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Shakespeare wrote for all of us, whereas Newton wrote for the happy few

An interview with Alexandru Buium

 

Alexandru Buium was born in Bucharest in 1955. He received his M.S. and Ph.D. from the University of Bucharest, in 1980 and 1983. In Romania, he was a researcher at the National Institute for Scientific Technical Research (INCREST) and at the Institute of Mathematics of the Romanian Academy. After leaving the country in 1993, he worked at the Institute for Advanced Studies (Princeton), at the University of New Mexico at Albuquerque (where he is now a full professor) and at the University of Illinois (Urbana-Champaign). He visited Columbia University, University of Essen and Max Planck Institut für Mathematik und Physik in Bonn, where he also was a Humboldt fellow. He defines himself as a number theorist. He published over 50 research papers, spread in the most praized mathematical journals (as Annals of Mathematics, Inventiones Mathematicae, Duke Mathematical Journal, American Journal of Mathematics, Mathematische Annallen, Journal für die Reine und Angewandte Mathematik etc.) and 3 research monographs (Differential Algebra and Diophantine Geometry, Hermann, Paris, 1994; Differential Algebraic Groups of Finite Dimension, Lecture Notes in Math. 1506, Springer, 1992; Differential Function Fields and Moduli of Algebraic Varieties, Lecture Notes in Math. 1226, Springer 1994).

The interview that follows is mainly based on e-mail correspondence.

Liviu Ornea: You started in Bucharest. How was it?

Alexandru Buium: Bucharest (in the time I was a University student) was a great place to fall in love with mathematics and learn it the way one should: without a lot of pressure, systematically, and in depth. Then, in the 80's I was a researcher at INCREST (more precisely at the mathematical department of that research institute). INCREST was a little miracle in Romania of those years that would astonish all our foreign visitors: pure research without mandatory teaching, very active seminars, strong interaction with the students and the professors at the University. Of course it was a miracle at the edge of disaster, continuously threatened with annihilation. Its survival in the 80's and subsequent transfiguration in the early 90's (when the math department of INCREST became again what it was before 1974: The Institute of Mathematics of the Romanian Academy) is one of the most amazing blessings for the Romanian mathematics.

L.O.: You just said "Romanian mathematics". You mean only "mathematics made in Romania" or more than this? Is there a "national style" in mathematics?

A.B.: Various mathematical schools have their own style, of course. But my feeling is there is no deep relation between these mathematical styles and national cultural values as perceived in the humanities, for instance. The style of a mathematical school seems to me to be most often the product of the style of its leaders.

L.O.: Talking about leaders, who were the most influential persons in your becoming a mathematician?

A.B.: Nicolae Radu, Lucian Badescu, Paltin Ionescu, Alexandru Dimca. They taught me mathematics. Wonderful people.

L.O.: And who are the ones you admire most now?

A.B.: I'm a number theorist and here are some of the gods of number theory (in alphabetical order): Bombieri, Deligne, Faltings, Manin, Mazur, Tate, Wiles.

L.O.: I know you read classics (both mathematics and literature) and I saw quite a lot in your personal library. Would you elaborate a little on this topic?

A.B.: I think reading the classics of mathematics is enormously profitable. It is very hard for me to read them thoroughly; this seems to be the job of a scholar in the history of mathematics. Indeed the notations used by the classics are so much different from ours, the precise meaning of concepts differs from ours, etc. But the general lines of the arguments are sometimes clear and the joy of understanding is enormous. I remember I read in Romania some of the papers of Poincaré and Fuchs (from around 1900); as I discovered later, many people in my generation read them, drawing inspiration from them. Then I read some 18-th century mathematics. I read parts of Legendre's treatise on number theory (where he claims the first proof of the quadratic reciprocity law) and I was amazed to see the amount of "numerical experiments and examples" that he seems fit to reproduce there. It's like trying to convince by showing numbers instead of proofs. Legendre's proof of the quadratic reciprocity law was not complete, by the way.

