Explaining his research to a layperson is not as easy as it sounds: computational mathematician Florian Heß on parallels with car manufacturing, the irony of history – and an unsolved million-euro question.
QUESTION: You recently completed a six-year project funded by the German Research Foundation (DFG), the title of which I’m afraid I didn’t understand. Could you explain to me – as someone with no background in the subject – what ‘Algorithmic Methods for Arithmetic Surfaces and Regular, Minimal Models’ actually means?
HESS: The difficulty in mathematics often lies in the fact that there is a very long history behind it. These are topics that have been researched for 200 or 300 years, or even longer, and the work builds on previous research. What others worked out a long time ago does not become obsolete, but forms the foundation for further research. And to understand the cutting edge, you often have to go relatively far back down to the foundations. This isn’t the case everywhere, but it certainly is in number theory and geometry – the synthesis of which provided the questions for our project. Much of it can be traced back to the Greeks; for example, most people are familiar with Pythagoras’ theorem from their school days. But certain questions then require tools that become increasingly complex and abstract – and that isn’t something you can grasp straight away.
QUESTION: So would you first have to send anyone interested on a basic mathematics course?
HESS: Indeed, it can take quite some time to make a research project like the one we’ve just completed understandable. But this isn’t just the case for non-specialists when it comes to mathematics: if you think of a car, for example, it’s similar. You can drive it, of course, but you don’t know the details of how it actually works on the inside, or how to use the tools in car manufacturing or in a garage. And the more computers there are in cars, the more complicated and opaque it all becomes. It’s basically the same in mathematics. It all builds up. A conceptual framework with tools that becomes very extensive.
QUESTION: Although the situation may be similar in many disciplines – why does mathematics seem particularly elusive to many people?
HESS: Because, moreover, it doesn’t necessarily have to be anchored in the material world or in real-life objects. It is a mental construct where one no longer has to take technical limitations or physical tangibility into account. For example, things that take place in higher dimensions cannot, after all, be imagined in three-dimensional space. Mathematics ventures into conceptual realms that are decoupled from the world of sensory perception.
QUESTION: When it becomes this abstract, is it particularly difficult not only to explain mathematical research but also to transfer its results? Or is the focus initially less on practical applications?
HESS: It varies. Sometimes it is simply a matter of open questions within mathematics itself that need to be resolved at a theoretical level. However, there are also applied questions – for example, from physics, structural analysis or economics – where mathematical solution methods can also be used to achieve results. In such cases, the aim is, of course, to make these specific applied problems manageable.
QUESTION: Hardly anyone would question the complexity of mathematical research. Do you also feel that its relevance is clear to everyone?
HESS: Not necessarily. When most people think of relevance, they probably first think of the benefit for specific applications. In applied mathematics, where mathematical methods are used to solve real-world problems, the purpose is certainly clearer.
QUESTION: Does it have to be applied?
HESS: The English mathematician and number theorist Hardy, for example, did not consider this desirable. In 1940, he was proud that number theory was a science free from any harmful influence of the real world; created solely for its own sake, for the sake of its own beauty. The irony of the story is that it was precisely at this time and in line with this very trend that a branch of mathematics developed within number theory which now – in the age of computers – forms the basis of internet security and is therefore to be regarded as highly relevant from today’s practical perspective. Of course, nobody could have imagined that back then. Cryptography, encryption and digital signatures are based on this – on a branch of mathematics that was developed without any practical application in mind. However, the relevance of mathematical research often also lies in its internal mathematical applications to problems that are only indirectly related to original fields of application such as physics.
QUESTION: And computers also play an important role in your research project – think algorithmic methods?
HESS: Exactly. Computers are also the element that unites our entire Algebra research group: how can we tackle certain problems using computers? As recently as the 19th century, mathematicians were still doing an enormous amount of calculations by hand. The ‘calculation masters’ of that era pushed it to the limit; there were people who spent years working through logarithm tables. Incidentally, a computer could do that today in a single second. In the first half of the 20th century, theoretical development then made great strides. And since computers have been around, it has become fashionable to return to more experimental calculations: how do things behave, and what laws might apply?
QUESTION: So are we re-examining things anew? Or are we trying to transfer them from the analogue to the digital realm as accurately as possible?
