Thank you very much. It's my great pleasure. It's an honour to do this, teach much literature lecture, as was said.
And it's my great pleasure also to have been introduced by John Bull,
with whom I have worked so many times over the past years about matters of policy of science.
I'm not sure which was my last visit to Oxford, but I remember vividly giving a seminar here in 2008 on the invitation of Terry Lyons and the subject,
which since then has grown to proportions that I thought would be impossible.
Anyway, it's great to be back here, especially in this new,
beautiful building and to do this exercise in which one has to address an audience
ranging from end of high school to distinguished university professor at the same time.
So let's go on with recalling some things about mathematics now with these kind of particular science mathematicians know it, of course,
then mathematicians have to be aware and I like to remind of an event which was not unnoticed by some people a few years ago when the Wall
Street Journal decided to make a ranking of all the possible jobs on the face of the earth that they could think of and what came out first.
Of course, you may be guessing if I am quoting these statistics, it may be good news for the mathematician.
And indeed it was mathematician number one here.
Of course, they put the poor lumberjack class and some of you may know some lumberjacks and feel depressed.
But don't these kind of rankings we know is total crap.
But it's was here. The first rank, the first ranking, the first place has some truth in it for sure.
So mathematician number one and 2014, they did it again and they put again mathematician at the first place.
Now there are a company called Career Cars to introspection of what will be the jobs of the future and so on.
And they have this point that mathematician is very well adapted to a very moving future
because by definition mathematics can adapt to any kind of problem and situation,
and also a world in which is much and much more dominated by changes of and processing of information or driven through computers,
which is a way to bring mathematics to real life, so to speak.
What was the definition that our colleagues from Wall Street Journal use?
That's interesting.
Mathematician is somebody who applies mathematical theories and formulas to teach or solve problems in a business, educational or industrial climate.
No, that's good. In particular, the fact that they use the word solve problems because mathematics was invented to solve problems,
was created or discovered to solve problems, and has been doing so for many, many years, for thousands of years,
arguably and still there is something missing for the definition to be complete because of course,
mathematician is also somebody who creates new mathematics,
because mathematics is not just something which is given and that you apply,
it is also something which is in constant evolution, continuous evolution.
It's good to remind that round the earth now there are maybe a couple of hundreds
of thousands of mathematicians and maybe as many theorems proven per year.
It's a very rapidly evolving field, also driven by interaction with other sciences and driven by interaction with the technology.
Or more and more.
It has been estimated a few years ago that mathematical research was accounting for 16% of the gross value of gross value added in the UK economy.
There was a study quantifying this. When the results arrived, many mathematicians said, Wow, this, how did they do this study?
And so then in the end was not so bad to the the way they handled the process and so on,
which is very complicated because we have all kind of indirect effects.
Was not so bad actually in France we are doing the same study nowadays as usual will see the different terminology,
different procedure and so on, make it different results.
But one thing for sure whatever the precise result, mathematics is playing a very important role in today's economy, in today's technology.
And this role is increasing and increasing year after year.
And also people are getting it now, for instance, in culture, in the data industry and so on.
I remember a few years ago a big conference in San Diego with a joint mathematical meeting.
In one of the in one of the sessions, there was the director of research of some big studio making movies, some major.
And he was saying to an audience of delighted mathematicians, we should pay you guys royalties for every blockbuster which is out in Hollywood,
given the amount of mathematical research that to use all the equations, new equations,
numerical schemes and so on, it plays a more and more important role in the movie industry also.
Well speaking, I am noticing that the power is not here.
So there may be. If we don't do something, it might turn off the middle of the talk.
Hold on. It's just talking. It's just a fragile contact.
Okay. So the technological problem.
Yes, one important think about mathematics also, and that the reason on this slide, which I will not go through every detail, is very impressionistic.
There are figures from the past and also figures from the distant birth past, because we also live with them.
Every mathematician knows we are friends with the mathematician from the past using their ideas and they are still there.
For instance, I put full year, I put during the year there is Shannon, there is LA Plus and so on.
One of the most important theorems in mathematics is the central limit theorem.
It says Whenever you are looking at an experiment, material is repeated a large number of times, a random experiment repeated many, many times.
And you want to know which is the deviation of the observed statistics, respect to the theoretical statistics,
respect to the mean this fluctuations are gushing with a large probability of being small and the small probability of being large.
And the first proof of this was by Laplace, beginning of the 19th century.
