## The Calderón problem for connections

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##### Authors

Cekić, Mihajlo

##### Advisors

Paternain, Gabriel

##### Date

2017-10-03##### Awarding Institution

University of Cambridge

##### Author Affiliation

Department of Pure Mathematics and Mathematical Statistics

##### Qualification

Doctor of Philosophy (PhD)

##### Language

English

##### Type

Thesis

##### Metadata

Show full item record##### Citation

Cekić, M. (2017). The Calderón problem for connections (Doctoral thesis). https://doi.org/10.17863/CAM.13753

##### Abstract

This thesis is concerned with the inverse problem of determining a
unitary connection $A$ on a Hermitian vector bundle $E$ of rank $m$ over
a compact Riemannian manifold $(M, g)$ from the Dirichlet-to-Neumann
(DN) map $\Lambda_A$ of the associated connection Laplacian $d_A^*d_A$.
The connection is to be determined up to a unitary gauge equivalence
equal to the identity at the boundary.
In our first approach to the problem, we restrict our attention to
conformally transversally anisotropic (cylindrical) manifolds $M \Subset
\mathbb{R}\times M_0$. Our strategy can be described as follows: we
construct the special Complex Geometric Optics solutions oscillating in
the vertical direction, that concentrate near geodesics and use their
density in an integral identity to reduce the problem to a suitable
$X$-ray transform on $M_0$. The construction is based on our proof of
existence of Gaussian Beams on $M_0$, which are a family of smooth
approximate solutions to $d_A^*d_Au = 0$ depending on a parameter $\tau
\in \mathbb{R}$, bounded in $L^2$ norm and concentrating in measure
along geodesics when $\tau \to \infty$, whereas the small remainder
(that makes the solution exact) can be shown to exist by using suitable
Carleman estimates.
In the case $m = 1$, we prove the recovery of the connection given the
injectivity of the $X$-ray transform on $0$ and $1$-forms on $M_0$. For
$m > 1$ and $M_0$ simple we reduce the problem to a certain two
dimensional $\textit{new non-abelian ray transform}$.
In our second approach, we assume that the connection $A$ is a
$\textit{Yang-Mills connection}$ and no additional assumption on $M$. We
construct a global gauge for $A$ (possibly singular at some points) that
ties well with the DN map and in which the Yang-Mills equations become
elliptic. By using the unique continuation property for elliptic systems
and the fact that the singular set is suitably small, we are able to
propagate the gauges globally. For the case $m = 1$ we are able to
reconstruct the connection, whereas for $m > 1$ we are forced to make
the technical assumption that $(M, g)$ is analytic in order to prove the
recovery.
Finally, in both approaches we are using the vital fact that is proved
in this work: $\Lambda_A$ is a pseudodifferential operator of order $1$
acting on sections of $E|_{\partial M}$, whose full symbol determines
the full Taylor expansion of $A$ at the boundary.

##### Keywords

Geometric Inverse Problems, Analysis of PDEs, Differential Geometry, Calderon problem, X-ray transform, Magnetic Schrodinger equation, Inverse Problems, Dirichlet-to-Neumann map, Semiclassical pseudodifferential operators, Carleman estimates, Complex Geometric Optics, Yang-Mills, Unique Continuation Property, Inverse Boundary Value problem

##### Sponsorship

Trinity College, University of Cambridge

##### Identifiers

This record's DOI: https://doi.org/10.17863/CAM.13753

##### Rights

No Creative Commons licence (All rights reserved)