This project treats algebraic (and analytic) actions on affine varieties from the viewpoint of smooth (differentiable) transformation groups. The subject of smooth transformation groups has been strongly influenced by the following two central problems: LINEARITY PROBLEM. Which groups act nonlinearly on affine space of some dimension? (An action is linear if it is conjugate to an action defined by a representation of the group.) FIXED POINT PROBLEM. Which groups act without fixed points on affine space of some dimension? The tools used to settle these problems and the further problems they generated have been important themes in the field of smooth transformation groups. During the last few years researchers from algebraic transformation groups as well as those from smooth transformation groups have recognized the importance of these two questions in the algebraic category, and in that case the groups dealt with should be reductive (which includes as a special case the finite groups). Petrie will continue his work in this direction, bringing topological methods to bear on algebraic questions.
