Research+Notes
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Polynomial Dynamics in Per_n(0):
We introduce a program for showing Per_n(0) is connected, as well as its non-escaping set M_n (analog of the "Mandelbrot set" in Per_1(0)).
MSRI 5/20/22, part of the Complex Dynamics program's Friday seminar (recording available through this link)]
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Dynamical Gaskets, Groups, Games
11/08/21 Michigan Complex Dynamics
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Elastic Graphs for Main Molecule Matings: (D-Powell-Winarski-Yang) http://arxiv.org/abs/2010.11382
Left: Michigan Dynamics Seminar 2/16/21
Right: Annotated Slides
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MC Escher slides
Slides to the left: UM Math club talk 2/25/21
Slides to the right: Annotated Slides
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Almost Every Mating is a Carpet: (with I. Park, in progress)
Insung Park and I quantify the likelihood of two PCF quadratic maps that are mateable to be a carpet.
above: mating loci of f=14/31 and f'=19/31. 14/31 is the first instance of a map with CP(f)=1--in the sense that no other 5-cycle has CP(f)=1, nor does any 3 or 4 cycle. There are more 6-cycles with CP(f)=1, and as n gets large, this is the generic case. The high carpet probability is reflected in parameter space by being surrounded by a"fuzzy" region. Contrast this with the mating locus of 19/31 which has an entire limb's worth of parameters that correspond to non-carpet matings. Shown between pink and green.
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above: video courtesty of Cheritat of a minimal (in terms of post-critical set size) possible Sierpinski Carpet mating, with 8=3+5 post-critical points; the mating of the airplane and the kokopelli in the 1/4-limb with crtiical value 7/31.
Coming soon:
-notes on Per_n
-notes on carpet matings and conformal dimension
A few talks from zoom-based graduate student seminars, that haven't been lost to the sands of time:
Coarse Geometry and Hyperbolicity 9/29 (Bers-Teichmueller)
Survey of Singular Perturbations (MSRI COMD Jr Seminar)