S. O. Gorchinskiy, V. G. Danilov, S. Yu. Dobrokhotov, V. V. Kozlov, V. E. Nazaikinskii, N. N. Nefedov, D. O. Orlov, S. A. Stepin, I. A. Taimanov, D. V. Treschev, N. A. Tyurin, A. T. Fomenko, S. E. Yakush, “Andrei Igorevich Shafarevich (on his sixtieth birthday)”, Russian Math. Surveys, 79:3 (2024), 553

The corresponding member of the Russian Academy of Sciences, dean of the Faculty of Mechanics and Mathematics at Lomonosov Moscow State University Andrei Igorevich Shafarevich observed his sixtieth birthday on 1 September 2023.

He was born in 1963 in Moscow, in a family of mathematicians: his father Igor Rostislavovich Shafarevich was a prominent Russian researcher, political writer, and thinker, and his mother Nina Ivanovna taught mathematics in the Moscow Engineering Physics Institute for many years.

In 1981 Andrei Shafarevich enrolled in the Faculty of Physics at Moscow State University as the door to the Faculty of Mechanics and Mathematics was closed to him because of the public position and human rights activity of his father, who had been dismissed from Moscow University shortly before. However, Andrei’s penchant for mathematics and the example of his parents determined his choice of scientific career, which started at the Department of Mathematics of the Faculty of Physics. At that time Petr Nikolaevich Zhevandrov taught classes on general mathematics subjects at the Faculty, while Sergei Yur’evich Dobrokhotov read there a special course on asymptotic methods. Meeting these former students of Viktor Pavlovich Maslov was a landmark in Andrei’s life: in his second year at the university he got interested in their range of problems and anchored his life in science with Maslov’s scientific school. At the Department of Mathematics he prepared his diploma thesis; its topic had been proposed by Maslov, who became Shafarevich’s scientific advisor.

His first research paper was written in co-authorship with Dobrokhotov and published in Doklady Akademii Nauk¹ when he was in his fifth year at the university. Both Maslov and Dobrokhotov were advisors of his Ph.D. thesis Asymptotic solutions of evolution problems with rapidly varying coefficients and localized initial conditions. After defending the thesis Andrei Shafarevich worked for several years in the Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences, but in 1994 he moved to a permanent job at Moscow State University. There he developed from a lecturer at the Department of Differential Geometry to a dean of the Faculty of Mechanics and Mathematics, continuing to combine teaching and administrative work with research at the Institute for Problems in Mechanics. In 1999 he defended his D.Sc. thesis, Localized solutions of the Navier–Stokes equations at the Steklov Mathematical Institute of Russian Academy of Sciences. With time his research results got widely appreciated by the academic environment: in 2016 he was elected a corresponding member of the Russian Academy of Sciences, the Division of Mathematical Sciences. In 2019 he was elected a dean of the Faculty of Mechanics and Mathematics at Moscow State University, where, as his first step, he initiated the introduction of the new specialization “Fundamental mathematics and mathematical physics”. This reflected an increased demand for a return to the classical relationships between mathematics and physics. In 2022, as a well-known expert in the field, Andrei Shafarevich was appointed the head of the Department of Mathematical Physics of the Steklov Mathematical Institute.

His area of research interests is quite wide and lies at the interface of quantum mechanics, hydrodynamics, and mathematical physics, including an important component of the latter, the asymptotic and geometric theory of linear and nonlinear partial differential equations. One of Andrei Shafarevich’s main results was his solution of an important problem stated by Maslov and concerning the multiphase asymptotics for equations of hydrodynamics. In solving this problem Shafarevich developed significantly the methods of analysis of asymptotic solutions of nonlinear partial differential equations and established connections between such solutions and topological invariants of vector fields and Liouville foliations. He made a profound contribution to the construction and study of singular and localized asymptotic solutions of the Navier–Stokes equations and equations of magnetic hydrodynamics. He found the asymptotic behaviour of waves propagating on singular spaces, in particular, on geometric graphs and hybrid manifolds. It turned out on the way that this asymptotic behaviour is closely connected with some well-known problems in number theory.

