Nous sommes tous des eleves d'Henri Cartan / Souvenirs des Seminaire Cartans au quotidien

0:00 It is a great pleasure for me to participate in the celebration of Henri Catin's 100th birthday and to congratulate him and Mrs. Catin with all my heart. Seven years ago, on June 14, 1997, the Société Mathématique de France organized a special day in honor of Henri Catin. I gave a lecture emphasizing in particular the close cooperation between Paris and Münster between Ari Katang and my teacher Heinrich Benke. The lecture was published in the Gazette in October 1997. Today, Ari Katang and I are still the same people, except that both of us are seven years older now. Therefore, some overlap between my lecture of 1997 and the present one is unavoidable. Ten years ago, on the occasion of Henri Cartan's 90th birthday, the German Mathematical Society elected Cartan as honorary member. Martin Grötsche, president of the society in 1994, wrote to him, And I read from the French translation of Grotschel's letter published in the last issue of the Gazette. Dear Mr. Catton, this poem is for me a great pleasure and a great honor to announce to you that the board of directors of the association of German mathematicians has elected you as a member of honor. We hope that you will accept this election and that we will be able, through it, to show the links who have joined you for a long time to German mathematicians. 63 years ago, in 1931, you came to Münster to give conferences. C'est ainsi qu'ont débuté vos étroites relations scientifiques et personnelles avec Heinrich Boehnke et son école d'analyse complexe, avec les nombreux mathématiciens qui en sont issus et même encore avec les disciples de ceux-là.

2:30 On the occasion of Boehnke's 80th birthday on October 8, 1978, Henri Cartin gave a beautiful dinner speech. Sadly enough Heinrich Boehnke unexpectedly could not attend the dinner because of illness. He died one year later. I read from Cartan's dinner speech as printed by Springer under the title Quelques Souvenirs par Henri Cartan. The meetings between Germans and French were not very frequent at that time. This young professor of 32 years had decided to make the University of Münster a living center for mathematics and he had grouped around him a circle of young researchers of two sexes. For them, he had organized the visit of a young Frenchman of 26 years and had conceived a full-time work program. Since the visitors had to give, within a week, four one-hour conferences in German and one conference in French, without mentioning German speakers and a forest excursion. Henri Catin spoke about many friends he had met in Münster for the first time in this dinner speech, in particular about Peter Thun. An assistant of Boehnke at the time 25 years old. In the dinner speech, Cartan said about Thun, Ce devait etre pour lui et moi le debut d'une collaboration scientifique puis d'une longue amitié. Boehnke and Thun wrote the famous report Theorie der Funktionen mehrerer komplexer Veränderlichen, which was published by Springer in 1934. A new edition edited by Reinhold Remmert appeared in 1970 with a new preface by Benke and Thulen and with many articles by various authors to update the book.

5:00 Thulen lived from 1907 to 1996. His 80th birthday was celebrated in Fribourg, Switzerland in 1987. Henri Cartin lectured sur quelques progrès dans la théorie des fonctions analytiques des variables complexes entre 1930 et 1950. This lecture was published in 1991 in the volume Miscellanea Mathematica dedicated to Dr. Heinz Götze of Springer Verlag, who lived from 1912 to 2001. Dr. Goetze was a great friend of mathematics and of mathematicians. This is a good occasion to remember the dinners given by Dr. Goetze in Paris and Bonn when Cartank's collected papers appeared in 1979, Sers collected papers in 1986 and mine in 1987. But now I use Cartank's paper in the volume dedicated to Dr. Goetze. To do some mathematics, beginning with mathematics of the time more than 70 years ago when Frank Cartan first came to Münster. And I follow exactly Cartan. So, let D be a bounded domain and then C2, the space of two complex variables, and G of D is group of automorphism. Then Reinhardt in 1921, Mathematische Annalen, studied domains with the following properties, the origin is in the domain and the domain is stable under special linear maps written down there where lambda and mu have absolute value one. And gave there the classification of all Reinhardt domains, admitting an automorphism which does not tick the origin. So admitting an additional automorphism.

