Open Algebraic Varieties and Group Actions

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My field of research is Algebraic Geometry, the study of objects defined by polynomial equations. An example is the circle, defined by the equation x^2+y^2=1. Is this the same object as the hyperbola xy=1? The answer is less obvious than it may look, it is yes when we allow complex numbers in the solution, no when we allow only real numbers. Of particular interest to me are the flat affine spaces and some very fundamental questions about them. If in the plane we look at a line and a parabola, then they are actually the same (technically: isomorphic) objects in Algebraic Geometry and there is a symmetry (technically: an automorphism) of the plane carrying one into the other. Understanding such phenomena fully in higher dimensions is an extremely difficult task requiring sophisticated techniques from diverse and highly developed mathematical disciplines such as Algebra, Geometry, Topology and Logic. My past, present and future research aims at deepening this as yet incomplete understanding. It has been published in high quality jornals.