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Sa'ar Hersonsky - Electrical Networks and Stephenson's Conjecture

The Riemann Mapping Theorem asserts that any simply connected planar domain which is not the whole of it, can be mapped by a conformal homeomorphism onto the open unit disk. After normalization, this map is unique and is called the Riemann mapping. In the 90's, Ken Stephenson, motivated by a circle packing approximation scheme suggested by Thurston (and first proved to converge by Rodin-Sullivan), predicted that the Riemann Mapping may be approximated by a different scheme, i.e., by a sequence of finite networks endowed with particular choices of conductance constants. These networks are naturally defined in terms of the contact ...
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