7:30 And he gave the complete classification up to equivalence. And the result is the following. There are his first bi-cylinder, of course, a Cartesian product of two disks. And the dimension of the automorphism group is 3 plus 3 equals 6, and of course everybody here knows the structure of this automorphism group. Yeah, and the classification goes as follows. Besides the disk, there are these domains given by equations x absolute value square plus y absolute to the power a is less than 1, where a is positive. And, of course, for A equal 2, we have the complex disc in two complex variables, where the dimension of the automorphism group is 8. And for A different from 2, we have an automorphism group of dimension 4, which is written down explicitly by Thulen. And studied them, which are stable under the special automorphism, sending simply the vector xy into its multiple with lambda, for lambda absolute value one, and H. Cartan in 1930 had proved a theorem. That if you have a bi-holomorphic map of two circle domains sending the origin into the origin, then F is linear. Actually, Cartan proved this in arbitrary dimension. And this result was, as Cartan says, motivated by a paper of Boehnke, where Boehnke had done the same. Proof the same, but under additional assumptions. So Boehmke liked this rather simple proof of Cartan very much, and this, I think, was the origin of the visit of Cartan to Münster and the beginning of a long relationship.

10:00 Well, it also led to a paper by two authors with the same name, Elie and Henri Catin, in 1931, published a paper on the classification of all circle domains, which admit an automorphism, like in the case of Toulon, an automorphism which moves the origin. And so then, of course, one has all the Toulon domains, but they are Reinhardt domains and admit a bigger automorphism group, depending on two parameters, lambda and mu, as we explained. And then Elie Katang and Henri Katang introduce in 31 these domains, delta, alpha. Their x and y are both of absolute value less than one. But also the distance in the hyperbolic sense between x and y is less than alpha, where alpha between 0 and 1. And there the dimension of the automorphism group is 3 and Elie and Henri Cartin write it down. Then the interest of Henri Cartin in this area. Continuous, and as I said, all what I report here is from the paper of Katang dedicated to Thulen in the book dedicated to Dr. Götze. So, for example, Katang proved in 1936 that if you have a bounded domain in CN, then the group of automorphisms of this domain is a real Lie group. published the famous paper Sur les domaines bornés homogènes de l'espace de l'invariable complexe. And here he showed that up to dimension 3 all homogeneous bounded domains are also symmetric. In the dimensions beginning with 4 He could only classify the symmetric homogeneous domains, and later, for dimension 4, Piatecki-Shapiro discovered a homogeneous domain which is not symmetric.

12:30 But this quote here in recent years refers to the introduction which Elie Katan gave in 1935 to his paper, namely that He used progress by Reinhard Caratheodori, Henri Cartan, and Thulen. At the International Congress in 1932, Elie Cartan gave a plenary talk on compact homogeneous symmetric complex manifolds. Let's say they were classified. And you know that they are very closely related to the bounded homogeneous symmetric domains. One knows in all cases the structure written down in terms of Lie groups. I put down both transparencies immediately and because Katang and Tullin wrote an important paper in Mathematischer Nahlen in 1932, but one year later then they met in Münster and I was told that they had very many discussions at the dinner table, at the dinner given for Katang in 1931 and that they almost finished the paper there. In Munster. So now the results of this paper are described by Katang in the Tulun paper using already modern terminology. Namely Katang mentions the definition of sign manifolds. We heard this this morning in Demayi's lecture. And so I can hear very brief. The Stein manifold is defined using the ring of all holomorphic functions defined on the manifold by introducing the holomorphic convex closure of a compact set K.

15:00 It was also written down this morning. The holomorphic closure is the set of all x in capital X of our connected complex manifold so that for all f in h of x we have, one can say, the maximum principle that f of x in absolute value is less or equal to the supremum of f of y on k. And a Stein manifold is introduced by Stein in 1951 by several properties, one and another one and two. And one means the holomorphic function separate x. So for two different points we always have holomorphic function with different values in these points. And then secondly here if k is compact then also this holomorphic convex closure is compact, holomorphic convexity. That one can give the definition in such a short form. This was proved by Grauert in 1955. And Rammert in 1956, as Katang reports, proved that X is Stein if and only if X is embeddable in CN for some big N. Yeah, today at dinner I think I heard that dann und nur dann should be written D-A-N-N-N. Like if, e, f, f, i, f, f, yeah, so. Today at lunch I heard this. Dinner is still coming, yeah, so. Okay, one direction is trivial. Name is direction from embeddability to Stein. So this, there's another chapter in Katang's paper for Tulin, but this I, ah, sorry, I... There is one, I have to finish this. There is already the transparency. They investigate here in the Cartan-Tullin paper in 1932 domains which are schlicht over cn, domains et al. they lie over cn.

