Computations of the cohomological Brauer group of some algebraic stacks

The theme of this dissertation is the Brauer group of algebraic stacks. Antieau and Meier showed that if $k$ is an algebraically closed field of $\\on{char} k \\ne 2$, then $\\Br(\\ms{M}_{1,1,k}) = 0$, where $\\ms{M}_{1,1}$ is the moduli stack of elliptic curves. We show that if $\\on{char} k = 2$ then $\\Br(\\ms{M}_{1,1,k}) = \\Z/(2)$. In another direction, we compute the cohomological Brauer group of $\\G_{m}$-gerbes; this is an analogue of a result of Gabber which computes the cohomological Brauer group of Brauer-Severi schemes. We also discuss two kinds of algebraic stacks $X$ for which not all torsion classes in $\\H_{\\et}^{2}(X,\\G_{m})$ are represented by Azumaya algebras on $X$ (i.e. $\\Br \\ne \\Br'$).