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Harmonic determinants and unique continuation
preprint (2018), arXiv:1803.09182.

PDF 
arXiv 
We give partial answers to the following question: if F is an m by m matrix on R^n satisfying a second order linear elliptic equation, does det F satisfy the strong unique
continuation property? We give counterexamples in the case when the operator is a general
nondiagonal operator and also for some diagonal operators. Positive results are obtained
when n = 1 and any m, when n = 2 for the LaplaceBeltrami operator and also twisted with
a YangMills connection. Reductions to special cases when n = 2 are obtained. The last
section considers an application to the Calderón problem in 2D based on recent techniques.

A contribution to the Calderón problem for YangMills connections
preprint (2017), arXiv:1704.01362.

PDF 
arXiv 
We consider the problem of identifying a unitary YangMills connection ∇ on a Hermitian vector bundle from the DirichlettoNeumann (DN) map of the connection Laplacian ∇∗∇ over compact Riemannian manifolds with boundary. We establish such uniqueness of the connection up to a gauge equivalence in the case of line bundles in the smooth category and for the higher rank case in the analytic category. Furthermore, we prove that on the restriction of the vector bundle to the boundary the DN map is an elliptic pseudodifferential operator of order one, whose full symbol determines the complete Taylor series of an arbitrary connection and a metric (also of an associated potential) at the boundary.

Calderón problem for connections
Comm. Partial Differential Equations 42 (2017), no. 11, 17811836.

PDF 
arXiv 
In this paper we consider the problem of identifying a connection ∇ on a vector
bundle up to gauge equivalence from the DirichlettoNeumann map of the connection
Laplacian ∇∗∇ over conformally transversally anisotropic (CTA) manifolds.
This was proved in [ 1] for line bundles in the case of the transversal manifold
being simple – we generalise this result to the case where the transversal manifold
only has an injective ray transform. Moreover, the construction of suitable
Gaussian beam solutions on vector bundles is given for the case of the connection
Laplacian and a potential, following the works of [ 2]. Consequently, this enables us
to construct the Complex Geometrical Optics (CGO) solutions and prove our main
uniqueness result. Finally, we prove the recovery of a flat connection in general from
the DN map, up to gauge equivalence, using an argument relating the Cauchy data
of the connection Laplacian and the holonomy.
