 So I gave the title and I'll cite you two languages. Two limit free for today. Looking at the order, I suppose it's better if I speak English. Okay, I don't have one. I will speak very... I'll give only very few words about her life of God and Dick. It's not the purpose of this talk. And I just mentioned that a few days ago I received a movie which has been made by some colleagues in Toulouse about God and Dick. But in this movie there is no new document about God and Dick himself. And I recommend for people who are not aware that it's a so-called God and Dick circle site which is maintained by the Lashnevs. And it gives a lot of information about God and Dick, his document and so on. Everything which has not been sent so by God and Dick himself. I'm sorry, because I checked that website, but the website seems to be done. Did it appear? Yes. I mean I checked a few days ago because I was trying to get the standard conjecture and the website seems to be done. Okay, okay, okay. But in the secret archive of our institute there was also some document about God and Dick. Okay, so just a few words about the life of God and Dick. What? God and Dick? What was the joke? God and Dick was born on the 28th of March of 1928 in Berlin. And as a result we don't have a physical proof of his existence because all the documents disappeared at the end of the Second World War. And Dick made some difficulty with him because he was stateless and so on. So his family background, I mean his father was two names. Shapira or Shapiro, I don't know, and Tanarov. By birth he was Shapiro or Shapiro, which means that he is born in a very small, special place. All Shapiro come from a sort of geographical area, which is now a triple poem between Ukraine, Belarus and Russia. And so this Shapiro is a Jewish name and these were very, very obedient Jews and sometimes called the fool of God. And sometimes, what should we call them? What else I call them now? Hasidim or what? Okay, so his father started this career at the age of 13 or 14 in the first Russian Revolution, 1905, by enlisting in some military anarchist group in Ukraine. And they fought against the Tsar and they resisted two years after the collapse of the 1905 Revolution. They remained for two years in the forest. And finally they were all caught in 1907, something like that. And they were all sentenced to death, but Tanarov was considered too young and irresponsible, so he was not shot. He was among the few people who were not shot at the time, but he spent many years in jail. And he left jail during the First World War around 1915 or 1916 and just before the Soviet Revolution, first of all, the Menshevik Revolution, Soviet Revolution. And he was in St. Petersburg. He participated at these events, but there is some kind of confusion because there is another political figure with the same name as him, which is often confused with the father. He brought him himself in to obey the confusion. Okay, so in 1917 he participated to the revolution in St. Petersburg but when the Bolshevik war and Lenin killed all his political friends and political enemies, and so Gotenik's father, who was Shapiro Tanarov, was again sentenced to death if I'm correct. And so, as Gotenik mentioned to me sometimes, my father has been sentenced to death twice in Russia, one by the Tsar, one by Lenin. It is a rather good description of the life of his father. So his father was very active, so after that his father was very active in various revolutionary groups. He was in Berlin with Rosa Luxemburg. He was in Budapest with Belakoun. He was in many places. And I remember mentioning to him, oh, your father would deserve to be the who is who of revolution. And he did not know what was who. So I explained to him, my father would have been unhappy to be the who is who. No, I mean the who is who of revolution. And then after many events, he spent many years in Berlin, from 1920 to 1930 approximately, and then he met the Ankar Gotenik, who is the mother of Alexander Gotenik, who was a young German lady, who had already one daughter and who was, I mean, publishing in the leftist newspaper, the German leftist newspaper at the time. She was a feminist and a socialist. And then they met and they lived together for a while and Gotenik was born out of the union. But then when Hitler came, they had to leave. Of course, they were not welcome in the Nazi rule. So both father and mother, so I repeat, his mother was, well, he was called Sasha, or Alexander if you want, and his mother was called Ankar Gotenik. So Gotenik is the name of his mother. And I checked that in the north of Germany, Gotenik is a quite common name, family name. So they both left Germany around 1932 at the advent of the Nazi rule, and they went in different places. But then Gotenik was left behind. He was left behind in the foster family near Hamburg. The people who took care of Gotenik were, I mean, the head of the family was a previous Lutheran pastor, Lutheran Reverend who had left the church and who was kind of an archist and who had a private school in the manner of what was recommended by Bertrand Röthl, Bertrand Röthl at the time, was a fond of education principle. And then this private school was run along similar principle. And then a number of Jewish children were fostered in this school. So it lasted for some years, but in 1938, everything became too dangerous. This one, of course, was of course a suspicion by the Nazis and the Gestapo, and then he had to get rid of Gotenik. So he managed to get into contact with a Gotenik mother who was in Paris at the time and through the French consulate in Hamburg, that was 1938, remember, shortly before Second World War. So they managed to come into contact with his mother and Gotenik was put in training at Hamburg for 12 hours of training to Paris. With the only instruction, someone is waiting for him in Paris. So it was his mother. He hardly remember his mother at the time when his mother left him when he was four or something like that. But then he was reunited with his mother. But his mother, there was a half-sister and half-sister remained in Germany. She was in a kind of religious school, Jewish children kept by a religious school which was not too common, and she managed to survive during the Second World War. But Gotenik was reunited with his mother, but not with his father. His father as well as his mother had been participated to the Civil War in Spain. And so the mother came back, shortly before the collapse of the Republicans in Spain and the victory of the fascist. But Gotenik's father remained behind, and it is told that he was a member of the Duhuti column. Duhuti column was a small army from the anarchist in Barcelona who tried to get hold of Madrid from the fascist. They failed. And it seems that Gotenik's father was part of this small army and it would be amusing to see where the Gotenik's father and Simon they met because they were at the same time among the same political groups in Barcelona. At least if someone wants to write a play, I recommend it. It should be a good play. Okay, so Gotenik's father was as after the collapse of the Republicans in Spain, there were a number of refugees, very large number of refugees, who