17:30 Someone writes down cn and they have local by. Local bi-holomorphic coverings and Cartang and Tullin want to approach the theorem in modern terminology with Stein that x is Stein if you have a domain et al. then x is Stein even only if x is a domain of holomorphy and in this Cartang-Tullin paper this is proof for the finite case. But every point in Cien has only finitely many inverse images in the domain et al. lying over part of the Cien. For the general results, one needs famous papers of Ocker, where also work of Bremermann and Norgay enters. And all this can be studied very well in Oka's collected papers published by Springer in 1984, with very interesting commentaries by Henri Cartan. So this is not a mathematical lecture, therefore I use these papers to read. Yeah, I'm sorry. But in Cartan's Quelques Souvenirs, So the dinner speech in 1978. He reports about his second visit to Münster in 1938. Betatun had left Germany and had accepted a position in Ecuador. The name of the assistant with whom Boehnke worked at that time actively was Karl Stein. There were not many students and the political atmosphere was depressing. Karl Stein, 1913 to 2000. And I. Both are from Hamm, a town near Münster. Stein was taught mathematics in secondary school by my father for six years before he began his studies with Benke in Münster in 1932 and received his Ph.D. degree in 1936.

20:00 Catton came to Munich and lectured sur les travaux de Karl Stein. He reported in particular about Stein's Habilitationsschrift of 1940, which concerns Cousin's second problem. The title, translated into English, is Topological Conditions for the Existence of Holomorphic Functions with a Given Zero Divisor. But I will come to this later in connection with the famous theorem B of Cartan and Jean-Pierre Serres. During the war, the friendship between Cartan and Boehnke was not interrupted. Boehnke, for example, received a mathematical letter from Oka in December 1940 and informed Cartan about it. In 1943, Cartan's brother, Louis, was deported to Germany. About this tragedy, Kartang says in Kelke Souvenir addressed to Boehnke, Je ne peux pas non plus oublier toutes les démarches que vous avez faites durant les années 1943 et 1944 pour tenter de retrouver la trace de mon frère Louis, déporté en Allemagne au mois de février 1943 et qui ne devait jamais revenir. Already in 1946, Cartan came to Ober-Wolfach, where he met Boehnke again after eight years. An extract of the Ober-Wolfach guestbook can be found on page 41 of the Gazette of June 1992. There you can see that Kartang participated actively in a concert, Heidenbach-Beethoven, on November 1, 1946, and that he lectured on November 4, 1946. His lecture was on Galois theory for noncommutative fields. By the way, Oberwolfach will celebrate its 60th birthday on July 2 this year. Please observe that Oberwolfach was two years old when Kartang visited it in 1946. Also Eresmann and Heinz Hopf came to Oberwolfach in 1946.

22:30 In 1949 Kartang visited Münster again and met Benke, Stein, Grauert and Remmert. I had begun my studies in Münster, in a terribly destroyed city and university, already in November 1945. When Cartan came to Münster, I was in Zurich working with Heinz Hopf. In his address during the reception on the Journée de l'honneur d'Henri Cartan on June 14, 1997, Reinhold Remmert reported about the tremendous impact on the Münster School of Kartang's talk on fiber bundles and coherent analytic sheaves. Grauert and Stein were present. I quote from Remmert's address published in the Gazette of October 1997. After the talk, there was a reception. On this occasion, Kartang gave a toast à l'Europe. Which at that time sounded like a cryptic message for us German students and Reinhard closes his address of 1997 as follows Dear Professor Kattan, it is my privilege today to thank you on behalf of many German colleagues for your great efforts in the late 40s and early 50s to reconciliate mathematicians on both sides of the Rhine River. Your never-forgotten toast à l'Europe is still a message for the future. I am one of the German mathematicians whom Katan gave courage and strength after the war. I wish to underline every word which Remmert said. For example, Katan reported about my thesis written under the influence of Boehnke and Hopf and which came out in 1950 in the Bobacki seminar of December 1953. The title was Functions et variétés algébroïdes, d'après F. Herzebourg. In the thesis, I had introduced complex spaces of dimension 2 and described the resolution of their singularities.

