May 24, 2017: AMI Spring congress - Wageningen

Lees het hele artikel van Christian Jongeneel hier

of via www.technischweekblad.nl

May 21, 2015: AMI Spring congress - Utrecht

Hij past chaostheorie toe om typische ingenieursproblemen op te lossen: van de bouw van windmolens op zee tot verbetering van de atoomkrachtmicroscoop (AFM). De Israëlische civieltechnicus Oded Gottlieb hield daarover een key note lecture tijdens het 3TU.AMI Spring Congress in Utrecht op 21 mei.

‘Ik ben een ingenieur die van wiskunde houdt’, zegt Gottlieb veelbetekenend. Dat is tamelijk bijzonder, wil hij maar zeggen. Want meestal is het andersom: ‘De meeste mathematische bollebozen in de ingenieurswetenschappen zijn wiskundigen die geen aanstelling konden krijgen in hun eigen vakgebied en daarom maar zijn uitgeweken.’

Gottlieb studeerde civiele techniek aan de technische universiteit Technion in Haifa (ook wel het ‘MIT van Israël’ genoemd). Hij was ervan overtuigd dat hij in de industrie zou belanden, net als zijn vader. Maar het liep anders. Gottlieb werd tegen het eind van zijn studie verliefd: op de chaostheorie.

‘Ik las het boek Instabilities and catastrophes in science and engineering van Michael Thompson. Mijn supervisor aan Technion kwam ermee aanzetten. In mijn undergraduate studie begin jaren tachtig had ik helemaal niets over chaos geleerd. Ik was gegrepen door de global effecten die chaos in de natuur kan hebben. Dat een zwaaiende hand heel ver weg uiteindelijk invloed kan hebben op het raam waar we hier nu doorheen kijken.’

Chaostheorie was begin jaren tachtig nog een jong vak, twintig jaar eerder geboren met het beroemde artikel Deterministic Nonperiodic Flow (1963) van de Amerikaanse meteoroloog Edward Lorenz. Hij had bij toeval ontdekt dat een miniem verschil in de begintoestand van de atmosfeer naar verloop van tijd totaal ander weer oplevert (de beroemde metafoor van de vlinder die in Brazilië met zijn vleugels klapt en daarmee maanden later in Texas een tornado veroorzaakt).

Dat artikel van Lorenz veroorzaakte een aardverschuiving in de exacte wetenschappen. De aanname was tot die tijd namelijk dat kleine veranderingen kleine gevolgen hebben. Dat bleek dus niet te kloppen; er bestaan fysische systemen (zoals stromingen in vloeistoffen of de beweging van een slinger) die na enige tijd onvoorspelbaar zijn, hoe goed de computer ook rekent.

Gottlieb: ‘Pas in de jaren zeventig drong chaostheorie door in de ingenieurswetenschappen, vooral in de elektrotechniek, mechanica en lucht- en ruimtevaarttechniek.’ Daarna bereikte het de informatica in de vorm van de bekende fractals, waarmee databestanden konden worden gecomprimeerd.

Omdat er in Israël in de jaren tachtig nog nauwelijks onderzoek plaatsvond naar chaos besloten Gottlieb en zijn vrouw naar de Verenigde Staten te gaan – de liefde voor de nieuwe theorie zat diep. Gottlieb promoveerde er op het onderwerp en werd postdoc aan MIT. En toen vroeg zijn oude universiteit in Haifa of hij daar hoogleraar wilde worden in het nieuwe vakgebied. ‘Ik heb daar lang over nagedacht’, zegt Gottlieb. Hij was tenslotte ingenieur en wilde eigenlijk naar het bedrijfsleven.

En nu is hij een van ’s werelds grootste specialisten in de toepassing van chaostheorie op typische ingenieursterreinen. Bij de bouw van windmolens op zee bijvoorbeeld. Hoe diep moeten die in de zeebodem verankerd zijn om ze bij harde wind verantwoord te laten doordraaien? Chaostheorie speelt een belangrijke rol bij de beantwoording van die vraag.

Of neem de atoomkrachtmicroscoop (atomic force microscope, AFM). De ‘gewone’ variant brengt atomaire ‘landschappen’ in kaart doordat de naald van de microscoop de Vanderwaalskrachten tussen atomen meet. Door de naald op een slimme manier ‘chaotisch’ over het materiaal te laten bewegen, kan de AFM veel meer materiaaleigenschappen meten dan op de traditionele manier. ‘Soms gaat het er in mijn onderzoek om hoe chaos te vermijden is, maar soms ook – zoals bij de AFM – hoe je chaos juist kunt gebruiken.’

Dat hij een ingenieur is in de wereld van wiskundigen heeft een groot voordeel, bekent Gottlieb: hij kan makkelijker aan geld komen voor zijn onderzoek dan collega’s. Niet alleen de industrie subsidieert zijn onderzoek regelmatig, ook (semi-)overheidsfinanciers vragen meer en meer naar concrete toepassingen voor ze geld beschikbaar stellen. ‘Voor zuiver wiskundigen is het veel moeilijker om onderzoeksgeld binnen te halen dan voor mij.’

Een nadeel heeft dat natuurlijk ook, erkent hij: hij is voor zijn onderzoek afhankelijk van die (vaak kortetermijn-)interesses van zijn financiers. ‘Toch heb ik gelukkig altijd kunnen doen wat ik wilde: ik bedenk mijn eigen wetenschappelijke problemen en schrijf daar mijn eigen onderzoeksvoorstellen bij. Maar ik kan inderdaad niet alles onderzoeken wat ik wil. Zo zou ik me graag storten op onderzoek naar het menselijk hart: bij hartfalen speelt chaostheorie een belangrijke rol, het vertoont gelijkenissen met de chaotische processen bij het kapseizen van een schip.’

Hoewel iedere technische universiteit tegenwoordig ten minste één chaosspecialist in huis heeft, blijft chaostheorie een beperkt onderdeel van de ingenieurswetenschappen. ‘Dat is niet zo raar, want het is uiteindelijk een subdomein van een subdomein. Het meeste onderzoek vindt nog steeds plaats binnen lineaire dynamische systemen. Niet-lineaire dynamische systemen vormen een kleiner domein; chaotische systemen een nog kleiner.’

Techniekstudenten die interesse hebben in chaos raadt hij het boek aan van Francis J. Moon: Chaotic Vibrations: An Introduction for Applied Scientists and Engineers. ‘De inleiding is voor bachelorstudenten uitstekend te volgen, je hebt er eigenlijk alleen middelbareschoolwiskunde en wat kennis van lineaire algebra voor nodig.’ Voor leken maakte de BBC vijf jaar geleden een goede film van een uur: The secret life of chaos.

Die film heeft overigens nog wel een zweem van de romantische connotatie die chaostheorie vooral in de jaren zeventig had: chaos als ‘bevrijding’ van de wetmatigheid van de natuurwetenschap. Maar wat dat aangaat verschilt chaostheorie principieel van kwantummechanica met zijn indeterministische aspecten. ‘Chaotische processen zijn deterministisch en niet stochastisch. Een minieme verandering in de begintoestand van een systeem heeft uiteindelijk grote gevolgen, maar het blijft deterministisch.’

Photo by Marc Blommaert

Mr.Gottlieb applies chaos theory to solve typical engineering problems, from building windmills at sea to the improvement of the atomic powered microscope (AFM). The Israeli civil engineer Oded Gottlieb was invited by 3.TU.AMI to hold a keynote lecture during the Spring Congress in Utrecht on the 21st of May.

“I’m an engineer who loves mathematics”, says Gottlieb significantly. That is rather special, he means to say. Because mostly the reverse holds true: most mathematical whizzes in engineering science are mathematicians who were not able to get an appointment in their own field and therefore switched.

Gotllieb studied civil engineering at the technical university of Haifa, Technion (sometimes called the “MIT of Israel”) . He was convinced he would end up in industry just like his father. But it turned out differently. At the end of his study he fell in love: with chaos theory.

“I read the book ‘Instabilities and catastrophes in science and engineering’ of Michael Thompson, which my supervisor at Technion turned up with. During my undergraduate education at the beginning of the eighties I had never studied anything about chaos theory. I was seized by the global effects which chaos could have in nature, that a waving hand very far away in the end can have influence on a window we are looking through here.”

At the beginning of the eighties chaos theory was a new discipline, borne twenty years before with the famous article ‘Deterministic Non periodic Flow’ (1963) of the American meteorologist Edward Lorentz. He discovered by accident that a tiny difference in the initial condition of the atmosphere can bring about completely different weather after some time (the famous metaphor of the butterfly flapping its wings in Brazil - affecting the weather pattern – causing a tornado in Texas a few months later.

The article of Lorentz caused a landslide in the exact science. At the time one thought that small changes would result in small consequences. This was not true, there exist physical systems (such as flows in liquids or motions of a pendulum) which are unpredictable, no matter how well the computer calculates.

Gottlieb: “Since the seventies chaos theory entered into the engineering science, especially into electrical engineering, mechanics and aerospace engineering. After that it was used in computer science with its famous fractals with which its databases could be compressed.”

Because hardly any research was done in chaos theory in Israel, Gottlieb and his wife decided to move to the United States – the love for the new theory was deep. Gottlieb obtained his PhD degree on the subject and got a MIT-postdoc. And then his former university at Haifa offered him a professorship in this new discipline. “I’ve thought about it for a long time” Gottlieb said . After all, he was an engineer and actually preferred industry.

And now he is one of the greatest specialists in the application of chaos theory in typical engineering domains, e.g. windmill installations at sea. How deep do these have to be anchored in the seabed, so that they will turn safely in stormy weather? Chaos theory plays an important role in answering this question.

Or consider the atomic force microscope. The ‘common’ device pictures atomic landscapes because the needle of de microscope measures the “Vanderwaals” forces between the atoms. By moving the needle in a smart chaotic way across the material the microscope can disclose much more material properties than in the traditional way. “So my research is sometimes driven by avoiding chaos and sometimes - in case of the microscope - by using chaos.”

To be an engineer in a mathematical environment has a great advantage remarks Gottlieb: it makes fund raising for research easier. Not only industry endows his research regularly, but also (semi)governmental financiers are asking more and more for tangible applications before making money available. For pure mathematicians fund raising is much harder to do.

Still there is also a disadvantage, he admits: for his research he depends on the (short-term) vision/interest of his financiers. “But fortunately I have always been able to do what I wanted: I devise my own research problems and write my own research proposals for them. But apparently I cannot investigate everything. So I like to focus on research of the human heart: In heart failure chaos is playing an important role, there is a resemblance between these processes and those of the capsizing of a ship.”

Although nowadays every technical university will have at least one chaos specialist, the chaos theory still is a limited research field within engineering sciences. “That is not strange, because in the end it is a subdomain of a subdomain. Most research still takes place within the linear dynamical regime, not-linear dynamical systems form a smaller domain; chaotic systems even smaller still.