And this thing is still an extremely important theorem, still used in many cases, and so on.
And conversely, every all of these things,
all of these contributions get enriched and get new developments in the tides of current world and current technology.
Let's take one example, just one for years, and that is so here comes full year, but the same time as the plus four.
He worked on the full year transform and full year analysis, of which the two basic equations are the ones appearing on the left of for you here.
It's written here in very compact form. Just suffice to know that the theory is a theory on which you can write books of hundreds of pages,
but also which can be summarised in a very small number of equations in a very compact form, and still is extremely useful, extremely applied.
And that's the strength of mathematics, because it is based on abstraction.
It can be applied to a number of concrete situations.
It also draws inspiration from a number of concrete situations.
So we have this kind of branching the real world going in many different ways to the abstract concept.
I considered going back in many different ways to the real world. The full technique of looking,
which are which are the frequencies in the given signal were first applied by Fujii to study the evolution of the heat in the material.
So you had a piece of metal, you are heating some place.
You want to know how the heat would propagate, how it would cooled down and so on.
And it has been used to so many different things now we use for geneticists to study the songs of birds because it's a signal,
even though so different from the signal for you we use for geneticists, for images.
This is famous picture of beautiful Lena used in all studies of image processing and this is the beautiful fully transform of Lena.
And this is used as a way to benchmark, to test,
to communicate for geneticists is used worldwide whenever there is need of analysing and that is in the variations of the signal in the image.
And it's used also in medical devices like this when you are having a scanner are amending and
it's basically doing a free transform of your body and that's how you get some information.
Then you have to play the full year formulas to invert the fully transform and get back to the information here.
How to get back from the fully transformed to Lena. That's a mathematical problem.
How to get back from your fully transform to the images of your your organs.
That's also a problem. And this gets mixed with all kinds of new techniques.
Of course, at the last, the International Congress of Mathematicians in World, there was plenary talk by Immanuel Kant,
this French mathematician working in Stanford,
specialist of what is called as various techniques related to machine learning, artificial intelligence, so to speak.
And the sparsity here.
And he explain how by working with people from some hospital humanas trying to write mathematical techniques based on free transform analysis,
he was able to divide by eight.
The time that takes somebody to be in the scanner to get the information collected with important consequences,
in particular when it comes to curing children.
And this is, of course, the blending of old phooey techniques with new techniques coming in this case from mixture of statistics,
geometry, whatever, in one of these experiments, in one of these other parts of this of his talking can,
this was describing as results of work he had done in collaboration with 3000 other people in which from a fully transform image of some organism,
somebody you delete 98% of the information and still able to reconstruct it to such accuracy
that you cannot distinguish the original and the modification with your bare eyes.
So these are techniques that show how you can blend the old and the new mathematics in this and
also show that it is constantly renewed to adapt to new constraints and goods and technologies.
So that's for commenting on the status of the mathematical knowledge and the mathematical science.
Now, this so far was a kind of rosy picture, but the life of a mathematician is not so rosy.
It is suffering every day and failure every day.
I remember very well our our common friend thinking of John Craig Evans from Berkeley saying
exactly these words every day is a failure because you are searching and it doesn't work.
It doesn't work and it doesn't work. And still, in the end, when you walk, when you look back, say, ten year period of ten years,
you're amazed to see how much has been done and how much progress has been made.
Let's because mathematics as any science is globally cooperative.
And so there's a lot and a lot of progress done very fast through the joint efforts of everybody.
But then the scale of the individual, it's permanent suffering, so to speak.
Well, still, it is quite thrilling. If you do not have the suffering, then you are not happy to win.
So here we are in the way that the typical way of working, thinking and.
That's finding and waiting for plane carriage to whisper some height in our ears, you know, how will this occur?
Sometimes there are these experiments in which the solution seems to be coming.
Who knows from where? Ramanujan used to say that there was some goddess whispering to his ear during sleep or something.
I personally experienced a couple of wonderful, wonderful things.
One is I said in the book that they will describe in more detail later.
So imagine that I was it was in Princeton and there I was some evening working alone, trying to fix some proof that doesn't work.
I had already announced the result publicly, so I was under monster pressure to fix the damn proof.
And there was this problem. Well, this problem comes from.
I have to fix this one. The M.
2 a.m. Three M nothing occurs. No solution.
I in the end gets go to bed completely desperate.
And in the morning just wake. What is it?
And then there is in my in my in my brain really the voicing put the second term on the other side and take for you transform.
