Andrey I. Shafarevich

1. Life and career

Andrey I. Shafarevich was born on 1 September 1963. In 1981 he entered the Faculty of Physics of Lomonosov Moscow State University, from which he graduated in 1987 at the Department of Mathematics with a diploma thesis “Asymptotic solutions of equations of a stratified fluid with rapidly varying coefficients” written under the supervision of S. Yu. Dobrokhotov and V. P. Maslov. In the same year he entered the postgraduate programme of the Faculty of Physics, completing it in 1990.

In 1990 he defended his candidate dissertation “Asymptotic solutions of evolution problems with rapidly varying coefficients and localized initial conditions” (advisors S. Yu. Dobrokhotov, V. P. Maslov and A. Kh. Khrgian). After the postgraduate programme he joined the Laboratory of Mathematical Methods in Mechanics of the Institute for Problems in Mechanics of the Russian Academy of Sciences (headed by Academician V. P. Maslov), and in 1994 he moved to the Department of Differential Geometry and Applications of the Faculty of Mechanics and Mathematics of MSU (headed by Academician A. T. Fomenko).

In 1999 he defended his doctoral dissertation “Localized solutions of the Navier–Stokes equations”, and since 2002 he has been Professor at the Department.

At the Department he has taught the compulsory courses “Classical Differential Geometry”, “Differential Geometry and Topology”, “Differential-Geometric and Topological Methods” and “Mathematical Methods in Economics”, the natural-science courses “Geometric Structures in Quantum Mechanics” and “Geometric Methods in Quantum Mechanics”, and special courses on mathematical physics and on geometric aspects of classical and quantum physics.

Part-time he works at the Moscow Institute of Physics and Technology (Faculty of Nano-, Bio-, Information and Cognitive Technologies; until 2006 the Institute of Natural Sciences and Ecology at the Kurchatov Institute), where he teaches the compulsory courses “Mathematical Analysis”, “Linear Algebra and Geometry”, “Equations of Mathematical Physics” and “Group Theory”, and a special course on non-linear partial differential equations. A. I. Shafarevich is a member of the methodological council of the Faculty of Mechanics and Mathematics of MSU, of the academic council of MIPT and of the dissertation council of MIEM. At various times he has also worked at schools (school No. 16, the Lyceum “Second School”, school No. 2030).

2. Degrees and positions

3. Research

The main scientific interests of A. I. Shafarevich lie in mathematical physics, the asymptotic and geometric theory of linear and non-linear partial differential equations, quantum mechanics and hydrodynamics. He solved the problem of multiphase asymptotics for the equations of hydrodynamics posed by V. P. Maslov and widely discussed in the literature. In doing so, methods for studying asymptotic solutions of non-linear partial differential equations were substantially developed; connections of such solutions with topological invariants of vector fields and Liouville foliations were established, and a method of describing asymptotics by means of equations on graphs was developed. Various applications to mechanics were obtained, in particular vortex filaments, films and narrow peaks in an incompressible fluid were studied.

A. I. Shafarevich has also worked on the theory of quasiclassical quantization of invariant isotropic manifolds of Hamiltonian systems. In this area he obtained diverse results, in particular he described spectral series corresponding to trajectories and invariant tori in resonance systems and to one-dimensional singular sets of partially integrable systems, and described quasiclassical spectral series of operators with point potentials.

Among his other results are the description of singular and localized solutions of the linearized Navier–Stokes equations and of the magnetic dynamo problem, and the construction of a counterexample to the diffusion approximation method.

4. Publications (selected)

Full list of publications — 298 records (department page and ISTINA profile)

1. A. I. Shafarevich. Localized solutions of the Navier–Stokes equations. Doctoral dissertation, Moscow, MSU, 1999.
2. Asymptotic solutions of evolution problems with rapidly varying coefficients and localized initial conditions. Candidate dissertation, Moscow, MSU, 1990.
3. A. I. Shafarevich et al. Asymptotic properties of classical dynamical systems in quantum problems on singular spaces. Russian Journal of Mathematical Physics (2010).
4. M. A. Nikulin, F. Yu. Popelensky, A. I. Shafarevich. Asymptotic behaviour of energy levels of a quantum free particle in an elliptic sector. Physica Scripta, vol. 99, no. 1 (2024), 015207.
5. A. T. Fomenko, A. I. Shafarevich, V. A. Kibkalo. Main recent directions and achievements of the Chair of Differential Geometry and Applications. Moscow University Mathematics Bulletin, 79:6 (2024), pp. 283–295.

The full list of publications (298 records) is given on the Full list of publications page.

5. Links and sources

• Russian personal page: dfgm.math.msu.su/people/shafarevich/.
• ISTINA profile: istina.msu.ru.

Compiled from the department page dfgm.math.msu.su/people/shafarevich/ (biography, research interests, teaching, publications).