Microlocal analysis of global hyperbolic propagators on manifolds without boundary
The main goal of this thesis is to construct explicitly, modulo smooth terms, propagators for physically meaningful hyperbolic partial differential equations (PDEs) and systems of PDEs on closed manifolds without boundary, and to do this in a global (i.e. as a single oscillatory integral) and invari...
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| Format: | Doctoral or Postdoctoral Thesis |
| Language: | English |
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UCL (University College London)
2020
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| Online Access: | https://discovery.ucl.ac.uk/id/eprint/10100081/1/main_PhDthesis.pdf https://discovery.ucl.ac.uk/id/eprint/10100081/ |
| Summary: | The main goal of this thesis is to construct explicitly, modulo smooth terms, propagators for physically meaningful hyperbolic partial differential equations (PDEs) and systems of PDEs on closed manifolds without boundary, and to do this in a global (i.e. as a single oscillatory integral) and invariant (under changes of local coordinates and any gauge transformations that may be present) fashion. The crucial element in our approach is the use of complex-valued, as opposed to real-valued, phase functions — an idea proposed in the nineties by Laptev, Safarov and Vassiliev. It is known that one cannot achieve a construction global in time using a real-valued phase function due to obstructions brought about by caustics; however the use of a complex-valued one makes it possible to circumvent such obstructions. This is the subject of the first part of the thesis, where we study the global propagator for the wave equation on a closed Riemannian manifold of dimension d ≥ 2 and the global propagator for the massless Dirac equation on a closed orientable Riemannian 3-manifold. Our results allow us to compute, as an application, the third local Weyl coefficient for the massless Dirac operator. A natural way to obtain a system of PDEs on a manifold is to vary a suitably defined sesquilinear form. In the second part of the thesis, we study first order sesquilinear forms acting on sections of the trivial C^m-bundle over a smooth d-manifold. Thanks to the interplay of techniques from analysis, geometry and topology, we achieve a classification of these forms up to GL(m, C) gauge equivalence in the special case of d = 4 and m = 2. Finally, in the last chapter we develop a Lorentzian analogue of the theory of elasticity. We analyse the resulting nonlinear field equations for general Lorentzian 4-manifolds, and provide explicit solutions for the Minkowski spacetime. |
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