  This dissertation represents work on three different subjects relating to
quantum gravity and the AdS/CFT correspondence.
  First, we review a holographic computation of the one-loop corrections to the
Weyl anomaly on Ricci flat backgrounds in six dimensions. This allows us to
determine the correction to one linear combination of the anomaly coefficients.
Then, we will show that these corrections may be obtained from the
six-dimensional superconformal index.
  The second section covers consistent truncations on the Lunin-Maldacena
background. We show how to restore minimal supersymmetry to this background by
determining the reduction ansatz which includes the graviton and a gauge field
of 5d $\mathcal{N} = 2$ supergravity. We also discuss our previously
unpublished attempt to construct a truncation which includes a scalar field
corresponding to the $\beta$-deformation parameter of the dual field theory. We
show that if such a solution exists, it must differ drastically from the LM
background.
  Finally, we discuss higher-derivative corrections to black hole solutions and
the weak gravity conjecture in a few settings. First we compute
higher-derivative corrections to the extremality bounds for black holes which
are charged under an arbitrary number of $U(1)$ gauge fields, and determine the
constraints implied by the WGC. Next we consider the higher-derivative
corrections to thermodynamic quantities for charged black holes in Anti-de
Sitter space. We confirm and clarify a previously noted relationship between
the shift to the extremality bound and the shift to the Wald entropy. We also
show that if the shift in the Wald entropy is assumed to be positive, then the
coefficient of the $R_{\mu \nu \rho \sigma} R^{\mu \nu \rho \sigma}$ term in
the effective Lagrangian must be positive as well.
