AfterMath
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
1
AfterMath
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)
2
AfterMath
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.
3
AfterMath
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 highschool 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
4
(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 [email protected] .)
AfterMath
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.
5
AfterMath
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:
Readers
Senior Lecturers:
Lecturers:
Temp Lecturers
Senior Tutor:
Tutor:
Temp Tutors:
Graduate Assistants:
6
(A.L. Blakers & H.C. Levey)
(J.P.O. Silberstein, D.G. Hurley, N.U. Prabhu)
(F. Gamblin, R.J. Storer, E.W. Bowen, P.E. Wynter)
(B. Briner, B. Hume, M. Hood, W.S. Falk, B.S. Niven)
(J. Billings, R.L. Duty)
(R. List)
(R.M. Coghlan)
(E.J. Giles, K.G. McNaughton, R.A.H. Jackson, C.L. Jarvis, P.N.Kennedy )
(U.N. Blat, K. Freeman)
AfterMath
Top Five Reasons Not to do A PhD
Hello. I’m Sophie Ambrose, a mildmannered 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.
2.
3.
4.
Honours was bad enough, I don’t want more of it!
I’m sick of uni, it's time I grew up and a got a real
job.
Who wants to spend the rest of their life hanging
around mathematicians? (Or statisticians)
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
7
AfterMath
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
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.)
Divides Crossword
1
2
3
5
6
7
9
4
8
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?
8
AfterMath
Modern Security, by Jon Cohen
Meet 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!
9
AfterMath
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
10
AfterMath
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.
2.
3.
Bob publishes his public key
Alice encrypts her message using Bob’s public key
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.
2.
3.
The number 1
A prime number
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:
11
AfterMath
Modern Security, by Jon Cohen cont.
251959084756578934940271832400483985714292821262040320277771378360436620207075955562640185258807844069182
906412495150821892985591491761845028084891200728449926873928072877767359714183472702618963750149718246911
650776133798590957000973304597488084284017974291006424586918171951187461215151726546322822168699875491824
224336372590851418654620435767984233871847744479207399342365848238242811981638150106748104516603773060562
016196762561338441436038339044149526344321901146575444541784240209246165157233507787077498171257724679629
26386356373289912154831438167899885040445364023527381951378636564391212010397122822120720357
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.
2.
3.
4.
5.
6.
Alice and Bob publish their respective public keys
Alice encrypts the plaintext message with her private key
Alice encrypts the message with Bob’s public key
Alice sends the message to Bob
Bob decrypts the message with his private key
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.
12
AfterMath
Research In Probability And Statistics
Research 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.
omits co-authors who visited UWA for intermediate terms,
of about 1-3 months.)
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, C. (2002) Worldwide research output in probability
and statistics: an update. The Canadian Journal of Statistics
30, 329-342.
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
Genest, C. (1999) Probability and statistics: A tale of two
worlds? The Canadian Journal of Statistics 27, 421-444.
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
13
AfterMath
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”,
14
“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
AfterMath
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
Dr Martin Hazelton
Mr Frank Yeomans
9380 3356
9380 3460
9380 3384
[email protected]
[email protected]
[email protected]
15
AfterMath
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: [email protected]
Fax: (08) 9380 1028
16
AfterMath
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.
a
Proof. Suppose that 2 =
for some integers a and b,
b
whose greatest common divisor is 1.
2
a
Then,  b  = 2 and hence a 2 = 2b 2 . Since b 2 is an integer,
 
2
a 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 a 2 = 4c 2 = 2b 2 . Thus b 2 = 2c 2 . Since c 2 is an
integer, b 2 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
17
AfterMath
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).
A set X of k positive integers consists of sociable numbers if
there exists a labelling x1 , x 2 , K , x k of the elements of X such
xi +1 is equal to the sum of the proper factors of xi , for
all i ∈ {1, …, k – 1}, and x1 is equal to the sum of the
that
proper factors of x k .
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
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.
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.
18
AfterMath
World News in Mathematics
While 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 quadannual 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.
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 3dimensional 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
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!
19