Last summer, while dining in Bucharest at COS (Casa Oamenilor de Stiinta, i.e. The House of Scientists), professor Marcus remarked that the careers of Alexandra Bellow and my career are strikingly different: Alexandra arrived quickly at mathematical maturity while I struggled for many years before becoming mathematically mature and professor Marcus wondered why. I was caught by surprise and gave a short explanation, too short to explain the discrepancy. Now, while reading his article, many memories sprang into my mind, memories of that period of time in the middle of the twentieth century when I was a student in mathematics in Bucharest. In response to professor Marcus' article I wrote a letter to him giving him a more ample explanation and he suggested that I publish its content under the above title in an article to be submitted for the volume in honor of professor Ionescu Tulcea, of Libertas Mathematica.
In 1953, I was 16 and about to enter my last school year. Instead I decided to study during the summer, pass all necessary examinations and then go to the university. So in the fall of 1953 I enrolled as a first year student in the Faculty of Mathematics and Physics of the University of Bucharest. The analysis freshman course was taught by C.I. Ionescu Tulcea who was a man of imposing presence. For this course we had five hours of lectures per week plus four hours of seminar, which for my group, was conducted by one of Ionescu Tulcea's assistants, namely Solomon Marcus.
The analysis course was, as described by Alexandra Bellow, "elegant and economical". It began with the Peano axioms and from them it was built solidly with definitions, propositions and theorems in the Bourbaki style. That was exactly the time when books by Nicolas Bourbaki were being published in Paris, books which of course could not even be seen in our Faculty's library. The course, intended to make us live our time in mathematics, flowed impeccably, always forward without any revisions, and Ionescu Tulcea never used notes for his lectures. The densely packed course covered a lot of material with maximum rigor. There were no pictures or diagrams and of course it was very abstract. As Alexandra Bellow wrote, we were dumbfounded. Myself much more than her since I was younger and I jumped over my last year in school. The concepts could only be grasped after a lot of hard thinking followed by a fleeting, hard won moment of joy, when the specific concept became clear.
Fortunately for us, the course was coupled with an excellent seminar lead by S. Marcus who usually came with long lists of problems and some of these were open. We had the opportunity to see that mathematics is a body of knowledge which is alive and to which we could contribute by adding something of our own. One of those open problems in Marcus' list attracted my attention and this lead to my first mathematical publication, an eight page paper published in 1956 in the Scientific Bulletin, Mathematical and Physics Section, article presented by Miron Nicolescu.
It is hard to imagine that we could have assimilated the rich, dense and very abstract material of the course without the help of this seminar. As in his lectures professor Ionescu Tulcea never presented any problems I asked him why. He replied that instead of working to solve problems we need to study memoirs. This was new and rather disconcerting. All the more so because at that time the library of the Faculty of Mathematics and Physics in Bucharest was very poor. In fact there was not even a book in that library which could have eased our work for this course. Once professor Ionescu Tulcea gave me an issue of a journal which of course contained memoirs. I now guess he probably meant that to come closer to main stream mathematics, problems are not enough and memoirs are of help. This was however not further discussed and I only made this reflection many years later.
In the sophomore year, analysis was taught by Miron Nicolescu and again the seminar was led by S. Marcus. Professor Nicolescu's course was not taught in the Bourbaki style, drawings were drawn on the board which made the material easier to grasp. The years which followed 1955-1958 did not have this precious quality of balance between theory and creative exercise in the study of mathematics. The courses which I most vividly recollect from my years as a student in Bucharest were the ones taught by Ionescu Tulcea (1953-1954), Miron Nicolescu (1954-1955), Dan Barbilian (1956-1957) and Tudor Ganea (1957-1958).
In 1956 Samuel Eilenberg came to Bucharest to attend the 4th Congress of Romanian Mathematicians held that year in Bucharest and Ganea had an opportunity to discuss mathematics with Eilenberg. Professor Ganea's course in the academic year 1957-1958 was about the axioms for homology and the uniqueness theorem, as developed by Eilenberg and Steenrod in their book Foundations of Algebraic topology, published in 1952. Ganea's course, taught five years after the book's publication, was meant to bring the teaching of topology in Bucharest into the middle of the twentieth century. That course was however our first brush with algebraic topology. We had no previous experience about homology or homotopy groups, no opportunity to calculate even simple homology or homotopy groups before. What we saw there was an elegant axiomatical construction and the theorem to which this construction led. Ganea was a very intelligent person and at the end he fully grasped the results of his course. I recall his phrase: "It seems to me that the material in this course looks like a coat which is too large for you." And how right he was! We should have first comprehended concretely what those groups meant and how important these invariants are.