Engineering students having an interest in chaos, are advised by Gottlieb to read the book of Francis J. Moon: ‘Chaotic Vibrations: An Introduction for applied Scientists and Engineers’ . “The introduction is very accessible for bachelor students, actually you only need secondary school mathematics and some knowledge of linear algebra.” Five years ago the BBC created a good one hour movie for laymen: ‘The secret life of chaos’.

This film has a touch of the romantic connotation which the chaos theory had in the seventies: chaos as liberation of the rule of science.

But with regard to this, chaos theory principally differs from quantum theory.

(interview vertaald door E. van Elderen)

“I do not really know how to give shape to my retirement”, Arjeh Cohen admits. “I still have so many things to do.” That is the story of his life: a portrait of a busy man.

Text Anouck Vrouwe, May 2014

“I am quite picky about the mathematical problems I work on. But every now and then I decide that this one is mine. And then I will not let go. I guess that is my strength.” Arjeh Cohen shows an article that is about to be published. “This story, for example, started 25 years ago, in 1989. All those years, the problem stayed on my mind. I worked on it every now and then, together with a colleague. And finally, we solved it. The result was not as great as we had hoped for, but I am really happy we brought it to its conclusion.”

He chose to study math because of its beauty, not because he was planning to become a mathematician. But a mathematician he became. And a good one too. Arjeh Cohen, now Professor Emeritus Discrete Mathematics, gave his farewell speech in a packed auditorium at Eindhoven University of Technology. “I never thought of myself as a mathematician. At university, there were guys in my year who were a lot brighter than I am. But slowly, they disappeared, they followed other interests. And I turned out to be the go-getter with a career in science”, Cohen explains.

He once tried to do something else, after finishing his dissertation at Utrecht University. Finite complex reflection groups was the subject of Cohen’s dissertation. His supervisor was Tonny Springer – a respected mathematician, one of the world’s greatest experts on the theory on algebraic groups. “My dissertation was a tough job. Things were different then. My supervisor threw me in at the deep end. You make it or break it anyway, was his vision. I did not discuss my research topic with him more than three times. There was no room for failure. I saw good people drop out. I struggled a lot, but I made it. After this period of very abstract math, I decided it was time to do something useful.”

Cohen applied to a job at the environmental department of Openbaar Lichaam Rijnmond – a former authority between the city council of Rotterdam and the Province of South Holland. He refers to it as his traineeship. “I modeled the impact of the industrial environment on citizens. I learned a lot about applied mathematics, like statistics and numerical mathematics. But it was not my world. I disliked the way politicians took decisions without proper knowledge of the problem, and I disliked how officials made their way up by acting important.”

So when a job opened up at Discrete Mathematics in Enschede, Cohen was more than happy to apply. “So far for the excursion. I really liked my new job, the concreteness of it. It was the start of my career, I found my favorite kind of math.” Soon after, Cohen changed jobs again. CWI, the Dutch research institute for mathematics and computer science, started a new research group in Discrete Mathematics.

“That was when I started doing useful things with computers”, Cohen understates. His first love is the geometry of the classification of mathematical groups, like Lie groups. “That work will last – some of the things I have proven may still be used in next centuries.” The computer was the second important theme in Cohen’s career. “Doing computer algebra is a challenge, and I liked it. You cannot fool a computer; you have to fully understand a mathematical problem to be able to program it. If you take 1,2,3,4,5 for example: that is a list of numbers, but it is a way of ordering them too. A human is aware of these two meanings at the same time, but a computer is not: if that ascending order is important to you, you will have to tell the computer that this is more than numbers. You have to be very explicit and precise, and that made working with computers good practice for me.”

During his time at CWI, Cohen helped lay the foundation for CAN, the foundation Computer Algebra Nederland. The goal was to stimulate and coordinate the use of computer algebra systems in education and research. “We became the central distributor of computer algebra systems, like Mathematica and Maple. But we also promoted their use, by teaching people how to use them.” At Eindhoven University of Technology, Cohen shaped the university's vision on the use of computers in education. “To give all the freshmen a laptop was one of my ideas, for example. That was quite extraordinary in those days.” And after his retirement Cohen will develop an interactive book on calculus for Sowiso, an e-learning platform for mathematics. “I am convinced that adaptive learning will become more and more important; it makes it possible to educate people at their own speed and level.”

In his own work, Cohen used the computing power of computers for his proofs a lot. “One of my best results was a search for a special small subgroup in the biggest exceptional group in group theory, E8. I had a Macintosh at home, and I managed to break the problem down to a solvable set of linear equations. I remember how I looked at the result of the calculations – I saw the numbers and immediately realized that the solution of my problem presented itself. It is unforgettable if your logic turns out to do the trick.”

So the Eureka-moment really does exist? Cohen: “Sure. I remember one time at Utrecht Central Station – the moment I left the train, just before my foot touched the platform. I immediately knew I found the solution to the problem I had been working on for quite some time.”

Another thing Cohen is really proud of is his book Diagram Geometry, which he finished last year with co-author Francis Buekenhout. “It took quite some time to mature. It describes the most important things I worked on for all these years.” Group-related incidence geometry is a wonderful subject. Cohen tells how he loves the wonderful objects that arise from just a simple set of points, lines and some rules. “It is fascinating.”

Since 1992, Cohen has been a full professor of Discrete Mathematics in Eindhoven. In his inaugural speech What is the point of algebra? he pointed out that algebra is a world in itself, where ‘finite groups like the Monster live’. He stated that his world holds enough challenges and rewards to forget about the real world. But Cohen was not a mathematician who simply closed his door and did his math. Though he did not like the political games at the Openbaar Lichaam Rijnmond, he became very active in management duties in Eindhoven. He was dean at the Department of Mathematics and Computer Science from 2009 till 2013. Thoughtfully: “It was not my ambition to become dean, not at all. For your math, it is better to stay out of governance as much as possible. You will publish more if you do, and get more praise. But to be able to do so, you need a healthy and well organized department. It never felt like I had a choice; it was important work that needed to be done and there were not many people who could do it.” Though becoming dean might not have been his first choice, Cohen's colleagues say he did a good job: he was an attentive and straight manager. He left a healthy department behind. One of the things he professionalized is the fundraising of the department. “Mathematicians always think they only have to do great math to get grants. But that is not enough anymore. The competition has hardened: you have to sell your ideas – also to people who are not experts in math. We hired professional trainers to polish our presentations. That may sound expensive, but it pays for itself in the long run.”

Cohen was also involved in national initiatives to improve the position of mathematics in The Netherlands, like Platform Wiskunde Nederland (PWN) en AMI, the collaboration of the three technical universities. “I believe in cooperation. Take Mastermath; the universities bundled their forces to organize the best courses for master students. That is good for all of us.”

One of his other priorities as dean was to increase the number of women in the department. His farewell reception showed how men still dominate the field: his wife, professor in Medical Psychology, was the only female professor at his farewell reception. “A lot of people noticed. We do have two female professors at our department”, Cohen says, “and we are working hard to increase their numbers. That will not happen on its own, you have to actively search for women. In general, women do not put themselves at the forefront as much as men.” All his 21 PhD-students were male too. “But luckily, I still supervise two PhDs – and one of them is female.”

Cohen has been phasing out activities in preparation of retirement, but he is as busy as before. He works on an interactive book on calculus, has some papers in preparation and supervises his last PhDs. “I don’t really know how I will shape my retirement. It is weird. But I do like the idea of spending more time with my grandchildren.” A short laugh: he realizes very well that he will have to plan such precious time in his overloaded schedule. Retired or not – Cohen is still that energetic man he has always been.

CV - Arjeh Cohen (1949, Haifa)

1971 – Master degree, Utrecht University

1975 – Dissertation ‘Finite Complex Reflection Groups’, Utrecht University

1975 – Researcher at Openbaar Lichaam Rijnmond

1976 – Researcher at University of Twente

1979 – Researcher at CWI, Amsterdam and Professor of Discrete Mathematics, Utrecht University

1992 – Full professor of Discrete Mathematics at Eindhoven University of Technology

2009 – Dean of the Department of Mathematics and Computer Science

*Note: thanks to Onno Boxma for his help *

June 13, 2013: AMI Springfestival - TUDelft

**Mathematics and numbers**There are two answers to the question why Gerhard Frey chose to study mathematics. One is he liked the subject at school. But then again, he also liked history. The other answer is that with mathematics you can discover things just by thinking and you don’t have to learn things by heart. So with mathematics he could be the laziest man of the world.

We interview Gerhard Frey before the 3TU.AMI Springfestival held at the TU Delft, where he will be the key note speaker with a presentation with the intriguing question: Can Mathematics keep a secret? Gerhard’s specialty is number theory. “During the start of my study at the University of Tubingen I had a beginner course by the quite famous Peter Roquette. He sparked my interest in number theory. Number theory starts very simple. You use your fingers to count and you learn about numbers, calculation, prime numbers and polynomials in school. In number theory you want to find relations between numbers with properties. For example: Euclid stated that there are an infinite number of prime numbers and he gave a beautiful and simple proof. The other extreme is Fermat’s last theorem that is also very simple to state but it was very hard to prove it. In number theory you need a lot of theory to prove simple things. It is fascinating. Everyone is interested in numbers. People playing a Sudoku are playing with numbers. Number theory proved to have many big challenges for me, but if I maybe had another instructor at my beginner’s courses I might have ended up in another field of mathematics.”

**Fermat’s last theorem**Peter Roquette was Gerhard’s supervisor during his PhD work in Heidelberg. “After my Diploma (master) I had the choice to go into the industry, work at a school or continue at the university. Nowadays it is quite difficult for young people to get a position at the university, but at that time it easier”, tells Gerhard. “From the 1960’s onwards until the end of the last century it was a golden age for number theory. We proved so many results . It was nice to be part of a revolution.” When we ask him why so many mathematicians were fascinated by Fermat’s last theorem, Gerhard says that he has to disappoint us. “It is a very simple question. Everyone can understand it. Still once every week I receive a letter, by someone who claims to have found a proof that only takes two pages, but it is always nonsense. But for a long time you didn’t have the power to attack it. The famous mathematician Gauss tried to solve it. He asked himself: Why should it be so important? I can think of similar questions once a week.” It took 358 years to publish the proof. Gerhard explains: “During the last 200 years people started to develop theories that would hopefully help to prove the theorem. The key word is algebraic number theory. Fermat’s last theorem was a great motivation to develop this theory. But only in the sixties and seventies of the last century theories were raised to give a method on how to solve it. You have to find the right method and structure for the problem to apply the theory . My fascination with Fermat’s last law was to find a structure that we could apply. Algebraic manipulations of formulas gave no way to solve the equation. We took a totally new approach and attached Galois representations to Fermat's Last Theorem. This turned out to be an important step in the final proof by Andrew Wiles.”

