<p>Isotropy representations provide powerful tools for understanding the classification of equivariant principal bundles over the $2$-sphere. We consider a $\Gamma$-equivariant principal $G$-bundle over $S^2$ with structural group $G$ a compact connected Lie group, and $\Gamma \subset SO(3)$ a finite group acting linearly on $S^2.$ Let $X$ be a topological space and $\Gamma$ be a group acting on $X.$ An isotropy subgroup is defined by $\Gamma_x = \{\gamma \in \Gamma \lvert \gamma x=x\}.$ Assume $X$ is a $\Gamma$-space and $A$ is the orbit space of $X$. Let $\varphi: A\rightarrow X$ be a continuous map with $\pi \circ \varphi = 1_A$. An isotropy groupoid is defined by $\mathfrak{I} = \{(\gamma,a) \in \Gamma\times A \lvert \ \gamma \in \Gamma_{\varphi(a)}\}.$ An isotropy representation of $\mathfrak{I}$ is a continuous map $\iota : \mathfrak{I} \rightarrow G$ such that the restriction map $\mathfrak{I}_a \rightarrow G$ is a group homomorphism. $\Gamma$- equivariant principal $G$-bundles are studied in two steps; \begin{enumerate} [1)] \item the restriction of an equivariant bundle to the $\Gamma$ equivariant 1-skeleton $X \subset S^2$ where $\mathfrak{I}$ is isotropy representation of $X$ over singular set of the $\Gamma$-sets in $S^2$ \item the underlying $G$-bundle $\xi$ over $S^2$ determined by $c(\xi)\in \pi_2(BG).$ \end{enumerate}</p> / Master of Science (MSc)

The Vlasov-Poisson system is most commonly used to model the movement of charged particles in a plasma or of stars in a galaxy. It consists of a kinetic equation known as the Vlasov equation coupled with a force determined by the Poisson equation. The system in Euclidean space is well-known and has been extensively studied under various assumptions. In this paper, we derive the Vlasov-Poisson equations assuming the particles exist only on the 2-sphere, then take an in-depth look at particles which initially lie along a great circle of the sphere. We show that any great circle is an invariant set of the equations of motion and prove that the total energy, number of particles, and entropy of the system are conserved for circular initial distributions. / Graduate