Department of Mathematical Physics

The Department of Mathematical Physics was founded in 1969 by Academician of the USSR Academy of Sciences Prof. V. S. Vladimirov. The first research staff of the Department included several members from the Department of Theoretical Physics (S. S. Khoruzhii and B. M. Stepanov) and from the Department of Differential Equations (V. P. Mikhailov, A. A. Dezin, V. N. Maslennikova, V. S. Vinogradov, and PhD student A. K. Gushchin) of the Steklov Mathematical Institute. In 1970 the Laboratory of Partial Differential Equations was founded in the Department under the direction of V. P. Mikhailov.

Now the research staff of the Department includes 10 permanent researchers: Corresponding Member of the Russian Academy of Sciences Prof. I. V. Volovich (Head of the Department), Professors Yu. N. Drozhzhinov, A. K. Gushchin, M. O. Katanaev, S. V. Kozyrev, N. G. Marchyuk, A. N. Pechen, A. G. Sergeev, V. V. Zharinov, and Dr. A. S. Trushechkin. PhD student: N. B. Il'in.

The main current research directions at the Department are the following:

• P-adic mathematical physics (V. S. Vladimirov, I. V. Volovich, S. V. Kozyrev, E. I. Zelenov),
• Quantum information and theory of open quantum systems (I. V. Volovich, S. V. Kozyrev, A. N. Pechen, A. S. Trushechkin),
• Boundary value problems for equations of mathematical physics, including non-local equations (V. S. Vladimirov, A. K. Gushchin, V. P. Mikhailov),
• Multidimensional complex analysis, multidimensional and one-dimensional Tauberian theory for distributions (Yu. N. Drozhzhinov, B. I. Zavialov),
• Applications of the multidimensional complex analysis to quantum field theory and gauge field theory (A. G. Sergeev, R. V. Palvelev),
• Algebro-geometric approach to differential equations (V. V. Zharinov),
• Gravity models and the geometric theory of defects (M. O. Katanaev),
• Mathematical models in biology (I. V. Volovich, S. V. Kozyrev),
• Equations of relativistic field theory, Dirac equation, Clifford algebras (N. G. Marchyuk),
• Kinetic theory and non-equilibrium processes, Loschmidt's paradox, quantum mechanics in bounded domains, mathematical theory of nanosystems (I. V. Volovich, S. V. Kozyrev, A. N. Pechen, A. S. Trushechkin),
• Quantum control (A. N. Pechen).

The most fundamental results obtained by the Department's members include:

• Proof and application in the axiomatic field theory of the Bogolyubov-Vladimirov 'finite covariance' theorem (N. N. Bogolyubov, V. S. Vladimirov);
• Proof of the theorem on the C-convex hull (V. S. Vladimirov);
• Development of p-adic mathematical physics, including p-adic quantum mechanics and p-adic string theory (V. S. Vladimirov, I. V. Volovich, S. V. Kozyrev, E. I. Zelenov);
• Derivation of necessary and sufficient conditions for automodel asymptotic in quantum field theory (N. N. Bogolyubov, V. S. Vladimirov);
• Development of Tauberian theory for distributions of many variables (V. S. Vladimirov, Yu. N. Drozhzhinov, B. I. Zavialov);
• Development of the geometric theory of defects in solids (I. V. Volovich, M. O. Katanaev);
• Construction of the master field describing the limit of the large gauge group in gauge theory (I. V. Volovich);
• Construction of new solutions of non-linear partial differential equations of supergravity. Estimation of the probability of the creation of black holes and wormholes in the high energy scattering of particles (I. V. Volovich);
• Derivation of a necessary and sufficient condition for the existence of a mean square limit at the boundary of solutions of linear second-order ellyptic differential equations (V. P. Mikhailov);
• Proof of the theorems on the stabilization and quasistabilization for long times of the solutions of boundary value problems for linear parabolic and hyperbolic second-order differential equations (A. K. Gushchin, V. P. Mikhailov);
• Proof of a compact version of the extended future tube conjecture (A. G. Sergeev, jointly with German mathematician P. Heinzner). Later a student of A. G. Sergeev, Chinese mathematician Zhou Xiangyu proved this conjecture in the general setting;
• Twistor quantization of loop spaces of compact Lie groups (A. G. Sergeev, some results were obtained jointly with A. Popov and Bulgarian mathematician J. Davidov) and quantization of the universal Teichmueller space (A. G. Sergeev);
• Mathematical description of the adiabatic limit in the Ginzburg-Landau and Seiberg-Witten equations (A. G. Sergeev);
• Unification and extension in terms of a special functional space of the fundamental properties (such as belonging to a corresponding local Sobolev space and Holder continuity) of generalized solutions of linear second-order elliptic equations with measurable and bounded coefficients. Derivation of global estimates for the solution of the Dirichlet problem with a quadratically integrable boundary function which describe its behavior near the boundary (A. K. Gushchin);
• Classification of global solutions of equations for two-dimensional gravity (M. O. Katanaev);
• Construction of a deductive scheme of quantum mechanics on a line (A. A. Dezin);
• Development of the theory of p-adic wavelets and applications of p-adic analysis in the theory of spin glasses and other complex systems (S. V. Kozyrev);
• Generalization of the Dirac equation possessing an additional symmetry with respect to the pseudo-unitary, symplectic, or spinor group (N. G. Marchyuk);
• Derivation of equations for higher order corrections to the stochastic weak coupling limit of open quantum systems (I. V. Volovich and A. N. Pechen);
• Development of a quantum white noise technique for the analysis of the low density limit for open quantum systems (A. N. Pechen, in part jointly with L. Accardi and I. V. Volovich);
• Proof of the existence of second order traps for a wide class of quantum control systems (A. N. Pechen and D. J. Tannor).

