Russian Mathematical Surveys, 2023, Volume 78, Issue 1, Pages 203–204 DOI: https://doi.org/10.4213/rm10097e (Mi rm10097)

DOI: https://doi.org/10.4213/rm10097e

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The prominent mathematician, author of fundamental works on algebraic geometry, mathematical physics, complex and real algebraic geometry Igor Moiseevich Krichever passed away at the age of 72 on 1 December 2022 after a serious illness. His life path and his contributions to mathematics were the topics of the tribute paper “Igor Mosiseevich Krichever (on his 70th birthday)”, Uspekhi Matematichekikh Nauk, 76:4 (2021), 183–193.

The first cycle of Krichever’s papers concerned a well-known problem in algebraic topology, actions of compact Lie groups on smooth manifolds. In this area he introduced the $G$-equivariant Hirzebruch genus taking values in the ring of formal series, derived a localization formula for this genus and, as a consequence, obtained a theorem on the rigidity of this genus for $S^1$-manifolds with vanishing first Chern class, that is, for Calabi–Yau $S^1$-manifolds. This result gained much response and found a number of applications. Since then the term ‘Krichever genus’ has been used in the literature.

Krichever’s results at the crossroads of algebraic geometry and mathematical physics received the widest recognition. His work determined largely the shape of the state of the art soliton theory and underlies the new language which uses the concepts from real and complex geometry of algebraic curves to describe solutions of well-known equations in the theory of integrable systems, mathematical and theoretical physics. Krichever’s construction of finite-gap solutions of the Kadomtsev–Petviashvili equations brought S. P. Novikov to his famous conjecture concerning the classical Schottky problem and was used in T. Shiota’s solution (1986) of this problem. The statements of the classical Schottky problem and its contemporary generalizations involve the Jacobi and Prym varieties of algebraic curves. In the early 2000s Krichever proposed to characterize Jacobi and Prym varieties, instead of soliton equations, in terms of linear problems associated with these equations. This approach enabled him to find a new solution to the Schottky problem for Jacobians, solve the corresponding problem for Prym varieties, and prove Welters’ trisecant conjecture, which is well known in algebraic geometry.

Krichever’s important achievement was to extend the algebro-geometric method of integration to systems where some or all of the variables are discrete. In a series of papers (some of which were written with co-authors) he integrated the discrete Peierls–Frölich model and examined perturbations of this model. He developed the algebro-geometric averaging theory for two-dimensional integrable equations of soliton theory, deduced Whitham’s equations for multiphase solutions of equations of Kadomtsev–Petviashvili type, and presented a construction of their exact solutions. In conjunction with other authors, he revealed the connections of the Seiberg–Witten solution of the equations of supersymmetric gauge theoried for $N=2$ with the theory of integrable system. In this direction, in 1994 Krichever introduced the quasi-classical tau function in maximal generality. The results of this work underlie the modern method of topological recursion, where ‘Whitham–Krichever differentials’ are now known.

Krichever was one of the most respected active mathematician of our time. As a leader in several topical research areas, he was a plenary speaker at the Internatonal congress of mathematicians in 2022. He was a deputy editor-in-chief of the journal Funtktsional’nyi Analiz i ego Prilozheniya^1[x]1Translated into English as Funcational Analysis and its Applications., a member of the editorial board of the journal Uspekhi Matematichskikh Nauk^2[x]2Translated into English as Russian Mathematical Surveys., a member of the board of the Moscow Mathematical Society. He distinguished himself as a remarkable organizer and research supervisor at the responsible position of the scientific director of A. A. Kharkevich Institute for Information Transmission Problems of the Russian Academy of Sciences and, subsequently, as the director of the Skoltech Center for Advanced Studies.

The entire mathematical community, all persons who knew and collaborated with Igor Krichever will retain a grateful memory of him in their hearts.

Citation in format AMSBIB

\Bibitem{BucNovTai23}
\by V.~M.~Buchstaber, S.~P.~Novikov, I.~A.~Taimanov
\paper Igor Moiseevich Krichever (obituary)
\jour Russian Math. Surveys
\yr 2023
\vol 78
\issue 1
\pages 203--204
\mathnet{http://mi.mathnet.ru//eng/rm10097}
\crossref{https://doi.org/10.4213/rm10097e}
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