were not welcome in France, not at all welcome in France. And while if you look more carefully, you will see that France was already half fascist in 1939, half fascist, not really fascist, but half fascist. And Stalin worsened the situation by making this alliance visible. And so the French government had a good excuse to expand the Communists for all positions and putting them in jail and so on. And you suppose that Gotenik, they did not make much difference between Gotenik's father, who was a Russian anarchist and communist, of course. And so Gotenik's father was sent to a detention camp and this detention camp was created around 37 in France, maybe a little bit, but that's certainly 37. And a number of refugees were taken into this camp. They were not death camp like the Nazi camp, but they were not very present. And so, and Gotenik, but Gotenik was not the United, so he hardly met his father. Hardly met his father, if I'm correct. And so, but then he was with his mother and they were both taken this time, when the Second World War started, as enemy citizens. They were both German by law. We're enemy citizens. So they were put in a camp and they survived with some difficulty. It would be too long to tell what the story is, but what he's known is that he was in MAD, in MAD, and there was a camp and he was allowed to, his mother was not allowed to leave the camp, but he was allowed to go out to attend high school. And then, when things became worse and when the French collapse in June 1940, it was not much head for these people. They were for a few weeks out of jail, but put in another day immediately after that. And then Gotendick's mother survived with much difficulty in various camps. And Gotendick himself was taken as a student in a protestant private school, in the Russian Mosul, you know, which was a major place of resistance, resistance of the protestant against the Nazis. Some people commented to me 10 years later, we forged the armies of the king with a forge and Hitler was not works. That was the spirit of these people. So Gotendick managed to, at least he had rather regular teaching in this protestant high school, and it was there, I mean, it's a history lesson. But so he managed to get his baccalaureate and then after that he was a student in Montpellier and it was clear that he knew already more mathematics than most of his teachers there. And he had his teacher recommended to him that said that, well, if you have any questions, ask Mr. Lebesgue and he has all the answers. And after this recommendation, Gotendick rebuilt his own version of Lebesgue integration in a very, very general form, ignorant of everything which has been known that the textbook by Sacks existed, I mean the reason that she existed, but he was not the way of that. So he reconstructed that. And then a good stroke of luck was that a friend of Henri Cator and Andre Veil was called Magna. He was not an academic, he was part of the bureaucracy of the national education. So his task was to go around France just after the war and to spot good, very promising young student who needed support. And so he was going around in various high schools and his mission was to discover young talents and to provide them with proper fellowships. And that's what has happened with Gotendick. So he got the fellowship out of this one. And if you read now the exchange of letters between Henri Cator and Andre Veil, they are very, very fine. This is mentioned in this exchange of letters. So Gotendick completed his so-called distance in Montpellier. And then in 48 he was given this fellowship which enabled him to go to Paris. And then he went, he had a recommendation to Carton Ely, because one of his teachers in Montpellier did, well, I suppose, master's thesis or something like that with Colbert. Don't remember what was the exact data of that kind of master's thesis with Ely Cator. And so he had Gotendick a recommendation for Ely Cator. When he arrived in Paris he discovered that Ely Cator was almost died in 1951, and he was already very bad. But that's the success of Henri Cator. So in 1948, Gotendick arrived in Paris and attended a way of recommendation to Carton the father, or Carton the son, doesn't matter. And then he attended the Carton Seminar for the first year of the famous Carton Seminar, 1948 to 1949. And that was topology, and he confessed and sheathed for any confession. After that he confessed that he understood very little. But nevertheless people were impressed because in the back of the audience there was a young man very brave and very bold who asked many questions to Carton. People who have known Carton quite forward can imagine the contrast between the personality of Gotendick who was like a young dog at the time and there are pictures of him at the time. He looks actually young dog. And Carton the father, Carton the son was a very formal protestant. But finally he did not go very well but at the end of the year there was a discussion between Carton and maybe there is something in his publishers and Henri Bey said, obviously it's not proper for him to stay with you but we will send him to Nancy. At the time in Nancy there was Giorgione and Laurent Schwart and other people as well but two major, two more towering figures where Giorgione and Schwart. Schwart had just begun his career as a founding father of distribution and so Gotendick came to Nancy and the end of Giorgione is a position of the generalized love and integration and you can imagine what people who have met Giorgione can imagine. Young man, young man. This is all Lord, this is all Lord. This is all Lord. If you want something serious I can give you something serious. And he had just finished to write a paper with Laurent Schwart on the functional analysis foundation for distribution theory and at the end of the paper there were 14 major questions and you say, young man, try your teas on that. And according to the legend it took him a few weeks or a few months to solve all these problems and a few months later he had already something published. Okay, and then now I will stop for that because one of my games today is to exist on the first phase of the career of Grotendijk which is less known as a second phase and which ought to be better known but for that I have a document which is a schistématique des principaux travaux mathématiques de Grotendijk. That's a paper he wrote in one kind of CD he wrote in 1972 and you say that there is a copy in the 5th century. So, a document on how he got it but I think it was at the time he was applying for a position at Collège de France 72,000 feet. I will not tell the story which is entangled and complicated it's not my purpose today. But then Grotendijk gives his own account his own overview of his work which is a very, very interesting document. So, the first section is functional analysis and I will start with something. So, functional analysis, what was it? There was the first discovery of distributions by Laurent Schwarz and Soboleff. Some people mentioned earlier paper by Soboleff in 19, so this is around 1947 or 48 and there is a few papers by Soboleff 10 years before but there is a difference. Soboleff published very highly technical papers