V. S. Vladimirov

Mathematical Problems of the Uniform-Speed Theory of Transport (in Russian).

Moscow, Izdat. Akad. Nauk SSSR, 1961, 158 pp. (Trudy Mat. Inst. Steklov., Vol. 61)

V. S. Vladimirov

Methods in the Theory of Functions of Several Complex Variables (in Russian).

Moscow, Nauka, 1964, 411 pp.

V. S. Vladimirov

Equations of Mathematical Physics (in Russian).

Moscow, Nauka, 1967, 436 pp.

Ibid., 2nd ed., Recasted and revised, Moscow, Nauka, 1971, 512 pp.

Ibid., 3rd ed., Moscow, Nauka, 1976, 527 pp.

Ibid., 4th ed., Edited and extended, Moscow, Nauka, 1981, 512 pp.

Ibid., 5th ed., Extended, Moscow, Nauka, 1988, 512 pp.

V. S. Vladimirov

Generalized Functions in Mathematical Physics (in Russian).

Moscow, Nauka, 1976, 280 pp. (in series Current Problems in Physics and Technology)

Ibid., 2nd ed., Extended and revised, Moscow, Nauka, 1979, 318 pp.

V. S. Vladimirov

Methods of the Theory of Generalized Functions (in English).

London, Taylor & Francis, 2002, 311 pp. ISBN 0415273560.

V. S. Vladimirov, I. V. Volovich, and E. I. Zelenov

P-adic Analysis and Mathematical Physics (in Russian).

Moscow, Nauka, Fiz. Mat. Lit., 1994, 352 pp.

V. S. Vladimirov, I. V. Volovich, and E. I. Zelenov

P-adic Analysis and Mathematical Physics (in English).

World Scientific Pub Co Inc, 1995, 456 pp. ISBN 9810208804.

V. S. Vladimirov, Yu. N. Drozhzhinov, and B. I. Zav'yalov

Multidimensional Tauberian Theorems for Generalized Functions (in Russian).

Moscow, Nauka, 1986, 304 pp.

V. S. Vladimirov, Yu. N. Drozhzhinov, and B. I. Zav'yalov

Tauberian Theorems for Generalized Functions (in English).

Springer, 1988, 308 pp. ISBN 9027723834.

V. S. Vladimirov and V. V. Zharinov

Equations of Mathematical Physics. Textbook (in Russian).

Moscow, Fiz. Mat. Lit., 2000, 399 pp.

