In the first part of the talk (based on joint work with L. Coburn, J. Sj\"ostrand, and F. White) we discuss continuity conditions or Toeplitz operators acting on spaces of entire functions with quadratic exponential weights (Bargmann spaces), in connection with a conjecture by C. Berger and L. Coburn, relating Toeplitz and Weyl quantizations. In the second part of the talk (based on joint work in progress with J. Sj\"ostrand), we discuss elements of a semiglobal approach to analytic second microlocalization with respect to a hypersurface, in the semiclassical case, based on the study of the heat evolution semigroup for large times. We describe properties of the associated exponentially weighted spaces of holomorphic functions with ($h$--dependent) plurisubharmonic exponents and construct asymptotic Bergman projections in such spaces.

Let $X \subset \mathbb R^d$ be an unbounded open set. We wish to understand how decay of solutions to the wave equation on $X$ is related to the geometry of $X$.

When $\mathbb R^d \setminus X$ is bounded, this is the celebrated obstacle scattering problem. Then a particularly favorable geometric assumption, going back to the original work of Morawetz, is that the obstacle is star shaped. We adapt this assumption to the study of waveguides, which are domains bounded in some directions and unbounded in others, such as tubes or wires. We prove sharp wave decay rates for various waveguides, including the example of a disk removed from a straight planar waveguide, that is to say $X = ((-1,1) \times \mathbb R) \setminus D$, where $D$ is a closed disk contained in $(-1,1) \times \mathbb R$. Our results are based on establishing estimates and pole-free regions for the resolvent of the Laplacian near the continuous spectrum.

This talk is based on joint work with Tanya Christiansen.

The focus of this talk is (families of) widely separated configurations of points in Euclidean space, and the analysis of certain associated elliptic operators.  Families of widely separated point configurations generally display `multi-scale' or equivalently `clustering’ behaviour. The different scales are the relative sizes of the lengths d(p_i, p_j) between the points in the configuration. We use blow-up techniques to resolve such families and to study certain associated Laplace-type operators.  The main result is a set-up in which these elliptic families can be inverted uniformly.  The motivation for this study is the construction of corners of the compactified moduli space of monopoles, and if time permits, this application will be briefly discussed.