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.

We continue the discussion of iteration theory for forcing, with the ultimate goal to study countable support iterations of proper forcing.