====== Michael Dobbins (Binghamton) ======

====== When Conformal Maps Get Squished ======

===== Abstract for the Combinatorics Seminar 2019 September 3 =====

A map is called conformal when it preserves angles. Consider a sequence of simple closed curves C<sub>t</sub> in the plane that get squished to a path. That is, C<sub>t</sub> consists of a pair of curves E<sub>t</sub> and W<sub>t</sub> that converge to the same curve in Fréchet distance. By the Riemann Mapping Theorem, there is a conformal map from the interior of C<sub>t</sub> to the upper half-plane. Preserving angles comes at the cost of distorting scale, and as C<sub>t</sub> gets squished, this map becomes highly distorted. I will show in this talk that, for appropriate choice of conformal map, the image of any arbitrarily small neighborhood of a single point covers the entire upper half-plane in the limit.


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