The Versatility of Integrability

Celebrating Igor Krichever's 60th Birthday

A Conference on Integrable Systems in Algebra,
Geometry, and Physics

Columbia University, May 4–7, 2011

Short Biography of I.M. Krichever

Igor KricheverIgor Krichever was born in Samara (Kuyibyshev) 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.

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

Reprinted (with minor editions) from Moscow Mathematical Journal, 11(2), (2011)

Publications of I.M. Krichever

  1. Bordisms of groups acting freely on spheres. (PDF: Russian)
    Uspehi Mat. Nauk 26:6(162) (1971), 245–246.
  2. Formulae for the fixed points of an action of the group Zp. (PDF: Russian)
    Uspehi Mat. Nauk 28:1(169) (1973), 237–238 (with S.M. Gusein-Zade).
  3. Actions of finite cyclic groups on quasicomplex manifolds. (PDF: Russian, English)
    Mat. Sb. (N.S.). 90(132):2 (1973), 306–319.
  4. On invariance of the characteristic classes for the manifolds of the homotopy type of CP(n). (PDF: Russian)
    Uspehi Mat. Nauk 28:5(173) (1973), 245–246.
  5. A remark on the paper "Actions of finite cyclic groups on quasicomplex manifolds'' . (PDF: Russian, English)
    Mat. Sb. (N.S.). 95(137):1(9) (1974), 146–147.
  6. Formal groups and Atiyah-Hirzebruch formula. (PDF: Russian, English)
    Izv. Akad. Nauk SSSR 38:6, (1974), 1289–1304.
  7. Equivariant Hirzebruch genera. The Atiya-Hirzebruch formula. (PDF: Russian)
    Uspehi Mat. Nauk 30:1(181) (1975), 243–244.
  8. The potentials with zero reflection coefficient on the background of the finite-gap potentials. (PDF: Russian, English)
    Funkcional. Anal. i Priložen 9:2 (1975), 77–78.
  9. Obstructions to the existence of S1-actions. Bordisms of branched coverings. (PDF: Russian, English)
    Izv. Akad. Nauk SSSR Ser. Mat. 40:4 (1976), 828–844.
  10. An algebraic-geometric construction of the Zaharov-Šabat equations and their periodic solutions.
    Dokl. Akad. Nauk SSSR 227:2 (1976), 291–294.
  11. The Schrödinger equation in a periodic field and Riemann surfaces. (PDF: English)
    Dokl. Akad. Nauk SSSR 229:1 (1976), 15–18 (with B.A. Dubrovin, S.P. Novikov).
  12. Algebraic curves and commuting matrix differential operators. (PDF: Russian, English)
    Funkcional. Anal. i Priložen 10:2 (1976), 75–76.
  13. Integration of nonlinear equations by the methods of algebraic geometry. (PDF: Russian, English)
    Funkcional. Anal. i Priložen 11:1 (1977), 15–31.
  14. Geometry of Riemann surfaces and non-linear differential equations. (PDF: Russian)
    Uspehi Mat. Nauk 32:1(193) (1977), 229–230 (with B.A. Dubrovin).
  15. Methods of algebraic geometry in the theory of nonlinear equations. (PDF: Russian, English)
    Uspehi Mat. Nauk 32:6(198) (1977), 183–208.
  16. Rational solutions of the Kadomcev-Petviašvili equation and the integrable systems of N particles on a line. (PDF: Russian, English)
    Funkcional. Anal. i Priložen. 12:1 (1978), 76–78.
  17. Rational solutions of the Kadomcev-Petviašvili equation. (PDF: Russian)
    Uspehi Mat. Nauk 33:2(200) (1978), 227–228.
  18. Commutative rings of ordinary linear differential operators. (PDF: Russian, English)
    Funktsional. Anal. i Prilozhen. 12:3 (1978), 20–31.
  19. Holomorphic vector bundles over Riemann surfaces and the Kadomcev-Petviašvili equation. I. (PDF: Russian, English)
    Funktsional. Anal. i Prilozhen. 12:4 (1978), 41–52 (with S.P. Novikov).
