Topology of real algebraic plane curves

The topology of real algebraic curves studies the relation between the algebraic properties of a given equation and the geometric properties of the curve determined by this equation. In this thesis, two different topological classifications of non-singular real algebraic curves will be discussed. The first one is "Topological classification", that is, understanding what are the possible sets (up to homeomorphism) of real points of such curves. The other is "Isotopic classification", that is, understanding how such a curve can lie in the real projective plane. In other words, it is to examine the relative positions of components of the curve. The second problem is known in the literature as "Hilbert's 16th problem". Our aim is to understand these problems according to the degree of such curves, and try to present the known results in a systematic way. To summarize in outline: In the first chapter, some basic definitions, theorems and examples of algebraic geometry related to affine/projective space, affine/projective algebraic varieties are given in order to understand the content of the thesis. In the second chapter, various topological properties of non-singular real algebraic plane curves, which are special cases of algebraic varieties, are presented systematically. In addition, some construction methods of such curves are mentioned. In the last chapter, the complex version of the results given in the previous one, that is the results derived from the complexification of real algebraic plane curves are presented.