final message. Even with the complexity of this system, it was broken by the English mathematician Alan Turing. The ability to break the U-Boat messages was a significant factor in the eventual Allied victory. Essentially, a cryptography system’s security is based around the security of the key. The harder it is to guess the key from a piece of ciphertext, the better the system is. As a result, the communicating parties need to keep their keys secure – if the enemy finds out the key, the system is worthless (similarly, leaving the key to a high tech safe lying around negates the security provided by the safe)! It is not unusual for diplomats to have keys couriered half way around the world to the person with whom they wish to communicate. Exchange Problems Let’s return to Alice and Bob, armed with some knowledge of cryptography. We now know that it is possible for them to engage in secure communications utilizing elaborate cryptographic routines. But we haven’t solved the problem of how they can communicate securely without ever meeting each other. Why not? Well, the basic protocol followed would be for Alice to pick a key and send that key to Bob so that any subsequent communication could be encrypted with that key. It is not possible for them to utilize a courier service because they do not want anyone to know that they are communicating so the only other alternative is to send the key across the communications channel. But this is not wise because there is nothing stopping Eve from just making a copy of the key as it goes through the channel and using it to break any messages encrypted with it. So, how do we get around this problem? How can two people who have never met before transmit a key across an insecure channel? The solution proposed was extremely novel and completely contrary to established cryptography dogma. A little more cryptography history is needed to fully appreciate its impact. Since the First World War, cryptography has been considered a military technology. The major modern players have been the National Security Agency (NSA) in America and the Government Communications Headquarters (GCHQ) in the UK. Any information related to either cryptography or cryptanalysis (the study of how to break encryption systems) was considered highly classified and fiercely protected (Especially during the Cold War). Armed guards would transport keys between communicating parties. The mere thought of allowing any unauthorized access to a key was unthinkable. The little knowledge that existed outside of intelligence organizations was mostly related to ROTN and slightly trickier systems – coming nowhere close to the sophistication of military schemes. Indeed, public knowledge was mostly on the level of children’s puzzle books. This was the scene when a mathematics graduate named Whitfield Diffie first thought of the key exchange problem. Far from being concerned with aiding industrial espionage, Diffie was concerned with how to secure privacy in an increasingly electronic world. Being a civilian Diffie had no access to government knowledge in the area. However, the problem plagued him and he spent much of the early 1970’s traveling around the United States trying to learn all he could about cryptography. It was not until he met up with an engineer named Martin Hellman that a solution appeared. In 1976 they published a journal article which changed the face of cryptography forever. In it, they described a system which, would later come to be known as the Diffie-Hellman Key Exchange Protocol and form the foundation for modern cryptography. Splitting the key What made the discovery so revolutionary? It broke from hundreds of years of tradition in cryptography by asserting that the key would be published! But how can this be if the key provides all the security? All previous systems were based on so-called symmetric encryption. What this means is that the same key is used in both the encryption and the decryption process. What Diffie

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Modern Security, by Jon Cohen cont and Hellman did was to split the key into two parts – a Public Key which Alice is free to publish anywhere she wants and a Private Key which must be kept secure. How do these two keys work together? The trick is that anyone can encrypt a message with a public key but only the person with the corresponding private key can decrypt the message. This has been likened to a postbox – anyone can put messages into a postbox (i.e. the post flap serves as the public key) but only the person with the correct key to the box (i.e. the private key) can open it and read the messages. It is now possible for Alice to send messages to Bob securely by following these steps:

1. Bob publishes his public key 2. Alice encrypts her message using Bob’s public key 3. Alice sends the encrypted message to Bob

