Igor Krichever

Professor
Department of Mathematics
Columbia University
New York , NY 10027

Research Area

Algebraic Geometry, Integrable Systems.

Seminar

Math physics seminar.

Main Scientific Achievements

  1. General algebraic-geometrical construction of exact periodic and quasi-periodic solutions of non-linear integrable systems of the soliton theory.
  2. Solution of the classification problem of commuting ordinary differential operators. Effectivization of the results of twenties on the classification of such operators of co-prime orders.
  3. Application of the algebraic-geometrical methods of soliton theory for the problems of the solid state physics. Solution of the Peierls model and investigation of its perturbation.
  4. Construction of Floque spectral theory of periodic two-dimensional operators. The proof with its help the density of finite-gap solutions of the KP-2 equation in the space of all periodic solutions.
  5. Construction of the algebraic-geometrical perturbation theory for two-dimensional integrable systems. Construction of the exact solutions of the Whitham equations.
  6. Construction (with S.P.Novikov) theory of operator quantization of closed bosonic strings; Fourier-Laurent theory on Riemann surfaces of arbitrary genus; generalization of the Virasoro, Heisenberg and Kac-Moody algebras.
  7. Application of the Baker-Akhiezer functions for Atiyah-Hirzerbruch rigidity problem for multiplicative genera of manifolds. The proof with the help of methods of formal groups the rigidity property for the generalized elliptic genus for manifolds with zero first Chern class.
  8. The introduction of the tau-function for the universal Whitham hierarchy and a proof that this function coincides with the partition function of the topological field theory models. A proof that the tau-function of the dispersionless 2D Toda hierarchy is a generating function for conformal map of a simply-connected domain bounded bounded by smoooth curve.
  9. A construction of universal Hamiltonian approach to the soliton equations and construction of the action-angle variables.
  10. A theory of zero-curvature and isomonodromy equations on algebraic curves.
  11. Proof of Welter's trisecant conjecture, and solution of characterization problem for Prym varieties.

Honors and Awards
2022 Plenary talk ICM-22 (Virtual)
2003 Plenary talk at ICMP-2003, Lisbon
1997 Invited address AMS meeting, Urbana
1994-1996 Russian State science fellowship prize for outstanding scientists
1990 Invited address ICM-90, Tokyo
1990 The prize of the Mathematical Section of USSR Academy of Sciences