**The importance of conferences**Gerhard’s career changed from the theoretical side to the more applied side, when he started to work for the Institute for Experimental Mathematics in 1990 in Essen. There were four chairs. Gerhard had the chair for number theory. The other chairs were on group theory, computer science and digital communication. “It was a big luck to work with professor Han Vinck. He is an engineer with a great background in mathematics. We became interested in coding theory. In one conference we held in Essen somebody spoke about cryptography and a participant asked questions about public key cryptography. We saw immediately that we could apply the whole machinery we used for solving Fermat’s last theorem to this question. Again we found the structure that was behind and we understood that our expertise could help to provide useful theory . So the next 15 years I led a double life, working both in theoretical and in applied mathematics. We had the good luck to attract many very clever young people who wished to work in this new area of cryptography.”

Gerhard stresses the importance of conferences for the work of a mathematician. “It is important to travel around the world and to exchange problems and questions. As a mathematician you don’t have a lab or machines, so you are free to go to conferences that are important to improve your ideas.”

In 2005 Gerhard's work had consequences for the electronic passport in Germany. “Our institute was involved in developing the mathematical background. We helped to find criteria for the right elliptic curves, which are the backbone of the theory, which were put in use in the passport. This passport is a very complex thing, with a lot of protocol. When a passport and a machine communicate they ask each other in milliseconds to solve a mathematical problem. This can be done only if both passport and machine have certain knowledge and thus it helps to establish whether it is the right passport and the right machine. ”

**Answers in your sleep**Gerhard emphasizes the role of intuition in arriving to solutions for difficult problems. “If you speak to a mathematician, you will often hear: I have the feeling that this can’t be true. Experience is of course very important to get this intuition. I often got an answer to my work problems during sleep, but, alas, only to find out in most cases in the morning that the answer was completely wrong! I had a thirty minute bike ride to and from work and that was a good way to think ideas through or get new ones. Input that comes from discussion with others is also very important. So during the day you work on calculation and read papers, but the best ideas come when you’re not working!”

When asked what the biggest challenges in number theory are, Gerhard quotes the famous mathematician David Hilbert “We must know and we will know.” He thinks that there are many unknown areas and old and new challenges to be solved. “For example complexity theory. It tells you how difficult an algorithm has to be to solve a problem, and whether a “machine” can find a solution or not to a particular problem. It’s sometimes a bit like the book Hitchhiker’s guide to the Galaxy by Isaac Asimov, that you have to build a gigantic machine to get an answer to a problem and the output is trivial like 42. One of the big theoretical classical challenges is of course the Riemann hypothesis. That is also one of the 10 Clay Mathematics Institute Millennium Prize Problems, who let you win a million dollar if you solve one of the problems. But in number theory there are also many seemingly small problems, who need a lot of structure to solve. Many of them could be done if we would understand the Galois group of rational numbers. “

**Young talent**In Gerhard’s opinion there are many talented people in high schools, and universities offer good personal contact with the students to develop these talents. “The hard thing is the PhD thesis, which will determine whether you are more suited for a career in industry or in research.” Gerhard sees a big pool of talent both male and female, but thinks we should do more to give them more security. “It is a rat race. People have to communicate about their ideas in papers and at conferences, but at the same time there is a lot of competition. They often have a contract for 2 years and already have to apply for a new position after one year.” He thinks it’s a shame that till now there are very few women with a chair in mathematics, whereas in Germany the student numbers are 50 percent male and 50 percent female. This postdoc period is difficult. The woman often follows the man and it is also the time to start a family. “We should give them more support.”

**No stress**Gerhard now enjoys the fact that he no longer has the stress of having a chair and the responsibility for the scientific career of young people. “I still write papers, go to conferences and have discussions with friends, but it is more relaxed now.” He looks back on a great career. “My biggest success was the attack on the Fermat theorem, that Wiles found proof for. Later we developed a method for elliptic curves that are still used in public key cryptography. A lot of the guidelines of national security agencies go back to the work we did in this area. I started out in the highest theoretical field, with the lowest possibilities for applications. Through contacts and discussions with electrical engineers, who were designing chips, we came up with new mathematical methods that have great applications in cryptography.”

Quantum computing is a new area that will propose big challenges. Gerhard says: “We know from physics that quantum computing will work in principle and that it will break public key systems used today in milliseconds if a big enough computer will be built. Young colleagues will have to find other mathematical problems that will be resistant to quantum computing. During the next 20 years quantum computers may be there, but that is a story that we have been telling already for a long time. The difficulty is how to build and engineer such a computer. It is like hydrogen fusion. We know from the sun that it can be a source of energy, and from the 1940’s onward people have been trying to build such a system. But up till now, not a microVolt has come out. Maybe quantum computing will follow a similar path, because it is difficult to build. But as well it may be otherwise...”

**Cautious**Coming back to the question “Can Mathematics keep a secret?”, Gerhard proposes the situation that the two interviewers want to exchange a secret, with him present in the room and him hearing the exchange and knowing the method of how we exchange the secret. Public key crypto give means to build up such a secret connection. “This is an important method that is useful in everyday applications like smart cards, but also when your prime minister wants to have a telephone conversation with the president of the USA via a public net.” But there can always be leaks in the security if one is not careful. Gerhard remembers that thieves followed drivers of luxurious cars and recorded the beep that was given to open and close the door. And played that beep later to open the car door: not enough mathematics was used. So: Yes, mathematics can keep a secret, if you are cautious!

May 24, 2017: AMI Spring congress - Wageningen

Lees het hele artikel van Christian Jongeneel hier

of via www.technischweekblad.nl

May 21, 2015: AMI Spring congress - Utrecht

Hij past chaostheorie toe om typische ingenieursproblemen op te lossen: van de bouw van windmolens op zee tot verbetering van de atoomkrachtmicroscoop (AFM). De Israëlische civieltechnicus Oded Gottlieb hield daarover een key note lecture tijdens het 3TU.AMI Spring Congress in Utrecht op 21 mei.

‘Ik ben een ingenieur die van wiskunde houdt’, zegt Gottlieb veelbetekenend. Dat is tamelijk bijzonder, wil hij maar zeggen. Want meestal is het andersom: ‘De meeste mathematische bollebozen in de ingenieurswetenschappen zijn wiskundigen die geen aanstelling konden krijgen in hun eigen vakgebied en daarom maar zijn uitgeweken.’

Gottlieb studeerde civiele techniek aan de technische universiteit Technion in Haifa (ook wel het ‘MIT van Israël’ genoemd). Hij was ervan overtuigd dat hij in de industrie zou belanden, net als zijn vader. Maar het liep anders. Gottlieb werd tegen het eind van zijn studie verliefd: op de chaostheorie.

‘Ik las het boek Instabilities and catastrophes in science and engineering van Michael Thompson. Mijn supervisor aan Technion kwam ermee aanzetten. In mijn undergraduate studie begin jaren tachtig had ik helemaal niets over chaos geleerd. Ik was gegrepen door de global effecten die chaos in de natuur kan hebben. Dat een zwaaiende hand heel ver weg uiteindelijk invloed kan hebben op het raam waar we hier nu doorheen kijken.’

Chaostheorie was begin jaren tachtig nog een jong vak, twintig jaar eerder geboren met het beroemde artikel Deterministic Nonperiodic Flow (1963) van de Amerikaanse meteoroloog Edward Lorenz. Hij had bij toeval ontdekt dat een miniem verschil in de begintoestand van de atmosfeer naar verloop van tijd totaal ander weer oplevert (de beroemde metafoor van de vlinder die in Brazilië met zijn vleugels klapt en daarmee maanden later in Texas een tornado veroorzaakt).

Dat artikel van Lorenz veroorzaakte een aardverschuiving in de exacte wetenschappen. De aanname was tot die tijd namelijk dat kleine veranderingen kleine gevolgen hebben. Dat bleek dus niet te kloppen; er bestaan fysische systemen (zoals stromingen in vloeistoffen of de beweging van een slinger) die na enige tijd onvoorspelbaar zijn, hoe goed de computer ook rekent.

Gottlieb: ‘Pas in de jaren zeventig drong chaostheorie door in de ingenieurswetenschappen, vooral in de elektrotechniek, mechanica en lucht- en ruimtevaarttechniek.’ Daarna bereikte het de informatica in de vorm van de bekende fractals, waarmee databestanden konden worden gecomprimeerd.

Omdat er in Israël in de jaren tachtig nog nauwelijks onderzoek plaatsvond naar chaos besloten Gottlieb en zijn vrouw naar de Verenigde Staten te gaan – de liefde voor de nieuwe theorie zat diep. Gottlieb promoveerde er op het onderwerp en werd postdoc aan MIT. En toen vroeg zijn oude universiteit in Haifa of hij daar hoogleraar wilde worden in het nieuwe vakgebied. ‘Ik heb daar lang over nagedacht’, zegt Gottlieb. Hij was tenslotte ingenieur en wilde eigenlijk naar het bedrijfsleven.

En nu is hij een van ’s werelds grootste specialisten in de toepassing van chaostheorie op typische ingenieursterreinen. Bij de bouw van windmolens op zee bijvoorbeeld. Hoe diep moeten die in de zeebodem verankerd zijn om ze bij harde wind verantwoord te laten doordraaien? Chaostheorie speelt een belangrijke rol bij de beantwoording van die vraag.

Of neem de atoomkrachtmicroscoop (atomic force microscope, AFM). De ‘gewone’ variant brengt atomaire ‘landschappen’ in kaart doordat de naald van de microscoop de Vanderwaalskrachten tussen atomen meet. Door de naald op een slimme manier ‘chaotisch’ over het materiaal te laten bewegen, kan de AFM veel meer materiaaleigenschappen meten dan op de traditionele manier. ‘Soms gaat het er in mijn onderzoek om hoe chaos te vermijden is, maar soms ook – zoals bij de AFM – hoe je chaos juist kunt gebruiken.’

Dat hij een ingenieur is in de wereld van wiskundigen heeft een groot voordeel, bekent Gottlieb: hij kan makkelijker aan geld komen voor zijn onderzoek dan collega’s. Niet alleen de industrie subsidieert zijn onderzoek regelmatig, ook (semi-)overheidsfinanciers vragen meer en meer naar concrete toepassingen voor ze geld beschikbaar stellen. ‘Voor zuiver wiskundigen is het veel moeilijker om onderzoeksgeld binnen te halen dan voor mij.’

Een nadeel heeft dat natuurlijk ook, erkent hij: hij is voor zijn onderzoek afhankelijk van die (vaak kortetermijn-)interesses van zijn financiers. ‘Toch heb ik gelukkig altijd kunnen doen wat ik wilde: ik bedenk mijn eigen wetenschappelijke problemen en schrijf daar mijn eigen onderzoeksvoorstellen bij. Maar ik kan inderdaad niet alles onderzoeken wat ik wil. Zo zou ik me graag storten op onderzoek naar het menselijk hart: bij hartfalen speelt chaostheorie een belangrijke rol, het vertoont gelijkenissen met de chaotische processen bij het kapseizen van een schip.’