I read parts of Lagrange's amazing "Theorie des Fonctions" which is "differential algebra avant la lettre" (I should say that differential algebra, invented by Ritt some 150 years after Lagrange, was to become my first love in my student years.) Later I read parts of Newton's Principia. I was amazed to see how aware he was of the philosophical problems raised by his approach; we tend not to feel these problems because our whole scientific education is based on his Principia so we take Principia for granted.

I went back in time and read then some Archimedes and Euclid. Amazing, again, to see the extent to which their preoccupation with rigor makes them our contemporaries. We feel they are closer to us in time than Newton, I think. The same applies to humanities: we all feel that Euripides, say, is closer to us in time than Dante or Shakespeare. And Thucidides is closer to us than, say, Gibbon. In the meantime I bought most of these classical mathematical books and many more (Euler, Kepler, Cardano, etc. But also Einstein, Weyl, Frege, Gödel...). I like to skim through them from time to time and understand various arguments. I don't spend a lot of time trying to read them in detail, though; that would be considerably harder.

L.O.: Let's talk a bit about your research. How do you choose your subjects?

A.B.: I'm a very single minded person. I have a small number of mathematical obsessions and I'm letting myself carried away by them. I'm not a "problem solver". The few problems that I have solved I have solved by accident. I hope there will be more accidents but I'm not going to try to make them happen...

L.O.: "Not a problem solver", yet you participated in the International Mathematical Olympiad during high school.

A.B.: Solving research problems is so much harder than solving Olympiad problems.... I'm not surprised I'm not as successful...

L.O.: You say you are "single minded", but still, you had some collaborations.

A.B.: Yes, I had a beautiful collaboration with Felipe Voloch in Austin. We have a number of joint papers and it was great interacting with him: I learned a lot in the process. These were my first papers on characteristic p; I owe him my introduction to this fascinating world. I also had a very exciting collaboration with Anand Pillay in Urbana. Anand is a logician and our paper is at the borderline between logic and complex analysis. I came to realize how subtle the mathematics of the logicians is, contrary to the general misconception, I would say. Another collaboration which I enjoyed a lot was with my former student Mugurel Barcau.

L.O.: Do you have a favorite result?

A.B.: I guess one of my favorite results is my arithmetic "Theorem of the kernel" (Inventiones Mathematicae, 1995). I came to it after I tried hard to come up with a "right definition" for the "derivative of an integer". The search for this definition was all the fun; it took most of my year in Princeton (1993/94). The theorem itself came as a bonus, or rather as an "unavoidable accident". I remember walking rather often in the evening with my friend Florian Pop (who was in Princeton during that year) and telling him each time about my last definition of the derivative of an integer. He, in turn would introduce me little by little into the world of number theory. He would evaluate and criticize my definitions in the light of his number theoretic intuition. In the end the right definition turned out to be simpler than anything that I had tried before: the derivative of an integer n with respect to a prime p is (n-n^p)/p, which is an integer by Fermat's little theorem.

L.O.: You said you enjoyed collaborating with Mugurel Barcau. In general, how do you select and how do you work with your students? How many can you advise? Do you meet periodically, how often?

A.B.: The description that follows is based on my experience at the University of New Mexico, University of Illinois, and Columbia University. The graduate students who have passed their "qualifying examinations" approach a potential PhD adviser and, if accepted by him or her, start working towards their PhD thesis. The number of PhD students per faculty member can vary dramatically. But apparently 1 or 2 students is the ideal number. The qualifier exams are generally written exams (although Columbia used to have oral exams). At University of New Mexico there are generally 3 exams in algebra, real analysis, and complex analysis respectively; all 3 are at a level slightly higher than the usual corresponding undergraduate courses at the University of Bucharest. University of Illinois had an additional exam in a more specialized field (like, say, algebraic number theory at the level of Lang's book). So passing a qualifying exam doesn't really mean that the student knows enough to start doing research. These students will take a number of topic courses and individual study courses. Most of the topic courses that I taught at University of New Mexico (on elliptic curves, modular forms, zeta functions, p-adic integration) were mostly designed for the students who were working with me; at each point in time I had in average 3 students. Right now I have 2. What I usually do (and most of the people I know do the same) is to assign a problem to each student and meet with him or her, say, once or twice a week to discuss progress (and suggest new ideas). Many times I have to modify the original problem or change the problem altogether.