And that was it. That was the beginning of the solution.
So powerful is the unconscious brain, you know, continuing to work even through your thinking you have abandoned.
Where does this come from? That's key problem.
And we would pay a lot to know how to trigger this.
Now, even when you have the good idea comes another problem, which is convince your colleagues.
Books of both history of science are full of stories, tragic stories like Galois or able, for instance,
of scientists who did not manage to convince their peers that they were looking in the right direction.
This one is a symbol of something even more compelling.
This is Ignaz Zimmer was the first one to understand the importance of cleaning for medical doctors.
And, you know, if you're a medical doctor and you are doing a dissection of a corpse,
and just later that day you are helping some woman to give birth, it's good to wash hands in the in between.
Well, it looks like so obvious now, but it was not that the time would be tremendous death rates in the hospitals.
And this guy just became crazy, literally trying and not managing to convince his colleagues about this, doing experiments, whatever.
So this is an example, of course, science in the end,
correct the mistakes and people understood it was all this movement about bacteria, about hygiene and so on.
But it may be longer than at the scale of an individual and scientists who practice are not so much better than other people for changing ideas.
Science is very good at changing theory, but scientists often get used to a theory and fall in love with it and are very
reluctant when somebody tells them no know that's not the right way anyway.
So convincing the peers is another of these obstacles.
And then another thing that occurs to you.
Another you will not see plague, but disease is the problem of people asking you to tell the future.
It can be jollies. Can you profess to villain what will be mathematics in 2015?
Wow. What the [INAUDIBLE] do I know? I don't answer this on the on the television, but sometimes I answer with this.
Could my point, Caryn, before I read the could by Poincaré,
let me say that it may be annoying when you have a journalist asking you to do this predicting job,
but it's also annoying when it's a government or institution,
for instance, asking you to justify for the use of the money that you receive and predicting what results you can prove in one year,
two years, three years, four years, and so on. What do we know?
Most of the good results will rise without you forcing it by surprise, and you should never try to try and predict too much.
So here's his point, Kerry. Somebody some journalist is asking him now we are in 1900s what is the what will be the science of the 20th century?
And Kerry answers, Monsieur, your letter is coming to me, arriving to me after some time.
If in 1801. Had asked to.
Any scientist what would be science of the 19th century?
How much nonsense he would have said good. Heaven.
This thought forbids me to answer.
I think that surprising results shall be obtained.
And that's precisely the reason why I cannot say anything about them.
Because if I could foresee them. What surprise would there remain in them?
So please excuse my silence. So this spring, a record they said they were with used, saying that you can never predict the future of the science.
Now if you think a bit more and that will be the subject of the talk,
I think that surprising results shall be obtained is more than just playing with the concepts and words.
It is also asserting belief in the fact that there will always be inventive people
discovering new things that they are teachers their masters have not predicted.
And that point is the fundamental problem of higher education.
Too many people in the audience, it is obvious. But let's we are said this the important thing of education and the problem of master and student
is to train the student in such a way that he can have new ideas that the master did not have.
So and I put on these slides some examples of people who, with just a few ideas,
revolutionised story of 20th century and people that Poincaré could never have imagined the contributions
or people who are emblematic of all these things which happened in the world of ideas of 20th century.
I also put the picture of a university here.
I'm sorry, I should have put Oxford when coming here, but I put a picture of famous university to remind of what they just said.
What makes the value of the University of Ohio institution of something is not the fact that students there
are told the best science of the day is the fact that the students will one day go out and do science,
which is better than what they were taught that the principal problem, the most important and nobody really knows how to do this,
but still we have to solve this problem here in the right it start with him.
Alan Turing as we know it as the Hollywood movie around him.
And Turing is remarkable and exemplifies first power of ideas, of course.
And as we know, one of the main fathers of computer theory, but also how much it can change history.
We know there was his role in Second World War.
He was involved in the deciphering of the Enigma code, in particular the one for submarines.
And when you read days and accounts of story there, the role of Turing and so on,
it is clear first that there would not have been an efficient solution to solve this without his particular input.
And second, that without this, the story of Second World War could have been completely different.
In 1944, the Normandy event would not have been possible.
The war would have lasted probably at least two more years.
Anything was possible. It's amazing to think that during the four last years of the of the war,
the Allied Coalition was able to read all the messages of the Germans, and still it was so darn difficult.
So let's imagine what it would have been without that single idea.