combined, it appeared a new method for solving the kind of partial differential equation and Soboleff shortly after that was taken into the military industrial complex of Soviet Union and he did not publish much about that because his work was a classified work. And the other end Grotendijk Schwarz, Schwarz I can admit that Schwarz never really understood the Soboleff space it's true that he never did not want to do it he did not know how to use it but the point is that Laurent Schwarz the genius of Laurent Schwarz was to create obvious seeds to take something which was there that no one saw it before and to make it a very efficient and simple tool so this solution is a very simple tool but it provided immense immense progress and I can say that for instance 10 years later, 5 years later when I was a student the group of my generation which was Breua, Malgrange and Martin and other people we all learned from our master Laurent Schwarz this new technique and during 5 years we were really ahead of everyone in various fields because we knew this new tool since the tool is easy within 5 years many people absorbed it and to became benefit of it but that was a special genius of Laurent Schwarz who made simple discovery simple definition simple discovery and to discover simple but very efficient tool so I will not enter into the discussion about making so-called efforts for physics but then, distributions get fun get fun also no, no, no there was also somebody before so-called ok I come through ok so what has happened this, well Gelfand understood immediately what he could do with that and if you compare the book published by Laurent Schwarz himself about distribution which is rather high-brow and quite formal quite, quite formal and if you compare the books published by Gelfand and his collaborator at the same time you see the difference the book by Gelfand and his collaborator are very practical they explain to you what you deal with this method in various situations they give you Fourier transform and so on at least in my opinion the best discovery of Laurent Schwarz has been a very convenient Fourier transformation the rest was more or less known but Fourier transformation is really his creation and it's such an efficient tool such a beautiful tool ok again, simple once you have it, it's obvious but you have to know it so and this will come again so ok that Laurent Schwarz and Gelfand they resulted on some abstract definition of abstract vector spaces locally convex etc and I remember a comment by Gelfand claiming that I will do, I will do distribution theory without all these first some statement of Gelfand of this kind so in one of his books but then Laurent Schwarz and Gelfand discovered that they were lacking two foundations for the general tools of functional analysis locally convex vector spaces and the outcome the final outcome was a book by Boba Key on that subject five chapters comes out four plus an additional chapter for hyper-spaces so and Giotten here was very instrumental in generalizing the basic result of of Banach to non-norm spaces Banach space is a space which has one norm and the locally convex spaces are spaces which have a family of norms that's the main difference but then there was this question and then just by for knowing one for knowing one also we know but we will not tell you the story of functional analysis Giotten has written a very very documented book about that so you should mention Wiener you should mention for Norma many people but the point is that they were asking a few questions and as I said Götendijk thought but then it's interesting that immediately after that Götendijk was interested in the problem which was connected with that and a question which was asked to him by Lohr-Schwarz Lohr-Schwarz was very fond of a theorem which is called the kernel theorem theorem of kernels which is something that every physicist will admit that if you have a functional operator you can always write it as an internal operator an operator acting on functions with suitable restriction property provided that you admit that K is not an ordinary function but a generalized function for instance Dirac knew that it is formula so the identity operator certainly not an integral operator but if we invent a generalized function like the Dirac function which is properly understood as a distribution issue so every operator can be represented as an integral operator many benefits of that Schwarz himself gave a rather complicated proof asking on various explicit calculations which were not, well it was convincing but then he was not happy with his own proof and what he wanted to what he wanted to explore is the idea that what is the meaning of tensor product for an analogist is the following a function on one variable X and another function variable Y if you take the product with independent variable X and Y that's a function of two variables and it's properly represented by the tensor product it has all the properties that you expect in algebra from a tensor product so Schwarz knew that and he knew that there was a possibility of reducing his theory of kernels compared to properties of tensor product of space but the question was the question by Laurent Schwarz to Rothenlich was a rather typical one can you develop a theory of tensor product valid not in finite dimensional space as usual but for infinite dimensional space that was a program asked by Laurent Schwarz to Rothenlich and typically with Rothenlich he went very high in the sky very high in the sky that was his method to go very high in the sky he did not try to understand the explicit calculation done by Laurent Schwarz while using Fourier series was not very difficult but he did not want but he tried to invent a general theory and then to his greatest mean he came back to Schwarz he wanted to live but I have not one notion of tensor product I have two notions which which we are not the same but two notions so he was very embarrassed he was very embarrassed and Schwarz also was very embarrassed until they both discovered that the clue was to invent a framework where the two tensor product would coincide for people who are familiar algebraic homological algebra one of them was left exact the other one was right exact and then if they coincide it is exact and for what it is exact which is of course what you can expect the best that you can expect for finite dimensional spaces the tensor product is exact if you have an exact sequence you tensor it by your space it remains exact and is all the consequences so but then and in more general terms I mean when Sehr invented invented flat modules in the 50s it was exactly if it is a purple to have a tensor product which is exact on both sides ok which respect the exactness of exact sequence and so what gotenic invented is something in functionalism which has exactly the same property as flat modules in algebraic and algebraic geometry very important to so I just mentioned this because I just want to mention that there is an analytic continuation in his mind from what he did in analysis to what he did in algebra is exactly the same spirit the same method ok so he invented invented two tensor product and later on he invented even