A classical problem in Galois theory is a strong variant of
the Inverse Galois Problem: "What finite groups arise as the Galois
group of a finite Galois extension of the rational numbers, if you
impose the additional condition that the extension can only ramify at
finite set of primes?" This question is wide open in almost every
nonabelian case, and one reason is our lack of knowledge about how to
find number fields with prescribed ramification at fixed primes. While
such fields are often constructed to answer arithmetic questions,
there is currently no known way to systematically construct such
extensions in full generality.

However, there are some inspiring programs that are gaining ground on
this front. One method, initiated by Chenevier, is to construct such
number fields using Galois representations and their associated
automorphic representations via the Langlands correspondence. We will
explain the method, show how some recent advances in these subfields
allow us to gain some additional control over the number fields
constructed, and indicate how this brings us closer to our goal. As a
application, we will show how one can use this knowledge to study the
arithmetic of curves over number fields.

We discuss two step iterations of proper forcing.

 We prove the existence of GSpin-valued Galois representations corresponding to regularalgebraic cuspidal automoprhic representations of general symplectic groups under simplifyinglocal hypotheses. This is joint work with Arno Kret.

In this talk, we discuss fast iterative solvers for the large sparse linear systems resulting from the stochastic Galerkin discretization of stochastic partial differential equations. A block triangular preconditioner is introduced and applied to the Krylov subspace methods, including the generalized minimum residual method and the generalized preconditioned conjugate gradient method. This preconditioner utilizes the special structures of the stochastic Galerkin matrices to achieve high efficiency. Spectral bounds for the preconditioned matrix are provided for convergence analysis. The preconditioner system can be solved approximately by optimal multigrid solver. Numerical results demonstrate the efficiency and robustness of the proposed block preconditioner, especially for stochastic problems with large variance.