School of Mathematical Sciences

Department Colloquium

Schreiber 006, 12:15

Pennsylvania State University

Abstract:The work of A. N. Kolmogorov, J. Moser and V.I. Arnold revolutionized classical mechanics half a century ago by showing that simple quasi-periodic motions are persistent and ubiquitous in general Hamiltonian systems contrary to the belief going back to Poincare that fully ``chaotic'' behavior should be the rule.

KAM method found a variety of other applications. I will discuss one of those developments that appeared in the last few years in a series of joint papers with Danijela Damjanovic.

We use KAM type scheme for a purpose that from the first glance looks as an antithesis to its original use: we consider certain dynamical systems with multi-dimensional ``time'' and orbit structure of either high complexity (hyperbolic and partially hyperbolic behavior) or, even more surprisingly, intermediate complexity (parabolic behavior) and show for that orbit structure persistence in a strong sense (differentiable rigidity). From the point of view of functional analysis we solve certain overdetermined systems of nonlinear functional equations. Tools used in the proofs include Fourier analysis and unitary representations of semi-simple Lie groups.

Coffee will be served at 12:00 before the lecture

at Schreiber building 006