I have heard about "Galois theory for schemes" in this note.
I haven't read it yet and I know Galois theory and a litle bit about schemes (as what it is, some properties like seperated, irreducible, connected...) but I can't imagine how these two topics combine with each other.
Could you please explain for me what is Galois theory for schemes and what is it role in modern mathematics.
Thanks.
Galois theory of schemes studies finite étale morphisms. This is the first step to étale cohomology, which is a vast and extremely rich area of mathematics with many applications. The "main theorem" of Galois theory for schemes classifies the finite etale coverings of a connected scheme $X$ in terms of its fundamental group $π(X)$. For a connected scheme $X$ there exists a profinite group $π_1(X)$ – the fundamental group of $X$, which is uniquely determined up to isomorphism, such that the category of finite étale coverings is equivalent to the category of finite permutation representations of $π_1(X)$. A full discussion is given in SGA1 (So read SGA1, including the introduction). The profinite group $π_1(X)$ is also called the étale fundamental group of the connected scheme X.
A further topic, highly interesting for you, then will be Grothendieck's anabelian geometry, which also is very important for Mochizuki's papers on the abc-conjecture. For an explanation and the role of the abc-conjecture in modern mathematics, fortunately, many people have written "survey articles" recently.