The to the left of Turing is Paul Elder, the most prolific mathematician of the 20th century,
also emblematic of the fact that ideas and creativity to such a geopolitical issue, social issue.
And there was a tremendous event also during Second World War, a huge, huge migration of ideas of the idea ecosystem from Europe to America,
with all these very bright Jews who were expelled and their friends and laboratories coming and so on,
some of them extremely dedicated, some of them changing the whole of science.
If this was one of these and as according to the facts we know of his biography, there was this guy with no home, no family,
apart from his mother, no possessions, no car, no job, nothing but a suitcase with a few belongings and ideas, ideas, ideas.
And there were many like this during the. Wondering to have been changing the face of the world ecosystem of ideas during the 20th century.
To the left of FDR, she's Leo Szilard, not so well known, but the key person.
Also, during the same period, Silla was the first human being to have the idea of a nuclear chain reaction.
And he told how this occurred.
This was in London. You had read the account of a lecture by Rutherford, the father of atomic physics, the father of nuclear physics.
And Rutherford was saying something like, Yes, there is some energy in the atom, but it's so tiny.
Anybody who would want to extract something from it would be talking moonshine.
I'd be a dreamer and so on. And Szilard was very annoyed by this and just by contradiction spirit.
He wanted to prove him wrong. So thinking, thinking, thinking.
No solution, no solution. And one day crossing the street and staring at some light, whatever green light becomes red.
He was the idea chain reaction and he had the idea of the principle.
And so this was 1933 and see how how this could could change things.
Just at the time he had the he had the feeling he knew that German's physics, the most advanced of the time,
was the best position to devise this weapon and so frightened that the idea would get into the wrong.
This is the patent which Fermi and Szilard made for nuclear reactor.
So Szilard also was the guy who wrote the letter for Einstein, the letter which was sent to the President of the United States,
and in the end was the start of a program that involved more than 100,000 people.
Just one idea at the beginning, and here we see on these examples, the power of ideas can be just one brain with the right configuration,
and then it is multiplied by the power of collaboration in technology projects and scientific projects.
So here are some of these people were in there is this element in some of the for instance, in the Szilard story, which I also already told,
which is the fact that it's unpredictable when the illumination will come in science and research,
we all notice the small illuminations, the big illuminations, all rippling carrying.
The famous text wrote about some of his illuminations for which he was famous.
In one of these stories he says he tells about like more emitted peaceful mashup here.
That is when he puts the foot on the stairs. That is to come back in some bus.
Then comes the elimination. And there was nothing related to the to the problem he was striving to solve in this bus.
But just the idea was there as he was doing something completely different.
Here is another passage of the same point. Read text.
Disgusted by my failure, I went to spend a few days on the seaside thinking something completely different.
One day while walking on the cliff.
The idea came to me always with the same features very brief, sudden and immediate certainty that arithmetic trends,
forms of indefinite ternary quadratic forms are identical to those of non-Euclidean geometries.
A few remarks on this. So this is part of a long text about it.
First, even for people in the audience who don't know about quadratic forms, they see that there is nothing comparable to walking on the cliff.
So in some popular representation, there are these ideas of the eureka moment coming, like when you have this guy Archimedes in his bath.
And then the idea for the for the Archimedes thrust on Newton.
The apple in this legend. And then the idea for gravitation and so on.
And the idea being triggered by something which has some relation with the problem.
These are usually legends. Here it is completely different.
There is no relation whatsoever between the grief and the illumination just taking you by surprise.
And that's often the case. The second is that Frank insists on the fact that he's further disgusted by his figure,
which means long period of working hard on the problem, then making a break, and then the rosy illumination.
The process described by Poincare is not pure work, not pure illumination.
It's alternating between both. And that's also typical, I guess, by the way, I see there are some young people listening.
If someday, you know, you have this big math exam to prepare and you need to go and meet your friends at the pub some evening,
you can tell your parents you're playing the point. I remember to think that.
And maybe it will work fine, especially if you say it was told to you by the French professor anyway.
So the final remark on the text is that here, when Kerry put the big words without explanation and he says in the text,
they will be technical words, but don't worry, you don't need to understand them.
Only the circumstances matter. And I think that's one of the reasons why the text was successful if plan Kerry had tried to expand it.
Okay, let's do it slowly. We'll explain to you what is required to form that.
A ternary cratic form. What is it? Indefinite. I will explain you.