L.O.: As far as I understand, the students you had were not really collaborators?

A.B.: I was very lucky to have an exceptional PhD student, Mugurel Barcau, who graduated from the University of Bucharest before coming to the US. He had a very broad and deep mathematical background and our interaction was more like between two collaborators. This is not typical, though. In most cases the PhD adviser will invest a lot of time and energy in coaching the PhD student, especially talking to him about the content of the special courses that the student is taking.

L.O.: Do you require publication in peer reviewed journals prior to accepting the thesis?

A.B.: PhD advisers do not require publication of the results prior to thesis defense; but of course, thesis results need to be good enough to be publishable in an average journal.

L.O.: Let's talk about the way our activity is appreciated. What is a professor supposed to do in order to honestly serve the community and to deserve his salary?

A.B.: Research university professors in the US are expected to have contributions in 3 areas: research, teaching and service (in this order). By definition "service" means service to the profession and to the community. You are asking about the service to the community issue. This is a delicate and interesting aspect of our lives. Society expects us to be in the middle of public debates (example: on teaching in high schools), of various public efforts (example being on various committees, panels), to take a stand on various public issues, to be part of the democratic life. Some of my colleagues (and friends from other universities) actually do that and I'm quite impressed by what they're doing; I wish I were able to do more of these things. Maybe I will.

Mathematical research in the US is funded by Universities (through salaries which are, as a rule, bigger for research university professors than for college professors) and the Federal Government (through grants, mainly NSF and NSA grants). The Institutes (in Princeton, Berkeley, etc) are mostly funded through federal money (mainly NSF) and they need to periodically convince the federal agencies that they deserve the money; this is not a formal task and they are taking it very seriously.

L.O.: But what about you, in particular, which is your publication rythm?

A.B.: The number of papers per year that I tend to publish is 1 or 2. That's not much. I don't have ideas for more and I could not afford to publish less.

L.O.: You have a very good mathematical record. Does mathematics take all your time? What do you like to do besides mathematics?

A.B.: I like to read fiction, for instance, but I feel that's complementary to my doing math. The same goes for listening to music and watching my favorite movies.

L.O.: But do you see any connection between them? I mean, is it possible to be a good mathematician and to have no feeling for what usually is called "culture"?

A.B.: I think the quality of what a mathematician (or scientist) is doing has nothing to do with his or her involvement with humanities. This is unfortunate and I wish I were wrong.

L.O.: So, you seem to agree that it is not shameful that a humanist intellectual know nothing about, say, Newton's laws, but it is almost a scandal that a scientist never read Shakespeare. Why this?

A.B.: One possible answer: Shakespeare wrote for all of us whereas Newton wrote for the happy few.

L.O.: Now, that we are here, let's end with some classical question: would you give me a list with your favorite books and movies?

A.B.: I tell you what first comes into my mind, without trying to think too much. Once I start thinking the list is going to change and I'm not going to be able to pin it down.

L.O.: That's exactly what I want, the first list.

A.B.: Books: Goethe, Wilhelm Meister's Apprenticeship; Thomas Mann, Doctor Faustus; Ernesto Sabato, The Angel of Darkness; Alejo Carpentier, The Lost Steps; Michail Bulgakov, Master and Margarita; Bertold Brecht, The Caucasian Chalk Circle; Eugen Ionescu, Exit the King; Boris Vian, L'Écume des jours; Tony Kushner, Angels in America; Homer, Oddisey.

Favorite movies: Coppola, One from the heart; Wenders, Paris, Texas; Antonioni, Blow up; Tarkovski, The Mirror; Kubrick, Eyes wide shut; Rafelson, Five easy pieces; (I had to look up the name of the director for this one) Altman, Nashville; Truffaut, La nuit Americaine; Losey, Go-between; Scorsese, Mean streets.

Ad Astra • Volume 3, Issue 1, 2004 • Focus
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