Ramification theory is a deep and very extended subject in mathematics which appears in complex analysis (Riemann surface theory), in algebraic topology (covering map via the Euler-Poincaré characteristic), in algebra (group actions on rings), in algebraic geometry (unramified, étale cover)…
We first propose to give a little introduction on ramification theory, in algebraic number theory. More precisely, we will explain how the ramification groups measure the complexity with which the arithmetic of Z passes though number field extensions. Then, we will see how we can generalize this theory for group actions on rings. Finally, we will give an idea of ramification theory on a more general picture: actions of group schemes on schemes, via a definition of ramification groups. In particular, we will propose a definition of higher ramification groups as generalization of the ramification groups in number theory (joint work with Prof. Ted Chinburg). We will also present slice theorems that we were able to obtain in this context.