HESS: It’s actually quite rare to question things that have already been proven. But there are many open questions. One famous question, which is quite old and dates back to Riemann in the mid-19th century, concerns the zeroes of a particular function – it’s rather complicated. Riemann also provided a conjectural answer at the time, which states that these zeroes essentially all lie on a straight line. Computers have now calculated that this is the case for the first trillion zeroes – but the proof that it is always the case is still missing. This is one of the greatest unsolved problems in mathematics, not least because many principles in number theory depend on it. If you solve it, you’ll receive a prize of one million US dollars from the Clay Foundation and be famous for all time.
QUESTION: Are you working on that too?
HESS: No.
QUESTION: That’s a shame.
HESS: (Laughs) Generally speaking, people think that a solution is probably still out of reach for the time being. It’s a tricky one. I don’t really concern myself all that much with theoretical proofs of such problems. But they do lead you to experiment. When you’ve got trillions of zeros on a straight line, you’re inclined to think it must be right – but who knows… if you were to find another zero using a computer, you’d have disproved the current conjecture and might then be able to prove it by hand.
QUESTION: Are you a Computing Science expert as well?
HESS: I studied it as a minor subject and I’m certainly keen on computer science; that helps. And when the task is 3 to the power of 120,000 – the result is a number with more than 57,000 digits – I’m certainly grateful that I don’t have to calculate it by hand. You do have to teach the computer first, though, how to calculate something like that as quickly as possible using all sorts of tricks and techniques. It only takes milliseconds. Even when it comes to displaying the irrational number pi, for example, to the 3,000th decimal place. (Types) Here, the number doesn’t even fit on my computer screen any more – and I know there’s something after the last digit, 6, so there’s an inaccuracy there.
QUESTION: So even with a computer, you can’t determine pi exactly, because the number never ends.
HESS: In numerical calculations, we’re satisfied if the first 20 decimal places are correct. For example, when specifying the length of a bridge in metres, six decimal places should suffice; that already amounts to one thousandth of a millimetre. In our field of computational mathematics, it’s different; we need greater precision. Because sometimes these numbers are converted back into whole numbers, which must, of course, be correct.
QUESTION: When it comes to multiplication – is that perhaps the ‘complex multiplication’ that you’ll now be working on in your new DFG project?
HESS: It has to do with mathematical objects in which multiplication by complex numbers occurs. The square root of -1 plays a role here. If you come from the Stone Age, you know one, two, three – you can count, after all. Whether it’s animal skins or bears… A mental progression then leads to 0 as a number. The next step is to solve equations. But x + 7 = 3 – that doesn’t work at all. That’s how negative numbers come into play. Then you move from whole numbers to fractions, decimals, and later on to radical expressions. Take the square root of 2, for example. If you multiply this number by itself – that is, square it – you get 2 again. Sometimes, however, there are equations that cannot be solved even with these numbers. And what have we been doing over the past 2,000 years? Whenever an equation couldn’t be solved, we’ve expanded the number line. We’ve created a new type of number – such as the square root of -1 – and then everything was fine. That’s how complex numbers came about, which can also be multiplied together.
QUESTION: It’s just a matter of reaching the right level of abstraction.
HESS: Yes, and that was around the 18th century. Then it was said that the square root of -1 is i, and i squared equals -1. Even back then there was quite strong opposition; people weren’t as quick to go along with it as they perhaps were with other things, because it seemed too abstract to them. That’s why it’s called ‘i’, the ‘imaginary unit’ – and it seems to be less readily accepted.
QUESTION: I’m actually finding it a bit difficult to follow you at this point…
HESS: And why is that? It may well be because the basic rules of the game aren’t – or weren’t – entirely clear. What are the boundaries, what is permissible, and what isn’t? This became clearer from the end of the 19th century onwards, and nowadays it’s no problem at all to carry out calculations with these numbers.
QUESTION: And does that feature in your new DFG project?
HESS: It’s based, amongst other things, on complex numbers – let’s put it that way. Such numbers are used in many fields, such as electrical engineering; they’re, so to speak, commonplace. At school, however, the subject – if it’s covered at all – often causes headaches, probably for similar reasons to those in the 18th century.
QUESTION: And how do you explain to your children what you do?
HESS: (Smiles) I don’t explain it to my four-year-old daughter at all. I can hardly explain it to my nine-year-old son either. You realise, especially with children, that it’s just not really possible. You notice it in yourself too – you can only ever proceed step by step. You can only ever take one step further, based on what you already know. As soon as there are two or more steps, you can’t keep up. This incremental nature – the fact that you’re left behind as soon as the gap becomes too wide – is precisely what characterises mathematics.
Interview: Deike Stolz