V. S. Vladimirov, V. P. Mikhailov, et al.

Collected Problems on Equations of Mathematical Physics. Textbook (in Russian).

Moscow, Nauka, 1974, 272 pp.

Ibid., 2nd ed., Extended and revised, Moscow, Nauka, 1982, 256 pp.

Ibid., 3rd ed., Revised, Moscow, Fiz. Mat. Lit., 2001, 288 pp.

V. S. Vladimirov (Editor)

A collection of Problems on the Equations of Mathematical Physics (in English).

Springer, 1986, 288 pp. ISBN 3540166475.

L. Accardi, Y. G. Lu, I. V. Volovich

Quantum Theory and Its Stochastic Limit (in English).

Berlin, Springer-Verlag, 2002.

M. Ohya, I. V. Volovich

Mathematical Foundations of Quantum Information and Computation and Its Applications to Nano- and Bio-systems (in English).

Springer, 2011.

A. A. Dezin

Invariant Differential Operators and Boundary Value Problems (in Russian).

Moscow, Izdat. Akad. Nauk SSSR, 1962, 88 pp. (Trudy Mat. Inst. Steklov, vol. 68)

A. A. Dezin

General Questions of the Theory of Boundary Value Problems (in Russian).

Moscow, Nauka, 1980, 207 pp.

A. A. Dezin

Multidimensional analysis and discrete models (in Russian).

Moscow, Nauka, 1990, 238 pp.

A. A. Dezin

Differential Operator Equations: A Method of Model Operators in the Theory of Boundary Value Problems (in Russian).

Moscow, Nauka, 2000, 175 pp. (Trudy Mat. Inst. Steklov., Vol. 229).

V. V. Zharinov

Distributive Lattices and their Applications in Complex Analysis (in Russian).

Moscow, Nauka, 1983, 80 pp. (Trudy Mat. Inst. Steklov., 162)

V. V. Zharinov

Lecture Notes on Geometrical Aspects of Partial Differential Equations (in English).

Singapore, World Scientific, 1992. 360 pp. (Ser. on Soviet and East-European Mathematics; Vol. 9)

N. G. Marchuk

Fields Theory Equations and Clifford Algebras (in Russian).

RHD, Moscow-Izhevsk, 302 pages, 2009.

V. P. Mikhailov

Partial Differential Equations. Textbook (in Russian).

Moscow, Nauka, 1976, 391 pp.

Ibid., 2nd ed., Recasted and revised, Moscow, Nauka, 1983, 424 pp.

A. G. Sergeev

Kahler Geometry of Loop Spaces.

Moscow, MCCME, 2001, 128 pp. (Ser. Modern Mathematical Physics, Problems and Methods, Vol. 4)

A. G. Sergeev

Kahler Geometry of Loop Spaces (Nagoya Math. Lectures, vol. 7) (in English).

Nagoya, Nagoya Univ., 2008. 226  pp.

A. G. Sergeev

Kahler Geometry of Loop Spaces (MSJ Memoirs, 23) (in English).

Tokyo, Mathematical Society of Japan, 2010. 212 pp. ISBN: 978-4-931469-60-0.

A. G. Sergeev

Vortices and SeibergВ–Witten equations (Nagoya Math. Lect., vol. 5) (in English).

Nagoya, Nagoya Univ., 2002. 87 pp.

A. V. Domrin, A. G. Sergeev

Lectures on Complex Analysis. Part I.

Steklov Mathematical Institute, Moscow, 2004. ISBN: 5-98419-007-9, http://www.mi.ras.ru/books/pdf/ser1.pdf

A. V. Domrin, A. G. Sergeev

Lectures on Complex Analysis. Part II.

Steklov Mathematical Institute, Moscow, 2004. ISBN: 5-98419-008-7, http://www.mi.ras.ru/books/pdf/ser2.pdf

S. S. Khoruzhii

Introduction to Algebraic Quantum Field Theory.

Moscow, Nauka, 1986, 304 pp.

S. V. Kozyrev

Methods and applications of ultrametric and p-adic analysis: from wavelet theory to biophysics.

Modern Problems of Mathematics, 12, Steklov Math. Inst., Russian Academy of Sciences, Moscow, 2008, 170 pp.