Thinking about my university studies in Bucharest I have difficulty in saying that they prepared me for becoming a mathematician. The only more substantial experience of what active mathematics is about came during my last year as a student and during the summer of 1958. That year I had to think about a topic for my thesis. After five years we could graduate provided that we pass the "State Examination" covering the main subjects we studied, and present a thesis on a topic of our choice. As we had the possibility of buying good mathematics books written in Russian by Russian mathematicians from the bookstore Cartea Rusa (The Russian Book) I bought M.A. Naimark's book Normirovanie koltsa (Normed Rings) and I began to read it. Ciprian Foias was then a young assistant in the analysis group, a very talented and already very active mathematician. I told him that I was reading Naimark's book and he mentioned he had an open problem on normed rings. Thus I began to work with him but as he was only an assistant, I had to have an official supervisor who had to be a professor and Octav Onicescu kindly accepted to be my official supervisor. By the middle of the summer of 1958 I finished writing my Diploma thesis: a theorem of decomposition of an anti-symmetric Banach algebra into a direct sum of two symmetric Banach algebras which Foias and myself proved. This was an original work and Foias suggested that we publish it together. This is probably what we would have done under different circumstances.
Like all graduates in my class, at the end of that summer we were expecting our official job assignment. Mine turned out to be a position as a teacher in an elementary school in Urziceni, a small town of a few thousand people in the Ialomita County, Romania, located at about 60 km from Bucharest. I chose not to go and instead I gave private mathematical lessons. There were two Merit Diplomas in my class: Ion Cuculescu's and mine. But my Diploma de Merit was of no help in obtaining a job which could have opened up for me the possibility of becoming an active mathematician.
After my emigration together with my husband's family to Canada in 1961, my husband and I taught at Dalhousie University in Halifax, Nova Scotia, where we both found jobs, Norbert as an assistant professor and me as a lecturer. Dalhousie University had at that time a strong reputation in Law. But the university had a small department in mathematics and no graduate school. We taught there for two years and very much enjoyed the fact that the professors we met there were not only mathematicians but also historians or literary people. We all shared a common lounge. These were cultured people and later on when we went to places with large mathematical departments we missed this opportunity of meeting people from other departments.
But Dalhousie had no graduate school in mathematics and hence we had no opportunity to grow in our fields there. And we were longing for more mathematics. We decided to apply for fellowships in order to complete our Ph.D's. Norbert would have liked to go to Princeton which was an excellent idea. But at that time Princeton did not admit women in its Ph.D. program. We tried to find a solution so that I too could be enrolled in a graduate school. Since Rutgers University is close to Princeton we had the idea of combining Rutgers for me and Princeton for Norbert. But we realized that we did not have the means for this. In the two years at Dalhousie we managed to put aside some money, though not enough for this option. Indeed, Princeton was and still is an expensive place to live. Furthermore commuting between Rutgers and Princeton meant having a car and wasting time for driving between the two places. So this option was dropped. I was offered a teaching assistantship at Berkeley and Norbert a research assistantship so we decided to go to Berkeley. As we later found out, Berkeley was also an expensive place to live. All the money I earned as a teaching assistant went for paying the rent of our small one bedroom apartment and we began to dip regularly into our savings from Dalhousie. At that time Berkeley was like an industry producing Ph.D's in mathematics. In Julia Robinson's course which I attended that year, we must have been close to sixty graduate students, perhaps even more. I was assigned an advisor but to see him I had to stay in line. I had weekly assignments which were compulsory, a thing which I never had to do since I finished my high school years. It felt like going back to high school. Furthermore Norbert had already passed his preliminary examinations for his doctorate in Bucharest and was anxious to begin to work on his thesis. In Princeton this would have been possible. Neither of us was prepared for this atmosphere. Although younger than Norbert, I too lost five years. We were both older and had very different expectations than our fellow graduate students.
