Geometric wave propagator on Riemannian manifolds

Abstract: We study the propagator of the wave equation on a closed Riemannian manifold M. We propose a geometric approach to the construction of the propagator as a single oscillatory integral global both in space and in time with a distinguished complex-valued phase function. This enables us to provide a global invariant definition of the full symbol of the propagator - a scalar function on the cotangent bundle - and an algorithm for the explicit calculation of its homogeneous components. The central part of the talk is devoted to the detailed analysis of the subprincipal symbol; in particular, we derive its explicit small time asymptotic expansion. We present a general geometric construction that allows one to visualise topological obstructions and describe their circumvention with the use of a complex-valued phase function. We illustrate the general framework with explicit examples in dimension two.

high energy physicsmathematical physicsanalysis of PDEsspectral theory

Audience: researchers in the topic

Scattering, microlocal analysis and renormalisation

Series comments: Mondays: aarhusuniversity.zoom.us/j/9036772485 Meeting ID: 903 677 2485

Thursdays: us02web.zoom.us/j/6094800950 Meeting ID: 609 480 0950

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Links to the slides can be found here: drive.google.com/file/d/1uu6ZvU6zlGalJpZR7ZDi7igsvBMBylOA/view