M.Sc. (Moscow State University)
Ph.D. (Steklov Math.l Inst. of Academy of Sciences of USSR)
Habilitation Doctor Degree (Landau Inst. for Theoretical Physics)
My general area of research is mathematical physics and its applications. During the past decade I have worked on the theory of tensor invariants of dynamical systems and systems of partial differential equations. I defined and studied rings of cohomologies of dynamical systems. These tensor invariants distinguish integrable dynamical systems from the generic ergodic systems.
I introduced an extended concept of integrability of Hamiltonian systems that generalizes the classical Liouville concept and includes Hamiltonian systems with invariant tori Tq of arbitrary dimension q : k < q £ 2k on a symplectic manifolds M2k (the Liouville concept allows only tori of dimension k and less).
In the area of partial differential equations, I found effective tensorial necessary conditions for the existence of Hamiltonian structures and conservation laws. These results have led to the effective necessary conditions for separability of systems of partial differential equations (the Courant.s problem).
Recently, I have been working along the directions above on the applications of the theory of tensor invariants to problems of plasma equilibrium. These problems arise in astrophysics and in controlled thermonuclear fusion. I have derived two new families of exact solutions to the plasma equilibrium equations which possess axial and helical symmetries. The solutions model the astrophysical jets, solar prominencies and the external and internal kink modes of plasma in tokamaks. I have introduced a new method of symmetry transforms for constructing the ideal magnetohydrodynamics equilibria.