Danijela Damjanovic: Full, parametric and transversal local rigidity for smooth group actions: guided tour of examples and a general result

  I will discuss the question concerning classification of small smooth perturbations of Lie group actions on smooth manifolds.The starting point is the classical result obtained via KAM (Kolmogorov-Arnold-Moser) method for Diophantine circle rotations. I will present a general result for Lie group actions in which the role of Diophantine conditions is taken by certain cohomological assumptions. Several classes of algebraic examples will be discussed, where the cohomological conditions can be checked with the use of representation theory. I will also list several examples where the general result may apply, and dynamical issues which arise in studying the cohomological properties of actions.