Eve now knows as much as Alice does about communicating with Bob. That is, she knows what encryption system is being utilized and what Bob’s public key is. However, she cannot feasibly read the plaintext message that Alice sent. How this is possible, depends on the existence of a certain type of mathematical function. In theory, it is possible to derive the private key from the public key. So, all Eve needs to do is work out Bob’s private key, based on his public key, and she can then read any messages sent to Bob. In practice, her job is a lot harder than it sounds because Diffie and Hellman specified that the encryption process must be a trapdoor one-way function. A one-way function is a process which is easy to carry out but extremely difficult to reverse. This is analogous to breaking a dinner plate – breaking it is easy, putting it back together is almost impossible. As far as the new encryption system is concerned, encrypting is the easy bit and decrypting is the hard bit. However, it would not have been wise for Diffie and Hellman to suggest the use of a one way function because then Bob would have a hard time decrypting any messages which are sent to him. Instead, they added something called a “trapdoor” to the function. This is an extra piece of information which makes reversing the function a lot easier. In the case of the dinner plate scenario, if you were given a video of the plate being broken, along with the broken pieces, playing the tape backwards in slow motion would make the job of gluing the plate back together a lot easier. The trapdoor is an integral part of Bob’s private key and is essentially how he can easily decrypt messages which are sent to him. Eve doesn’t have the trapdoor though, so her life is considerably more difficult. Putting it into practice Diffie and Hellman did not actually provide such a function in their paper. Indeed, many people dismissed the paper because they thought it impossible for such a function to ever be devised. It was a few years later that the cryptography community was again rocked by three young academics, Ronald Rivest, Adi Shamir and Lenard Adleman, who devised just such a function. Their scheme would quickly become known as the RSA algorithm and forms the basis of modern e-commerce security. We first need to go through a little more maths before understanding RSA. The set of numbers 1,2,3,… (the … means that the list goes on forever) is known as the natural numbers. A divisor of a natural number is another natural number that leaves no remainder upon division. So, for example, 2 is a divisor of 4 because 4/2 = 2 + 0. However, 3 is not a divisor of 4 because 4/3 = 1 + ⅓. A prime number is a natural number whose only divisors are 1 and itself. The first few prime numbers are: 2,3,5,7,11 and 13. (Challenge: Prove that there are infinitely many prime numbers). It is a classical result of number theory (the branch of mathematics which deals with the properties of natural numbers) that natural numbers are of three forms: 1. The number 1 2. A prime number 3. A product of prime numbers (This is called a composite number. Every composite number has a unique way in which it can be

decomposed into a product of primes) We are now ready to describe the heart of the RSA scheme. This relies on the apparent difficulty of factoring composite numbers (that is, working out which primes were multiplied together to give that composite). I say “apparent” because no one has yet been able to prove that factorization really is hard. (How you prove a problem to be hard is a whole other story which we will not get into here). The basic idea behind the system is to generate two large primes, p and q, and to multiply them together. Their product, pq, effectively serves as the public key, while p and q make up the private key. This initially seems extremely insecure. After all, all Eve needs to do is factor pq into p and q and she has broken the system! In practice, however, this is very difficult to do. The RSA system mandates that p and q are of the order of hundreds, or even thousands, of digits. Factoring their product with today’s mathematical techniques would take a supercomputer longer than the known lifetime of the universe! There is even a prize of US$200,000 if you can factorize the following 617 digit RSA key:

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Modern Security, by Jon Cohen cont. 25195908475657893494027183240048398571429282126204032027777137836043662020707595556264018525880784406918290641249515082189298559149176184502808489120072844992687392807287776735971418347270261896375014971824691165077613379859095700097330459748808428401797429100642458691817195118746121515172654632282216869987549182422433637259085141865462043576798423387184774447920739934236584823824281198163815010674810451660377306056201619676256133844143603833904414952634432190114657544454178424020924616515723350778707749817125772467962926386356373289912154831438167899885040445364023527381951378636564391212010397122822120720357 Problem Solved? Alice and Bob now have a way to exchange messages securely using the RSA encryption system. However, we have not solved the second of our problems – how does Bob know that it is really Alice who sent the message to him? After all, Eve has access to Bob’s public key and the communications channel so nothing is stopping her from pretending to be Alice! However, we can solve this problem with the tools we have developed so far, with one small catch. The catch is that there is another half to the Diffie Hellman protocol. It is also possible to encrypt messages with a private key and decrypt it with the corresponding public key. This works as a form of digital signature as follows:

1. Alice and Bob publish their respective public keys 2. Alice encrypts the plaintext message with her private key 3. Alice encrypts the message with Bob’s public key 4. Alice sends the message to Bob 5. Bob decrypts the message with his private key 6. Bob decrypts the message with Alice’s public key

Eve does not have access to Alice’s private key, so if Bob receives a message from Eve, step 6 would produce gibberish and Bob would know that the message is not from Alice! Where to from here? The description of cryptography provided by this article has been necessarily broad and vague. The field utilizes many different branches of mathematics from number theory to geometry. As such, a major in pure or discrete mathematics would provide the perfect basis for a deeper understanding of the subject. It is also a good idea to include a few computer science units as well to provide the necessary background on implementation issues. There are a couple of highly enjoyable historical accounts of cryptography. The first is by Simon Singh and is called The Code Book. This chronicles “The science of secrecy from Ancient Egypt to Quantum Cryptography”. The second book is by Stephen Levy and is called Crypto. This provides a detailed discussion of the development of public key cryptography and reads more like a spy novel than a historical account. Highly recommended! Also recommended is the novel by Neal Stephenson – Cryptonomicon.