  20. Algebraic curves and nonlinear difference equations. (PDF: Russian, English)
    Uspekhi Mat. Nauk 33:4(202) (1978), 215–216.
  21. Holomorphic bundles and the Kadomcev-Petviašvili equation. (PDF: Russian)
    Uspehi Mat. Nauk 33:5(203) (1978), 209–211. (with S.P. Novikov)
  22. Holomorphic bundles and nonlinear equations. Finite-gap solutions of rank 2. (PDF: English)
    Dokl. Akad. Nauk SSSR 247:1 (1979), 33–37 (with S.P. Novikov).
  23. Rational solutions of duality equations for Yang-Mills fields. (PDF: Russian, English)
    Funktsional. Anal. i Prilozhen. 13:4 (1979), 81–82.
  24. On the rational solutions of the Zaharov-Šabat equations and completely integrable systems of N particles on the line. (PDF: Russian, English).
    Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 84 (1979), 117–130.
  25. Methods of algebraic geometry in contemporary mathematical physics.
    Soviet Sci. Rev. Sect. C: Math. Phys. Rev. 1 (1980), 1–54 (with V.G. Drinfel'd, Yu.I. Manin, S.P. Novikov).
  26. The inverse problem method and holomorphic bundles on Riemann surfaces.
    Soviet Sci. Rev. Sect. C: Math. Phys. Rev. 1 (1980), 5–26 (with S.P. Novikov).
  27. An analogue of the d'Alembert formula for the equations of a principal chiral field and the sine-Gordon equation.
    Dokl. Akad. Nauk SSSR 253:2 (1980), 288–292.
  28. Self-similar solutions of equations of Korteweg-de Vries type. (PDF: Russian, English)
    Funktsional. Anal. i Prilozhen. 14:3 (1980), 83–84.
  29. Holomorphic bundles over algebraic curves, and nonlinear equations. (PDF: Russian, English)
    Uspekhi Mat. Nauk 35:6(216) (1980), 47–68 (with S.P. Novikov).
  30. Elliptic solutions of the Kadomcev-Petviašvili equations, and integrable systems of particles. (PDF: Russian, English)
    Funktsional. Anal. i Prilozhen. 14:4 (1980), 45–54.
  31. The periodic non-Abelian Toda chain and its two-dimensional generalization. (PDF: Russian, English)
    Uspekhi Mat. Nauk 36:2(218) (1981), 72–77.
  32. The Baxter equations and algebraic geometry. (PDF: Russian, English)
    Funktsional. Anal. i Prilozhen. 15:2 (1981), 22–35.
  33. Holomorphic bundles and nonlinear equations. (PDF: English)
    Physica D: Nonlinear Phenomena 3:1–2 (1981), 267–293 (with S.P. Novikov).
  34. The spectral theory of difference operators, algebraic geometry, and Peierls model. (PDF: Russian)
    Uspekhi Mat. Nauk 37:2(224) (1982), 259–260.
  35. Algebro-geometric spectral theory of the Schrödinger difference operator and the Peierls model.
    Dokl. Akad. Nauk SSSR 265:5 (1982), 1054–1058.
  36. Discrete Peierls models with exact solutions. (PDF: English)
    Soviet Phys. JETP 56:1 (1982), 212–225 (with S.A Brazovskii, I.E. Dzyaloshinskii).
  37. Exactly soluble Peierls models. (PDF: English)
    Phys. Let A 91:1 (1982), 40–42 (with S.A Brazovskii, I.E. Dzyaloshinskii).
  38. Commensurability effects in the discrete Peierls model. (PDF: English)
    Soviet Phys. JETP 56:4 (1982), 908–913 (with I.E. Dzyaloshinskii).
  39. The Peierls model. (PDF: Russian, English)
    Funktsional. Anal. i Prilozhen. 16:4 (1982), 10–26.
  40. Topological and algebraic-geometrical methods in contemporary mathematical physics.
    Soviet Science Reviews, Math. Phys. Rev. 3 (1982), 1–151 (with B.A. Dubrovin, S.P. Novikov).