25:00 And in this context, in his Bobacki lecture, Katang introduced the notion of espace analytique général de dimension n, And made it clear in which category I was working. Hans Grauert and Reinhold Remmert published a long paper, Komplexe Räume in Mathematische Annahmen in 1958, which they dedicated to Heinrich Boehnke in dankbarkeit und verehrung zum 60. Geburtstag. In the introduction of this paper, Grauert and Remmert report about the concepts of complex spaces. There is a concept by Cartan and Serre on the one hand where complex spaces are defined by having local models which are analytic subsets in some big space C to the capital N. For Benke and Stein, complex spaces of dimension n had as local models so-called analytically ramified covers of open sets of Cn. Grauert showed that the analytically ramified coverings of Benke and Stein can be given by algebraic functions. Which implies that there are complex spaces in the sense of Cartan and Serre. So these two concepts in Münster and Paris coincide, which is a good sign. Here is a photo which was taken by Cartan in Cartan's retirement party in 1975. It shows four students of Heinrich Boehnke, so from left to right, Karl Stein, then Reinhold Remmert, then Hans Grauert, and then myself. As I mentioned before, Karl Stein received his doctoral degree in 1936 and celebrated his golden anniversary of his degree, a very, very important event in Germany, in 1986. Everything which is 50 years ago is celebrated. Marriage or important papers and so on.

27:30 So Grauert, Remmert and I only studied after the war and are students of Karl Stein also. I celebrated my golden degree in Münster in 2000. Grauert and Remmert celebrated it in 2004, namely just two days ago. On Saturday, I was in Munster to congratulate. I was very surprised and touched when Jean-Pierre Bourguignon presented a letter of congratulations by Henri Cartan for Grauert and Remmert. Unfortunately, Grauert was unable to attend because of illness. Cartan received the first honorary degree of his life in 1952 from Munster. We should have celebrated his Golden Honorary Degree in 2002, but it is still a good time if we celebrate it today, and as a former Münster student, I congratulate him in the name of Münster University for his Golden Doctor Honoris Causa. Not so many people are able to celebrate a Golden Honorary Degree. In my case, I have to wait until 2030. Here is a photo, Benke on the left, and Cartan, taken in Oberwolfach in the early 60s when the old buildings of Oberwolfach still existed. I already quoted from Remmert's address in 1997 where he mentioned Cartan's visit to Münster in 1949 and the tremendous impact of Cartan's lecture. Remmert recalls, from that time on we were under the spell of the new way of doing several complex variables. In 1953 this French revolution with its motto, il faut facotiser, was already history. After the 1953 colloquium at Brussels, where Kartang and Sepp presented theorems A and B for Stein manifolds to a dumbfounded audience, Karl Stein commented tersely,

30:00 We have bows and arrows, the French have tanks. I don't like military terminology, but still it shows the difference. Yeah, I was in Princeton from 1952 to 54 and therefore I did not attend the Brussels Colloquium, but we in Princeton, Armand Borel, Kodaira, Spencer and I, knew very well from Paris theorems A and B and its implications. This is... Quoted from Katang's lecture in Brussels. It's exactly his text of theorem A and theorem B. I don't have to explain this here in details because this was done by Desmailly this morning. And anyhow, I like to concentrate on theorem B. Proofs are in the Katang seminar of 1951-52. Proofs for theorem A and theorem B for Q equal 1 occur essentially in the paper of Katang of 1950. Katang points out that the cohomological formulation of theorem B and the idea to study not only the case Q equal 1 but the case of an arbitrary positive integer Q are due to theorem. In my Cartan lecture of 97, I recall the two Cousin Problems, which play such an important role in the work of Cartan and Serre, Ochre and the Münster School. To understand more closely the development, just before the Brussels conference, I recommend to read the letters of Serre to Cartan. Which are published in the book Miscellanea Mathematica dedicated to Dr. Goetze under the title Le Petit Cousin. I only make now some remarks on Cousin 2. So in his Brussels lecture, Serre introduces the exact sequence of the constant sheaf of integers,