JOKES Three men are in a hot-air balloon. Soon, they find themselves lost in a canyon. One of the three men says, "I've got an idea. We can call for help in this canyon and the echo will carry our voices far." So he leans over the basket and yells out, "Helllloooooo! Where are we?" They hear the echo several times. Fifteen minutes later, they hear this echoing voice: "Helllloooooo! You're lost!" One of the men says, "That must have been a mathematician." Puzzled, one of the other men asks, "Why do you say that?" "For three reasons. One, he took a long time to answer; two, he was absolutely correct, and three, his answer was absolutely useless." Why did the mathematician call his dog Cauchy? Because it leaves a residue at every pole.

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Research In Probability And StatisticsResearch Rankings Measuring the real contribution of research output is a difficult task. Using a volume count of work published in journals universally recognised as being in the top rank should ensure there is a reasonably high association between volume and quality. Genest (1999) and (2002) made a scholarly analysis of international publication data for the two disciplines, Probability and Statistics, and it is heartening to find that UWA is the top Australian centre for probability research, and holds a high position in international rankings. Genest (1999) bases his analysis on numbers of papers published during 1986-1995 in nine leading probability journals. His analysis was extended in Genest (2002) to cover the period 1986-2000. Nominal page numbers are scaled to equivalent numbers for the “Annals” journals published by the Institute of Mathematical Statistics (U.S.A.). Genest's principal ranking statistic is the sum over all article pages times the number of authors, but he explores alternative indices and finds they yield similar outcomes. The data up to 1995 places Australia ninth in the world, and UWA is ranked 33rd out of rather more than 150 author institutions. No Australian institution appears in the top 25. Some institutions in the top 25 appear because they are represented by just one or two authors, at least one of whom was very productive. Genest (1999) adjusts for this by ranking institutions having at least five contributing authors, his criterion of a “fertile research environment”. On this basis UWA ranks 14th in the world, and it ranks first among institutions outside Europe and North America. The data up to 2000 ranks Australia world number six, and UWA moves up five places to 28th position. Genest (2002) modifies his definition of “fertile research environment” to require at least ten distinct contributing authors, and UWA then rates 10th in the world out of 165 institutions. In addition UWA maintains its leading position outside Europe/North America. An interesting subsidiary aspect of Genest’s data emphasizes the difficulty of trying to keep abreast of scholarly progress. His selected probability journals published a total of 146,833 pages during 1986-2000, ignoring other probability journals and the large volume of applied and theoretical research appearing in economics, finance, physics, biology and engineering journals. UWA authors contributing to this outcome comprise: (UWA teaching & research staff) A.J. Baddeley, T.C. Brown (left 1992), R.A. Maller, R.K. Milne, A.G. Pakes, V.T. Stefanov, P.G. Taylor (left 1990), S. Zhou (left 1997); (Research Fellows/students) Y.C. Chin, U. Hahn, G.F. Yeo. (The list

