Topology of real algebraic sets

A real algebraic set is a set of common zeros of a system of polynomials with real coefficients. The main subject of the talk will be a Nach theorem. The Hash theorem states that if $M$ is a smooth closed manifold embedded into $R^N$ then there exists a diffeomorphism $f: R^N \to R^N$ such that the image $f(M)$ is a connected component of a real algebraic set (provided that N is sufficiently large). Also it will be explained that one can actually choose f such that $f(M)$ will be just a real algebraic set. As an easy corollary we get that every smooth closed manifold is diffeomorphic to a real algebraic set.