Hoewel iedere technische universiteit tegenwoordig ten minste één chaosspecialist in huis heeft, blijft chaostheorie een beperkt onderdeel van de ingenieurswetenschappen. ‘Dat is niet zo raar, want het is uiteindelijk een subdomein van een subdomein. Het meeste onderzoek vindt nog steeds plaats binnen lineaire dynamische systemen. Niet-lineaire dynamische systemen vormen een kleiner domein; chaotische systemen een nog kleiner.’

Techniekstudenten die interesse hebben in chaos raadt hij het boek aan van Francis J. Moon: Chaotic Vibrations: An Introduction for Applied Scientists and Engineers. ‘De inleiding is voor bachelorstudenten uitstekend te volgen, je hebt er eigenlijk alleen middelbareschoolwiskunde en wat kennis van lineaire algebra voor nodig.’ Voor leken maakte de BBC vijf jaar geleden een goede film van een uur: The secret life of chaos.

Die film heeft overigens nog wel een zweem van de romantische connotatie die chaostheorie vooral in de jaren zeventig had: chaos als ‘bevrijding’ van de wetmatigheid van de natuurwetenschap. Maar wat dat aangaat verschilt chaostheorie principieel van kwantummechanica met zijn indeterministische aspecten. ‘Chaotische processen zijn deterministisch en niet stochastisch. Een minieme verandering in de begintoestand van een systeem heeft uiteindelijk grote gevolgen, maar het blijft deterministisch.’

Photo by Marc Blommaert

Mr.Gottlieb applies chaos theory to solve typical engineering problems, from building windmills at sea to the improvement of the atomic powered microscope (AFM). The Israeli civil engineer Oded Gottlieb was invited by 3.TU.AMI to hold a keynote lecture during the Spring Congress in Utrecht on the 21st of May.

“I’m an engineer who loves mathematics”, says Gottlieb significantly. That is rather special, he means to say. Because mostly the reverse holds true: most mathematical whizzes in engineering science are mathematicians who were not able to get an appointment in their own field and therefore switched.

Gotllieb studied civil engineering at the technical university of Haifa, Technion (sometimes called the “MIT of Israel”) . He was convinced he would end up in industry just like his father. But it turned out differently. At the end of his study he fell in love: with chaos theory.

“I read the book ‘Instabilities and catastrophes in science and engineering’ of Michael Thompson, which my supervisor at Technion turned up with. During my undergraduate education at the beginning of the eighties I had never studied anything about chaos theory. I was seized by the global effects which chaos could have in nature, that a waving hand very far away in the end can have influence on a window we are looking through here.”

At the beginning of the eighties chaos theory was a new discipline, borne twenty years before with the famous article ‘Deterministic Non periodic Flow’ (1963) of the American meteorologist Edward Lorentz. He discovered by accident that a tiny difference in the initial condition of the atmosphere can bring about completely different weather after some time (the famous metaphor of the butterfly flapping its wings in Brazil - affecting the weather pattern – causing a tornado in Texas a few months later.

The article of Lorentz caused a landslide in the exact science. At the time one thought that small changes would result in small consequences. This was not true, there exist physical systems (such as flows in liquids or motions of a pendulum) which are unpredictable, no matter how well the computer calculates.

Gottlieb: “Since the seventies chaos theory entered into the engineering science, especially into electrical engineering, mechanics and aerospace engineering. After that it was used in computer science with its famous fractals with which its databases could be compressed.”

Because hardly any research was done in chaos theory in Israel, Gottlieb and his wife decided to move to the United States – the love for the new theory was deep. Gottlieb obtained his PhD degree on the subject and got a MIT-postdoc. And then his former university at Haifa offered him a professorship in this new discipline. “I’ve thought about it for a long time” Gottlieb said . After all, he was an engineer and actually preferred industry.

And now he is one of the greatest specialists in the application of chaos theory in typical engineering domains, e.g. windmill installations at sea. How deep do these have to be anchored in the seabed, so that they will turn safely in stormy weather? Chaos theory plays an important role in answering this question.

Or consider the atomic force microscope. The ‘common’ device pictures atomic landscapes because the needle of de microscope measures the “Vanderwaals” forces between the atoms. By moving the needle in a smart chaotic way across the material the microscope can disclose much more material properties than in the traditional way. “So my research is sometimes driven by avoiding chaos and sometimes - in case of the microscope - by using chaos.”

To be an engineer in a mathematical environment has a great advantage remarks Gottlieb: it makes fund raising for research easier. Not only industry endows his research regularly, but also (semi)governmental financiers are asking more and more for tangible applications before making money available. For pure mathematicians fund raising is much harder to do.

Still there is also a disadvantage, he admits: for his research he depends on the (short-term) vision/interest of his financiers. “But fortunately I have always been able to do what I wanted: I devise my own research problems and write my own research proposals for them. But apparently I cannot investigate everything. So I like to focus on research of the human heart: In heart failure chaos is playing an important role, there is a resemblance between these processes and those of the capsizing of a ship.”

Although nowadays every technical university will have at least one chaos specialist, the chaos theory still is a limited research field within engineering sciences. “That is not strange, because in the end it is a subdomain of a subdomain. Most research still takes place within the linear dynamical regime, not-linear dynamical systems form a smaller domain; chaotic systems even smaller still.

Engineering students having an interest in chaos, are advised by Gottlieb to read the book of Francis J. Moon: ‘Chaotic Vibrations: An Introduction for applied Scientists and Engineers’ . “The introduction is very accessible for bachelor students, actually you only need secondary school mathematics and some knowledge of linear algebra.” Five years ago the BBC created a good one hour movie for laymen: ‘The secret life of chaos’.

This film has a touch of the romantic connotation which the chaos theory had in the seventies: chaos as liberation of the rule of science.

But with regard to this, chaos theory principally differs from quantum theory.

(interview vertaald door E. van Elderen)

“I do not really know how to give shape to my retirement”, Arjeh Cohen admits. “I still have so many things to do.” That is the story of his life: a portrait of a busy man.

Text Anouck Vrouwe, May 2014

“I am quite picky about the mathematical problems I work on. But every now and then I decide that this one is mine. And then I will not let go. I guess that is my strength.” Arjeh Cohen shows an article that is about to be published. “This story, for example, started 25 years ago, in 1989. All those years, the problem stayed on my mind. I worked on it every now and then, together with a colleague. And finally, we solved it. The result was not as great as we had hoped for, but I am really happy we brought it to its conclusion.”

He chose to study math because of its beauty, not because he was planning to become a mathematician. But a mathematician he became. And a good one too. Arjeh Cohen, now Professor Emeritus Discrete Mathematics, gave his farewell speech in a packed auditorium at Eindhoven University of Technology. “I never thought of myself as a mathematician. At university, there were guys in my year who were a lot brighter than I am. But slowly, they disappeared, they followed other interests. And I turned out to be the go-getter with a career in science”, Cohen explains.

He once tried to do something else, after finishing his dissertation at Utrecht University. Finite complex reflection groups was the subject of Cohen’s dissertation. His supervisor was Tonny Springer – a respected mathematician, one of the world’s greatest experts on the theory on algebraic groups. “My dissertation was a tough job. Things were different then. My supervisor threw me in at the deep end. You make it or break it anyway, was his vision. I did not discuss my research topic with him more than three times. There was no room for failure. I saw good people drop out. I struggled a lot, but I made it. After this period of very abstract math, I decided it was time to do something useful.”

Cohen applied to a job at the environmental department of Openbaar Lichaam Rijnmond – a former authority between the city council of Rotterdam and the Province of South Holland. He refers to it as his traineeship. “I modeled the impact of the industrial environment on citizens. I learned a lot about applied mathematics, like statistics and numerical mathematics. But it was not my world. I disliked the way politicians took decisions without proper knowledge of the problem, and I disliked how officials made their way up by acting important.”

So when a job opened up at Discrete Mathematics in Enschede, Cohen was more than happy to apply. “So far for the excursion. I really liked my new job, the concreteness of it. It was the start of my career, I found my favorite kind of math.” Soon after, Cohen changed jobs again. CWI, the Dutch research institute for mathematics and computer science, started a new research group in Discrete Mathematics.

“That was when I started doing useful things with computers”, Cohen understates. His first love is the geometry of the classification of mathematical groups, like Lie groups. “That work will last – some of the things I have proven may still be used in next centuries.” The computer was the second important theme in Cohen’s career. “Doing computer algebra is a challenge, and I liked it. You cannot fool a computer; you have to fully understand a mathematical problem to be able to program it. If you take 1,2,3,4,5 for example: that is a list of numbers, but it is a way of ordering them too. A human is aware of these two meanings at the same time, but a computer is not: if that ascending order is important to you, you will have to tell the computer that this is more than numbers. You have to be very explicit and precise, and that made working with computers good practice for me.”

During his time at CWI, Cohen helped lay the foundation for CAN, the foundation Computer Algebra Nederland. The goal was to stimulate and coordinate the use of computer algebra systems in education and research. “We became the central distributor of computer algebra systems, like Mathematica and Maple. But we also promoted their use, by teaching people how to use them.” At Eindhoven University of Technology, Cohen shaped the university's vision on the use of computers in education. “To give all the freshmen a laptop was one of my ideas, for example. That was quite extraordinary in those days.” And after his retirement Cohen will develop an interactive book on calculus for Sowiso, an e-learning platform for mathematics. “I am convinced that adaptive learning will become more and more important; it makes it possible to educate people at their own speed and level.”

In his own work, Cohen used the computing power of computers for his proofs a lot. “One of my best results was a search for a special small subgroup in the biggest exceptional group in group theory, E8. I had a Macintosh at home, and I managed to break the problem down to a solvable set of linear equations. I remember how I looked at the result of the calculations – I saw the numbers and immediately realized that the solution of my problem presented itself. It is unforgettable if your logic turns out to do the trick.”

So the Eureka-moment really does exist? Cohen: “Sure. I remember one time at Utrecht Central Station – the moment I left the train, just before my foot touched the platform. I immediately knew I found the solution to the problem I had been working on for quite some time.”

Another thing Cohen is really proud of is his book Diagram Geometry, which he finished last year with co-author Francis Buekenhout. “It took quite some time to mature. It describes the most important things I worked on for all these years.” Group-related incidence geometry is a wonderful subject. Cohen tells how he loves the wonderful objects that arise from just a simple set of points, lines and some rules. “It is fascinating.”