omits co-authors who visited UWA for intermediate terms, of about 1-3 months.) Genest, C. (1999) Probability and statistics: A tale of two worlds? The Canadian Journal of Statistics 27, 421-444. Genest, C. (2002) Worldwide research output in probability and statistics: an update. The Canadian Journal of Statistics 30, 329-342. Econometrics and Finance Dr Jiti Gao, a senior lecturer in statistics, has research interests in the fast moving world of finance. He and collaborators are working on nonparametric and semiparametric approaches to econometric and financial modelling, currently supported by significant grants from the Australian Research Council and the Hong Kong Research Grants Council. Jiti has recently presented a review of his latest research results at the Australasian Meeting of the Econometric Society in Sydney, and will give a fuller account at the 2004 North-American Winter Meeting of the Econometric Society in San Diego early next year. Cars, Genes, and Applied Statistics For many statisticians, it is the wide applicability of their discipline which makes it so fascinating. This is certainly the case for Dr Martin Hazelton, a senior lecturer in statistics and Director of the Statistical Consulting Group. Martin has worked on statistical problems in a number of different fields, but is particularly interested in applications in road transport and genetic epidemiology. Both are fields of research that make considerable use of mathematics and statistics, and innovations in both areas have the potential to make a real difference to people’s lives. Road traffic congestion is a major problem in a great many cities across the world. Travel delays result in a loss of productivity, while traffic congestion also contributes significantly to greenhouse gas emissions. Traffic planners evaluate schemes intended to reduce traffic congestion using computer based models that make heavy use of statistics. In particular, statisticians can play a pivotal role in tuning the models using observed road traffic data. Genetic epidemiology is the study of the role of genetics in complex diseases. For many complex diseases, like asthma and cancer, both genetic and environmental factors play some part in determining whether or not a person is afflicted. In recent years there has been an explosion in the amount of available data, including the development of some large databases linking medical records for different members of the same family. Statisticians need new, sophisticated methods to analyse these huge data resources and pinpoint the most important risk factors for disease.

- Dr Robin Milne, A/Prof Tony Pakes, & Dr Martin Hazelton

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What’s happening in Pure Mathematics? Bertrand Russell, a famous British philosopher and mathematician of the twentieth century, defined pure mathematics as the “set of propositions”. However, practitioners in pure mathematics would define “pure mathematics” as that part of mathematical activity that is done without explicit or immediate consideration of direct application which is driven by curiosity and a desire to solve complex problems. The Science Citation Index (SCI) indicates the impact of a scientific result, and is one way to measure the influence of researchers on the world of science. The School of Mathematics and Statistics of UWA joined the group of the top 1% of institutions worldwide in terms of citation counts

in mathematics and statistics in 2002. Cheryl Praeger, the professor of mathematics of UWA, is an eminent mathematician in pure mathematics. During the last 10 years, she has been extremely productive and has published 86 research papers in important mathematical journals. In fact, she is ranked at 8th in the world in terms of papers published by mathematicians with at least 116 citations in the last 10 years! She is one of the most influential mathematicians during this period, and her articles have been cited more than 300 times - which gives her a ranking of 88 in mathematics world-wide. Praeger’s research lies in group theory and algebraic combinatorics. In these areas, her publication ranking is number 1 in both citations and number of published papers.

- Associate Professor Cai-Heng Li

What’s happening in Applied Mathematics?

The International Congress on Industrial and Applied Mathematics (ICIAM) is held every four years and is the most important general meeting, worldwide, for applied mathematicians, and covers the full spectrum of research topics in applied mathematics and its industrial applications. The 5th congress, ICIAM 2003, was held in Sydney, from July 7 to 11. Over 1600 delegates, from more than 60 countries, attended. There were six other cenferences embedded within ICIAM itself! There were over 1700 talks. Thirty-four of these were given by invited speakers, all of whom are world leaders in their areas of research. One of the invited talks was given by Professor Cheryl Praeger of the University of Western Australia's School of Mathematics and Statistics. The contributed talks (that is, all of the other talks) were squeezed into the congress week in blocks of parallel sessions, each with 43 talks going simultaneously! Among the multitude of talks presented were “Rhythms of the nervous system: biophysics and dynamical structure”,

“Random matrices and the Riemann zeta function”, “Separating salt and pepper: axial segregation of granular materials” and my favourite, “The mathematics and physics of body-surfing”. You’d think that with all these talks going on, the delegates would spend all of their congress time sitting and listening, but this is not the case. Your brain would explode if you tried to do that. Much of the time is spent catching up with old friends and collaborators and meeting new people in your area of research. Often the input of another point of view can lead to great insight into the solution of a problem. Sydney is a great place to visit and with the congress being based in and around the tourist area of Darling Harbour there were lots of things to do and see when the brain started to overheat, and lots of relaxed areas where serious discussions of research problems could take place. All in all, a very productive and satisfying week.

- Dr Des Hill

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Maths courses at UWA, and why they are useful!