32:30 the sheaf of local holomorphic functions, and the sheaf of local non-vanishing holomorphic functions. Where the map X, of course, sends F to e to the 2 pi i F. And then we have a long exact sequence. And the theorem B says that H1 XO and H2 XO are 0. So here one uses the dimension 2 in cohomology. They are zero, and therefore, because of this long exact sequence, h1 x0 star and h2 of xz are isomorphic. So the theorem B implies, yeah, so if x is Stein, then we have this isomorphism. So we can now interpret this in a topological way, because Oh, there's a mistake, H1XO star. Sorry, I was just totally confused because H1XO is zero. So, H1XO star is also the group of polymorphic complex line bundles. And a delta is a map which attaches to each complex line bundle its first Chern class. And over an arbitrary space, the group of isomorphism classes of topological line bundles is isomorphic to H2XZ. So this map here, which can also be defined for H1X if you write here something like C star. Continuous maps, local continuous maps into the complex numbers, non-vanishing, determines the group of continuous complex line bundles and then this is always an isomorphism for every space. A line bundle is the complex line bundle and is determined by its first Chern class. Now we have to tell also the story here of Karl Stein because

35:00 Now, maybe I'll come to this later. But, yeah, let me say this. Carl Stein investigated this also. So, he wanted to have topological conditions so that a Cousin problem is trivial, has a solution. Today we would say that a complex line bundle is trivial, and he used a result of Oka, a fundamental result of Oka, that a Cousin-2 problem is solvable if it has a continuous solution, which actually means this complex line bundle is trivial if the corresponding topological line bundle is trivial or if this cohomology class is trivial. We have investigated this cohomology class without knowing cohomology, and since the Cousin-2 problem is given by some divisor, under Poincaré isomorphism, this divisor corresponds to this cohomology class, so we had to express the vanishing of this cohomology class as closely as possible. By saying that all intersection numbers of two-dimensional cycles, also cycles modulo m, with this divisor are zero. So we can interpret this result also that over a Stein manifold the classification of complex line bundles So the structural group is C-star, but we have a holomorphic line bundle, and the classification of topological line bundles, structural group, again C-star, is equivalent. Now, Grauert in his Habilitation, published in two papers in Mathematische Annalen, proved that everywhere we have an arbitrary complex Lie group, The classification of principal bundles with structural group L and the classification of complex analytic bundles with structural group L are the same. This is not an isomorphism, it is a bijective. In this sense it is an isomorphism.

37:30 Or analytic maps into this complex Lie group, and here the sheaf of continuous maps. Of course, if the complex Lie group is C star, then this is Cousin 2, and if the complex Lie group is C, this is Cousin 1. And so we have such a result, such an Oka principle, one could say, begins with Oka, for an arbitrary complex Lie group. And in fact, Cartan taught about these results of Grauert in the Mexico conference, 1956, and Grauert thanks Cartan in his papers for advice and encouragement. I could only speak about a very small portion of Cartan's mathematical work. I concentrated on the early period which was so important for the cooperation Paris-Münster. During this day we heard about the impact of Cartan's seminars. This applies also to myself. I needed also Cartan's purely topological work on homotopy theory, etc., because I was very interested in cohomology operations. But now, at the end, I want to speak about Henri Catin, the European. The first European Congress of Mathematicians took place in Paris from July 6 to July 10, 1992. In Volume 1 of the proceedings, you find the list of committees. It begins as follows. Max Scarubi, Steering Committee, Chairman Henri Catin. And I was a member as a president of the European Mathematical Society, so I could observe how the Congress was prepared. In his opening speech to the Congress, Cartan called the Congress an event of great importance, showing that the mathematicians know the solidarity of the countries of Europe,

40:00 which are different in so many ways, but have a rich common heritage and a common future. Katan is especially glad that this first European Congress reunites the mathematicians from the two parts of Europe that were separated for such a long time. It was just the first big Congress after this sort of unification of Europe. Twelve years later we have witnessed the eastern enlargement of the European Union. Katang's 88th birthday was celebrated during the first European Congress in the residence of the German ambassador in the Palais Bois-Nez. In fact, my wife bought flowers, gave them to the German ambassador who presented them to Katang. The fourth European Congress begins today in Stockholm. On the occasion of the first European Congress, Henri Katang wrote again quelques souvenirs which appeared in the Gazette. He once more mentions his visit to Oberwolfach in 1946. Cartan reports about the foundation of the Association Européenne des Anciens-Youngs, European Association of Teachers, in Paris in 1956. He was president of the French section. As such, he took the initiative To invite participants from eight European countries to a meeting in Paris in October 1960, Emil Artin, Heinrich Behnke and I were the German members. The second meeting of this committee was in Dusseldorf in March 1962. As a result, the Livret Européen de l'étudiant, European Students Record, was published and distributed by the European Association of Teachers. The booklet contained a description of minimal requirements for basic courses. It was supposed to increase the mobility of students from one country to another. The professor at one university would mark in the booklet the contents of courses attended by the student. The professor at the next university would then be able to advise the student and tell him in which courses to enroll.

