The Topology and Arithmetic of Real Algebraic Varieties

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The principal objective of this work is to study connections between the topology and arithmetic of real algebraic varieties. The contributions of the investigator and his collaborator have uncovered such connections in the theory of real algebraic morphisms and complex structures on real algebraic varieties. He intends to continue his research in this direction. Another topic for further investigation is approximation of smooth manifolds by real algebraic varieties. Algebraic varieties are geometric objects which can be described by polynomial identities satisfied by the coordinates of their points. "Real" means that the coordinate numbers are limited to ordinary real numbers, avoiding the square root of minus one. While these are physically natural objects, paradoxically, the theory for treating them is in a far less satisfactory state than the one permitting imaginary numbers, those involving the square root of minus one. The investigator's efforts have been directed to remedying this situation and have had some gratifying successes.