Degree theory for G-equivariant gradient maps and its applications

Let G be any compact Lie group. The aim of this lecture is to present the degree theory for G-equivariant gradient maps and to point out some applications of this degree to Hamiltonian systems. We will start with some remarks concerning the Brouwer degree. Moreover, using the Brouwer degree, we will classify homotopy classes of continuous gradient maps. Next, we will present properties of the degree for G-equivariant gradient maps and, using this degree, classify homotopy classes of continuous G-equivariant gradient maps. Additionally, we will show how to compute this degree. Finally, we will study the existence of periodic solutions of Hénon-Heiles nad Yang-Mills Hamiltonian systems in a neighborhood of an isolated, degenerate stationary solution. Moreover, we are going to study continuation of nonstationary periodic solutions of autonomous Newtonian systems. We will finish this lecture with some remarks and open questions concerning the degree for G-equivariant gradient maps.