Microlocal Analysis and Geodesic X-Ray Transform Problems

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Abstract/Contents

Abstract: We restate main results on microlocal analysis including the definition and properties of scattering symbols and pseudodifferential operators. We also provide a brief overview on topics including microlocalization, operators on manifolds and propagation of singularity. Next, we discuss an application of microlocal analysis on geodesic X-ray transform problems. We will look at the scalar and one form cases of the problem and provide a relatively self contained treatment, with general curve families in the place of geodesics. Using a back projection operator and theory of scattering pseudodifferential operators, we show that for manifolds satisfying certain convexity requirements, the geodesic X-ray transform on scalars is injective and the geodesic X-ray transform on one forms is injective on functions satisfying the normal gauge.

Type of resource	text
Date created	[ca. June 2019]

Author	Zhu, Jiren
Advisor	Vasy, Andras

Subject	Department of Mathematics
Subject	School of Humanities and Sciences
Subject	Stanford University
Subject	microlocal analysis
Subject	geodesic X-ray transform
Subject	pseudodifferential operators
Genre	Thesis

Location	https://purl.stanford.edu/wt376wq5714

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Stanford University, Department of Mathematics, Honors Theses

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