  41. Algebraic geometry methods in the theory of the Baxter-Yang equations.
    Soviet Science Reviews, Math. Phys. Rev. 3 (1982), 53–81.
  42. The "Hessian" of integrals of the Korteweg-de Vries equation and perturbations of finite-gap solutions.
    Dokl. Akad. Nauk SSSR 270:6 (1983), 1312–1317.
  43. Sound and charge-density wave in the discrete Peierls model. (PDF: English)
    JETP 58:5 (1983), 1031–1040 (with I.E. Dzyaloshinskii).
  44. Nonlinear equations and elliptic curves. (PDF: Russian, English)
    Itogi Nauki i Tekhniki, Akad. Nauk SSSR, VINITI, 23 (1983), 79–136.
  45. The Laplace method, algebraic curves and nonlinear equations. (PDF: Russian, English)
    Funktsional. Anal. i Prilozhen. 18:3 (1984), 43–56.
  46. Two-dimensional periodic difference operators and algebraic geometry.
    Dokl. Akad. Nauk SSSR 285:1 (1985), 31–36.
  47. Integrable systems. I. (PDF: Russian)
    Itogi Nauki i Tekhniki, Akad. Nauk SSSR, VINITI, Dynamical Systems 4 (1985), 179–277 (with B.A. Dubrovin, S.P. Novikov).
  48. Two-dimensional periodic Schrödinger operators and Prym's θ-functions. (PDF: English)
    Geometry today (Rome, 1984), Progr. Math., 60 (1985), 283–301 (with A.P. Veselov, S.P. Novikov).
  49. Algebraic-geometrical methods in some problems of solid state physics.
    Proc. int. conf Dubna (1985) .
  50. Rational multisoliton solutions of the nonlinear Schrödinger equation.
    Dokl. Akad. Nauk SSSR 287:3 (1986), 606–610 (with V.M. Eleonskii, N.E. Kulagin).
  51. Wess-Zumino Lagrangians in chiral models and quantization of their constants. (PDF: English)
    Nuclear Phys. B 264:2–3 (1986), 415–422 (with M.A. Olshanetsky, A.M. Perelomov).
  52. The spectral theory of "finite-gap'' nonstationary Schrödinger operators. The nonstationary Peierls model. (PDF: Russian, English)
    Funktsional. Anal. i Prilozhen. 20:3 (1986), 42–54.
  53. Algebras of Virasoro type, Riemann surfaces and the structures of soliton theory. (PDF: Russian, English)
    Funktsional. Anal. i Prilozhen. 21:2 (1987), 46–63 (with S.P. Novikov).
  54. Algebras of Virasoro type, Riemann surfaces and strings in Minkowski space. (PDF: Russian, English)
    Funktsional. Anal. i Prilozhen. 21:4 (1987), 47–61 (with S.P. Novikov).
  55. Evolution of the Whitham zone in the Korteweg-de Vries theory. (PDF: English)
    Dokl. Akad. Nauk SSSR 295:2 (1987), 345–349 (with V.V. Avilov, S.P. Novikov).
  56. The averaging method for two-dimensional "integrable" equations. (PDF: Russian, English)
    Funktsional. Anal. i Prilozhen. 22:3 (1988), 37–52.
  57. Exact solutions of the time-dependent Schrödinger equation with self-consistent potentials. (PDF: Russian)
    Soviet J. Particles and Nuclei 19:3 (1988), 579–621 (with B.A. Dubrovin, T.M. Malanyuk, V.G. Makhan'kov).
  58. Virasoro-Gelfand-Fuks type algebras, Riemann surfaces, operator's theory of closed strings. (PDF: English)
    J. Geom. Phys. 5:4 (1988), 631–661 (with S.P. Novikov).
  59. A periodic problem for the Kadomtsev-Petviashvili equation.
    Dokl. Akad. Nauk SSSR 298:4 (1988), 802–807.
