Askold Georgievich Khovanskii This and the next issue of the Moscow Mathematical Journal is dedicated to a member of its editorial board, our friend and teacher Askold Georgievich Khovanskii, who turned sixty in June 2007. Askold Khovanskii put his indelible mark on many areas of mathematics: real and complex algebraic geometry, singularity theory, differential equations, topology. Attempting to reduce his results to a common denominator, one comes up with a single word: beauty. The “Khovanskii contribution” to mathematics could be described by the quote, “If a theorem has only one proof,—then probably this theorem is not well understood”, which many of us heard from him on various occasions. The Paladin of Mathematical Beauty is the title that Askold inherited from his teacher, Vladimir Arnold, and it is this noble title that he is bequeathing to his students. Askold created several elegant theories with such simple and elementary descriptions that one wonders how the classics could have missed them. One example is topological Galois theory. In the early 1960s, V. Arnold found a connection between the solvability of algebraic equations in radicals and the solvability of the monodromy groups of the corresponding algebraic functions, and gave a geometric proof of the Abel theorem. Khovanskii, then a graduate student of Arnold, developed these ideas in the context of differential Galois theory, also called Picard–Vessiot theory. He connected the non-solvability of differential equation in quadratures with the non-solvability of the corresponding monodromy group. At the same time, he extended topological Galois theory to a much larger class of functions, going far beyond the Picard–Vessiot theory. This class includes, in particular, analytic functions with dense sets of ramification points. In the 1970s, A. Kushnirenko obtained his famous result that expresses the multiplicity of a singular point of a function as the number of integer points under its Newton diagram. I. Gelfand suggested that the very involved proof of this fact should be simplified. D. Bernstein, an old friend of Khovanskii's, generalized Kushnirenko's result and discovered experimentally a geometric formula for the number of roots of polynomial systems through the mixed volumes of its Newton polytopes. This opened a heroic period of the theory of Newton polytopes and toric varieties. Askold found various proofs of Bernstein's formula; the number of proofs that he has obtained up to now is about fifteen (worthy of the Guinness Book of Records?), and each of them can be explained to an advanced high-school student in half an hour. As a development of these ideas, Khovanskii discovered remarkable connections between the geometry of convex polytopes and the theory of toric varieties, results that are now used in both algebraic geometry and combinatorics. Closely related is his work on the index of vector fields on manifolds with boundary and his generalizations of the Petrovski–Oleinik inequalities. Along the same lines is the famous Khovanskii–Pukhlikov formula—the first multidimensional generalization of the Euler–MacLaurin formula, which initiated a broad and very fruitful activity in the geometric community. There are at least half a dozen different refinements of this formula discovered by different people. Khovanskii is one of the few mathematicians who has generated a whole new theory. His is the fewnomial theory. This theory develops the classical “Cartesian rule” generalized by K. Sevastianov to multivariable functions. The general idea behind the theory is strikingly simple: real systems of equations with simple structure (e. g. few terms) have solutions with a simple structure (e. g. few points). The theory is based on the reduction of functional-Pfaffian equations to purely functional ones. The transcendental function in the system is replaced by a partial differential (Pfaffian) equation with polynomial coefficients, and then this Pfaffian equation is replaced by a polynomial one. The elimination procedure described above uses the so-called Rolle–Khovanskii theorem. It is incredible that a nontrivial result may be obtained by a mere application of Rolle's theorem to functions on a circle! The fewnomial theory has numerous applications, in particular, to Abelian integrals (the famous Varchenko–Khovanskii finiteness theorem) and to the Hilbert–Arnold problem about bifurcations of polycycles in generic finite parameter families. The recent theory of O-minimal sets is to a large extent motivated by Khovanskii's fewnomial theory. The large works of Khovanskii and D. Novikov aimed at solving another problem posed by Arnold founded a new branch of geometry, namely, convex-concave geometry. Askold has fundamental results in algebraic geometry and commutative algebra. Recently he returned to geometric Galois theory and extended it to multivariable functions. His results are always interdisciplinary, they relate different, even seemingly distant mathematical subjects, and they have an all-mathematical significance. Askold Khovanskii has also worked on practical applications of mathematics, including mathematical economics and nomography. Khovanskii is never satisfied with the fact that a theorem is formally proved. He thinks about the result until it becomes crystal clear. He can repeat the words of Pasternak:
Во всем мне хочется дойти which, in our English translation, read as:
In everything I seek to grasp Askold is an enthusiastic and brilliant teacher. In his presentations he comes directly to the point and always concentrates on the main ideas purified from technical details. He has “scientific children” working not only in various areas of pure mathematics, but also in applied mathematics and mechanics. He feels personal responsibility for maintaining the traditions of the Russian Mathematical School. Askold is a very open person who is always ready to help people. In the difficult days for Russia, he was the scientific director of the Moscow Mathematical Institute (MMI) founded and sponsored by the American Mathematical Society. The institute was able to support a large number of mathematicians in Russia. Later the MMI merged with the Independent University of Moscow, one of whose founding fathers and active teachers is Askold Khovanskii. Askold belongs to the famous kin of the Khovanskii princes. His ancestors go back to Gediminas, one of the key figures of the Lithuanian history 600 years ago. During all these six centuries, the princes Khovanskii played an important role in the life of Russia. The classical opera “Khovanshchina” by Mussorgsky is devoted to a rebellion headed by a Prince Khovanskii in the seventeenth century. Together with his wife Tatyana, Askold has completed and enlarged the book about his kin written by one of his ancestors. We stress here that within the democratic team of the IUM, Prince Khovanskii is one of the most democratic figures. We wish Askold Khovanskii many years of creative activity and many new brilliant students. We wish the University of Toronto and the IUM to benefit for a long time in the future of having Professor Khovanskii on their team.
V. Arnold, F. Borodich, O. Gelfond, Yu. Ilyashenko, |
Moscow Mathematical Journal |