We decided to return to Canada and try to obtain positions in Montreal so as to be able to study at McGill University. Once we were both offered jobs as lecturers at McGill we readily accepted them and initiated our Ph.D. studies there in 1964. At that time the most prominent faculty member of the mathematics department was Joachim Lambek, a specialist in ring theory, who accepted to supervise my work. He suggested that I follow his graduate course in category theory which he taught that year. Category theory was then a fashionable subject which attracted him and he wanted to work in this subject. Lambek is a very good expositor, with very polished lectures but we finished the course enriched with a few abstract notions and a few theorems but without any real perspective for future work. This course thus landed me in a subject which I did not choose and in which, due to the circumstances, I was going to work for a number of years. In the hands of a genius like Alexander Grothendieck, category theory was instrumental in creating a new vision in algebraic geometry. But this is not what I was exposed to at McGill. This is something which I later discovered on my own. One thing is to use category theory in order to construct a unified homology theory like Eilenberg and Steenrod did in their book, or introduce schemes and the etale cohomology as Grothendieck did for the purpose of finding suitable invariants for algebraic varieties over finite fields and in order to prove the Weil conjectures. A completely different thing is when people work on rather artificially created formal problems in category theory, as they seemed to me to be.
The time came when I decided that enough was enough. I told a friend of mine also working in category theory of my decision to search for richer mathematical topics. He said I was making a mistake, that I will never be able to publish again. My friend stopped publishing around 2005, exactly the time when I started to publish more.
But one thing is to decide to change direction and another is to see where to go. The 1978 International Congress of Mathematicians in Helsinki was of great help. From the lectures which I attended there, the lecture which impressed me most was Sergei Novikov's who was an invited plenary speaker at that ICM. He lectured on algebraic geometric methods used in understanding partial differential equations such as the Korteweg-de Vries (KdV for short) equation and I was absolutely fascinated by this symbiosis of algebraic geometry and partial differential equations. Unfortunately, the courses in geometry and in differential equations which I attended in Bucharest were rather dull and weak. So I started by first acquiring some background in differential equations.
The academic year 1978-1979 was our sabbatical year from the Universite de Montreal where we were both associate professors with tenure. Our decision was to spend this sabbatical year in Oxford. It so happened that Michael Atiyah's seminar that year was on Novikov's work. Attending this seminar was in many ways an extraordinary experience. I always cherished mathematics but it was in Oxford that for the first time in my life I was exposed to absolutely fascinating mathematics in the making.
Norbert thought that if one has a very powerful talent, then one does not have to strain, work is so much easier, and then it is really worthwhile doing it. But I saw in Oxford that Atiyah would come everyday early in the morning, run up the stairs, go to his office and work until noon. After lunch he would again run up the stairs to his office and work all afternoon. I saw that he protected his time, carefully selecting the lectures he attended. All this seemed to contradict Norbert's view. But Atiyah was very generous with his time in his seminar. You could ask whatever question would cross your mind. A question could come for example from an applied mathematician, numerical analyst, who was asking how a hole in his numerically drawn picture which he was not able to complete numerically, could be filled up. And Atiyah would reply: "Yes I know what the problem is. We have there a singularity which is a tacnode...". This illustrates well that there is actually no barrier between pure and applied mathematics; that good applied mathematics requires a good solid base in pure mathematics.
That year in Oxford was mathematically a rich, splendid experience. I attended there a beautiful course taught by Roger Penrose in General Relativity where again I saw a symbiosis between geometry and differential equations and I also followed a course on elliptic curves taught by Michael Eastwood, another very beautiful subject.
My years after Oxford were years of hard work. Since I was then an associate professor, I had classes to teach, students to coach, meetings to attend, take part in committee work,... If I wanted to grow mathematically I would need to work in the evenings and since I noticed that if my suppers were lighter I could do this, I became a vegetarian. Eventually I succeeded in using geometrical methods in differential equations and my work today is on the geometry of planar polynomial vector fields. But how I managed to do this is another story to be told elsewhere.
These memories surfaced after reading Solomon Marcus' article and Alexandra Bellow's recollections and they perhaps constitute a partial answer to the question of professor Marcus as to why Bellow's mathematical career and mine were so very different and why she reached mathematical maturity much earlier than me.
Departement de Mathematiques et de Statistiques
Universite de Montreal
Montreal QC H3C 3J7, Canada
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|Title Annotation:||Cassius Tocqueville Ionescu Tulcea|
|Date:||Nov 1, 2013|
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