Diffractive Propagation on Conic Manifolds

In this survey, we review some applications and extensions of the author's results with Richard Melrose on propagation of singularities for solutions to the wave equation on manifolds with conical singularities. These results mainly concern: the local decay of energy on noncompact manifolds with diffractive trapped orbits (joint work with Dean Baskin); singularities of the wave trace created by diffractive closed geodesics (joint work with G. Austin Ford); and the distribution of scattering resonances associated to such closed geodesics (joint work with Luc Hillairet).Comment: 15 pages; contribution to Seminaire Laurent Schwartz proceeding

Resolvent estimates with mild trapping

We discuss recent progress in understanding the effects of certain trapping geometries on cut-off resolvent estimates, and thus on the qualititative behavior of linear evolution equations. We focus on trapping that is unstable, so that strong resolvent estimates hold on the real axis, and large resonance-free regions can be shown to exist beyond it.Comment: 15 pages. For Journ\'ees EDP 2012 conference proceeding

Local smoothing for the Schr\"odinger equation with a prescribed loss

We consider a family of surfaces of revolution, each with a single periodic geodesic which is degenerately unstable. We prove a local smoothing estimate for solutions to the linear Schr\"odinger equation with a loss that depends on the degeneracy, and we construct explicit examples to show our estimate is saturated on a weak semiclassical time scale. As a byproduct of our proof, we obtain a cutoff resolvent estimate with a sharp polynomial loss.Comment: 26 pages, incorporated referee's comment