This talk will describe an analogue of a Dirichlet to Neumann
map for Poincar\'e-Einstein metrics, also known as asymptotically
hyperbolic or conformally compact Einstein metrics. An explicit
identification of the linearization of the map at the sphere will be
given for even interior dimensions, together with applications
to the structure of the map near the sphere and to a different proof of
the positive frequency conjecture of LeBrun which was resolved by Biquard.

The notion of topological field theory has stymied topologists partly because it assigns to spaces quantities that are multiplicative under disjoint union; traditional homological or homotopical constructions are additive. In this talk I will survey how the use of an old "multiplicative" object in topology ("the spectrum of
units" in the class of vector spaces) leads to a successful formulation of a simple (but non-trivial) 2-dimensional field theory (the "Verlinde ring" and its deformations) and to new topological results about the moduli space of vector bundles on a Riemann surface. This is based on joint work with Freed-Hopkins and with Woodward.

The design and analyis of numerical methods for the solution of PDE of the form du/dt + Lu = 0, where L is a constant-coefficient differential operator, is greatly simplified by the fact that, for many methods, a closed-form representation of the computed solution as a function of (x,t) is readily available. This is due in large part to the fact that for such methods, the matrix that represents a discretization of L is easily diagonalizable. For variable-coefficient problems, however, this simplification is not available. This talk describes an alternative approach to the variable-coefficient case that leads to a new numerical method, called a Krylov subspace spectral method, for which the computed solution can easily be represented as a function of (x,t). This function can be analytically differentiated with respect to t, resulting in new approaches to deferred correction and the solution of PDE that are second-order in time such as the telegraph equation. The basic idea is to use Gaussian quadrature in the spectral domain to compute components of the solution, rather than in the spatial domain as in traditional spectral methods. As the method is more accurate for problems where the operator L has smoother coefficients, approaches to preconditioning differential operators using unitary similarity transformations are also presented.