Complex Dynamics
Bernardo Da Costa, Koushik Ramachandran, Jingjing Qu, and I had a two semester learning seminar in complex analysis and potential theory.
A major theme in the Fall 2012 semester was Zalcman's Lemma.
In particular, we went through W. Bergweiler's beautiful proof of Ahlfor's five islands theorem.
The Spring 2013 semester, we have focused on complex dynamics using mainly Milnor's book.
We have also covered side topics like applications of complex dynamics to Newton's method,
and studying zeros of certain harmonic mappings.
We studied potential theory of the Julia set (the equilibrium measure as an ergodic measure),
and a bit of quasiconformal mapping--in order to see the proof of D. Sullivan's no-wandering-domain theorem.
On the experimental side, Bernardo made these nice pictures with the computer (of the Mandelbrot set and Julia set).
The choice of parameter is at the cusp of the "Julia set dichotomy" (so it is hard to tell whether the Julia set in the second picture is truly connected),
and in my opinion this is an aesthetic choice of parameter.
The Sea horse valley of the Mandelbrot set.
The Julia set for a choice of parameter near the sea horse valley.
A zoom in on the previous picture.
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