Explaining his research to a layperson is not so easy: computer mathematician Florian Heß on parallels with car construction, the irony of history - and an unresolved million-dollar question.
QUESTION: You recently completed a six-year project funded by the German Research Foundation, the title of which I unfortunately didn't understand. Could you explain to me, as a non-specialist, what "Algorithmic Methods for Arithmetic Surfaces and Regular, Minimal Models" is all about?
HESS: The difficulty in mathematics often lies in the fact that there is a very long history. These are things that have been researched for 200 or 300 years or even longer, and they continue to build on each other. What others have worked out a long time ago does not become obsolete, but forms the foundation for further research. And in order to understand the top, you often have to go relatively far down to the foundations. This is not the case everywhere, but certainly in number theory and geometry - the synthesis of which has provided the questions for our project. A lot of this goes back to the Greeks; most people know Pythagoras' theorem from school, for example. But certain questions then require tools that become more and more complicated and increasingly abstract - it's not easy to understand.
QUESTION: So you would first have to send interested people to a basic maths seminar?
HESS: It can indeed take some time to explain a research project like the one we have now completed. But it's not just maths that is difficult for laypeople: If you think of a car, for example, it's similar. You can operate it, but you don't know in detail how it actually works inside and how to use the tools in car construction or in the workshop. And the more computers there are in cars, the more complicated and opaque it becomes. It's basically the same in maths. It all builds up. It's a structure of ideas with tools that becomes very extensive.
QUESTION: Although it may be similar in many disciplines, why do many people find mathematics particularly difficult to grasp?
HESS: Because it doesn't necessarily have to be based on objects, on things from reality. It is an intellectual construction where no consideration is given to technical limitations or haptic tangibility. For example, things that take place in higher dimensions cannot be imagined in three-dimensional space. Mathematics enters intellectual realms that are decoupled from the visual world.
QUESTION: When it becomes so abstract, is it particularly difficult to explain mathematical research and transfer its results? Or is it initially less about an application reference?
HESS: That varies. Sometimes it's simply a matter of open internal mathematical questions that need to be solved on a theoretical level. But there are also applied questions, for example from physics, statics or economics, where mathematical solution methods can also be used to achieve something. The aim here would of course be to make these application questions manageable.
QUESTION: Hardly anyone questions the complexity of mathematical research. Do you also have the feeling that everyone realises its relevance?
HESS: Not necessarily. By relevance, most people probably first understand the benefits for specific applications. In the case of applied mathematics, where mathematical methods are used to solve real-world problems, the meaning and purpose are certainly clearer.
QUESTION: Does it have to be applied?
HESS: The English mathematician and number theorist Hardy, for example, did not find this desirable. In 1940, he was proud of the fact that number theory was a science free of any harmful influence from the real world; it was created solely for its own sake, for the sake of its own beauty. The irony of history is that it was precisely around this time and in the midst of this trend that mathematics developed in the field of number theory, which now - now that computers exist - forms the basis of internet security and is therefore highly relevant from today's application perspective. Of course, nobody would have dreamed of this back then. Cryptography, encryption and digital signatures are based on mathematics that was developed without any interest in application. However, the relevance of mathematical research often consists of its internal mathematical applications to questions 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 - keyword algorithmic methods?
HESS: Exactly. Computers are also the element that unites our entire algebra working group: how can computers be used to address certain issues? In the 19th century, mathematicians were still doing a huge amount of calculations by hand. The masters of arithmetic back then pushed it to the limit; there were people who spent years calculating logarithm tables. Incidentally, today a computer could do that in a second. In the first half of the 20th century, a lot of progress was made in the development of theories. And since the advent of computers, it has become fashionable to do more experimental calculations again: How do things behave, what laws might apply?
QUESTION: So do you scrutinise things again? Or do you try to transfer them as accurately as possible from analogue to digital?
