Spotlight
A Musing Mathematician
Mathematician Vladimir Pestov has come a long way from his birthplace in Siberia to a position as Reader in Mathematics at Victoria University. It's been a successful journey -- Pestov was one of two winners of the 1995 NZ Mathematical Society Research Award, and his citation highlighted his "creative and ingenious research in areas ranging from topological groups and Lie theory to the nonstandard analysis of superspace, in which he has solved long-standing open problems as well as demonstrating his breadth and depth of understanding and a gift for elegant and colourful exposition."
NZSM: When did you come to New Zealand?
VP: I left the then USSR in May 1991, just ten months after being allowed out of the country for the first time ever, and came as a Visiting Associate Professor to the University of Genoa, Italy, together with my wife and two children. From there I was invited as a visiting Lecturer to the University of Victoria, B.C., Canada, and while in Canada, I was offered a permanent position at VUW. We arrived in this country in January 1992, as your exemplary classical immigrants, with all our family belongings in two suitcases!NZSM: Why did you come here?
VP: In the old USSR I felt strongly that I was not needed. It is hard to explain, especially in view of the wide-spread myth here in the West -- and in NZ in particular -- about how advanced the research infrastructure in the old USSR was, and what an outstanding support was offered to the science by the State. Here is just one example. Tomsk State University, where I worked as an Associate Professor, was an old institution with a strong mathematical tradition -- Albert Einstein himself had published an article in the Journal of Tomsk University Mathematical Institute back in the 30s.In my day, nine staff members of the department of Mathematical Analysis shared one room, where I was entitled to one drawer in a desk, no more, no less. So much for the state support of academia!
Much worse, everything was governed by a set of perverse rules. To submit a paper to a foreign journal was practically prohibited, while the Soviet journals were concentrated in Moscow in the hands of a very few academic elite. If you did not belong to their clique, your article could be rejected outright. Here in the West, the academic system is open, friendly and fair. What particularly fascinates me, is that modern New Zealand is such a young society that a new settler, like myself, can participate in its formation.
NZSM: What is the current mathematical research scene in New Zealand like?
VP: There are a number of very active, top-class research mathematicians recognised worldwide: for example, Rob Goldblatt and Rod Downey here at Victoria, Gaven Martin at Auckland, plus the "imported brains" from the disintegrating Eastern bloc, such as Boris Pavlov, formerly from Leningrad, who is now at Auckland; and Wolfgang Vogel, originally from East Germany, working at Massey. And of course Vaughan Jones of Berkeley, one of the finest mathematicians of our time and an Auckland graduate, who spends a month or two out of each year at Auckland Uni as the Distinguished Alumni Professor, doing a great job for the Kiwi mathematicians. This list is, of course, by far incomplete.NZSM: What is the greatest problem facing the New Zealand mathematics scene?
VP: Perhaps it is still the brain drain. It is highly desirable for a bright student to head for North America or Europe to get a doctorate degree, but I would like to see greater numbers of them coming back eventually. The Information Superhighway turns the very remoteness of New Zealand from what was traditionally "the tyranny of distance" into a great asset. When this country implements the so-called MBone communication system within the next couple of years, we here will be able to "attend" the research seminars and talks given at the leading institutions worldwide without leaving our offices or homes. New Zealand is fast becoming an ideal location for a research mathematician, though not everyone realises that.NZSM:Tell us about your own research which led to the maths award.
VP: My major areas of interest are topology (the theory of shapes and continuous transformations), abstract analysis and mathematical theory of symmetry, created by Sophus Lie. In those areas I have published about 60 research papers. I will mention just one recent result of mine, proving a conjecture put forward back in 1972 by a French mathematician Adrien Douady.Many people remember from high school days what the polynomial functions are: they are the functions like f(x)=x2, or f(x, y)=xy, or else f(x, y,z)=x2+y2+z2-1, and so forth. If you consider, for a given polynomial function f=f(x, y,z, ...), depending on several variables, the collection of all values of arguments such that after plugging them in you get f=0, you obtain a geometric figure in a finite-dimensional space, called the locus of zeros for the function f and denoted by V(f).
For example, the locus of zeros of the polynomial of second degree f(x)=x2-1 consists of two points, called the roots of this polynomial, namely +1 and -1; the locus of zeros for the function f(x, y)=xy is the cross in the plane formed by the two coordinate axes; the locus of zeros of the function f(x, y,z)=x2+y2+z2-1 is a sphere of radius one centred at zero in the three-dimensional space, and so forth.
It is well known since the last century that not every figure can serve as the locus of zeros for a polynomial function. For example, while the empty circle is the locus of zeros for a function f(x, y)=x2-y2-1, the solid circle cannot be realized as the locus of zeros of any polynomial function!
My result can be put as follows: as the number of variables goes to infinity, the geometric restrictions on the shape of locus of zeros of a polynomial function become looser and looser to such an extent that, for a polynomial function depending on infinitely many variables, they do not exist any more. In particular, the same solid circle can be already made into the locus of zeros of a suitable polynomial function depending on infinitely many variables. This is potentially a very interesting phenomenon for the geometry of infinite dimensions, and it brings to life a number of interesting questions, like this one: how to measure the "strictness" of restrictions on the loci of zeros of polynomial functions, and what exactly happens to this numerical measure as the number of variables increases?
NZSM: How does a mathematician go about proving a theorem?
VP: When I came across Douady's conjecture, sitting on the carpet in my living room and browsing through a few old mathematical books -- something I love doing -- my brain got switched on in an instant. After about half an hour of deep concentration, I already knew in general terms how to prove the result (through cross-breeding an earlier theorem of Douady with a totally different area of mathematics, quantum group theory). Afterwards, it took me a couple of months of hard work to work out all the details. This time everything worked fine, but normally frustrations outnumber successes at the ratio 10 to 1.NZSM: What are your plans for the future?
VP:I plan to obtain my best results in pure mathematics within the next few years! It is no longer true that one's twenties are the best years for research, in particular because the "hygienic atmosphere of the New World", as Nabokov put it, helps to preserve one's brain in an excellent working condition for much longer! I am also getting excited by a perspective of occupying a niche in an area of applied research where my skills of a mathematician dealing with complex systems and their transformations could be readily transferred. Participating in a project of direct practical importance for this country would be great, and I consider it as one of future stages of my full integration in the society that proved to be so hospitable to me and my family.More about Vladimir Pestov's work can be found at his Web page:
http://www.vuw.ac.nz/maths/staff/Vladimir-Pestov.html
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