Modern life, as we know it, would be impossible were it not for mathematics. In our present rapidly changing society this need for mathematics will grow; students equipped with a knowledge of mathematics will be able to cope with the changing demands of society. Mathematics is a discipline which combines well with studies in an enormous number of other areas, including physics, engineering, computer science, chemistry, geology, accounting, agriculture, environmental science, human biology, commerce, economics, epidemiology, geography, archaeology, psychology, human movement, pharmacology, botany and zoology. Employment opportunities for mathematics graduates are better than those for many other disciplines. In a ranking of employment prospects of over thirty different subject areas, mathematics graduates rated in the top ten, ahead even of areas, which traditionally have high employment rates. The fields of employment are diverse, including private industry (mining, manufacturing, engineering, retailing, insurance, banking, computing, data processing), government institutions (CSIRO, Bureau of Meteorology, Australian Bureau of Statistics, SEC, Department of Agriculture, Department of Conservation and Land Management, Environmental Protection Authority, Water Authority of WA) and teaching, where there is always a need for teachers who specialise in mathematics. Employers value the key qualities of a mathematics graduate: the capacity for problem idealisation and problem solving. There are lots of ways you can study maths at UWA!

BACHELOR OF COMPUTER AND MATHEMATICAL SCIENCES (BCM) In this very flexible degree students combine studies in mathematics and computer science with studies in other disciplines including Economics, Commerce, Agricultural Economics, Geographical Information Systems, Environmental Engineering, Building Technology, Music, Philosophy, Linguistics, Japanese, German or Biochemistry. BACHELOR OF SCIENCE (BSc) In the BSc degree students can major in one or two of Applied Mathematics, Mathematical Statistics, Mathematical Sciences or Pure Mathematics, in conjunction with studies in a wide variety of disciplines in the physical, biological, social and earth sciences. BACHELOR OF ARTS (BA) Mathematics is available as a major in the BA degree, and combines well with studies in Economics, Psychology, Philosophy, Linguistics, Geography, Archaeology and Anthropology. MATHEMATICS AND ENGINEERING Mathematics units form an integral part of all courses in Engineering. Typically students in one of the BE courses take one unit of mathematics in first year (12 points) and one or two units in second year (6 - 12 points). In the combined Science/Engineering course students can include one first year unit (12 points) of mathematics, at least 18 points at second year and at least 24 points at third year, and can major in any one of Applied Mathematics, Mathematical Statistics, Mathematical Sciences or Pure Mathematics. The extra mathematics in the combined course programme greatly enhances the employment prospects of graduates and puts them in a strong position to understand many of the latest developments in engineering, science and technology. BACHELOR OF ECONOMICS (BEc) AND BACHELOR OF COMMERCE (BCom) Within the BEc and BCom degrees there are opportunities for combining Mathematics with majors in Econometrics, Economic Statistics or Quantitative Finance.

For More information please contact a first Year advisor Dr Jenny Hopwood 9380 3356 hopwood@maths.uwa.edu.au Dr Martin Hazelton 9380 3460 martin@maths.uwa.edu.au Mr Frank Yeomans 9380 3384 frank@maths.uwa.edu.au

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Degrees Beyond Graduation The University of Western Australia is one of Australia’s leading universities. Founded in 1913, it has over 15,000 students, including the vast majority of the top school leavers in the State. The University of Western Australia maintains world class standards of scholarship and teaching. School of Mathematics & Statistics The School has a teaching staff of 35 whose research interests cover most areas of pure mathematics, applied mathematics and probability & statistics. The School is ranked among the top 1% of institutions in the world in terms of research citation counts and 6th in the world for research in probability theory. Research is especially strong in the fields of dynamical systems, optimal control theory, industrial applications, spatial statistics, computationally intensive statistics, probability, stochastic processes, experimental design, group theory, combinatorics, linear analysis and topology. There are excellent computing and library facilities. Graduates of the School readily find employment in a wide spectrum of interesting occupations. Masters Degrees The Masters Programmes offered are

• Master of Mathematical and Statistical Science (Coursework)

• Master of Mathematical and Statistical Science (C/work & Dissertation)

• Master of Mathematical and Statistical Science (Research & Thesis)

• Master of Arts • Master of Science Education • Master of Financial Mathematics

The usual entry requirement is a Bachelor's or Honours degree. A programme normally takes from one to two years to complete, depending on qualifications, and it may be taken part-time.

Doctor of Philosophy (PhD) Supervised research is available on a wide range of topics. The usual entry requirement is a Honours or Master’s degree, or a Postgraduate Diploma. The programme can start at any time of the year.

Graduate Diplomas Graduate Diplomas in Science (GDipSc), in Financial Mathematics and in Computing & Mathematics (GDipCM) are offered. These are one year programmes and the normal entry requirement is a Bachelor’s degree.