What is the new code in geometry and so on? So much of the audience would have been lost even if Kerry was very pedagogical,
and the important fact, which is the elimination woman, would be buried in those explanations.
So the fact that Poincaré put the technical word without explanation, I think was important in the success of this writing.
And I did the same in my book to identify Living Theorem before the theorem, actually, without being aware of it, is another instance of these.
I knew the Poincare text, but while writing the book I did not realise he was applying to larger scale the same technique as Poincare.
It looked natural to me, you know.
We also knew this in mathematics very often that there is some guy coming and telling us, I have this great idea, let's do this and this.
And you tell him, yes, that's what I explained to you when?
Months ago, because you hear something and then you appropriate it to itself.
So for the record, by the way, when I wrote the book, the my original title was this Don't You?
Hammer and French publisher were not so happy.
They said, it's too explicit. Let's do something bit more mysterious.
Anyway, we had a discussion and they came out with this one and he said, That's good.
And then it was to the English publishers, and the English publisher said, Oh, to him, you know, you know, it's a bit mysterious.
We don't understand. And so they came back and said, Why not before the CRM?
They said, okay, let's do it. But the subtle details are always interesting when you go in the translations.
Anyway, in this book it's exactly the same.
I focussed on the circumstances. Nowadays there are very fine books about explaining about mathematics.
What is a science developed like? What's a theory, doing the history of the subject and so on.
But here I wanted something complementary with the daily life of a mathematician in his struggle, in the emotions.
And actually it was not my idea initially. It was because I met that other publisher.
This was in some dinner. This was before the feed's middle.
And the guy was saying, okay, why don't we try and publish a book together and so on.
Okay, I have ideas. I could write a book on entropy.
I am supposed to be one of the world experts. There are all these. It's fascinating, it's important and so on.
He didn't care the least, and the only thing he cared was, But what is it to be a mathematician?
What do you do? What do you seeing? How do you work? And so on. And so I was embarrassed how to describe this.
And then I thought after we spend part of our time explaining kids and so on, that science is an adventure.
So let's write it as a novel, as an adventure.
So I took a theorem of mine, a theorem charged with lot of emotion, theorem, which had been a difficult story.
And I wrote how it was the genesis of the theorem like adventure novel.
But everything in there was true. And so that's what is behind the singularity.
When we talk of a theorem, it's after publication and one's the theorem is ready to walk and to be used.
But all the process that is before that the life of the researcher, the mathematician, goes behind the scenes.
And that's a long process and how you're going to look for it.
What will you be trying to prove? This also is important and this is part of what you learn with the experience.
So the first chapter of the book is, I like to say, is like for conviction as a discussion between me and my former student,
Clement Wolf, and the main collaborator at the time.
And we are discussing about some problem and in the discussion because he remembers some conversation from two years before with somebody.
I remember another conversation from somebody else from last year.
We put this together and said, but this maybe has to do with the land of dumping.
It was completely not our subject initially.
It was not what we wanted to work on was different.
But then we said, okay, let's try. It was coincidence.
Let's try and go for it. And then so long story first decide which is a statement.
We're going to prove how it's related to physics, how are we going to change it, and so on.
Two years and a half in various places,
large part of it why I'm in Princeton and he's in Paris and communicating by email many times in which we are making mistakes occurring many,
several quality at several times in which there is conversation with somebody that takes us out of the predicament and so on.
And so I told this all in impressionistic way, like this difficulty,
the difficulty how we solved it and so on, and also with all the environment, how we do it.
So there is a description of that night that you are spending alone with your cup of tea and working and working endless computations.
But there's also that moment in which you are discussing with your colleague and the colleague doesn't agree with you,
and this would give you a new idea.
Or there is a moment in which your first attempt of publication gets rejected and you are reading the rejection letter.
And so suicide. Or there's the moment in which you are putting everything together and you are so excited and so forth.
That's the life that the real life of a mathematician and as the lives of all other human beings, it's full of emotions.
Also, I put the formulas straight and as they were in the language as it is.
So the object, which is the theorem, which is the hero of the book because we are witnessing the way it develops into something organised
start from chaos becomes organised little by little in such a way that you can publish it in the end.
And so we are witnessing the various stages with extracts of the paper, with scribblings, with formulas and so on, and it was a bit of a gamble.
There is this famous quote by Hawking, Stephen Hawking saying.
Well, actually, advice from his editor don't put any formulas.
Each formula will divide by two the number of readers. If the rule was really true, there would not have been a single reader for the book.