more he was not aware at the time that he invented a of from Norman by the name of I had invented Chatham had invented that develop similar ideas but the difference is that for Chatham it was operators in Hilbert space he wanted to have control of various classes of operator Hilbert space representing operators as a tensor product in the same way but then but he was not interested in modern art spaces ok so that was great and then came out of that the thesis of gotenic when Thies, when Laurent Schwarz and Deux-Denis where this thesis advisor mentioned often that when it was time for him to present the thesis he has already six papers each one which was a very good thesis something happened to me more recently was one of my PhD he had already three papers when he came to to defend his thesis but then he had six papers all were very relevant, very important but all very abstract and then finally what was chosen for his thesis is a combination of two of these papers published by the American Mathematical Society a fact book of 400 pages when I see a fact book with gotenic it's always a fact book he was writing at legs and then but then there was two parts and this is interesting to understand this way of thinking the second part is about about nuclear spaces why nuclear because of course it was motivated by kernel which is the same as nucleus so nuclear spaces and the first part was a very important problem stated by by Banach which is the following if you have a Banach space is it true that any so called compact operator completely continues in some terminology can be obtained as a limit of operator of finite tank can a certain be the limit of finite decomposition finite sum of if I add G I Y so it was a very famous problem and typically the first part of the thesis of gotenic which is certainly less relevant today that it was at a time developed 14 equivalent formulations for this concept of Banach and it was I think in what he had in mind was to invent many many many valians and at the end it was very typical of his way instead of solving one problem he put it in a very big cloud of problems and one of them is accessible one of them can be attacked so by first he put all these problems together and he knows how to go from one to the other one and he very carefully is a logical connection that you know these problems and then at the end one is easier than the other one he solved it so he solved everything but this time it failed it failed so this thesis this is a part of his thesis that I never studied very seriously I have to confess the second part from nuclear spaces I studied very carefully but not this part and what happens from years later that the conjecture was defeated on discovered counter examples first of all there was very in the 70 only in the 70 a very very complicated counter example a very complicated Banach space with a very complicated family of operators but then finally there was found a very simple solution among so if you consider what are possible Banach spaces well Hilbert space is certainly a Banach space now it is known that if you consider a Banach space if all continuous operator in a given Banach space they have a suitable norm and this is another Banach space so out of any Banach space X you can build L of X which is a center full compounded linear operator of a suitable norm take a Hilbert space very natural take the space very familiar this space contradicts the conjecture and so I think the solution is really in the space of botany but he did not he did not but so I don't remember who did that it was in the 70s was a very great progress so as you see that's part of the limitation of botany limitation of botany is major method is to go in the sky it's like an eagle sometimes people compare forms to eagle he is an eagle flying very high in the sky and then like in modern military technology at all at the time he was a stuka a German stuka flight stuka plane and so attacking from high in the sky and finding his place ok so but then the nuclear space is well I mean immediately understood by Gelfand and Gelfand took great advantage of this development in many ways and especially what Gelfand did one of the main contributions of Gelfand is to understand that there was a connection with probability field and the so-called mean law sphere which has been very very important later and especially in the program of so-called constructive quantum field theory which was very important one is a generalization of Fourier transform from finite dimensional spaces to in some infinite dimensional spaces but the nuclear spaces have the they are infinite dimension but there was a very close to finite dimension in a given nuclear space you can very well approximate the whole space by subspaces of finite dimension in increasing order and that's one of their property so Gelfand understood that and at that time there was a major problem to develop integration theory for the purpose of probability theory in infinite dimensional spaces and many people or other people contributed the Soviet school was very active and that was very successful and then that will set you the truth of goal 13 but then it's not yet the end of his career in functional analysis there was remember he was born in 28 he went to Paris in 48 at the age of 20 he defended his thesis in 53 at the age of 25 at 53 well since he was stateless because his parents his father was a Russian but was denied a Russian citizenship for political reason his mother was a German but was denied German citizenship by the Nazis for similar political reason and he was stateless so at that time it was not easy for someone who was stateless to find a position in front and he became better but at that time it was almost impossible as a result until the 70s had to travel on the so-called non-Send Passport which is a document issued by the United Nations to help people who are stateless and during the first and second world war they were due to archipelago many people remained stateless when the border changed extensively but as somewhere I learned that mentioned to me was born in a warf-warf-warf-vif then back a town which changed four or five times which is in recent days you have seen the report of what happened in the vif extreme west of Ukraine and somewhere I learned that mentioned to me that his grandmother never went out of her kitchen but changed four times of that personality she had been Polish Russian little Ukraine so yes so yeah so God and it was stateless which did not help for him and so to find a position when he had a fellowship from the French government as I mentioned before but a more common position was difficult so he went to South America and in Sao Paulo that was a town where Sao Paulo was a very active center still today still today but at the time it was a very active center and many prominent visitors like Oskar Zariski some Italian mathematician some Italian mathematician Andrei and many many important mathematicians of the time came to visit Sao Paulo and they managed a niche for Gotenik so Gotenik was hired for one or two years there and he gave a full course on his subject the nuclear topological vector spaces and this the notes for this lecture have