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Thesis Format



Doctor of Philosophy




Shafikov, Rasul


The goal of this dissertation is to prove two results which are essentially independent, but which do connect to each other via their direct applications to approximation theory, symplectic geometry, topology and Banach algebras. First we show that every smooth totally real compact surface in complex Euclidean space of dimension 2 with finitely many isolated singular points of the open Whitney umbrella type is locally polynomially convex. The second result is a characterization of the rational convexity of a general class of totally real compact immersions in complex Euclidean space of dimension n..

Summary for Lay Audience

In this dissertation we prove two original results that are of great interest for their applications to the theory of approximation of continuous functions. These two results unveil deep connections to other area of mathematics, such as symplectic geometry, Banach algebras and topology. More precisely, we study some geometric properties of a class of objects of complex space. The first result (Chapter 3) essentially establishes one of those properties (which we call Polynomial convexity) to a certain class of objects. The other one (Chapter 4) provides a characterization of a different kind of objects with respect to another type of geometric property (named Rational convexity).

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