Algebraic group actions and the structure of affine algebraic varieties

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1.Linearization ProblemWe have had some results on the Cancellation Problem which is closely related to the Linearization Problem.(1)There exists an infinite dimensional family of affine pseudo-planes without cancellation property. Furthermore, tom Dieck surfaces are characterized as affine pseudo-planes with nontrivial actions of an algebraic torus.(2)The kernel of a triangular derivation with a slice is a polynomial ring.2.Generalized Jacobian Problem(1)We studied the Jacobian Problem for smooth affine surfaces with A^1-fibrations, especially, affine pseudo-planes. We succeeded in making clear of the structure of affine pseudo-planes to some extent with joint work with R.V.Gurjar and P.Russell, but could not resolve the Generalized Jacobian Problem for affine pseudo-planes.(2)Miyanishi showed that the Generalized Jacobian Problem for affine pseudo-planes has an affirmative answer if they satisfy the generalized Sard property.