  60. New method of finding dynamic solutions in the Peierls model. (PDF: English)
    Soviet Phys. JETP 94:7 (1988), 344–354 (with I.E. Dzyaloshinskii, J. Chronek).
  61. Algebraic-geometry methods in soliton theory.
    Soliton theory: a survey of results., Nonlinear Sci. Theory Appl., Chapter 14 (1990), 354–400 (with P.G. Grinevich).
  62. Algebras of Virasoro type, the energy-momentum tensor, and operator expansions on Riemann surfaces. (PDF: Russian, English)
    Funktsional. Anal. i Prilozhen. 23:1 (1989), 24–40 (with S.P. Novikov).
  63. Spectral theory of two-dimensional periodic operators and its applications. (PDF: Russian, English)
    Uspekhi Mat. Nauk 44:2(266) (1989), 121–184.
  64. On Heisenberg relations for the ordinary linear differential operators.
    ETH Zürich preprint (1990).
  65. Generalized elliptic genera and Baker-Akhiezer functions. (PDF: Russian, English)
    Mat. Zametki 47:2 (1990), 34–45.
  66. Riemann surfaces, operator fields, strings. Analogues of the Fourier-Laurent bases. (PDF: English)
    Physics and mathematics of strings World Scientific (1990), 356–388 (with S.P. Novikov).
  67. The averaging procedure for the soliton-like solutions of integrable systems.
    Mechanics, analysis and geometry: 200 years after Lagrange North-Holland (1991), 99–125.
  68. The periodic problems for two-dimensional integrable systems.
    Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990) Math. Soc. Japan, Tokyo (1991), 1353–1362.
  69. Multiphase solutions of the Benjamin-Ono equation and their averaging. (PDF: Russian, English)
    Mat. Zametki 49:6 (1991), 42–58 (with S.Yu. Dobrokhotov).
  70. The dispersionless Lax equations and topological minimal models. (PDF: English)
    Comm. Math. Phys. 143:2 (1992), 415–429.
  71. Whitham theory for integrable systems and topological quantum field theories.
    New symmetry principles in quantum field theory (Cargèse, 1991) Plenum (1992), 309–327.
  72. Perturbation Theory in Periodic Problems for Two-Dimensional Integrable Systems.
    Sov. Sci. Rev., Sect. C, Math. Phys. Rev. 9:2 (1992), 1–103.
  73. The Cauchy problem for doubly periodic solutions of KP-II equation.
    Important developments in soliton theory Springer (1993), 123–146.
  74. Vector addition theorems and Baker-Akhiezer functions. (PDF: Russian, English)
    Teoret. Mat. Fiz. 94:2 (1993), 200–212 (with V.M. Bukhshtaber).
  75. The τ-function of the universal Whitham hierarchy, matrix models and topological field theories. (PDF: English, arXiv: hep-th/9205110)
    Comm. Pure Appl. Math. 47:4 (1994), 437–475.
  76. Algebrogeometric two-dimensional operators with self-consistent potentials. (PDF: Russian, English)
    Funktsional. Anal. i Prilozhen. 28:1 (1994), 26–40.
  77. Elliptic solutions of nonlinear integrable equations and related topics. (PDF: English)
    Acta Appl. Math. 36:1–2 (1994), 7–25.
  78. Finite genus solutions to the Ablowitz-Ladik equations. (PDF: English)
    Comm. Pure Appl. Math. 48:12 (1995), 1369–1440 (with P.D Miller, N.M. Ercolani, and C.D. Levermore).
  79. General rational reductions of the Kadomtsev-Petviashvili hierarchy and their symmetries. (PDF: Russian, English)
    Funktsional. Anal. i Prilozhen. 29:2 (1995), 1–8.
  80. Spin generalization of the Calogero-Moser system and the matrix KP equation. (PDF: English, arXiv: hep-th/9411160)
    Amer. Math. Soc. Transl. Ser. 2 170 (1995), 83–119 (with O. Babelon, E. Billey, M. Talon).
  81. Linear operators with self-consistent coefficients and rational reductions of KP hierarchy. (PDF: English)
    Phys. D 87:1–4 (1995), 14–19.