Since 1992, Cohen has been a full professor of Discrete Mathematics in Eindhoven. In his inaugural speech What is the point of algebra? he pointed out that algebra is a world in itself, where ‘finite groups like the Monster live’. He stated that his world holds enough challenges and rewards to forget about the real world. But Cohen was not a mathematician who simply closed his door and did his math. Though he did not like the political games at the Openbaar Lichaam Rijnmond, he became very active in management duties in Eindhoven. He was dean at the Department of Mathematics and Computer Science from 2009 till 2013. Thoughtfully: “It was not my ambition to become dean, not at all. For your math, it is better to stay out of governance as much as possible. You will publish more if you do, and get more praise. But to be able to do so, you need a healthy and well organized department. It never felt like I had a choice; it was important work that needed to be done and there were not many people who could do it.” Though becoming dean might not have been his first choice, Cohen's colleagues say he did a good job: he was an attentive and straight manager. He left a healthy department behind. One of the things he professionalized is the fundraising of the department. “Mathematicians always think they only have to do great math to get grants. But that is not enough anymore. The competition has hardened: you have to sell your ideas – also to people who are not experts in math. We hired professional trainers to polish our presentations. That may sound expensive, but it pays for itself in the long run.”

Cohen was also involved in national initiatives to improve the position of mathematics in The Netherlands, like Platform Wiskunde Nederland (PWN) en AMI, the collaboration of the three technical universities. “I believe in cooperation. Take Mastermath; the universities bundled their forces to organize the best courses for master students. That is good for all of us.”

One of his other priorities as dean was to increase the number of women in the department. His farewell reception showed how men still dominate the field: his wife, professor in Medical Psychology, was the only female professor at his farewell reception. “A lot of people noticed. We do have two female professors at our department”, Cohen says, “and we are working hard to increase their numbers. That will not happen on its own, you have to actively search for women. In general, women do not put themselves at the forefront as much as men.” All his 21 PhD-students were male too. “But luckily, I still supervise two PhDs – and one of them is female.”

Cohen has been phasing out activities in preparation of retirement, but he is as busy as before. He works on an interactive book on calculus, has some papers in preparation and supervises his last PhDs. “I don’t really know how I will shape my retirement. It is weird. But I do like the idea of spending more time with my grandchildren.” A short laugh: he realizes very well that he will have to plan such precious time in his overloaded schedule. Retired or not – Cohen is still that energetic man he has always been.

CV - Arjeh Cohen (1949, Haifa)

1971 – Master degree, Utrecht University

1975 – Dissertation ‘Finite Complex Reflection Groups’, Utrecht University

1975 – Researcher at Openbaar Lichaam Rijnmond

1976 – Researcher at University of Twente

1979 – Researcher at CWI, Amsterdam and Professor of Discrete Mathematics, Utrecht University

1992 – Full professor of Discrete Mathematics at Eindhoven University of Technology

2009 – Dean of the Department of Mathematics and Computer Science

*Note: thanks to Onno Boxma for his help *

June 13, 2013: AMI Springfestival - TUDelft

**Mathematics and numbers**There are two answers to the question why Gerhard Frey chose to study mathematics. One is he liked the subject at school. But then again, he also liked history. The other answer is that with mathematics you can discover things just by thinking and you don’t have to learn things by heart. So with mathematics he could be the laziest man of the world.

We interview Gerhard Frey before the 3TU.AMI Springfestival held at the TU Delft, where he will be the key note speaker with a presentation with the intriguing question: Can Mathematics keep a secret? Gerhard’s specialty is number theory. “During the start of my study at the University of Tubingen I had a beginner course by the quite famous Peter Roquette. He sparked my interest in number theory. Number theory starts very simple. You use your fingers to count and you learn about numbers, calculation, prime numbers and polynomials in school. In number theory you want to find relations between numbers with properties. For example: Euclid stated that there are an infinite number of prime numbers and he gave a beautiful and simple proof. The other extreme is Fermat’s last theorem that is also very simple to state but it was very hard to prove it. In number theory you need a lot of theory to prove simple things. It is fascinating. Everyone is interested in numbers. People playing a Sudoku are playing with numbers. Number theory proved to have many big challenges for me, but if I maybe had another instructor at my beginner’s courses I might have ended up in another field of mathematics.”

**Fermat’s last theorem**Peter Roquette was Gerhard’s supervisor during his PhD work in Heidelberg. “After my Diploma (master) I had the choice to go into the industry, work at a school or continue at the university. Nowadays it is quite difficult for young people to get a position at the university, but at that time it easier”, tells Gerhard. “From the 1960’s onwards until the end of the last century it was a golden age for number theory. We proved so many results . It was nice to be part of a revolution.” When we ask him why so many mathematicians were fascinated by Fermat’s last theorem, Gerhard says that he has to disappoint us. “It is a very simple question. Everyone can understand it. Still once every week I receive a letter, by someone who claims to have found a proof that only takes two pages, but it is always nonsense. But for a long time you didn’t have the power to attack it. The famous mathematician Gauss tried to solve it. He asked himself: Why should it be so important? I can think of similar questions once a week.” It took 358 years to publish the proof. Gerhard explains: “During the last 200 years people started to develop theories that would hopefully help to prove the theorem. The key word is algebraic number theory. Fermat’s last theorem was a great motivation to develop this theory. But only in the sixties and seventies of the last century theories were raised to give a method on how to solve it. You have to find the right method and structure for the problem to apply the theory . My fascination with Fermat’s last law was to find a structure that we could apply. Algebraic manipulations of formulas gave no way to solve the equation. We took a totally new approach and attached Galois representations to Fermat's Last Theorem. This turned out to be an important step in the final proof by Andrew Wiles.”

**The importance of conferences**Gerhard’s career changed from the theoretical side to the more applied side, when he started to work for the Institute for Experimental Mathematics in 1990 in Essen. There were four chairs. Gerhard had the chair for number theory. The other chairs were on group theory, computer science and digital communication. “It was a big luck to work with professor Han Vinck. He is an engineer with a great background in mathematics. We became interested in coding theory. In one conference we held in Essen somebody spoke about cryptography and a participant asked questions about public key cryptography. We saw immediately that we could apply the whole machinery we used for solving Fermat’s last theorem to this question. Again we found the structure that was behind and we understood that our expertise could help to provide useful theory . So the next 15 years I led a double life, working both in theoretical and in applied mathematics. We had the good luck to attract many very clever young people who wished to work in this new area of cryptography.”

Gerhard stresses the importance of conferences for the work of a mathematician. “It is important to travel around the world and to exchange problems and questions. As a mathematician you don’t have a lab or machines, so you are free to go to conferences that are important to improve your ideas.”

In 2005 Gerhard's work had consequences for the electronic passport in Germany. “Our institute was involved in developing the mathematical background. We helped to find criteria for the right elliptic curves, which are the backbone of the theory, which were put in use in the passport. This passport is a very complex thing, with a lot of protocol. When a passport and a machine communicate they ask each other in milliseconds to solve a mathematical problem. This can be done only if both passport and machine have certain knowledge and thus it helps to establish whether it is the right passport and the right machine. ”

**Answers in your sleep**Gerhard emphasizes the role of intuition in arriving to solutions for difficult problems. “If you speak to a mathematician, you will often hear: I have the feeling that this can’t be true. Experience is of course very important to get this intuition. I often got an answer to my work problems during sleep, but, alas, only to find out in most cases in the morning that the answer was completely wrong! I had a thirty minute bike ride to and from work and that was a good way to think ideas through or get new ones. Input that comes from discussion with others is also very important. So during the day you work on calculation and read papers, but the best ideas come when you’re not working!”

When asked what the biggest challenges in number theory are, Gerhard quotes the famous mathematician David Hilbert “We must know and we will know.” He thinks that there are many unknown areas and old and new challenges to be solved. “For example complexity theory. It tells you how difficult an algorithm has to be to solve a problem, and whether a “machine” can find a solution or not to a particular problem. It’s sometimes a bit like the book Hitchhiker’s guide to the Galaxy by Isaac Asimov, that you have to build a gigantic machine to get an answer to a problem and the output is trivial like 42. One of the big theoretical classical challenges is of course the Riemann hypothesis. That is also one of the 10 Clay Mathematics Institute Millennium Prize Problems, who let you win a million dollar if you solve one of the problems. But in number theory there are also many seemingly small problems, who need a lot of structure to solve. Many of them could be done if we would understand the Galois group of rational numbers. “

**Young talent**In Gerhard’s opinion there are many talented people in high schools, and universities offer good personal contact with the students to develop these talents. “The hard thing is the PhD thesis, which will determine whether you are more suited for a career in industry or in research.” Gerhard sees a big pool of talent both male and female, but thinks we should do more to give them more security. “It is a rat race. People have to communicate about their ideas in papers and at conferences, but at the same time there is a lot of competition. They often have a contract for 2 years and already have to apply for a new position after one year.” He thinks it’s a shame that till now there are very few women with a chair in mathematics, whereas in Germany the student numbers are 50 percent male and 50 percent female. This postdoc period is difficult. The woman often follows the man and it is also the time to start a family. “We should give them more support.”

**No stress**Gerhard now enjoys the fact that he no longer has the stress of having a chair and the responsibility for the scientific career of young people. “I still write papers, go to conferences and have discussions with friends, but it is more relaxed now.” He looks back on a great career. “My biggest success was the attack on the Fermat theorem, that Wiles found proof for. Later we developed a method for elliptic curves that are still used in public key cryptography. A lot of the guidelines of national security agencies go back to the work we did in this area. I started out in the highest theoretical field, with the lowest possibilities for applications. Through contacts and discussions with electrical engineers, who were designing chips, we came up with new mathematical methods that have great applications in cryptography.”

Quantum computing is a new area that will propose big challenges. Gerhard says: “We know from physics that quantum computing will work in principle and that it will break public key systems used today in milliseconds if a big enough computer will be built. Young colleagues will have to find other mathematical problems that will be resistant to quantum computing. During the next 20 years quantum computers may be there, but that is a story that we have been telling already for a long time. The difficulty is how to build and engineer such a computer. It is like hydrogen fusion. We know from the sun that it can be a source of energy, and from the 1940’s onward people have been trying to build such a system. But up till now, not a microVolt has come out. Maybe quantum computing will follow a similar path, because it is difficult to build. But as well it may be otherwise...”

**Cautious**Coming back to the question “Can Mathematics keep a secret?”, Gerhard proposes the situation that the two interviewers want to exchange a secret, with him present in the room and him hearing the exchange and knowing the method of how we exchange the secret. Public key crypto give means to build up such a secret connection. “This is an important method that is useful in everyday applications like smart cards, but also when your prime minister wants to have a telephone conversation with the president of the USA via a public net.” But there can always be leaks in the security if one is not careful. Gerhard remembers that thieves followed drivers of luxurious cars and recorded the beep that was given to open and close the door. And played that beep later to open the car door: not enough mathematics was used. So: Yes, mathematics can keep a secret, if you are cautious!

