I. Krichever

Igor Krichever

Our friend and colleague, a remarkable mathematician Igor Krichever is 60.

Igor was born and spent his childhood in Taganrog. His mathematical talent has become apparent quite early. Having graduated from primary school, he moved to Moscow and entered the famous Kolmogorov Boarding School #18. In 1967 Igor became a silver medalist of the International mathematical olympiad. The same year he started his studies at the mech-math department of the Moscow State (Lomonosov) University.

In MSU, Igor's student works, diploma thesis and PhD thesis were guided by S.P. Novikov. The close collaboration between I.M. Krichever and S.P. Novikov continues up to now. They are co-authors of about 20 joint works on integrable systems, string theory, algebraic geometry and topology methods in modern mathematical and theoretical physics. I.M. Krichever is a remarkable representative of Novikov's scientific school.

Krichever's works of the graduate and post graduate period (1971–75) are related to topology. He obtained important results in the frame of Novikov's program of application of the formal group in cobordisms to the study of manifolds with a group action. Krichever's results on computation of the Hirzebruch genera were a break-through in this direction and strongly influenced consequent investigations.

In 1974 Novikov published his famous article “A periodic problem for the Korteweg–de Vries equations” having initiated the epoch of algebraic geometry methods in the theory of integrable systems and in the spectral theory of periodic linear operators. This theory was completed within half a year by Novikov, Dubrovin, Matveev, and Its. Then Lax, McKean, and Van Moerbeke also contributed to this development in 1975. Igor has done the next important step in this direction. He completed the classical Burchnall–Chaundy theory of ordinary commuting rank one linear differential operators and obtained effective theta-functional formulae for the coefficients. Most important, he developed effective methods to solve the periodic problem for the Kadomtsev–Petviashvily (KP) equation. Simultaneously he invented a fruitful notion of Baker–Akhieser functions. Since then he is working permanently in the theory of nonlinear completely integrable equations.

All these results are related to the rank one pairs of commuting operators. In 1961 J. Dixmier has constructed certain rank 2 and 3 pairs of commuting differential operators with polynomial coefficients which define a nonsingular genus one curve. In their article, Burcnall and Chaundy claimed in 1929 that the classification problem for higher rank pairs is very difficult. Using contemporary ideas of algebraic geometry and soliton theory V. Drinfeld and D. Mumford obtained interesting though ineffective results in this direction. Krichever has elaborated the technique of vector-valued Baker–Akhieser functions for this problem. Jointly with Novikov he applied it to the KP equation and to the Burchnall–Chaundy problem. They developed the technique of deformation of the Tyurin parameters to solve these problems. It was developed later by P.G. Grinevich, O.I. Mokhov, and A.E. Mironov. It turned out that certain “higher rank” solutions to KP equations are closely related to a new universal partial differential equation now called the Krichever–Novikov equation.

By 1982 Igor possessed a powerful analytic and algebraic-geometrical technique based on Riemann surfaces, Baker–Akhieser functions, Witham equations, and masterly used that technique. This enabled him, during the consequent long period, to approach many difficult physical and mathematical problems and to obtain important results. Among them are the Yang–Baxter equation and Peierls model, Ruijsenaars–Schneider model and Toda chain, addition theorems in the theory of abelian functions, Darboux–Egorov metrics and associativity equations, commuting difference operators and holomorphic vector bundles, the theory of discrete integrable systems, Fourier–Laurent expansions on Riemann surfaces with applications to conformal field theory. Many of those works are carried out in collaboration with Novikov, Buchstaber, Zabrodin, and others.

In 2001 Krichever created a general theory of Lax operators with the spectral parameter on Riemann surfaces and developed the Hamiltonian theory of the corresponding Lax and zero curvature equations. Besides resolving the problem of constructing such Lax operators this theory absorbed a wide scope of known results on Hitchin systems, integrable gyroscopes, integrable cases of motion of a rigid body in fluid, and led to further developments in these directions. In particular, in 2006–2007 it has led to the theory of Lax operator algebras—a new type of current algebras on Riemann surfaces. This work was closely followed by a same general theory of higher genus equations of isomonodromic deformations. By these works Igor contributed a lot to the foundations of the theory of Lax and zero curvature equation. In particular, these works provide a short way of introducing the beginners into the theory of integrable systems. Let us stress that those theories go back to the joint works of Krichever and Novikov on holomorphic vector bundles and integrable nonlinear equations.

Krichever has contributed a lot to application of methods of the theory of integrable systems to the classical and contemporary problems of geometry and topology. In elaboration of his early results he has shown in 1990 that the genus given by a Baker–Akhieser function possesses a fundamental rigidity property on the SU-manyfolds, and that all rigid genera are particular cases of this one. This genus is well-known as the Krichever genus now. In 2005 he has crucially improved the solution of Riemann–Schottky problem and the characterization of Prym varieties, having continued later these works in collaboration with T. Shiota and S. Grushevsky.

Nowadays Krichever is a leading researcher of the Landau Institute for Theoretical Physics and of the Institute for the Information Transmission Problems, and a professor of the Columbia University, a member of Executive Committees of the European Mathematical Society and of the Moscow Mathematical Society, one of the world leaders in the field of integrable systems. Several objects of contemporary mathematics bear his name. Besides Krichever genus and Krichever–Novikov equation these are Krichever–Novikov algebras, Krichever construction relating Baker–Akhieser functions to the Sato Grassmanian, Buchstaber–Krichever functional equation.

Igor is an extremely friendly person with a great human charm. He is a good and open minded colleague who is always ready to support an interesting mathematical idea. He is one of those who always conduct physical ideas to mathematical community and vise versa. Some of us also know him as an excellent table tennis player and an experienced rafter.

Igor is a great friend of the Independent University and a member of the editorial board of our Journal.

We cordially congratulate Igor and wish him many happy returns of the day.

V. M. Buchstaber, L. O. Chekhov, S. Yu. Dobrokhotov, S. M. Gusein-Zade, Yu. S. Ilyashenko, S. M. Natanzon, S. P. Novikov, G. I. Olshanski, A. K. Pogrebkov, O. K. Sheinman, S. B. Shlosman, M. A. Tsfasman

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