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Professor Arieh Iserles – Computational mathematics

Arieh Iserles

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Ernest Raetz
Welcome to a La Trobe University podcast. I'm your host Ernest Raetz and my guest today is Professor Arieh Iserles of Cambridge University, one of the world's most highly esteemed mathematicians in computational and applied mathematics. He has been described as the leading figure in the modern field of differential equations. He is also widely respected for his ability to put a human face on this complex field, and he's currently visiting Australia to collaborate with mathematicians from La Trobe University.
Arieh Iserles
Differential equations are mathematical concepts that link motion to position of systems. Essentially they allow us a probe to understand the behaviour of systems that change across time and across space. Since the time of Isaac Newton, they have been the key instrument in using mathematics to understand the world. Not only physics and engineering, but just basing increasingly biology, social sciences, information and other areas. What brought me to that? Well, differential equations are arguably the one area that attracts the largest number of mathematicians, and I regard them fascinating. It allows a very, very easy contact between things which are mathematically deep and profound, and actual, real-life applications.
Ernest Raetz
Are mathematicians born, or are they nurtured?
Arieh Iserles
I don't know to which extent I'm representative of all mathematicians. And definitely this sort of a question you should ask evolutionary psychologists. I believe that there is a mixture of the two and definitely there is a genetic component.& But there is also a matter of nurture, of opportunity and actually of wanting to do it. What attracted me was sort of a youthful fascination with computers, more than mathematics. But once I wanted to study computers I understood that I need to understand mathematics first. Somehow, after a while, I fell in love with mathematics.
Ernest Raetz
Are computers driving a lot of modern maths these days? I mean, there was an era many years ago when a lot of students wanted to be scientists, but somehow things like physics and those basic sort of sciences seem to have waned a bit in interest. Is computers sort of kept that core discipline of maths going and re-charged it?
Arieh Iserles
Yes. But I believe that it happens at two different levels. One is that using computers can do things in mathematics that were unimaginable a generation ago. We can compute and not only computing numbers, but computing mathematical concepts. This allows us to do calculations and to develop expressions that would have been unimaginable, even in my youth. But also, I believe in a much more profound sense, that not only computers, but this whole second industrial revolution – the first industrial revolution was based on manufacturing stuff and physics and things following from physics.  The second industrial revolution is all about information.  It is not only computers – it is the internet, it is Google, it is the human genome project, it is information satellites, it is GPS and so on. And this creates a whole raft of absolutely fascinating mathematical problems. Mathematics is playing a central role. It is impossible to imagine, say, the internet without mathematics. In very, very deep mathematical ideas.
Ernest Raetz
How does something like a mathematical equation help deliver so much more information?
Arieh Iserles
Well, I can give you two examples, none, sadly of my work. These are really the two, I believe, mathematical ideas that completely changed the world, as we now know it. And one goes back two hundred years ago. To Carl Frederick Gauss. He had an idea which was completely bizarre. He never bothered to publish it. It was published after his death and this is what we now know as a fast formula transfer. José Fourier was a famous French mathematician, who in order to understand the heat flow, developed what we now know as the Fourier transfer, that represents information in what we now understand is a vegetable matter. Actually not only a mathematician, he was a very good friend of Napoleon Bonaparte, one of his coterie of scientists surrounding him when Napoleon conquered Egypt. The idea of the Fourier transfer – think about sound. You can represent sound simply the way we are listening to it but you can also represent it digitally. So we can break it into some basic components, and then you can re-synthesise it from these components, again into the sound to which we are listening, which is the way for example, the sound is broadcast on digital radio. It is first translated into something else, which is its Fourier transfer. And then it is reconstituted and we can listen to it, and the advantage of this is that this is much, much more robust. You are transmitting much less information – this information is not corrupted during transmission, you reconstitute exactly what you coded. And fast Fourier transfer allows us to transmit information very, very fast. Essentially in every television set, there is a small microchip using the fast Fourier transfer. This changed the world. No fast Fourier transfer – no internet, or maybe the sort of internet when you are waiting for half an hour for each page to download. The second idea, again without this, you don't have internet commerce, you don't have Amazon, you don't have apps shops, you don't have anything – is the RSA code. And again, a few years ago, three gentlemen Rivest, Shamir and another man, Hans Adelman, were sitting and discussion a very, very arcane problem in number theory and graph theory, and they came across an idea, and this idea is the basis of the transmitting securely of information on the internet. So, whenever you have a website which is HTTPS, this S for secret, or secure – this is using RSA coding. You are sending information – this wonderful trick is that you are encoding information in a way which is open to everybody.
Ernest Raetz
RSA means?
Arieh Iserles