May 24, 2017: AMI Spring congress - Wageningen

Lees het hele artikel van Christian Jongeneel hier

of via www.technischweekblad.nl

May 21, 2015: AMI Spring congress - Utrecht

Hij past chaostheorie toe om typische ingenieursproblemen op te lossen: van de bouw van windmolens op zee tot verbetering van de atoomkrachtmicroscoop (AFM). De Israëlische civieltechnicus Oded Gottlieb hield daarover een key note lecture tijdens het 3TU.AMI Spring Congress in Utrecht op 21 mei.

‘Ik ben een ingenieur die van wiskunde houdt’, zegt Gottlieb veelbetekenend. Dat is tamelijk bijzonder, wil hij maar zeggen. Want meestal is het andersom: ‘De meeste mathematische bollebozen in de ingenieurswetenschappen zijn wiskundigen die geen aanstelling konden krijgen in hun eigen vakgebied en daarom maar zijn uitgeweken.’

Gottlieb studeerde civiele techniek aan de technische universiteit Technion in Haifa (ook wel het ‘MIT van Israël’ genoemd). Hij was ervan overtuigd dat hij in de industrie zou belanden, net als zijn vader. Maar het liep anders. Gottlieb werd tegen het eind van zijn studie verliefd: op de chaostheorie.

‘Ik las het boek Instabilities and catastrophes in science and engineering van Michael Thompson. Mijn supervisor aan Technion kwam ermee aanzetten. In mijn undergraduate studie begin jaren tachtig had ik helemaal niets over chaos geleerd. Ik was gegrepen door de global effecten die chaos in de natuur kan hebben. Dat een zwaaiende hand heel ver weg uiteindelijk invloed kan hebben op het raam waar we hier nu doorheen kijken.’

Chaostheorie was begin jaren tachtig nog een jong vak, twintig jaar eerder geboren met het beroemde artikel Deterministic Nonperiodic Flow (1963) van de Amerikaanse meteoroloog Edward Lorenz. Hij had bij toeval ontdekt dat een miniem verschil in de begintoestand van de atmosfeer naar verloop van tijd totaal ander weer oplevert (de beroemde metafoor van de vlinder die in Brazilië met zijn vleugels klapt en daarmee maanden later in Texas een tornado veroorzaakt).

Dat artikel van Lorenz veroorzaakte een aardverschuiving in de exacte wetenschappen. De aanname was tot die tijd namelijk dat kleine veranderingen kleine gevolgen hebben. Dat bleek dus niet te kloppen; er bestaan fysische systemen (zoals stromingen in vloeistoffen of de beweging van een slinger) die na enige tijd onvoorspelbaar zijn, hoe goed de computer ook rekent.

Gottlieb: ‘Pas in de jaren zeventig drong chaostheorie door in de ingenieurswetenschappen, vooral in de elektrotechniek, mechanica en lucht- en ruimtevaarttechniek.’ Daarna bereikte het de informatica in de vorm van de bekende fractals, waarmee databestanden konden worden gecomprimeerd.

Omdat er in Israël in de jaren tachtig nog nauwelijks onderzoek plaatsvond naar chaos besloten Gottlieb en zijn vrouw naar de Verenigde Staten te gaan – de liefde voor de nieuwe theorie zat diep. Gottlieb promoveerde er op het onderwerp en werd postdoc aan MIT. En toen vroeg zijn oude universiteit in Haifa of hij daar hoogleraar wilde worden in het nieuwe vakgebied. ‘Ik heb daar lang over nagedacht’, zegt Gottlieb. Hij was tenslotte ingenieur en wilde eigenlijk naar het bedrijfsleven.

En nu is hij een van ’s werelds grootste specialisten in de toepassing van chaostheorie op typische ingenieursterreinen. Bij de bouw van windmolens op zee bijvoorbeeld. Hoe diep moeten die in de zeebodem verankerd zijn om ze bij harde wind verantwoord te laten doordraaien? Chaostheorie speelt een belangrijke rol bij de beantwoording van die vraag.

Of neem de atoomkrachtmicroscoop (atomic force microscope, AFM). De ‘gewone’ variant brengt atomaire ‘landschappen’ in kaart doordat de naald van de microscoop de Vanderwaalskrachten tussen atomen meet. Door de naald op een slimme manier ‘chaotisch’ over het materiaal te laten bewegen, kan de AFM veel meer materiaaleigenschappen meten dan op de traditionele manier. ‘Soms gaat het er in mijn onderzoek om hoe chaos te vermijden is, maar soms ook – zoals bij de AFM – hoe je chaos juist kunt gebruiken.’

Dat hij een ingenieur is in de wereld van wiskundigen heeft een groot voordeel, bekent Gottlieb: hij kan makkelijker aan geld komen voor zijn onderzoek dan collega’s. Niet alleen de industrie subsidieert zijn onderzoek regelmatig, ook (semi-)overheidsfinanciers vragen meer en meer naar concrete toepassingen voor ze geld beschikbaar stellen. ‘Voor zuiver wiskundigen is het veel moeilijker om onderzoeksgeld binnen te halen dan voor mij.’

Een nadeel heeft dat natuurlijk ook, erkent hij: hij is voor zijn onderzoek afhankelijk van die (vaak kortetermijn-)interesses van zijn financiers. ‘Toch heb ik gelukkig altijd kunnen doen wat ik wilde: ik bedenk mijn eigen wetenschappelijke problemen en schrijf daar mijn eigen onderzoeksvoorstellen bij. Maar ik kan inderdaad niet alles onderzoeken wat ik wil. Zo zou ik me graag storten op onderzoek naar het menselijk hart: bij hartfalen speelt chaostheorie een belangrijke rol, het vertoont gelijkenissen met de chaotische processen bij het kapseizen van een schip.’

Hoewel iedere technische universiteit tegenwoordig ten minste één chaosspecialist in huis heeft, blijft chaostheorie een beperkt onderdeel van de ingenieurswetenschappen. ‘Dat is niet zo raar, want het is uiteindelijk een subdomein van een subdomein. Het meeste onderzoek vindt nog steeds plaats binnen lineaire dynamische systemen. Niet-lineaire dynamische systemen vormen een kleiner domein; chaotische systemen een nog kleiner.’

Techniekstudenten die interesse hebben in chaos raadt hij het boek aan van Francis J. Moon: Chaotic Vibrations: An Introduction for Applied Scientists and Engineers. ‘De inleiding is voor bachelorstudenten uitstekend te volgen, je hebt er eigenlijk alleen middelbareschoolwiskunde en wat kennis van lineaire algebra voor nodig.’ Voor leken maakte de BBC vijf jaar geleden een goede film van een uur: The secret life of chaos.

Die film heeft overigens nog wel een zweem van de romantische connotatie die chaostheorie vooral in de jaren zeventig had: chaos als ‘bevrijding’ van de wetmatigheid van de natuurwetenschap. Maar wat dat aangaat verschilt chaostheorie principieel van kwantummechanica met zijn indeterministische aspecten. ‘Chaotische processen zijn deterministisch en niet stochastisch. Een minieme verandering in de begintoestand van een systeem heeft uiteindelijk grote gevolgen, maar het blijft deterministisch.’

Photo by Marc Blommaert

Mr.Gottlieb applies chaos theory to solve typical engineering problems, from building windmills at sea to the improvement of the atomic powered microscope (AFM). The Israeli civil engineer Oded Gottlieb was invited by 3.TU.AMI to hold a keynote lecture during the Spring Congress in Utrecht on the 21st of May.

“I’m an engineer who loves mathematics”, says Gottlieb significantly. That is rather special, he means to say. Because mostly the reverse holds true: most mathematical whizzes in engineering science are mathematicians who were not able to get an appointment in their own field and therefore switched.

Gotllieb studied civil engineering at the technical university of Haifa, Technion (sometimes called the “MIT of Israel”) . He was convinced he would end up in industry just like his father. But it turned out differently. At the end of his study he fell in love: with chaos theory.

“I read the book ‘Instabilities and catastrophes in science and engineering’ of Michael Thompson, which my supervisor at Technion turned up with. During my undergraduate education at the beginning of the eighties I had never studied anything about chaos theory. I was seized by the global effects which chaos could have in nature, that a waving hand very far away in the end can have influence on a window we are looking through here.”

At the beginning of the eighties chaos theory was a new discipline, borne twenty years before with the famous article ‘Deterministic Non periodic Flow’ (1963) of the American meteorologist Edward Lorentz. He discovered by accident that a tiny difference in the initial condition of the atmosphere can bring about completely different weather after some time (the famous metaphor of the butterfly flapping its wings in Brazil - affecting the weather pattern – causing a tornado in Texas a few months later.

The article of Lorentz caused a landslide in the exact science. At the time one thought that small changes would result in small consequences. This was not true, there exist physical systems (such as flows in liquids or motions of a pendulum) which are unpredictable, no matter how well the computer calculates.

Gottlieb: “Since the seventies chaos theory entered into the engineering science, especially into electrical engineering, mechanics and aerospace engineering. After that it was used in computer science with its famous fractals with which its databases could be compressed.”

Because hardly any research was done in chaos theory in Israel, Gottlieb and his wife decided to move to the United States – the love for the new theory was deep. Gottlieb obtained his PhD degree on the subject and got a MIT-postdoc. And then his former university at Haifa offered him a professorship in this new discipline. “I’ve thought about it for a long time” Gottlieb said . After all, he was an engineer and actually preferred industry.

And now he is one of the greatest specialists in the application of chaos theory in typical engineering domains, e.g. windmill installations at sea. How deep do these have to be anchored in the seabed, so that they will turn safely in stormy weather? Chaos theory plays an important role in answering this question.

Or consider the atomic force microscope. The ‘common’ device pictures atomic landscapes because the needle of de microscope measures the “Vanderwaals” forces between the atoms. By moving the needle in a smart chaotic way across the material the microscope can disclose much more material properties than in the traditional way. “So my research is sometimes driven by avoiding chaos and sometimes - in case of the microscope - by using chaos.”

To be an engineer in a mathematical environment has a great advantage remarks Gottlieb: it makes fund raising for research easier. Not only industry endows his research regularly, but also (semi)governmental financiers are asking more and more for tangible applications before making money available. For pure mathematicians fund raising is much harder to do.

Still there is also a disadvantage, he admits: for his research he depends on the (short-term) vision/interest of his financiers. “But fortunately I have always been able to do what I wanted: I devise my own research problems and write my own research proposals for them. But apparently I cannot investigate everything. So I like to focus on research of the human heart: In heart failure chaos is playing an important role, there is a resemblance between these processes and those of the capsizing of a ship.”

Although nowadays every technical university will have at least one chaos specialist, the chaos theory still is a limited research field within engineering sciences. “That is not strange, because in the end it is a subdomain of a subdomain. Most research still takes place within the linear dynamical regime, not-linear dynamical systems form a smaller domain; chaotic systems even smaller still.

Engineering students having an interest in chaos, are advised by Gottlieb to read the book of Francis J. Moon: ‘Chaotic Vibrations: An Introduction for applied Scientists and Engineers’ . “The introduction is very accessible for bachelor students, actually you only need secondary school mathematics and some knowledge of linear algebra.” Five years ago the BBC created a good one hour movie for laymen: ‘The secret life of chaos’.