Maybe a few molecules of reader someplace.
Given the amount of formulas which I put, some readers told me to spend as much time reading the formulas as reading the text.
Just contemplating, by the way, that the first one of the first book I am aware of,
which has been entirely for Bruno Jensen, 30 Typesetting in Tech, maybe a little.
Did you write books in tech? No. So this was this was quite an experiment.
Anyway, what is the story which I told before that let me mention that near the publication of the book, in which, as I said, the hero is the theorem.
I became aware of this amazing book by Francis before.
I guess he's a Cambridge scholar read plenty in which, in his own words, the hero is an idea and that ideas mathematical pontification of economy.
And we see through it the story of the Soviet economy from the boom of the fifties to the collapse of the nineties.
Let's see here.
And here it is told in such a way that we see the idea interacting, the abstract idea interacting in many ways with the humans and the times.
It works beautifully at times. It's a disaster in control of each one of my personal heroes.
It goes to the first hero of the of the book and the hero.
So there was this idea that there is this idea and see it's interference with the world of humans.
No. A little bit. Just a little bit of mathematics.
What is it that we proved, by the way, which I learned?
Which time should I aim at stopping? Okay.
Super. So in the book, we are working on the premise of mathematical physics and problems related to plasma physics.
Problem is about stability. Problem is about relaxation.
Stability is such an important property in the world.
And we are we have every kind of stability arises around us.
But the most important are the ones coming from friction.
We learn in school here about the rules of Galileo that if there is a ball and you kick this ball, we go on forever straight with constant speed.
Of course you do the experiment cross and of course the ball starts after a few metres.
So what you are being taught in school is false? Well, it's because of the friction.
And friction is mostly powerful force for stability.
Friction is related to irreversible phenomena.
Phenomena in which something is dissipated like you are hitting something in the heat is cooling down and
all kinds of phenomena in which there is something which doesn't go very perfectly or not perfectly at all.
Something is lost. Some information is lost. Disorder is increasing.
Entropy is increasing. As a physicist see at all scale, there is another powerful force of stability.
The fact that we are all feeling the extraction of the earth, keeping us down in place when by the end we saw it when there was this.
What was it like? Landed on this asteroid some time ago.
Almost no gravity made it so unstable, slumped at crazy distances from the asteroid.
But now in gas, there is some analogue of the friction.
There is this phenomena of increase of entropy in which I worked and many people worked a theoretical or practical level.
But the plasma is like a miraculous, perfect, perfect state of matter in which there is no friction and nothing irreversible, almost nothing.
Plasma is when you have imposed such conditions that the electrons have been ripped away from the nuclei.
These electrons, they interact with the good old rules like they are negatively charged.
So they repair each other.
They are very light, so they move very quickly and it it's important, technologically speaking and theoretically to understand how plasma works.
So it's a problem which is grounded in physics. And even though there is no entropy increase,
Landau in the forties suggested that there was some stability mechanism there and he argued using the Vlasov equation,
which is at the basis of plasma physics. So where a physicist can see all these beautiful different states for plasma mathematician,
we see that primarily it's this beautiful equation and that the equation you have to work on to understand this stability.
It's the same equation, by the way, which is used in galactic dynamics and also in certain instances of true dynamics.
So the same problems of stability arise in these areas.
And to convince people not to use mathematical reasoning and the good old for you transform in particular.
And most of all he use the linear approximation equation is not linear,
but simplify it and turn it into a so-called linear equation in which the response is always proportional to the impact, roughly speaking.
And is it true that the Landau reasoning applies to the real world, to the real plasma?
This was not clear. Even there were arguments to suggest that this was force and there was some debate, some fixing.
Yes, the no mathematicians are like and this is this problem particularly that we solved with claim on showing that yes,
the result the reasoning of Londo does work for the nonlinear equation also.
But then this was quite more difficult in the linear than the linear case, where the linear case is couple of pages of computation.
The nonlinear took us something like 150 page of reasoning and proofs and, and competition and most of all demanded new ideas.
So what is it that the mathematician can do with physics problems, even with physics problems which are known experimentally to hold,
is to suggest to new interpretation and to make connection connections between various fields.
These are one of the most powerful things. Mathematicians can finding the abstract links and in particular and this was a great
surprise because neither Kremen nor I had any suspicion of these connections.
We find that the stability problem of the plasma by Londo was related to the stability problem of the solar system,
in particular in the version suggested by Kolmogorov in the 1950s.