Rivest, Shamir, Adelman. Yes. The code, how to encode information, the so-called public key code. Everybody knows it, but once it is encoded, all the computers in the world will not break it, except when you know, have the secret key. And this allows transmission of information, secret information, whether by banks, internet commerce and so on. And again, this is an idea that came from mathematics.
Ernest Raetz
We often talk about the fact that microchips can only get to a certain size and at some stage we're going to reach some sort of logical limit, maybe at a subatomic level or something, but in terms of maths, can they keep making stuff faster and faster all the time, or can you see yourself hitting a wall there?
Arieh Iserles
Well, there are two levels in which things are becoming fast. The level of hardware, of microchip, and the level of software. Hardware, although mathematicians are in the loop, that is mostly an engineering problem. How to place more and more circuits on a microchip, how to prevent it overheating, perhaps one day moving into quantum computing which will be a huge step change, speeding everything by a huge factor. We don't know. Many people are trying to do it, but still it is an open question whether this will work. But another level, which is the one with which I am more concerned, is the algorithmic level. OK, you have this microchip, you have this computer. It works at certain clock speed. How can we compute the same thing much, much faster on that computer, using more clever ideas how to compute? And there's a constant competition between hardware and software.
Ernest Raetz
We've had a report from Australia's chief scientists, saying that the number of students who are studying enabling sciences like maths and chemistry in this country, has flat-lined. Is this something that's happening in, say, Europe and America as well? And how does this compare to what's happening in developing nations like China?
Arieh Iserles
It is a huge mystery to me. For example, in the UK, the number of students going into mathematics, the so called stem subjects, science, technology, engineering and mathematics, is increasing in the last five, ten years. And certainly in mathematics, we have more students, not only in Cambridge. In Cambridge we never complain – we are always getting as many students as we want, probably. We could have got ten times as many, and they are excellent. But overall, in the entire system, the number of mathematicians is growing. In other countries in Europe, it is not the case. I am completely at a loss to know why is it that one decade in Holland, the number of students goes down, then goes up, and sometimes in the UK … and all these things are out of phase. So there might be some global trends but they will happen across decades, rather than years. But definitely, in the third world, and in developing countries and especially China, India but also Latin America and so on, there are more and more students studying mathematics. In Africa there are more and more students. I have been involved – and this is a wonderful thing, the African Institute of Mathematical Sciences near Cape Town, where they are bringing, each year, now sometimes fifty, sixty of the best graduates in mathematics in Africa, and giving them one year of very, very concentrated mathematical tuition, and this is done by volunteers from all over the world, giving their time. And this is a wonderful place And it works. And these kids are absolutely first class. You know, once you are giving an opportunity to people in Africa and high level tuition, they are just as good as students in Cambridge or École Polytechnique or Moscow State or Harvard. So definitely, there are more and more people in the world who are taking up mathematics and stem subjects, and this only is to be expected because more and more of the world is catching up with the technological development.
Ernest Raetz
What are you actually doing, while you are at La Trobe University?
Arieh Iserles
My visit now is as part of a larger research project which is sponsored together by European Union and Australia and New Zealand funding agencies, which is to develop computational methods that preserve the underlying structures of the objects we are computing. So for example, if we have a differential system that conserves energy, then you want to compute it. You want to compute it in such a way that the energy is also conserved. So you want to maintain, not only qualitative features which is what computation is all about, but also quantitative features of what you are computing. So this is the aim of the project, and we are all working on it from different ends. We are dealing with abstract constructs, and these abstract constructs can be relevant to many different things. For example, the same equation that José Fourier developed originally to explain heat flow, also explains the spread of epidemics. It also is used in financial mathematics to explain changes in the prices of derivatives. The same equation. A great deal of research is sitting in your office, working on your own or with your students. But occasionally you have to meet other people. You have to compare ideas, you have to brainstorm together.
Ernest Raetz
Maths is a good career for people? I mean, it sounds like there's a small community of you – you have to travel a fair bit.
Arieh Iserles
Well, part of it is travelling, but actually the great fun is sitting in your office and doing research. It is not a career for everybody and heaven forbid, there would be too many applicants for jobs, but you know, when you love it, it is wonderful. It is the greatest adventure on earth.
Ernest Raetz
That was Professor Arieh Iserles, Director of the Cambridge Centre for Analysis and of Britain's Engineering and Physical Science Council's funded Centre for Doctoral Training and Mathematical Analysis. That's all the time we have for the La Trobe University podcast today. If you have any questions, comments, or feedback about this podcast, or any other, then send us an email at podcast@latrobe.edu.au.

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