Further Information The School’s homepage http://www.maths.uwa.edu.au has links to many sources of information, including financial support. Information and assistance is also available from:

Director of Graduate Programmes School of Mathematics & Statistics

University of Western Australia 35 Stirling Highway, Crawley

Western Australia 6009

E-mail: gradprog@maths.uwa.edu.au Fax: (08) 9380 1028

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ba=2

Perfect Numbers and the Order of Pythagoras By Geoffrey Pearce - 2003 Maths Honours Student

A number equal to the sum of all its factors but itself is known in the present day as a ‘perfect number’. The origins of this name can be traced to the Pythagorean Order --- a reclusive, dedicated brotherhood of mathematicians in Ancient Greece, led by the philosopher Pythagoras. In

this setting, the word ‘perfect’ was weighted with religious passion: for the Pythagoreans, mathematical inspiration came from the pursuit of universal wisdom and its perceived embodiment in numbers. In this paper we examine the historical context of their mathematical philosophy and the few records of Pythagoras' life-story, which together provide an illuminating backdrop to the discovery of perfect numbers, and in turn, serve to explain the emergence of mathematics as an independent study. Although Pythagoras is remembered principally as a pioneering mathematician, he was an almost legendary figure in his own time - a prophet and a mystic, for whom mathematics was as much a religion as a scientific practice. He was born on the Isle of Samos in about 560 BC, at the end of a two-hundred year period of Greek colonisation throughout the Mediterranean. This expansion had brought new affluence and political power to the Greek empire, and was accompanied by vigorous artistic and intellectual activity, heralding the start of the Classical Age. Pythagoras chose the remote, politically isolated colony of Croton, in Italy, as a base for his religious Order. Initially, its ideologies reflected the changing intellectual climate - particularly the shift away from mythology as an explanation of universal phenomena. At their core were beliefs in the divine nature of the soul and its actualisation through knowledge, the importance of proportion and harmony, and also humanitarian concerns, such as ethics and virtue. In the tradition of a religious cult, Pythagoras and his followers led an austere lifestyle, observing strict codes of abstinence and secrecy; these bore a strong resemblance to doctrines of the Egyptian priesthood, and were probably absorbed by Pythagoras during travels in his earlier years. Given the ostensibly sacred beginnings of the Order, it is interesting to consider why mathematics was eventually to preoccupy the Pythagoreans. It is known that Pythagoras developed a lasting fascination for the subject during his childhood. Additionally, Greek secular culture had considerable influence - in particular, the Greeks saw great importance in aesthetic appeal; their architecture, sculpture, and even dramatic works show a deep concern for beauty of form and proportion. The mathematical flavour of these qualities was widely appreciated - in fact, after the fall of

Athens to the Persian invasion in roughly 490 BC, the city was rebuilt entirely to a geometric design. Scholars tended to regard mathematical truths with reverential awe: to them they seemed the purest manifestation of art and the divine. The Pythagoreans were drawn to this veneration of mathematics, and formalised the discipline by introducing consistency and accountability via the notions of rigour and proof. Mathematics, and especially numbers, became the centrepiece of the Pythagorean ideology - a fact that was eloquently expressed in the dictum: ‘everything is numbers’. Unlike the transient material world, numbers appeared ageless and immutable, and these qualities were indicative of divinity. The Pythagoreans’ doctrine refers implicitly to the em natural numbers - the positive integers - as opposed to all real numbers. Pythagoras had based this universal conception on a few observations of whole number relationships in the natural world - for instance, in the ratios of lengths of strings vibrating with harmonious musical intervals, or in the orbits of the planets. The Order held this faith with deep emotion, as is suggested by the tumult following the discovery of incommensurability - which implied that the length of the diagonal of a unit square could not be expressed as a ratio of whole numbers. The proof given here is remarkable for its simplicity and aesthetic qualities (which must have rendered the result all the more painful to the Pythagoreans). Confronted by this unsettling truth, the Order hastily divided into several sects, each of which tried in a different way to resolve the discovery in the light of Pythagoras’ convictions. Theorem: The length of the diagonal of a unit square (equal to the square root of 2) is irrational. Proof. Suppose that for some integers a and b, whose greatest common divisor is 1. Then, and hence 22 2ba = . Since 2b is an integer,

2a is even, and hence a is even. Since 2 divides a, 2 cannot divide b, and so b is therefore odd. Now, since a is even, there exists an integer c such that a = 2c. So 222 24 bca == . Thus 22 2cb = . Since 2c is an integer, 2b is even, and hence b is even. This contradicts the earlier result that b is odd. So 2 is not rational, and is therefore irrational.