In conjunction with his colleagues, Shafarevich obtained profound results in the theory of quasiclassic quantization of invariant isotropic manifolds for Hamiltonian systems. In a recent cycle of joint papers with Dobrokhotov and V. E. Nazaikinskii they developed new representations for the Maslov canonical operator. An important difference from its original description in canonical variables was the use of arbitrary coordinates on Lagrangian submanifolds, which in some situations allows one to apply this construction to the non-compact case as well. Using such representations, the class of problems in which the method of the canonical operators can be used is extended to include, in particular, some important physics problems. For instance, localized solutions of a wide class of (pseudo)differential equations can effectively be described, which opens new prospects for the applications of quasiclassical methods to applied problems.

An important part of Shafarevich’s research is the calculation of spectral asymptotics for non-selfadjoint operators, including Schrödinger-type equations with complex potentials. For operators of the last type he managed to describe the quasiclassical asymptotics of the spectrum in the complex plane by using a complex analogue of the Bohr–Sommerfeld quantization rule, which is a quantization rule of a Riemann surface of constant energy.

In recent years Andrei Shafarevich developed new methods for the construction of asymptotic formulae for waves propagating in media with characteristics rapidly varying in a neighbourhood of a set of positive codimension. In particular, he optained general results on such solutions of strictly hyperbolic systems.

Apart from intensive research, Andrei Shafarevich has for many years been engaged in teaching at Moscow State University and Moscow Institute for Physics and Technology. His lectures are very popular with students and are a model of presentation which is both profound and accessible in its form. We must also stress that he is quite enthusiastically involved in the teaching process, which evokes as a feedback the first flashes of research inspiration among his listeners and students engaged in joint or autonomous work. He often delivers talks and lectures at conferences and scientific schools at scientific centres in Russia and overseas. He supervises the research seminar “Differential operators on singular spaces, algebraically integrable systems, and quantization”, at the Faculty of Mechanics and Mathematics at Moscow State University, and is one of the supervisors of the seminar “Asymptotic methods in mathematical physics” at the Institute for Problems in Mechanics. Apart from teaching at universities, for many years he also taught mathematics in school no. 1189 under the auspices of the Kurchatov Institute and in the legendary ‘School no. 2’, where he trained prospective researchers.

Andrei Shafarevich is the editor-in-chief of the highly rated international journal Russian Journal of Mathematical Physics and a member of the editorial boards of Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematichaskaya², Vestnik Moskovskogo Universiteta. Seriya 1. Matematika. Mekhanika³, and Regulyarnaya i Khaoticheskaya Dinamika⁴. He published more that 170 research papers⁵, and was a scientific advisor to six Ph.D. dissertations.

Andrei Shafarevich is a lively and multifaceted person. His interests extend far beyond his research and pedagogical work. Raised in a family of academics, since his early years he got acquainted to a broad range of issues of literature, history, and philosophy. From a very young age, he went through many routes with his father, on foot and by kayak, and he has maintained this passion till this day by participating in hiking trips over the Moscow region on weekends and in many-day travels on his vacancies. For all his broad outlook and deep intelligence, he is easy to talk and responsive. He can be a genuine friend, to which many people throughout Russia and abroad can attest.

From our hearts we wish Andrei Shafarevich new successes in research, creative activity, happy life, and good health.

Citation: S. O. Gorchinskiy, V. G. Danilov, S. Yu. Dobrokhotov, V. V. Kozlov, V. E. Nazaikinskii, N. N. Nefedov, D. O. Orlov, S. A. Stepin, I. A. Taimanov, D. V. Treschev, N. A. Tyurin, A. T. Fomenko, S. E. Yakush, “Andrei Igorevich Shafarevich (on his sixtieth birthday)”, Russian Math. Surveys, 79:3 (2024), 553–556

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