Steklov Mathematical Institute,
Gubkin St.8, Moscow, 119991, Russia,
Corresponding member of Russian Academy of Sciences (2003)
PRINCIPAL FIELDS OF RESEARCH: mathematical physics, p-adic analysis, including mathematical problems in quantum field theory, string theory, gravity, quantum information, quantum optics.
150 articles; and two monographies:
1) "p-adic Analys and Mathematical Physics" (with V.S. Vladimirov, E.I.Zelenov), World Sci.1994
2) "Quantum Theory and its Stochastic Limit" (with L.Accardi, Yu.G.Lu, 2002, Springer-Verlag).
SOME MAJOR RESULTS:
1) P-ADIC ANALYSIS and MATHEMATICAL PHYSICS.
In 1987 we suggested p-adic sting theory and p-adic quantum mechanics and more generally application of p-adic analysis in mathematical physics. In 1989 with V.S.Vladimirov we developed a formulation of p-adic quantum harmonic oscillator and performed spectral harmonic analysis of this system. More generally, applications of p-adic analysis to the spacetime stracture at the Planck lengh and to the studying of complex systems were initiated. These works have stimulated a wide activity by many researchers.
2) THE STOCHASTIC LIMIT METHOD IN QUANTUM THEORY.
We have developed a new asymptotic method in quantum theory of dynamical systems with quantum white noise as the leading term (with L.Accardi and Y.G.Lu). The origin of the method comes back to the works of Bogoliubov, Van Hove, Prigogine. This method was has been used for studying time dependence in numerous physical models in quantum optics, solid states physics. In many important cases it permits to compute explicitly time dependent correlation functions. The method is exposed in the monography mentioned above. The method is used by many researchers. There were organized 3 international schools devoted to the method.
3) THEORY OF DEFECTS AND TORSION.
Geometric theory of defects in the elastic body with continuous distribution of dislocations and desclinations based on the 3 dimensional Euclidean theory with torsion was developed (with M.Katanaev).
4) There are several pure mathematical works published in mathematical journals including the proof of existence of solution to a nonlinear elliptic equation, theory of conservation laws for nonlinear equations, computation of Chern numbers for anti-Kahlerian manifolds, investigation of p-adic Brownian motion, theory of knots, studying of Levy laplacian on manifolds.
5) QUANTUM INFORMATION.
Last years I work on the problems of quantum information and computers. In particular on the problems of quantum decoherence and theory of algorithms. I have proposed to build a quantum computer from a single atom. With M.Ohya we have proposed an approach (which goes beyond of the standard paradigm of quantum computation) to the solution of NP-complete problem in polynomial time. A new approach to problems of quantum nonlocality, Bell`s inequalities, quantum cryptography was developed based on the studying of the spatial dependence of entangled states.
LECTURES AND TALKS ON QUANTUM INFORMATION AND QUANTUM COMPUTERS. I have given a semester course of lectures on "Quantum computers" in Moscow State University in 1998-1999 for students and researchers, gave a semester course of lectures on quantum computers in 2000 in Simon Fraser University, Vancouver, Canada and in Independent University in Moscow, 2002. Invited talks at International conferences and schools on Quantum information in Japan (1997, 1999, 2001), Canada (2000), England (2000), Italy (2000.2001,2002), Sweden (2000,2002).
Invited speaker at numerous international conferences including twice at the International congresses in Mathematical Physics. I have many joint papers with researchers from Italy, Japan, Canada, France, Germany. I have visited many countries, collaborate and giving lectures at Universities in Rome, Turin, Tokyo, Nagoya, London, Berlin, Wroclaw, Cambridge, Geneve,Goteborg, Kopenhagen, Nancy, Amsterdam, Nottingham, Bombay, New-York, Vaxjo (Sweden), Princeton and Harvard Universities in USA.
INVOLVEMENT in Europe Union projects:
the leader of the team in INTAS grants 96-0698 and 99-05-90.
Also CRDF grant (USA).
The leader of the RFFI projects for the last 8 years.