42:30 The booklet had ten sections for the first level, from fundamentals of algebra to introduction to the calculus of probability. For the second level, two versions of nine or ten sections were given, depending on whether pure or applied mathematics would be in the center of the studies. The booklet was not used very much. For me, it was often useful when reforms of the contents of courses were discussed. I had to report about the plans for this booklet in a meeting of the Executive Committee of the German Mathematical Society in Berlin in December 1961. The Berlin Wall had been built some months before, and as president of the Society I had to organize two separate meetings, one in East Berlin and one in West Berlin. There was no place in the world where all members of the Executive Committee could come together. Coming from West Germany I could cross a wall. To report about efforts to increase the mobility of students in Europe in a city where students could not move from one part to the other was depressing. Soon the German Mathematical Society was split when the Mathematische Gesellschaft der DDR was founded. Now we are in the Boulogne process. In all European countries, bachelor and master degrees are to be introduced. It is doubtful whether this is the harmonization we wanted. Cata, the European, was also active on the international level. Together with his father, he attended the International Congress of Mathematicians in Strasbourg in 1920. At the Bologna Congress in 1928 he was a member in his own right for the first time. He attended the Zurich Congress in 1932 where he met Heinrich Boehnke again. I assume that Elie Katang, Henri Katang and Heinrich Boehnke regretted the development in preceding years which led to the dissolution of the International Mathematical Union at this Congress.

45:00 The union had been founded in Strasbourg in 1920, only 12 years before. The international congresses of mathematicians went on, but were interrupted by the war. Oslo was in 1936, Cambridge, Massachusetts was planned for 1940 and took place only in 1950. Martin worked for the foundation of the new International Mathematical Union, which gradually became responsible for the organization of the international congresses. He was a member of the constitutive convention of the new IMU in 1950 in New York and of the first General Assembly in Rome in March 1952 before the Amsterdam Congress in September 1954. During the four-year period from 1963 to 1966, he was vice-president of the IMU. I was a member of the executive committee. Therefore, we met regularly for the meetings of this committee. In particular, we met in Dubna in 1966, where the General Assembly of the IMU was held preceding the Moscow conference. In March and April 1917, Cartan and I exchanged letters about the membership of the two German states in the IMU, in particular about the fact that both states, as a combined Germany, had more votes than the United States, for example. It is a correspondence interesting to read nowadays. Katang addressed the International Congress of Mathematicians in Nice in 1970 during its opening ceremony and announced the names of the Fields Medal winners. Sergei Novikov, one of the four winners, was unable to attend, indicating the political difficulties with which the International Mathematical Union had to struggle during this time. It is really a great pleasure and honor for me to speak today for Khatang's 100th birthday.

47:30 I gratefully remember many encounters with Ari Khatang. I would like to end my talk by mentioning four of these encounters. In 1955, Khatang visited Münster and attended my inauguration lecture, Habilitation. When preparing the lecture, I asked Beintke for advice. He said, this is very simple. The dean, who is a professor of pharmacy, should understand everything, and Henri Cartin should find it interesting. I already told this story many times before, but I like it so much that I had to repeat it here. Secondly, celebration of the 150th birthday of Weierstrass in Düsseldorf in 1965. Henri Cartin and I gave lectures. Henri Cartin, über den Vorbereitungssatz von Weierstrass. Henri Cartin attended the Bonn Arbeitstagung in 1971. Finally, as a special highlight for me, I mention my introduction as associé étranger de l'Académie des Sciences in 1990. I have the suspicion that Ari Katang belongs to those who propose me for this great honor. I'm sure that I can speak for many mathematicians in the world if I close by saying many thanks to Ari Katang. Thank you.