HESS: It's less often about questioning things that have already been proven. But there are many unanswered questions. One famous question, which is quite old and was posed by Riemann in the middle of the 19th century, concerns the zeros of a certain function - quite complicated. Riemann also provided a presumed answer, which states that these zeros essentially all lie on a straight line. A computer has now calculated that this is the case for the first trillion zeros - but there is no proof that this is always the case. This is one of the biggest outstanding problems in mathematics, especially because many laws in number theory depend on it. If you solve it, you get a million US dollars in prize money from the Clay Foundation and are famous forever.
QUESTION: Are you in on it too?
HESS: No.
QUEST. Too bad.
HESS: (Laughs) In general, people think that a solution for the current time is probably still out of reach. It's a hairy thing. I'm not so much concerned with theoretical proofs of such problems. But they get you experimenting. If you have trillions of zeros on a straight line, you're inclined to think it must be true - but who knows... if you found another zero with the computer, you would have disproved the previous assumption and could perhaps prove it by hand afterwards.
QUESTION: Are you also a computer scientist?
HESS: I had it as a minor subject and I'm definitely a computer scientist, which helps. And if the task is 3 to the power of 120,000 - the result is a number with more than 57,000 digits - I'm grateful that I don't have to work it out by hand. However, you first have to teach the computer to calculate something like this as quickly as possible using tricks. It only needs milliseconds to do it. Even when it comes to displaying the irregular number pi to the 3000th decimal place, for example. (Types) Here, the number no longer fits on my computer screen - and I know there's something after the last digit 6, so I have an inaccuracy in there.
QUESTION: So you can't calculate pi exactly even with a computer, because the number never ends.
HESS: In numerics, you'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 be enough, which means a thousandth of a millimetre. It's different for us in computer maths, we need greater precision. Because sometimes these numbers become whole numbers again, which should of course be correct.
QUESTION: If it's about multiplication - is that possibly the "complex multiplication" that you will now be working on in your new DFG project?
HESS: It has something to do with mathematical objects that involve multiplication with complex numbers. The root of -1 plays a role here. If you come from the Stone Age, you know one, two, three - you can count. Whether skins or bears... A mental increase then leads to 0 as a number. The next step is to solve equations. But x+7=3, that's not possible. So minus numbers come into play. Then you move from whole numbers to fractions, decimals and later to root expressions. For example, the root of 2. If you multiply this number by itself, i.e. square it, you get 2 again. Sometimes, however, there are equations that cannot be solved even with these numbers. And what have we done over the past 2000 years? Whenever an equation could not be solved, we expanded the number range. We built a new type of number - like the root of -1 - and then everything was fine. This is how complex numbers are created, which can also be multiplied together.
QUESTION: You just have to reach the right level of abstraction.
HESS: Yes, and that was around the 18th century. Then it was said that the root of -1 is called i, and i squared gives -1. Even then there was relatively strong opposition, people didn't go along with it as easily as perhaps with the other things, because it seemed too abstract to them. That's why it's called i, "imaginary unit" - and seems to be less easily acceptable.
QUESTION: At this point, I actually have a problem following you...
HESS: And why is that? It could also be because the basic rules of the game are or were not so clear. What is the framework, what is admissible, what is not? That became clear from the end of the 19th century, and now it's no problem to calculate with these figures.
QUESTION: And this is what your new DFG project is about?
HESS: Among other things, it is based on complex numbers, let's put it that way. Such numbers are used in many fields, for example in electrical engineering, and are commonplace, so to speak. In school, however, the subject often causes stomach ache, if it is dealt with at all, probably for similar reasons as in the 18th century.
QUESTION: And how do you explain to your children what you do?
HESS: (Smiles) Not at all to my four-year-old daughter. I can hardly explain it to my nine-year-old son either. Especially with children, you realise that somehow it doesn't work at all. You also notice it in yourself, 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 you take two or more steps, you don't realise it anymore. This incremental nature of being left behind as soon as the gap becomes too big - that's what characterises mathematics.
Interview: Deike Stolz