And the paradox, according to which sometimes something like a solar system behaves very well,
like planets will never collide, even though there is no physical rule to force them to behave well.
And also was related to a famous experiment done by Muhlenberg at Al-Quds, the plasma eco beautiful experiment.
You send some pulse electric poles in the plasma, so there will be some electric field created by the electric pulse.
And then you let the electric field behave as it wishes.
It will be done spontaneously,
after which time you send the second electric pulse at a different frequency and let it behave and vanish and then do nothing.
And you will measure a spontaneous response of the plasma one.
It was quiet. This is true that plasmas are slightly weird objects.
It also shows that plasma reacts with delay to some stimuli.
And this delay, we found, was crucial to explain why it still worked in the nonlinear world.
So the connection between these three paradoxical statements, the contributions of long, dark and forgotten man bed was a crucial insight of us.
And I told you the story in a few minutes is plain words.
This before that the world two years and a half of struggle and efforts and coincidences.
The relation with Kolmogorov, by the way,
the first one who suggested this was my colleague Images du Mehta from Lille after hearing me discussing with some
colleague about related facts and the reason why he had this idea that there was a relation was totally wrong.
I mean, he saw some pictures on my on the desk.
He said, Yeah, this reminds me of the chronograph thing. But the drawing that they had done was precisely the case in which did not work.
See these kind of coincidences. But in research it's important to be able to recognise when you have a piece of luck.
And so I thought, okay, let's see if we can understand this took me one year to understand just the link and so on.
Let's conclude by. Commenting on the list of the ingredients.
Basically, everything I took so far was just the inspiration is important.
Yes, it's that makes all the difference. But who knows how it comes?
We don't know how the solution come to you,
but we know the ingredients that are important and they will be seven ingredients which are so important in this job and these seven ingredients.
We all have an important place in the tale. The first is documentation.
Documentation can be the all libraries can be the virtual libraries and so on.
But there is no such thing as a mathematician working from scratch without any knowledge of the past.
I put the bonus formula here because it was detailed, but something typical of the New Deal posed by the actual information at some point in the work.
We need a formula for accurate derivatives of the composition of functions.
And I think, gosh, I think there's a formula.
Maybe 15 years ago I had something like this in my studies.
No idea what the formula was called. No idea where to look for it, whatever.
But then with search engine in a few seconds and the right keywords get the formula, the history, the range of application, anything you want.
So this is a new idea in the sense that getting or finding the information, when you have the idea, what you're looking for is not a problem.
But still you need to have the right ideas and it changes the way we we live with this.
But still, these libraries, all these results of experiments and so on are so important.
Constructions from the first the second ingredient is probably the most important, but is the most elusive.
It's the motivation. Nobody knows how people are motivated.
Sometimes it's true that they have this inspiration from a book that they read when they are a child.
Here I put a picture of Donald in Magic Land.
I was watching this when I was a little kid. I saw it much later.
I thought, Wow, is this naive? But still, I found it so fascinating.
Maybe this had a great influence in me becoming mathematician later.
Sometimes it's a teacher. It is not that all the career of Alan Turing was inspired from the book he read when he was ten years old.
Natural Wonders Every Child Should Know. So this is the power of impressing, impregnating the brain of young children sometimes.
And all countries in developed or developed countries currently are striving for this because they see motivation,
lack of motivation as the main threat on science.
Nowadays, if not enough young people do it, then who is going to make science?
And that's why many people are working beautiful books of popularising science and also some pedagogical experiments are being conducted.
Here's one blackout on bees done in the den in London with the with the researcher working with kids, ten year old children,
8 to 10 year old children from school on some scientific experiment which was published
in Biology Letters on animal behaviour showing that you can do science even when your
nine year old principal finding we discovered that bumblebees can use a combination of
colour and spatial relationships in deciding which colour from what to forage from.
We also discovered that science is cool and fun because you get to do stuff that no one has ever done before is cute, right?
Usually it is said that the principal findings should be something completely new, but on this particular example, it's a principal finding.
It's important that people discover and rediscover and motivation.
When you think of when you have a big achievement to make monster obstacles to solve a scientist.
If you don't have enormous motivation, you just don't do it.
The third is the ecosystem, the environment.
Whenever in story of mankind, you look for the places where innovations are made and that sometime, you know,
Persepolis, the most innovative city in the world, sometimes it's Paris, sometimes London, sometimes it's Budapest.
