Speaker: Dmitry Treschev (Steklov Mathematical Institute of RAS)

Date: Nov. 14 (Tuesday)

Beijing Time: 19:00-20:00（Moscow time: 14:00-15:00）

Zoom ID: 835 3144 4792 Password: 355096

Abstract:

I propose a new approach to the theory of normal forms for Hamiltonian ODE systems near a non-degenerate equilibrium position. The traditional normalization procedure is performed step-by-step: non-resonant terms in the expansion of the Hamiltonian function are removed first in the lowest degree, then in the next one and so on. I consider the space of all Hamiltonian functions with equilibrium position at the origin and construct a differential equation in this space. Solutions of this equation move Hamiltonian functions towards their normal forms. Shifts along the flow of this equation correspond to canonical coordinate changes. So, we have a continuous normalization procedure. The formal aspect of the theory presents no difficulties. The analytic aspect and the problems of convergence of series, as usual, non-trivial.

Biography:

Prof. Treschev received his doctor degree of physics and mathematics in 1992. At Moscow State University, he is since 1993 a leading researcher, since 1998 a professor, and since 2006 head of the Department of Theoretical Mechanics. At the Steklov Institute he became in 2005 a chief researcher and the deputy director for research and is since 2017 the director for research. Prof. Treschev deals with integrability and non-integrability, dynamical stability, KAM theory, separatrix splitting, averaging in slow-fast systems, chaos in Hamiltonian dynamics, Arnold diffusion, statistical mechanics, and ergodic theory. In 1995 Prof. Treschev was a Laureate of the State Prize of the Russian Federation for young scientists. In 2007 he was awarded the Lyapunov Prize. In 2002 he was an invited speaker with talk Continuous averaging in dynamical systems at the International Congress of Mathematicians in Beijing.