Real-analytic manifolds are studied very much in the last century until the time when people found the partition of unity on smooth manifolds makes the manifold theory very tractable. The group of real-analytic diffeomorphisms is the natural automorphism group of the real-analytic manifold. Because of the analytic continuation, there are no partition of unity by functions with support in balls. The germ at a point of a real-analytic diffeomorphism determines the diffeomorphism and hence the group of them looks rigid. However, the group of real-analytic diffeomorphisms is dense in the group of smooth diffeomorphisms and diffeomorphisms can exhibit all kinds of smooth stable dynamics. I would like to convince the audience that the group of real-analytic diffeomorphisms is a really interesting object.In the first course, I would like to review the theorem by Herman which says the identity component of the group of real analytic diffeomorphisms of the n-torus is simple, which gives a motivation to study the group for other manifolds. We also review several fundamental facts in the real analytic category.In the second course, we introduce the regimentation lemma which can play in the real analytic category the role of the partition of unity in the smooth category. For manifolds with nontrivial circle actions, we show that any real analytic diffeomorphism isotopic to the identity is homologous to a diffeomorphism which is an orbitwise rotation.In the third course, we state a lemma which says that the multiple actions of the standard action on the plane is a final (terminal) object in the category of circle actions. This lemma would imply that the identity component of the group of real analytic diffeomorphisms is perfect.[-]

Real-analytic manifolds are studied very much in the last century until the time when people found the partition of unity on smooth manifolds makes the manifold theory very tractable. The group of real-analytic diffeomorphisms is the natural automorphism group of the real-analytic manifold. Because of the analytic continuation, there are no partition of unity by functions with support in balls. The germ at a point of a real-analytic diffeomorphism determines the diffeomorphism and hence the group of them looks rigid. However, the group of real-analytic diffeomorphisms is dense in the group of smooth diffeomorphisms and diffeomorphisms can exhibit all kinds of smooth stable dynamics. I would like to convince the audience that the group of real-analytic diffeomorphisms is a really interesting object.In the first course, I would like to review the theorem by Herman which says the identity component of the group of real analytic diffeomorphisms of the n-torus is simple, which gives a motivation to study the group for other manifolds. We also review several fundamental facts in the real analytic category.In the second course, we introduce the regimentation lemma which can play in the real analytic category the role of the partition of unity in the smooth category. For manifolds with nontrivial circle actions, we show that any real analytic diffeomorphism isotopic to the identity is homologous to a diffeomorphism which is an orbitwise rotation.In the third course, we state a lemma which says that the multiple actions of the standard action on the plane is a final (terminal) object in the category of circle actions. This lemma would imply that the identity component of the group of real analytic diffeomorphisms is perfect.[-]

Real-analytic manifolds are studied very much in the last century until the time when people found the partition of unity on smooth manifolds makes the manifold theory very tractable. The group of real-analytic diffeomorphisms is the natural automorphism group of the real-analytic manifold. Because of the analytic continuation, there are no partition of unity by functions with support in balls. The germ at a point of a real-analytic diffeomorphism determines the diffeomorphism and hence the group of them looks rigid. However, the group of real-analytic diffeomorphisms is dense in the group of smooth diffeomorphisms and diffeomorphisms can exhibit all kinds of smooth stable dynamics. I would like to convince the audience that the group of real-analytic diffeomorphisms is a really interesting object.In the first course, I would like to review the theorem by Herman which says the identity component of the group of real analytic diffeomorphisms of the n-torus is simple, which gives a motivation to study the group for other manifolds. We also review several fundamental facts in the real analytic category.In the second course, we introduce the regimentation lemma which can play in the real analytic category the role of the partition of unity in the smooth category. For manifolds with nontrivial circle actions, we show that any real analytic diffeomorphism isotopic to the identity is homologous to a diffeomorphism which is an orbitwise rotation.In the third course, we state a lemma which says that the multiple actions of the standard action on the plane is a final (terminal) object in the category of circle actions. This lemma would imply that the identity component of the group of real analytic diffeomorphisms is perfect.[-]