Although the Pythagoreans had rejected mythology as a viable philosophy, their attitude to numbers showed clear signs of the mythological traditions of drama and symbolism. This was most evident in their attribution of ‘personalities’ to numbers. There are records of these for all numbers from one to nine: one was representative of peace

22

=

ba

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and tranquillity, having no lesser parts and being the symbol of identity, unity and existence. Two, however, was the origin of contrast, or disunity, and accordingly was an evil number - and so on, in a similar vein. Other symbolism alluded to a more concretely ‘number-theoretic’ conception - all odd numbers were thought to be male, and even numbers female; prime numbers were virile, whereas composites were effeminate. Inevitably, the Pythagoreans’ idealism led them to consider notions of numerical perfection. Several of the natural numbers exhibited this quality: ten was thought to be a perfect number, being the sum of the first four ‘geometrical numbers’. The number four was ‘the first mathematical power, … the most perfect of numbers; … which gives the human soul its eternal nature’ [1]. According to [2], three was considered perfect for being the first ‘real’ number (the indivisibility of one, and the female gender of two precluded these numbers from such a classification!) However, the numbers for which the title has held are those equal to the sum of all their factors but themselves. These perfect numbers were so named for being balanced between the ‘deficient’ numbers, whose proper factors have a smaller sum, and the ‘abundant’ numbers, whose proper factors add to give a larger number (this latter quality perhaps violated Pythagoras’ advocacy of a frugal lifestyle). Finding a method for discovering all such perfect numbers became a pressing objective. The Order failed at this (in fact, there is still no such method known today), but they did find the first four (which are 6, 28, 496 and 8128). Discovering these awakened an interest in related questions; for instance, whether odd perfect numbers could exist, or whether even perfect numbers in sequence ended alternately in six and eight. These two types are defined formally as follows: (Note that by the proper factors of a number x we mean all factors of x excluding x itself). Two positive integers x and y are called amicable if y is equal to the sum of the proper factors of x, and x is equal to the sum of the proper factors of y.

A set X of k positive integers consists of sociable numbers if there exists a labelling kxxx ,,, 21 K of the elements of X such

that 1+ix is equal to the sum of the proper factors of ix , for

all i ∈{1, …, k – 1}, and 1x is equal to the sum of the proper factors of kx . The smallest pair of amicable numbers is 220 and 284. These two were known to the Pythagoreans, who naturally considered them to be symbols of friendship. Thus, perfect numbers were so named as a result of a mystical, manifestly superstitious view of numbers. As it turned out, Pythagoras’ driving conviction, that all universal truths were embodied by the natural numbers, was fatally flawed; in addition, there are probably few mathematicians today who would consider the assignment of genders to numbers to be of paramount importance. However, the legacy of the Pythagoreans should not be underestimated. Aside from having made elemental discoveries in geometry and number theory, the Pythagoreans were the first to perceive the necessity for mathematical rigour and proof. Moreover, it was their passion for numbers themselves - in an abstract sense - that led to a formalised study of mathematics for its own sake. Perhaps we can therefore consider their bizarre-seeming mathematical world - in which numbers equal to the sum of their factors were worshipped like Gods - as representing the infancy of modern pure mathematics. 1. Pythagorean Numbers [online] http://members.tripod.com/~onespiritx/magick07.htm 2. Pythagoras [online] http://www.angelfire.com/weird2/andstrife/bios/pythagoras.html

PUZZLES The other day, our Chief Statistician, Dan, was having a drink at a bar and got chatting with the bartender, who mentioned that he had three children. Dan asked, "How old are they?" The bartender said, "Well, you've got a head for figures. Can you guess how old they are if I tell you that their ages add up to 13?" Dan said, "Nah, not enough information. You'll have to tell me more." The bartender replied, "Well, when you multiply their ages together, you get the street number of this bar." Dan trotted outside for a look at the street number and came back in. "Nup. You'll have to tell me more." The bartender grinned, "Righto. Well, the oldest one loves strawberry icecream." "Aahh," said Dan, "then that would make them ___, ___ and ___ years old." What are their ages? -------------------------------------------------------------------------------------------------------------------------------------------------------------- Two wizards get on a bus, and one says to the other "I have a positive number of children each of which is a positive number of years old. The sum of their ages is the number of this bus and the product of their ages is my age." To this the second wizard replies "perhaps if you told me your age and the number of children, I could work out their individual ages." The first wizard then replies "No, you could not." Now the second wizard says "Now I know your age." What is the number of the bus? Note: Wizards reason perfectly, can have any number of children, and can be any positive integer years old.