Whatever the it's always an environment.
It's not single individuals. And that's your problem.
Also,
when you are directing an Institute of Research I'm been director of Security Plenary for five years and have no problems to maintain the ecosystem.
You are within an ecosystem, in this case Paris higher education ecosystem,
but you're maintaining the small ecosystem inside so that the conversation and so on will trigger the new direction of research.
As it triggered the start of the project I was talking about and learned out the MP.
The next ingredient is the exchange when the birth of the new project is mainly ecosystem mixing.
But when you are started your. Project and you are working on it.
It may be that it's two individuals or three individuals, a small number, and then the communication between them can be a matter of vital importance.
In my case was. Most of it was done by email as we were on the two sides of the Atlantic with emails every day.
And in the end, it's like you have these two brains working together, and it's more than just the sum of the two.
It's like a localised brain.
Tim Gowers made these extraordinary experiments putting must project in which
hundreds of mathematicians collaborate together on a single on a single problem.
Is this a direction of research for the future?
Future way to do research? Is it an experiment which also is interesting in itself, in the way that decision process is organised and so on?
Who knows? But for sure it is interesting and shows new possibilities from communication by technologies.
The next ingredient which is so important, is constraint.
So it looks like paradoxical to see that constraints are important to creativity and research, but it soudan important in mathematics.
We live with this constant constraint of having something totally rigorous, and if there is one mistake, some praise, the whole proof will fall down.
And we need, of course, the rigour and so on. But it's not.
And it's no accident that mathematics is also a feeling which creativity so extraordinarily valued and praised.
To take an analogy, poetry is a form of art in which constraints are monster in all kinds of of cultures,
and still in which creativity and imagination is too much valued.
Here are two examples coming from other arts.
One example from music in which a crazy constraint imposed on himself by Ligeti to write a short piece in which the only key,
the only note is a path from the very last one forces him to do all kinds of variations and the rhythm and the on the strength and so on.
You don't get bored at all. But by listening to this piece, which is only made of A's, and the other one is this famous book by of Perec,
in which there is no each letter e has disappeared, even though this is a thick novel.
And of course the language in there doesn't resemble that of any other book written in French language, very particular.
And you have to find all these creative solutions to to go around the constraint.
Next ingredient are we talked about but let's recall it explicitly is a mixture of hard work,
systematic and elimination coming at some point which Poincaré described so well.
And final ingredient is tenacity because it's always almost always wrong.
And you throw your efforts in the in the dustbin.
With tenacity comes luck. We come together and you need some luck to do the good things.
Most importantly, to be able to recognise when you do have some luck.
So this makes a lot of ingredients and that's why it's very precious.
These ideas when they come into shape, it's cute this nice you maybe some people know this electronic drawing and I carrot.
This is what you are taught at school about how things work. Well, let's make the hypothesis, experiment, conclusion, whatever.
Think it's not the way it works in practice.
Practice is like this and you won't always find that someone did do science instrument breaks not expect the amazing results turn out to be crap.
What the [INAUDIBLE] is going on? Thinking yes.
No. Yeah, that's funny. This makes sense. They figured this out 50 years ago and then realise again and again.
So here the cartoonist made this great job of describing the experiment, the experience of one scientist.
But to get a more accurate picture, multiply this by the number of scientists and make all kinds of intertwining
together because these are all these stories or sort of interactions together.
It's a very chaotic and confused process from which has to come out something very rigorous and very sound.
That's tricky and that's the beauty of it. And two quotes to conclude about the nature of ideas,
how precious they are and what we should regard that we should consider them first is by Jefferson.
Jefferson is so useful because whenever you have an argument about intellectual property with some American,
because he's a founding father, you are quoting Jefferson and he can say nothing against this.
So what those deficiencies, if nature has made any seem less susceptible than all others of exclusive property,
it is the action of the thinking power called an idea which an individual may exclusively possess.
As long as he keeps it to himself.
But the moment it is divulged, it forces itself into the position of everyone, and the receiver cannot dispossess himself of it.
Its peculiar character, too, is that no one possesses the less because every other possesses the whole of it.
He who we see tonight here from me receives instruction himself without listening mind,
as he who likes his tapered mind receives light without darkening me.
In those days, politicians did speak to their own country.
This is for playing carried the last word. A thought is like a lightning strike in a long night.
But it is this lightning, which is everything.
And with this, I will conclude. Thank you.