  82. Algebraic-geometrical methods in the theory of integrable equations and their perturbations. (PDF: English)
    Acta Appl. Math. 39:1–3 (1995), 93–125.
  83. Spin generalization of the Ruijsenaars-Schneider model, the non-Abelian 2D Toda chain, and representations of the Sklyanin algebra. (PDF: Russian, English, arXiv: hep-th/9505039)
    Uspekhi Mat. Nauk 50:6(306) (1995), 3–56 (with A. Zabrodin).
  84. Integrability and Seiberg-Witten exact solution. (PDF: English, arXiv: hep-th/9505035)
    Phys. Lett. B 355:3–4 (1995), 466–474 (with A. Gorsky, A. Marshakov, A. Mironov, and A. Morozov).
  85. Multidimensional vector addition theorems and the Riemann theta functions. (PDF: English)
    Internat. Math. Res. Notices 10 (1996), 505–513 (with V. Buchstaber).
  86. Quantum integrable systems and classical discrete nonlinear dynamics. (PDF: English, arXiv: hep-th/9604080)
    Statistical models, Yang-Baxter equation and related topics, and Symmetry, statistical mechanical models and applications (Tianjin, 1995) World Sci. Publ., River Edge, NJ (1996), 211–227 (with O. Lipan, P. Wiegmann, and A. Zabrodin).
  87. On the integrable geometry of soliton equations and N=2 supersymmetric gauge theories. (PDF: English, arXiv: hep-th/9604199)
    J. Differential Geom. 45:N2 (1997), 349–389 (with D.H. Phong).
  88. Quantum integrable models and discrete classical Hirota equations. (PDF: English)
    Comm. Math. Phys. 188:2 (1997), 267–304 (with O. Lipan, P. Wiegmann, and A. Zabrodin).
  89. The effective prepotential of N=2 supersymmetric SU(Nc) gauge theories. (PDF: English, arXiv: hep-th/9609041)
    Nuclear Phys. B 489:1–2 (1997), 197–210 (with E. D'Hoker and D.H. Phong).
  90. The effective prepotential of N=2 supersymmetric SO(Nc) and Sp(Nc) gauge theories. (PDF: English, arXiv: hep-th/9609145)
    Nuclear Phys. B 489:1–2 (1997), 211–222 (with E. D'Hoker and D.H. Phong).
  91. The renormalization group equation in N=2 supersymmetric gauge theories. (PDF: English, arXiv: hep-th/9610156)
    Nuclear Phys. B 494:1–2 (1997), 89–104 (with E. D'Hoker and D.H. Phong).
  92. Algebraic-geometric n-orthogonal curvilinear coordinate systems and the solution of associativity equations. (PDF: Russian, English, arXiv: hep-th/9611158)
    Funktsional. Anal. i Prilozhen. 31:1 (1997), 32–50.
  93. Solitons in high-energy QCD. (PDF: English, arXiv: hep-th/9704079)
    Nuclear Physics B 505:1–2 (1997), 387–414 (with G.P. Korchemsky).
  94. Elliptic solutions to difference non-linear equations and related many-body problems. (PDF: English, arXiv: hep-th/9704090)
    Comm. Math. Phys. 193:2 (1998), 373–396 (with P. Wiegmann and A. Zabrodin).
  95. Symplectic forms in the theory of solitons. (PDF: English, arXiv: hep-th/9708170)
    Surveys in differential geometry: integral systems Int. Press, Boston, MA (1998), 239–313 (with D.H. Phong).
  96. Vacuum curves of elliptic L-operators and representations of Sklyanin algebra. (PDF: English, arXiv: solv-int/9801022)
    Moscow Seminar in Mathematical Physics Amer. Math. Soc. Transl. Ser. 2 (1999), 199–221 (with A. Zabrodin).
  97. Discrete analogues of the Darboux-Egorov metrics. (PDF: Russian, English, arXiv: hep-th/9905168)
    Tr. Mat. Inst. Steklova 255 (1999), 21–45 (with A.A. Akhmetshin and Yu.S. Volvovski).