never been the text the notes have not been they are published in memograph form but they have never been re-published in a more stable form as far as I know but when he published this I mean there was a comment by Deudonis there is no need to continue to work in this direction he has solved everything he has killed the subject so Gotenik gave this lecture in South America in Sao Paulo but he produced something more interesting he produced a new people a new people which was he produced a new paper which was called very modestly a short summary of the the theory resume a short summary of the theory of the metrical theoretical the theory of nuclear of a known species and this is a very very interesting people but which is almost forgotten except by a few people like Pizier who made this mathematical life out of this and so this is a very interesting paper very typical of Gotenik the first half is very abstract and that will be the transition of the second part for the rest he said I will discard the 14 different kinds of tensor product between Banner species that you cannot escape to mention 14 different ones well if you look carefully well of course he is very well aware of the from total property of tensor product is it exact or not and what he defines indeed is what came out after is the homological derived phantoms special phantoms between Banner species and then you take the derived phantoms in the sense of homological he did not know anything about homological but more or less at least he was totally aware of the spirit of it and so this is supposedly functional analysis but it's really the way of thinking is already very functional the name category the name of functional appears nowhere and he did not have the language to express that but he had all the tools and all the methods so that's the first part that's the first part and then the comment at the end he said after that there is two kinds of tensor product alpha prime and beta star something like that and I will show that the other thing this is my main theory this is my main theory when you look at this you doesn't see much for people who are not totally aware of all the development it's a little but if you look more carefully about this proof and so on you discover that there is a fantastic identity inequality the inequality is to take a real symmetric n by n matrix ok how do you measure the size of this matrix the size of the eigen value very simple symmetric real matrix can be put in diagonal form look at the size of the eigen value second measure take the maximum size of the element of the matrix which is not independent of the basis this one and there is a certain inequality I will not give it technically there is a certain inequality which relates the maximum suppose that Aij or element between 1 and minus 1 minus 1 and plus 1 and symmetric what can you say about the eigen value the size of the eigen values that's a point and a crucial point is that you have an identity that for given n by n matrix there is a certain identity is not difficult to show that's a standard analysis that the function on a compact space at this is maximum it's not more than that but the crucial thing is that the value found is independent of the dimension and that's a very efficient strategy to deal with problems in infinite dimensional spaces and this is the spirit of that came from probability I mean the work of Paul Levy and the Wiener I mean the limit for probability I can always be phrasing this way for given n you have a certain identity which if you are not if you don't ask many detailed questions it's easy to show that there exists a formula for certain kind and then if you are more precise then you can show that it's an element just an example if you take the n-dimensional space with the ordinary distance you can then involve I take well I take 1 and I take a sphere what is the radius around which the mass concentrates quite not difficult to show that in n-dimensional space if you take the sphere of radius square root of n so x1 squared plus xn squared is n then almost all the mass is concentrated same way if you take a sphere with high dimension if you take an equator an hyperplane most of the mass is located this is just geometry great to express the central limit for probability so this is typical if you take the radius square root of n I mean the estimate about the mass which is to be more precise in the neighbor of that is independent of n so we discover that but then so there is a very clever the very clever identity that you have to prove and what has happened is that when he finished his paper he gave me a copy of his manuscript and I wrote a report for the Boba-Kiss seminar and in the Boba-Kiss seminar there are two parts of my report I say the first one is cryptic and I say that's what he does things and I discovered a few pages what he does and so product is final and now I will give you the key and then the second part is I gave very explicit calculation with special functions and similar things Rotary was not I mean this is typical of him Rotary was not form of algebraic analytical details he always sought in large, in bold but he was not interested in very specific calculation whether number of theoretical analytical and then he was so when I wrote these people he was not very happy he said that was not what I had in mind I said I agree that is my way of looking at this and I suppose that this second part will be too more important with just a consequence of your recent I don't claim any new result of my work but finally when people develop these ideas it was exactly the kind of of an estimate it was needed but it was typical that the special function did not appeal to him well I mean the Laplace operator in general appealed to him but I would have solved it in practice harmonic functions and harmonic spherical harmonic and so on things which are very familiar to an analyst who someone doing doing a mathematical physics he was not very happy that was his way always looking very chaneled so that's the end and then as I said there was this paper was hardly well was not very, did not have much influence except that Pizier took it and developed his theory in which he is still very active and has a very good school working at that so but I doubt that Goten he could have been happy to see him so that's the end of his his work in functionalities now the next that I will try to discuss now in a few minutes is a transition how he got from these functionalities to algebraic geometry and it took place in a few years so this paper this final paper about metric properties of of a was written in 1954 and published in 1956 some signal as that at that time he was already back to France I think the roots of CNRS have changed enough to allow him for the position at CNRS at that time so already in the late 1950s he had a position at CNRS a few years before coming here because as you know this institute was created in 1958 not in this space they moved in 1962 that's the first part the first start of the IHS was 58 so Goten he managed to have a position at CNRS which was quite new there are very few foreigners to get it and he was not even a foreigner he was a stateless and then Gotenik