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World News in MathematicsWhile I sit here typing and drinking milo from my torus, I’m reflecting on what has been a big year in mathematics. In particular, the very first Abel prize was awarded, the quad-annual Fields Medal was announced, and the famous Poincaré Conjecture was solved (we think?). The “Abel Prize” is a new award, founded by the Norwegian Academy of Science and Letters, which promises to have the prestige and fame of a Nobel Prize (there are no Nobel Prizes for Mathematics you know). The Abel prize is named after the 19th century Norwegian mathematician, Niels Henrik Abel for which the “abelian groups” are named after1. So who would you pick, out of the world’s living mathematicians, to be the most deserving of such an honour? The first Abel Prize was given to the French mathematician Jean-Pierre Serre, who already has a Fields Medal to his name. Those who have only treaded lightly in the various fields of pure mathematics will know that Serre is a bit of a legend. He is most known for his work in Algebraic Toplogy and Number Theory. Every four years, the Fields Medal is awarded to between one and four mathematicians for outstanding work in their field. This award has an age limit however. No Fields Medal receipient can be forty years or older. The Fields Medal is undoubtably the most well-known and prestigious of the prizes in mathematics, and in 2002, the award was given to Laurent Lafforgue and Vladimir Voevodsky. Lafforgue was honored for making major advances in the "”Langlands Program”; an area of mathematics inspired by a set of conjectures made by Robert Langlands that describe deep connections between number theory, analysis, and group representation theory. Voevodsky was honored for developing a new cohomology theory for algebraic varieties, called “motivic cohomology”. One consequence of Voevodsky’s work is a proof of the Milnor Conjecture - one of the recent main open questions in algebraic K-theory. Lafforgue is a frenchman and is one of the only Fields Medalists of recent times to have an infinite Erdos number (at the time the prize was announced, he had not yet written a paper with another author). Voevodsky currently has a position at the Institute for Advanced Study in Princeton – working alongside names such as Bombieri, Bourgain, Deligne, Langlands, Borel and Selberg. Henri Poincaré is commonly touted, by armchair mathematician spectators, as one of the most brilliant who has ever lived. Some say he was the last “universalist”2, a true genius whose influence and impact on mathematics will never be surpassed. In 1904, Poincaré proposed the following problem: 1 A group G is abelian if for every two elements a and b in G, we have ab=ba. 2 Although, John von Neumann may hold this title!

Is every compact simply connected n-manifold homeomorphic to the n-sphere?

In three dimensions we can simplify this question to: Can every bounded hole-less shape in 3-dimensional space be deformed continuously into a sphere?

A few years ago, the Clay Mathematics Institute in Cambridge, Massachusetts, offered $1 million to anyone who could settle the Poincaré conjecture. After working for years in near seclusion and supporting himself largely on personal savings, Grigory Perelman of the Steklov Institute of Mathematics (Russia) announced that he has proved the conjecture – 99 years after it was proposed. He also claims to have proved the much broader Thurston geometrization conjecture, which considers all closed three-dimensional shapes. The n=1 case of the Poincaré Conjecture is trivial, the n=2 case was known to 19th century mathematicians, n=4 was proved by Freedman in 1982 (for which he was awarded the 1986 Fields medal), n=5 was demonstrated by Zeeman in 1961, n=6 was established by Stallings in 1962, and n≥7 was shown by Smale in 1966. So if what Perelman has done is correct, and it will take quite some time for that to be decided, Perelman will have solved the problem that many great mathematicians, including Poincaré, could not solve.

By John Bamberg

Joke There are two functions x, and ex walking down the street and they run into a differential operator. He turns x into 1 and wonders why ex is not scared and ex says “you cant hurt me I'm ex”. The two functions continue walking down the street and before long run into another differential operator. This time 1 vanishes to nothing and ex again says “you can’t hurt me I'm ex”. To which the differential operator replies

“Ah ha, but I'm y∂

∂”!!

- Michael Giudici