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Analysis Seminar

Friday, March 13, 2020
3:00pm to 4:00pm
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Linde Hall 255
Gravitational lensing, anti-Thurston maps, and planar graph embeddings
Lukas Geyer, Department of Mathematical Sciences, Montana State University,

A gravitational lens consisting of N point masses in a common lens plane can produce several images of a single light source. The corresponding lens equation turns out to be a fixed point equation for an anti-rational function, i.e., the complex conjugate of a rational function of degree N. In 2003, Rhie showed with an explicit construction that for every N>1 there exist lensing configurations producing 5N-5 images of a single light source, and she conjectured that this was the maximal number possible. Using techniques of complex dynamics, Khavinson and Neumann in 2006 proved Rhie's conjecture. In an attempt to classify equivalence classes of maximal lensing configurations, we show how these correspond to certain anti-Thurston maps, i.e., topological models of postcritically finite anti-rational maps, and how those in turn can be classified through certain equivalence classes of planar graph embeddings. As an application, we find several maximal lensing configurations which are not equivalent to Rhie's examples.

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