went to France and he decided he said what it was he decided to settle on the other hand as long as his mother was alive he lived with his mother and in the kind of very close connection they were living together she died in 1958 or 1959 some similar I remember shortly before she died I mean we had a private meeting of Boba Key and he was a member of Boba Key at the time and he came with his mother came with his mother I remember there was something like 58-59 he came with his mother she was already very weak and she died shortly after that and I remember we had a walk in the mountains together we were alone gotenik and myself and he confessed to me the dream of my mother was to write novels maybe it's what I have to do maybe it's what I have to do and I already did some doubts about what was this what is this vocation I said no later on later on but not at the moment not full of beautiful ideas but I remember this discussion and then but his mother died shortly after that early 1960 something like that I don't remember fushu and psychologically it was a great change for him he had been in a sense he was isolated with his mother he was a very very very close connection I would say and so it would be another talk to another talk to pick up the variation of gotenik with women another story not for today not for today and it's clear that after the death of his wife there was a great change but then gotenik was given a position at CNNS and I remember a discussion in the spring of 504 and he commented of course he cannot make anything to recruit both of us which took a few years to realize so gotenik was he collaborated with boba key for a number of years at least six or seven years and he was very active he had a very good influence on that he was very active which we have not published but it doesn't matter we have to develop things and finally when he left boba key it was partly out of misunderstanding because he had a serious misunderstanding that he wanted to force us to start again and he said set theory is no more the true framework is categorically the point is that we all agree with it we all knew categorically at the time whether it was captain whether it was sailors or bowels we all knew the advantages of category but we have already 15 volumes published and we did not want to start again so we have to make a compromise but he was quite unhappy about that and then the other one something which is unfortunate there was a crush personal crush between Andreve and Gotenny nothing serious Andreve could be quite nasty quite sarcastic and he made a comment in some mathematical discussion Andreve made a nasty comment and Gotenny was unhappy with that so he said I left immediately the room when was it? 60 or 61 I suppose according to my recollection 61 some of 61 I suppose I can locate that with my own recollection of other things but then he stayed with us for 60 years which was a great benefit but finally we could not return but what the reason of this was that there was a peace treaty a peace treaty which was the following Boba Key would develop called commutative algebra up to that point and you will start at that point so there will be no overlap basically no overlap so all the preliminary algebra which is needed for the AGA the module geometry is already a large part of it is contained in Boba Key but without the geometrical description and it was decided that when you really begin to do geometry that is not so much easier because on both sides who wrote the books for Boba Key who wrote the book for Goten D you don't need so it was easy to arrange but then I will just mention about the transition of Goten D so the transition of Goten D is here for me so I mentioned that he had these nuclear spaces which are they take a counterpart of so-called so-called flat modules one of the properties of flat modules the definition is that if you have an exact sequence you take a tensor product with a flat module it remains exact a consequence, a rather formal consequence is a so-called Q net fuel I mean how to calculate if you have two complexes if you take the tensor product of the two complexes if you have no special property it's quite complicated what is known is that you have a certain spectral sequence which is not very easy to manipulate but then if you make some assumption of flatness it's rather easy and formal to derive the form you want if you have this complex with commrology A this complex is commrology B take the tensor product of the complexes commrology is the tensor product of the complex which is nicely useful of course, out of generality he knew that there was something known that there was something like the Q net fuel and he understood what was the core of the proof and translated it immediately using these these nuclear spaces if you take a function in one variable in one variable you multiply it by tensor product to another one and you want to calculate the so-called the round commrology it tells you that in the suitable condition the round commrology of the product of spaces is the tensor product of the commrology of the spaces and then all the formal properties what you have shown is that not only how to get abstract category of nuclear spaces which are the right front of this property but he knew that many many examples fit with that so basically all standard functions space are flat places in nuclear spaces in this sense so for use he gave a version of this and I remember there was a seminar of Laurent Schradt devoted to the work of quantum in analysis and it was my task to give the position of that so I remember it very well I have to explain to explain this in the first seminar so and then what is discovered is that at the same time Dolbo made a breakthrough he understood that there was a D-bar commrology and that is D-bar commrology but he wanted to show what is known as Crankary Lemma for D-bar that something with D-bar close it locally then Dolbo knew that and he tried repeatedly to prove it but he never succeeded but he discovered the true notion and he developed in his thesis many many consequences of this which was still a very useful tool in function and value in differential geometry for complex analytic space so it was but then he was unable to prove the basic lemma and I recently had a discussion with him and he said yes, that's true, he did not prove so by then in some carton seminars there is a proof by CER which is credited to authenticate using exactly this functional analytical tool so then he developed the D-bar commrology a tool at the time for differential geometry and topology of complex spaces and then he developed this and then of course he immediately understood what he could do and at the time there were other difficult problems of analysis connected with the work of CER, the CER duality I mean the first version was quite a difficult analysis and ultimately all law enforcement provided the right method to do so so and then ultimately it was interesting oh ok, I may not give you something it was of course the peak of the activity of the day this was a part, in Paris area it was really the thing to do the thing to do around around carton seminar and he understood that he had the tools so he