Let $X$ be a smooth, projective curve defined over an algebraically closed field $k$ of characteristic zero. The study of vector bundles on $X$ plays a central role in modern algebraic geometry, particularly in the context of moduli theory. A vector bundle $\mathcal{E}$ of rank $r$ on $X$ is said to be stable if for every proper subbundle $\mathcal{F} \subset \mathcal{E}$, we have \[ \frac{\deg \mathcal{F}}{\operatorname{rk} \mathcal{F}} < \frac{\deg \mathcal{E}}{\operatorname{rk} \mathcal{E}}. \] This notion is central to constructing moduli spaces of vector bundles, such as the moduli space $\mathcal{M}_X(r,d)$ of stable bundles of rank $r$ and degree $d$. Now consider the sheaf of differential forms $\Omega^1_X$ and the canonical bundle $\omega_X = \Omega^1_X$. If $g$ is the genus of $X$, then $\deg \omega_X = 2g - 2$. Suppose $\pi: Y \to X$ is a finite Galois cover of degree $n$ with Galois group $G$. The associated local system is given by the representation \[ \rho: \pi_1(X) o \operatorname{Aut}(V), \] where $V$ is a finite-dimensional $\mathbb{Q}_\ell$-vector space. The monodromy of this local system encodes the action of the étale fundamental group on the fibers of the covering. We are often interested in the space of such representations up to isomorphism, which leads to the character variety \[ \mathcal{X}_G(X) = \operatorname{Hom}(\pi_1(X), G) \sslash G, \] where $\sslash$ denotes the GIT quotient. From an arithmetic point of view, the image of the absolute Galois group $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ in the automorphism group of a local system over $\mathbb{P}^1 \setminus \{0,1,\infty\}$ gives rise to a profound connection between number theory and topology. In particular, rigid local systems with finite monodromy provide candidates for realizing finite groups as Galois groups over $\mathbb{Q}$. For example, if $G$ is a finite simple group and one can construct a three-point cover of $\mathbb{P}^1$ with monodromy group $G$, then $G$ occurs regularly as a Galois group over $\mathbb{Q}(t)$. This leads to the study of Hurwitz spaces, denoted $\mathcal{H}_{g,n}$, which parametrize isomorphism classes of covers $f: Y \to \mathbb{P}^1$ of genus $g$ with fixed ramification data over $n$ branch points. \[ \text{Given } \sigma_1, \dots, \sigma_n \in G ext{ such that } \sigma_1 \cdots \sigma_n = 1 \text{ and } \langle \sigma_1, \dots, \sigma_n \rangle = G, \] \[ \text{we associate a point in } \mathcal{H}_{g,n}(G). \] Finally, the cohomology $H^1_c(U, \mathcal{L})$ of a local system $\mathcal{L}$ on a punctured curve $U$ reflects the variation of the system under deformation of the branch points. This is the setting of parabolic cohomology and of the work of Dettweiler and Wewers.