This film has a touch of the romantic connotation which the chaos theory had in the seventies: chaos as liberation of the rule of science.

But with regard to this, chaos theory principally differs from quantum theory.

(interview vertaald door E. van Elderen)

“I do not really know how to give shape to my retirement”, Arjeh Cohen admits. “I still have so many things to do.” That is the story of his life: a portrait of a busy man.

Text Anouck Vrouwe, May 2014

“I am quite picky about the mathematical problems I work on. But every now and then I decide that this one is mine. And then I will not let go. I guess that is my strength.” Arjeh Cohen shows an article that is about to be published. “This story, for example, started 25 years ago, in 1989. All those years, the problem stayed on my mind. I worked on it every now and then, together with a colleague. And finally, we solved it. The result was not as great as we had hoped for, but I am really happy we brought it to its conclusion.”

He chose to study math because of its beauty, not because he was planning to become a mathematician. But a mathematician he became. And a good one too. Arjeh Cohen, now Professor Emeritus Discrete Mathematics, gave his farewell speech in a packed auditorium at Eindhoven University of Technology. “I never thought of myself as a mathematician. At university, there were guys in my year who were a lot brighter than I am. But slowly, they disappeared, they followed other interests. And I turned out to be the go-getter with a career in science”, Cohen explains.

He once tried to do something else, after finishing his dissertation at Utrecht University. Finite complex reflection groups was the subject of Cohen’s dissertation. His supervisor was Tonny Springer – a respected mathematician, one of the world’s greatest experts on the theory on algebraic groups. “My dissertation was a tough job. Things were different then. My supervisor threw me in at the deep end. You make it or break it anyway, was his vision. I did not discuss my research topic with him more than three times. There was no room for failure. I saw good people drop out. I struggled a lot, but I made it. After this period of very abstract math, I decided it was time to do something useful.”

Cohen applied to a job at the environmental department of Openbaar Lichaam Rijnmond – a former authority between the city council of Rotterdam and the Province of South Holland. He refers to it as his traineeship. “I modeled the impact of the industrial environment on citizens. I learned a lot about applied mathematics, like statistics and numerical mathematics. But it was not my world. I disliked the way politicians took decisions without proper knowledge of the problem, and I disliked how officials made their way up by acting important.”

So when a job opened up at Discrete Mathematics in Enschede, Cohen was more than happy to apply. “So far for the excursion. I really liked my new job, the concreteness of it. It was the start of my career, I found my favorite kind of math.” Soon after, Cohen changed jobs again. CWI, the Dutch research institute for mathematics and computer science, started a new research group in Discrete Mathematics.

“That was when I started doing useful things with computers”, Cohen understates. His first love is the geometry of the classification of mathematical groups, like Lie groups. “That work will last – some of the things I have proven may still be used in next centuries.” The computer was the second important theme in Cohen’s career. “Doing computer algebra is a challenge, and I liked it. You cannot fool a computer; you have to fully understand a mathematical problem to be able to program it. If you take 1,2,3,4,5 for example: that is a list of numbers, but it is a way of ordering them too. A human is aware of these two meanings at the same time, but a computer is not: if that ascending order is important to you, you will have to tell the computer that this is more than numbers. You have to be very explicit and precise, and that made working with computers good practice for me.”

During his time at CWI, Cohen helped lay the foundation for CAN, the foundation Computer Algebra Nederland. The goal was to stimulate and coordinate the use of computer algebra systems in education and research. “We became the central distributor of computer algebra systems, like Mathematica and Maple. But we also promoted their use, by teaching people how to use them.” At Eindhoven University of Technology, Cohen shaped the university's vision on the use of computers in education. “To give all the freshmen a laptop was one of my ideas, for example. That was quite extraordinary in those days.” And after his retirement Cohen will develop an interactive book on calculus for Sowiso, an e-learning platform for mathematics. “I am convinced that adaptive learning will become more and more important; it makes it possible to educate people at their own speed and level.”

In his own work, Cohen used the computing power of computers for his proofs a lot. “One of my best results was a search for a special small subgroup in the biggest exceptional group in group theory, E8. I had a Macintosh at home, and I managed to break the problem down to a solvable set of linear equations. I remember how I looked at the result of the calculations – I saw the numbers and immediately realized that the solution of my problem presented itself. It is unforgettable if your logic turns out to do the trick.”

So the Eureka-moment really does exist? Cohen: “Sure. I remember one time at Utrecht Central Station – the moment I left the train, just before my foot touched the platform. I immediately knew I found the solution to the problem I had been working on for quite some time.”

Another thing Cohen is really proud of is his book Diagram Geometry, which he finished last year with co-author Francis Buekenhout. “It took quite some time to mature. It describes the most important things I worked on for all these years.” Group-related incidence geometry is a wonderful subject. Cohen tells how he loves the wonderful objects that arise from just a simple set of points, lines and some rules. “It is fascinating.”

Since 1992, Cohen has been a full professor of Discrete Mathematics in Eindhoven. In his inaugural speech What is the point of algebra? he pointed out that algebra is a world in itself, where ‘finite groups like the Monster live’. He stated that his world holds enough challenges and rewards to forget about the real world. But Cohen was not a mathematician who simply closed his door and did his math. Though he did not like the political games at the Openbaar Lichaam Rijnmond, he became very active in management duties in Eindhoven. He was dean at the Department of Mathematics and Computer Science from 2009 till 2013. Thoughtfully: “It was not my ambition to become dean, not at all. For your math, it is better to stay out of governance as much as possible. You will publish more if you do, and get more praise. But to be able to do so, you need a healthy and well organized department. It never felt like I had a choice; it was important work that needed to be done and there were not many people who could do it.” Though becoming dean might not have been his first choice, Cohen's colleagues say he did a good job: he was an attentive and straight manager. He left a healthy department behind. One of the things he professionalized is the fundraising of the department. “Mathematicians always think they only have to do great math to get grants. But that is not enough anymore. The competition has hardened: you have to sell your ideas – also to people who are not experts in math. We hired professional trainers to polish our presentations. That may sound expensive, but it pays for itself in the long run.”

Cohen was also involved in national initiatives to improve the position of mathematics in The Netherlands, like Platform Wiskunde Nederland (PWN) en AMI, the collaboration of the three technical universities. “I believe in cooperation. Take Mastermath; the universities bundled their forces to organize the best courses for master students. That is good for all of us.”

One of his other priorities as dean was to increase the number of women in the department. His farewell reception showed how men still dominate the field: his wife, professor in Medical Psychology, was the only female professor at his farewell reception. “A lot of people noticed. We do have two female professors at our department”, Cohen says, “and we are working hard to increase their numbers. That will not happen on its own, you have to actively search for women. In general, women do not put themselves at the forefront as much as men.” All his 21 PhD-students were male too. “But luckily, I still supervise two PhDs – and one of them is female.”

Cohen has been phasing out activities in preparation of retirement, but he is as busy as before. He works on an interactive book on calculus, has some papers in preparation and supervises his last PhDs. “I don’t really know how I will shape my retirement. It is weird. But I do like the idea of spending more time with my grandchildren.” A short laugh: he realizes very well that he will have to plan such precious time in his overloaded schedule. Retired or not – Cohen is still that energetic man he has always been.

CV - Arjeh Cohen (1949, Haifa)

1971 – Master degree, Utrecht University

1975 – Dissertation ‘Finite Complex Reflection Groups’, Utrecht University

1975 – Researcher at Openbaar Lichaam Rijnmond

1976 – Researcher at University of Twente

1979 – Researcher at CWI, Amsterdam and Professor of Discrete Mathematics, Utrecht University

1992 – Full professor of Discrete Mathematics at Eindhoven University of Technology

2009 – Dean of the Department of Mathematics and Computer Science

*Note: thanks to Onno Boxma for his help *

June 13, 2013: AMI Springfestival - TUDelft

**Mathematics and numbers**There are two answers to the question why Gerhard Frey chose to study mathematics. One is he liked the subject at school. But then again, he also liked history. The other answer is that with mathematics you can discover things just by thinking and you don’t have to learn things by heart. So with mathematics he could be the laziest man of the world.

We interview Gerhard Frey before the 3TU.AMI Springfestival held at the TU Delft, where he will be the key note speaker with a presentation with the intriguing question: Can Mathematics keep a secret? Gerhard’s specialty is number theory. “During the start of my study at the University of Tubingen I had a beginner course by the quite famous Peter Roquette. He sparked my interest in number theory. Number theory starts very simple. You use your fingers to count and you learn about numbers, calculation, prime numbers and polynomials in school. In number theory you want to find relations between numbers with properties. For example: Euclid stated that there are an infinite number of prime numbers and he gave a beautiful and simple proof. The other extreme is Fermat’s last theorem that is also very simple to state but it was very hard to prove it. In number theory you need a lot of theory to prove simple things. It is fascinating. Everyone is interested in numbers. People playing a Sudoku are playing with numbers. Number theory proved to have many big challenges for me, but if I maybe had another instructor at my beginner’s courses I might have ended up in another field of mathematics.”

**Fermat’s last theorem**Peter Roquette was Gerhard’s supervisor during his PhD work in Heidelberg. “After my Diploma (master) I had the choice to go into the industry, work at a school or continue at the university. Nowadays it is quite difficult for young people to get a position at the university, but at that time it easier”, tells Gerhard. “From the 1960’s onwards until the end of the last century it was a golden age for number theory. We proved so many results . It was nice to be part of a revolution.” When we ask him why so many mathematicians were fascinated by Fermat’s last theorem, Gerhard says that he has to disappoint us. “It is a very simple question. Everyone can understand it. Still once every week I receive a letter, by someone who claims to have found a proof that only takes two pages, but it is always nonsense. But for a long time you didn’t have the power to attack it. The famous mathematician Gauss tried to solve it. He asked himself: Why should it be so important? I can think of similar questions once a week.” It took 358 years to publish the proof. Gerhard explains: “During the last 200 years people started to develop theories that would hopefully help to prove the theorem. The key word is algebraic number theory. Fermat’s last theorem was a great motivation to develop this theory. But only in the sixties and seventies of the last century theories were raised to give a method on how to solve it. You have to find the right method and structure for the problem to apply the theory . My fascination with Fermat’s last law was to find a structure that we could apply. Algebraic manipulations of formulas gave no way to solve the equation. We took a totally new approach and attached Galois representations to Fermat's Last Theorem. This turned out to be an important step in the final proof by Andrew Wiles.”