published a few papers in this direction and then gradually he said ok but then one CER, the breakthrough was CER paper on Coen and Chief which shows how to mimic this method in functional analysis in many complex variables to apply them in pure action it was really the breakthrough and of course Goten Descartes he had all the tools to do that and typically again so, but there was a technical problem which is developed by Leurin by Leurin was very, had been developed by Leurin by Carton, CER and other people and Borrel also but they concentrated on final dimensional spaces at the time looking at infinite dimensional space what, calculus of variation of course it's calculus of variation but I mean the so-called direct method of calculus of variation but people are still hesitant so linear theory after Banner was well understood the non-linear theory was still full of mysteries so there was this this theory and so ok so Goten Descartes was aware of that and so there was people who had this technical difficulty of infinite dimensional space so for instance the thesis of CER consists of two parts one part which is foundation which is painful because the few of Leurin did not apply directly so they had to re-breed everything from one to re-breed everything and then the last part of course all the consequences might be the consequences about homotopygu etc so that was and then CER when he developed this method of so-called crayon sheaves had to face a new difficulty of course anyone who does reasonable analysis or topology assumes that all spaces are house north spaces not house north spaces do they really exist no one takes them seriously for a long time no one took them seriously but the great discovery of both then and then CER it was first the way to discover it is that the proper topology it was used in natural geometry was the so-called Zaris ketopology Zaris ketopology is certainly not house north not house north the neighborhood of if you have a neighborhood of the neighborhood is enormous and so if you have two points all neighbors of this point neighborhood of this point so far very far from being house north space no one of you reassures that the spaces were house north and CER I have to repeat another time to feed the screw for another kind of spaces and then of course it was a major problem anyone in the end of the fifties I know later that it was a major problem and got to solve it exactly by his ordinary way he did not try to prove that sheaf fury would extend or that under some conditions relaxing some conditions he took things from the sky he said what is needed to have a fury of sheaf we knew from algebra that to a normal logical algebra the notion of projective resolution is a very important one but to construct a projective resolution is very easy because a free module is projective and so we do not do that everywhere but if you believe in duality and for instance I don't know who would always insist if you have a diagram with hours in this side take the diagram with hours outside and so if you have a notion of projective which is very natural very easy to do now the dual notion of injective which is quite complicated but then if you wanted to apply the algebraic method to sheaf fury we knew that projective resolution would lead nowhere there are many reasons for that projective resolution would lead nowhere but what about injective resolution but injective resolution you need to build a certain class of sheafs called injective sheafs which is a certain property doesn't it well Kotenik did not try that he said I take not a given space I take the category of all sheafs what he called a topos after that it's one of his major discovery you replace the space as a category of all sheafs and you express everything in terms of the category of sheafs ok now what so next question if you want to have an injective sheaf and injective resolution what kind of general property should the category of sheafs satisfy then he developed this big paper so called torcu paper he developed this at length which is called AB 5 star whatever reason AB 5 star because there is AB 1, AB 2, AB 5 and then the area which is AB 5 and then he said ok assuming these very very general properties of a category I can do everything now end of the proof I take the specific category of sheafs and this AB 5 star transition is very easy to check no so we have many people to struggle with many particular cases to extend it over time so and this has been well we will not suppose that enough now I will just finish in a few minutes and then what he had he repeated this strategy many times many times and then every time he invented I mean he always invented a more abstract notion a new how to use it of course this has its limit this method has its limitations there is a famous story of maybe the whole net a discussion where someone asked I take a prime p and so on I suppose it was he is the only one who could do that but which prime number goten did not know 57 everyone knows that 57 is not the prime number I know all the prime number to a few hundred by arts but goten did not know and he did not have any appeal to him he never tried to do explicit calculation even with ordinary numbers a number p was a later to represent the prime with certain properties etc but well of course he knew 235 he did not know them he did not care for them he did not care for them so of course it is typical but it is typical but then goten did again when it was finished with that you have to understand the nightmare of a geometry in late 40s and 50s I myself I suppose I learned so full time the beginning of algebra and geometry first in the big book of Andreve Foundation of algebra and geometry which is not the most pleasant place to learn second I was aware of the paper by Zariski many difficulties understanding him and then which was a little better the book of Samuel and Zariski which is a very good exposition then Chevalier produced his own version Seier produced his own version and when it came to finish writing my thesis I was finished with a problem of what kind of foundation do I take because I needed some reason which belong to the foundation to the presentation by Andreve other one by Chevalier and I had to make that so I built a certain compromise in my thesis the first chapter described a certain compromise and then out of that I could use all the available tools and prove what I wanted to prove a certain duality which was a major question of Andreve which I solved in my thesis and I remember suddenly discussing about my thesis this time he said you knew the general definition of a scheme is to I knew but I was not the only one as you know when a new idea comes I mean there are a number of people who are very among each other to say whether it was whether it was Seier whether it was ultimately what doesn't matter we were all aware of what was needed and we had variance and I produced the definition by what I wanted to say what you should do in your thesis is to give the foundation so that you can say my dear no I want to finish my thesis I want to finish my thesis of course it's a compromise very unhappy compromise but at least it enables me to give the proof and I'm quite