Serre's conjecture on the modularity of Galois representations makes a connection between two-dimensional Galois representations and modular forms. A conjecture by Ash, Doud, and Pollack generalizes Serre's to higher-dimensional Galois representations. In this paper we discuss an explicit computational example supporting the generalized claim. An ambiguity in a calculation within the example is resolved using a method of complex approximation.

This thesis explores the connection between Crouzeix's conjecture and the convergence of the GMRES algorithm. GMRES is a popular iterative method for solving linear systems and is one of the many Krylov methods. Despite its popularity, the convergence of GMRES is not completely understood. While the spectrum can in some cases be a good indicator of convergence, it has been shown that in general, the spectrum does not provide sufficient information to fully explain the behavior of GMRES iterations. Other sets associated with a matrix that can also help predict convergence are the pseudospectrum and the numerical range. This work focuses on convergence bounds obtained by considering the latter. In particular, it focuses on the application of Crouzeix's conjecture, which relates the norm of a matrix polynomial to the size of that polynomial over the numerical range, to describing GMRES convergence.