  98. A generating formula for solutions of associativity equations. (PDF: Russian, English, arXiv: hep-th/9904028)
    Uspekhi Mat. Nauk 54:2(326) (1999), 167–168 (with A.A. Akhmetshin and Yu.S. Volvovski).
  99. Periodic and almost-periodic potentials in inverse problems. (PDF: English, arXiv: math-ph/0003004)
    Inverse Problems 15:6 (1999), 117–144 (with S.P. Novikov).
  100. Trivalent graphs and solitons. (PDF: Russian, English, arXiv: math-ph/0004009)
    Uspekhi Mat. Nauk 54:6(330) (1999), 149–150 (with S.P. Novikov).
  101. Elliptic solutions to difference nonlinear equations and nested Bethe ansatz equations. (PDF: English, arXiv: solv-int/9804016)
    Calogero-Moser-Sutherland models (Montréal, QC, 1997) CRM Ser. Math. Phys. Springer (2000), 249–271.
  102. Baker-Akhiezer functions and integrable systems.
    Integrability: The Seiberg-Witten and Whitham equations Gordon and Breach (2000), 1–22.
  103. Elliptic analog of the Toda lattice. (PDF: English, arXiv: hep-th/9909224)
    Internat. Math. Res. Notices 8 (2000), 383–412.
  104. Spin chain models with spectral curves from M theory. (PDF: English, arXiv: hep-th/9912180)
    Comm. Math. Phys. 213:3 (2000), 539–574 (with D.H. Phong).
  105. Holomorphic bundles and difference scalar operators: single-point constructions. (PDF: Russian, English, arXiv: math-ph/0004008)
    Uspekhi Mat. Nauk 55:1(331) (2000), 187–188 (with S.P. Novikov).
  106. Holomorphic bundles and commuting difference operators. Two-term constructions. (PDF: Russian, English)
    Uspekhi Mat. Nauk 55:3(333) (2000), 181–182 (with S.P. Novikov).
  107. The τ-function for analytic curves. (PDF: English, arXiv: hep-th/0005259)
    Random matrix models and their applications, Math. Sci. Res. Inst. Publ. 40 (2000), 285–299 (with I.K. Kostov, M. Mineev-Weinstein, P. Wiegmann, and A. Zabrodin).
  108. The periodic and open Toda lattice. (PDF: English, arXiv: hep-th/0010184)
    Mirror symmetry, IV (Montreal, QC, 2000), AMS/IP Stud. Adv. Math. 33 (2002), 139–158 (with K.L. Vaninsky).
  109. Vector bundles and Lax equations on algebraic curves. (PDF: English, arXiv: hep-th/0108110)
    Comm. Math. Phys. 229:2 (2002), 229–269.
  110. Isomonodromy equations on algebraic curves, canonical transformations and Whitham equations. (PDF: English, arXiv: hep-th/0112096)
    Mosc. Math. J. 2:4 (2002), 717–752.
  111. Spin chains with twisted monodromy. (PDF: English, arXiv: hep-th/0110098)
    J. Inst. Math. Jussieu 1:3 (2002), 477–492 (with D.H. Phong).
  112. Elliptic families of solutions of the Kadomtsev-Petviashvili equation, and the field analogue of the elliptic Calogero-Moser system. (PDF: Russian, English, arXiv: hep-th/0203192)
    Funktsional. Anal. i Prilozhen. 36:4 (2002), 1–17 (with A.A. Akhmetshin and Yu.S. Volvovski).
  113. A two-dimensionalized Toda chain, commuting difference operators, and holomorphic vector bundles. (PDF: Russian, English, arXiv: math-ph/0308019)
    Uspekhi Mat. Nauk 58:3 (2003), 51–88 (with S.P. Novikov).
  114. Integrable chains on algebraic curves. (PDF: English, arXiv: hep-th/0309255)
    Geometry, topology, and mathematical physics, Amer. Math. Soc. Transl. Ser. 2 212 (2004), 219–236.