confident that the proper foundation will be given my proof will translate immediately which was of course true and then again and again at the time a major breakthrough by Grotonik was this proof of the so-called Riemann Rohe it's a good fuel and it's all very short which is very very very great process I mean when I remember it's so good commenting on that I was totally surprised but then in this yes in this proof stories are music I was in Princeton that was fall 57 I was at the Institute as a postdoc at the Institute of Finance that he spent most of his the full term at the Institute at the time and Boral was there and he was not yet there but he created this and then Grotonik sent to to say at the time I mean the tax on the air for letters of A.I so he used very simple people with simple to type how many pages together it was 40 pages so we received this week after week and then I remember I beg from from Ser Ser was away for a couple of days I beg that he would lend to me these people and that I made a better copy I made a better copy of my typewriter and then it was distributed to the older people who are interested and then it was really really great full lot of new ideas especially Kefir he was invented there but I remember that at the time when we made an exposition of this we had this seminar which was finally written by Seine and Boral I don't remember the name of Seine and Boral Kefir he said it's too abstract we do what we have to do but we try to escape it and he was right of course we were wrong he was wrong so what has happened is that but at some point it's interesting why did Grotonik refuse to put his name on the paper because he was unhappy about a part of his proof in a part of his proof he was doing a certain explicit construction of algebraic geometry which we call now blow up which is a very important method but at the time it looked like a doctor and then Grotonik used this tool and he was unhappy no no no my proof is not natural my proof is not natural again typical it should come from directly from the sky I don't have to to play with this complicated construction and that's what he refused and this is not yet the final proof finally when he published under his own name the proof then he agreed of course but he had understood better the meaning of all notes he had no restriction to publish later so ok so I suppose it's time to start and thank you what's the conclusion what the conclusion is that Grotonik was special Grotonik was special but he was a genius but I don't recommend to ordinary people to follow Grotonik he knew his method he was maybe the only one who could use that kind of method very upset and also maybe another comment is about what happened there was a miracle in this place a miracle in this place in the 50's and 60's what happened there was an unexpected collaboration between 3 persons totally different Grotonik the prophet you don't need half one worker and Ser the very street logician might the rest of them might so if you look at the extent of later between Grotonik and Ser which is fantastic you will see that Grotonik goes into fantasy if I could play that I would etc no no no 57 is not one I recommend to you it's very interesting also what is interesting is that after that we all know that when Grotonik left this place around 70 he engaged into we didn't speak of the green at the time but it was of course the same spirit hippies and green combination of the green party with hippie and then he was interested and asked the occasion to leave this institute was supposedly well you know about that a question about financing of this institute when some army I will not enter into this discussion but then Grotonik was of course I suppose after he was deeply engaged into political activities but the truth and if you look at what I said from his father and his mother you would see that he was deeply politically engaged no from 1950 for 1947 when he started in 1968 or 1969 he did not pay any attention that was behind around him of course these were the very difficult times of the war in Vietnam between France and Vietnam and after that war between Vietnam and America came after that and then there was a war between France and Algeria at the time he did not pay any attention to that he did not pay any attention to that you can find in this later when he complained that well he said why don't you ask Carter to manage so that the young mathematician do not have to be laughed it makes no sense and I remember in 1965 when the gold was forced to run off by Mitterrand the first army in 1965 after the first day of the election I met Grotonik and he asked me is it true that France does not have no more president I said your conviction is an artist you may stay you have enough stay so you will be happy maybe you need your president for that he said and then I had to use ten minutes to explain to him what was the rule of what was the legal rule what was the constitution in 1965 you feel nothing about politics about other politics in highway about laws legalities and so on he was totally on the way of that and also interestingly enough he had one occasion when he had his first child was born when he was a student in Nancy from Islam lady complicated story but at least he saw what his name is hello so he enters he is very close he has been very close to him and then when he was on the way to marry with Mireille who was his wife when he was here I mean he made a fantasy to make a legal claim to have the custody of his first born son he was not even married in Mireille so he wanted so he came to us for advice and he said it doesn't look very good and he said but you know the French law allows me to be my own lawyer of course but I don't recommend it if your case was a very good case you could do that but since your case doesn't appear to be a very good case if you are a no lawyer you will be defeated so he came to the court and he said which is legal which is legal and he was totally ignored and later on when he was prosecuted in the 70s it was a very nasty occasion very nasty occasion and he was prosecuted for having given shelter to some Buddhist monk who was a kind of hippie and that's all and he goes on the occasion that he was hosting a foreign people without proper ID cards which was true so but then then he made his own well I remember a long long discussion so he came to Paris to ask for advice and help so Schwartz helped him and I remember 6 hours discussion convincing him that we are a good lawyer we know all these kind of cases and that so finally we made a compromise that the lawyer would describe the right play and then after that he would say a few words but now explaining his position which is what still exists in some new form but he said now for the legal conclusion I leave it to master so and so to say it but we had a long discussion to convince him that it was appropriate and final market that's another story but so just to explain that he was very difficult for him to to tackle with ordinary life social behavior was totally needed he was probably very naive but I think his feeling against power was authentic and well hooted right he didn't know how to handle it I would say thank you very much