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Poincare, Escher, Einstein, and Overdetermined Equations


Speaker: Rod Gover, Department of Mathematics, University of Auckland
Location: 1147 MSB
Start time: Fri, Mar 12 2010, 4:10PM

It seems that the artist M.C. Escher was much inspired by a certain view of hyperbolic 2-space, and in particular the effect on a tessellation of it by fish! Mathematically this conformal projection has been much exploited for the Poincare disk model of the hyperbolic plane and its higher dimensional analogues. Recently there has been considerable interest in curved generalizations of these structures (due to Fefferman and Graham). For these Poincare-Einstein manifolds a central problem is to relate the geometry of the boundary at infinity (which has a notion of angle but not length) to the structure of the interior, which is "Riemannian" and so both length and angle are well defined. This problem is linked to the ideas behind Maldacena's AdS/CFT correspondence in String theory. Of course the conformal projection favoured by Escher is lost, but it turns out that one can capture a key aspect of it in a certain natural overdetermined partial differential equation. This approach also leads to a natural way to extend the Poincare-Einstein programme, and new problems in geometric analysis.