**The importance of conferences**Gerhard’s career changed from the theoretical side to the more applied side, when he started to work for the Institute for Experimental Mathematics in 1990 in Essen. There were four chairs. Gerhard had the chair for number theory. The other chairs were on group theory, computer science and digital communication. “It was a big luck to work with professor Han Vinck. He is an engineer with a great background in mathematics. We became interested in coding theory. In one conference we held in Essen somebody spoke about cryptography and a participant asked questions about public key cryptography. We saw immediately that we could apply the whole machinery we used for solving Fermat’s last theorem to this question. Again we found the structure that was behind and we understood that our expertise could help to provide useful theory . So the next 15 years I led a double life, working both in theoretical and in applied mathematics. We had the good luck to attract many very clever young people who wished to work in this new area of cryptography.”

Gerhard stresses the importance of conferences for the work of a mathematician. “It is important to travel around the world and to exchange problems and questions. As a mathematician you don’t have a lab or machines, so you are free to go to conferences that are important to improve your ideas.”

In 2005 Gerhard's work had consequences for the electronic passport in Germany. “Our institute was involved in developing the mathematical background. We helped to find criteria for the right elliptic curves, which are the backbone of the theory, which were put in use in the passport. This passport is a very complex thing, with a lot of protocol. When a passport and a machine communicate they ask each other in milliseconds to solve a mathematical problem. This can be done only if both passport and machine have certain knowledge and thus it helps to establish whether it is the right passport and the right machine. ”

**Answers in your sleep**Gerhard emphasizes the role of intuition in arriving to solutions for difficult problems. “If you speak to a mathematician, you will often hear: I have the feeling that this can’t be true. Experience is of course very important to get this intuition. I often got an answer to my work problems during sleep, but, alas, only to find out in most cases in the morning that the answer was completely wrong! I had a thirty minute bike ride to and from work and that was a good way to think ideas through or get new ones. Input that comes from discussion with others is also very important. So during the day you work on calculation and read papers, but the best ideas come when you’re not working!”

When asked what the biggest challenges in number theory are, Gerhard quotes the famous mathematician David Hilbert “We must know and we will know.” He thinks that there are many unknown areas and old and new challenges to be solved. “For example complexity theory. It tells you how difficult an algorithm has to be to solve a problem, and whether a “machine” can find a solution or not to a particular problem. It’s sometimes a bit like the book Hitchhiker’s guide to the Galaxy by Isaac Asimov, that you have to build a gigantic machine to get an answer to a problem and the output is trivial like 42. One of the big theoretical classical challenges is of course the Riemann hypothesis. That is also one of the 10 Clay Mathematics Institute Millennium Prize Problems, who let you win a million dollar if you solve one of the problems. But in number theory there are also many seemingly small problems, who need a lot of structure to solve. Many of them could be done if we would understand the Galois group of rational numbers. “

**Young talent**In Gerhard’s opinion there are many talented people in high schools, and universities offer good personal contact with the students to develop these talents. “The hard thing is the PhD thesis, which will determine whether you are more suited for a career in industry or in research.” Gerhard sees a big pool of talent both male and female, but thinks we should do more to give them more security. “It is a rat race. People have to communicate about their ideas in papers and at conferences, but at the same time there is a lot of competition. They often have a contract for 2 years and already have to apply for a new position after one year.” He thinks it’s a shame that till now there are very few women with a chair in mathematics, whereas in Germany the student numbers are 50 percent male and 50 percent female. This postdoc period is difficult. The woman often follows the man and it is also the time to start a family. “We should give them more support.”

**No stress**Gerhard now enjoys the fact that he no longer has the stress of having a chair and the responsibility for the scientific career of young people. “I still write papers, go to conferences and have discussions with friends, but it is more relaxed now.” He looks back on a great career. “My biggest success was the attack on the Fermat theorem, that Wiles found proof for. Later we developed a method for elliptic curves that are still used in public key cryptography. A lot of the guidelines of national security agencies go back to the work we did in this area. I started out in the highest theoretical field, with the lowest possibilities for applications. Through contacts and discussions with electrical engineers, who were designing chips, we came up with new mathematical methods that have great applications in cryptography.”

Quantum computing is a new area that will propose big challenges. Gerhard says: “We know from physics that quantum computing will work in principle and that it will break public key systems used today in milliseconds if a big enough computer will be built. Young colleagues will have to find other mathematical problems that will be resistant to quantum computing. During the next 20 years quantum computers may be there, but that is a story that we have been telling already for a long time. The difficulty is how to build and engineer such a computer. It is like hydrogen fusion. We know from the sun that it can be a source of energy, and from the 1940’s onward people have been trying to build such a system. But up till now, not a microVolt has come out. Maybe quantum computing will follow a similar path, because it is difficult to build. But as well it may be otherwise...”

**Cautious**Coming back to the question “Can Mathematics keep a secret?”, Gerhard proposes the situation that the two interviewers want to exchange a secret, with him present in the room and him hearing the exchange and knowing the method of how we exchange the secret. Public key crypto give means to build up such a secret connection. “This is an important method that is useful in everyday applications like smart cards, but also when your prime minister wants to have a telephone conversation with the president of the USA via a public net.” But there can always be leaks in the security if one is not careful. Gerhard remembers that thieves followed drivers of luxurious cars and recorded the beep that was given to open and close the door. And played that beep later to open the car door: not enough mathematics was used. So: Yes, mathematics can keep a secret, if you are cautious!

May 24, 2017: AMI Spring congress - Wageningen

Lees het hele artikel van Christian Jongeneel hier

of via www.technischweekblad.nl

May 21, 2015: AMI Spring congress - Utrecht

Photo by Marc Blommaert

(interview vertaald door E. van Elderen)

Text Anouck Vrouwe, May 2014

1971 – Master degree, Utrecht University

1975 – Dissertation ‘Finite Complex Reflection Groups’, Utrecht University

1975 – Researcher at Openbaar Lichaam Rijnmond

1976 – Researcher at University of Twente

1979 – Researcher at CWI, Amsterdam and Professor of Discrete Mathematics, Utrecht University

1992 – Full professor of Discrete Mathematics at Eindhoven University of Technology

2009 – Dean of the Department of Mathematics and Computer Science

*Note: thanks to Onno Boxma for his help *

June 13, 2013: AMI Springfestival - TUDelft

**Mathematics and numbers**There are two answers to the question why Gerhard Frey chose to study mathematics. One is he liked the subject at school. But then again, he also liked history. The other answer is that with mathematics you can discover things just by thinking and you don’t have to learn things by heart. So with mathematics he could be the laziest man of the world.

**Fermat’s last theorem**Peter Roquette was Gerhard’s supervisor during his PhD work in Heidelberg. “After my Diploma (master) I had the choice to go into the industry, work at a school or continue at the university. Nowadays it is quite difficult for young people to get a position at the university, but at that time it easier”, tells Gerhard. “From the 1960’s onwards until the end of the last century it was a golden age for number theory. We proved so many results . It was nice to be part of a revolution.” When we ask him why so many mathematicians were fascinated by Fermat’s last theorem, Gerhard says that he has to disappoint us. “It is a very simple question. Everyone can understand it. Still once every week I receive a letter, by someone who claims to have found a proof that only takes two pages, but it is always nonsense. But for a long time you didn’t have the power to attack it. The famous mathematician Gauss tried to solve it. He asked himself: Why should it be so important? I can think of similar questions once a week.” It took 358 years to publish the proof. Gerhard explains: “During the last 200 years people started to develop theories that would hopefully help to prove the theorem. The key word is algebraic number theory. Fermat’s last theorem was a great motivation to develop this theory. But only in the sixties and seventies of the last century theories were raised to give a method on how to solve it. You have to find the right method and structure for the problem to apply the theory . My fascination with Fermat’s last law was to find a structure that we could apply. Algebraic manipulations of formulas gave no way to solve the equation. We took a totally new approach and attached Galois representations to Fermat's Last Theorem. This turned out to be an important step in the final proof by Andrew Wiles.”

**The importance of conferences**Gerhard’s career changed from the theoretical side to the more applied side, when he started to work for the Institute for Experimental Mathematics in 1990 in Essen. There were four chairs. Gerhard had the chair for number theory. The other chairs were on group theory, computer science and digital communication. “It was a big luck to work with professor Han Vinck. He is an engineer with a great background in mathematics. We became interested in coding theory. In one conference we held in Essen somebody spoke about cryptography and a participant asked questions about public key cryptography. We saw immediately that we could apply the whole machinery we used for solving Fermat’s last theorem to this question. Again we found the structure that was behind and we understood that our expertise could help to provide useful theory . So the next 15 years I led a double life, working both in theoretical and in applied mathematics. We had the good luck to attract many very clever young people who wished to work in this new area of cryptography.”

**Answers in your sleep**Gerhard emphasizes the role of intuition in arriving to solutions for difficult problems. “If you speak to a mathematician, you will often hear: I have the feeling that this can’t be true. Experience is of course very important to get this intuition. I often got an answer to my work problems during sleep, but, alas, only to find out in most cases in the morning that the answer was completely wrong! I had a thirty minute bike ride to and from work and that was a good way to think ideas through or get new ones. Input that comes from discussion with others is also very important. So during the day you work on calculation and read papers, but the best ideas come when you’re not working!”

**Young talent**In Gerhard’s opinion there are many talented people in high schools, and universities offer good personal contact with the students to develop these talents. “The hard thing is the PhD thesis, which will determine whether you are more suited for a career in industry or in research.” Gerhard sees a big pool of talent both male and female, but thinks we should do more to give them more security. “It is a rat race. People have to communicate about their ideas in papers and at conferences, but at the same time there is a lot of competition. They often have a contract for 2 years and already have to apply for a new position after one year.” He thinks it’s a shame that till now there are very few women with a chair in mathematics, whereas in Germany the student numbers are 50 percent male and 50 percent female. This postdoc period is difficult. The woman often follows the man and it is also the time to start a family. “We should give them more support.”

**No stress**Gerhard now enjoys the fact that he no longer has the stress of having a chair and the responsibility for the scientific career of young people. “I still write papers, go to conferences and have discussions with friends, but it is more relaxed now.” He looks back on a great career. “My biggest success was the attack on the Fermat theorem, that Wiles found proof for. Later we developed a method for elliptic curves that are still used in public key cryptography. A lot of the guidelines of national security agencies go back to the work we did in this area. I started out in the highest theoretical field, with the lowest possibilities for applications. Through contacts and discussions with electrical engineers, who were designing chips, we came up with new mathematical methods that have great applications in cryptography.”

**Cautious**Coming back to the question “Can Mathematics keep a secret?”, Gerhard proposes the situation that the two interviewers want to exchange a secret, with him present in the room and him hearing the exchange and knowing the method of how we exchange the secret. Public key crypto give means to build up such a secret connection. “This is an important method that is useful in everyday applications like smart cards, but also when your prime minister wants to have a telephone conversation with the president of the USA via a public net.” But there can always be leaks in the security if one is not careful. Gerhard remembers that thieves followed drivers of luxurious cars and recorded the beep that was given to open and close the door. And played that beep later to open the car door: not enough mathematics was used. So: Yes, mathematics can keep a secret, if you are cautious!