Mathematical Sciences: Embedding Theorems for CR Manifolds

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In this project the principal investigator will study three conjectures related to the embedding of CR manifolds in complex Euclidean space. The first conjecture involves embedding a smooth, compact, pseudoconvex CR manifold. The second conjecture involves embedding a neighborhood of a point of a smoothly bounded, integrable, almost complex manifold at which a Levi form has at least two negative eigenvalues. The third conjecture involves embedding a relatively compact, open set in a Stein manifold in an affine algebraic variety. The subject of several complex variables has deep and important ties with the geometry of n-space. Mathematicians can often exploit these connections in their study of classical problems in the theory of several complex variables. In this project the principal investigator will attempt to answer three important conjectures regarding whether various types of manifolds can be embedded in complex Euclidean space.