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The anisotropic Calderon inverse problem consists in recovering the metric of a compact connected Riemannian manifold with boundary from the knowledge of the Dirichlet-to-Neumann map at fixed energy. A fundamental result due to Lee and Uhlmann states that there is uniqueness in the analytic case. We shall present counterexamples to uniqueness in cases when:

1) The metric smooth in the interior of the manifold, but only H\”older continuous on one connected component of the boundary, with the Dirichlet and Neumann data being measured on the same proper subset of the boundary.
2) The metric is smooth everywhere and Dirichlet and Neumann data are measured on disjoint subsets of the boundary.

This is joint work with Thierry Daude (Cergy-Pontoise) and Francois Nicoleau (Nantes).

Abstract:

TBA

The Donaldson-Uhlenbeck-Yau theorem relates the existence of Hermitian-Yang-Mills connections over a compact Kahler manifold with algebraic stability of a holomorphic vector bundle. This has been extended by Bando-Siu in 1994 to a class of singular Hermitian-Yang-Mills connections on reflexive sheaves. We study tangent cones of these singular connections in the geometric analytic sense, and show that they can be characterized in terms of certain algebro-geometric invariants of reflexive sheaves. In a sense, this can be viewed as a “local” version of the Donaldson-Uhlenbeck-Yau correspondence.  Based on joint work with Xuemiao Chen (University of Maryland). 

Future schedule is found here: https://scholar.harvard.edu/gerig/seminar