Problems in affine algebraic geometry

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In "algebraic" geometry we study objects defined by algebraic (as opposed to simpler, e.g. linear, or more complicated, e.g. differentiable) equations. "Affine spaces" are familiar in this context mainly as friendly shelters for objects that really interest us, a sphere in 3-space for instance. Yet, when studied for their own sake, they pose surprisingly thorny and, despite a lot of effort, largely unresolved problems: What properties characterise them? What are their symmetries? What are the ways to embed one into another? These are the basic questions of "affine algebraic geometry". They often sound simple, but are addressed typically with highly sophisticated techniques from classical (projective) algebraic geometry.   Some of the interrelated problems to be studied in this proposal are: the linearization of torus actions on affine space, the structure of "exotic affine spaces" (algebraic varieties topologically indistinguishable from affine spaces), the study of affine surfaces with "simple topology" (e.g contractible ones), the structure of very "homogeneous" affine surfaces (having a large, very transitive automorphism group), the study of plane rational curves.   This research is a continuation of previous work and forms part of a large, lively international activity in this area.