  115. Laplacian growth and Whitham equations of soliton theory. (PDF: English, arXiv: nlin/0311005)
    Phys. D 198:1–2 (2004), 1–28 (with M. Mineev-Weinstein, P. Wiegmann, and A. Zabrodin).
  116. Analytic theory of difference equations with rational and elliptic coefficients and the Riemann-Hilbert problem. (PDF: Russian, English, arXiv: math-ph/0407018)
    Uspekhi Mat. Nauk 59:6 (2004), 111–150.
  117. Integrable Structure of the Dirichlet Boundary Problem in Multiply-Connected Domains. (PDF: English, arXiv: hep-th/0309010)
    Comm. Math. Phys. 259:1 (2005), 1–44 (with A. Marshakov and A. Zabrodin).
  118. Algebraic versus Liouville integrability of the soliton systems. (PDF: English)
    XIVth International Congress on Mathematical Physics World Sci. Publ (2005), 50–67.
  119. Characterizing Jacobians via flexes of the Kummer Variety. (PDF: English, arXiv: math/0502138)
    Math. Res. Lett. 13:1 (2006), 109–123 (with E. Arbarello and G. Marini).
  120. Conformal mappings and the Whitham equations. (PDF: English)
    Surveys in modern mathematics, London Math. Soc. Lecture Note Ser. Cambridge Univ. Press (2005), 316–327.
  121. Integrable linear equations and the Riemann-Schottky problem. (PDF: English, arXiv: math/0504192)
    Algebraic geometry and number theory, Progr. Math. 253 (2006), 497–514.
  122. Integrable equations, addition theorems, and the Riemann-Schottky problem. (PDF: Russian, English)
    Uspekhi Mat. Nauk 61:1 (2006), 25–84 (with V.M. Bukhshtaber).
  123. A characterization of Prym varieties. (PDF: English, arXiv: math/0506238)
    Int. Math. Res. Not. Art. ID 81476 (2006), 36pp.
  124. Lax operator algebras. (PDF: Russian, English, arXiv: math/0701648)
    Funktsional. Anal. i Prilozhen. 41:4 (2007), 46–59 (with O.K. Sheinman).
  125. On the scaling limit of a singular integral operator. (PDF: English, arXiv: 0708.4157)
    Geom. Dedicata 132 (2008), 121–134 (with D.H. Phong).
  126. Abelian solutions of the KP equation. (PDF: English, arXiv: 0804.0274)
    Geometry, topology, and mathematical physics, Amer. Math. Soc. Transl. Ser. 2 224 (2008), 173–191 (with T. Shiota).
  127. Seiberg-Witten Theory, Symplectic Forms, and Hamiltonian Theory of Solitons. (PDF: English, arXiv: hep-th/0212313)
    Superstring theory, Adv. Lect. Math. (ALM) 1 (2008), 124–177 (with E. D'Hoker and D.H. Phong).
  128. Abelian solutions of the soliton equations and Riemann-Schottky problems. (PDF: English)
    Uspekhi Mat. Nauk 63:6(384) (2008), 19–30.
  129. The universal Whitham hierarchy and the geometry of the moduli space of pointed Riemann surfaces. (PDF: English, arXiv: 0810.2139)
    Surv. Differ. Geom. 14 (2009), 111–129 (with S. Grushevsky).
  130. Integrable discrete Schrödinger equations and a characterization of Prym varieties by a pair of quadrisecants. (PDF: English, arXiv: 0705.2829)
    Duke Math. J. 152:2 (2010), 317–371 (with S. Grushevsky).
  131. Characterizing Jacobians via trisecants of the Kummer Variety. (PDF: English, arXiv: math/0605625)
    Ann. of Math. (2) 172:1 (2010), 485–516.
  132. Abelian solutions of the soliton equations and geometry of abelian varieties. (PDF: English, arXiv: 0804.0794)
    Liaison, Schottky problem and invariant theory, Progr. Math. 280 (2010), 197–222 (with T. Shiota).
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    (with D. Zakharov).