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Topology/Geometry Seminar

Circle homeomorphisms and shear coordinates with l^2 structure

Catherine Wolfram (MIT)

Location:  Online (Click on the Zoom link)
Date & time: Tuesday, 07 March 2023 at 4:00PM - 5:00PM

I will explain discrete coordinates called shears, indexed by the edges of the Farey tessellation, for homeomorphisms of the circle up to Mobius transformations. Separately I will introduce the Weil-Petersson class of circle homeomorphisms coming form Teichmuller theory. The main goal of the talk will be to understand the relationship between the Weil-Petersson class and spaces of circle homeomorphisms defined in terms of shears. Motivated by the situation for piecewise-Mobius homeomorphisms (corresponding to maps with finitely many nonzero shears), I will define a simple variant of shear coordinates called diamond shears (one diamond shear corresponds to four regular shears). I will show that the class of circle homeomorphisms with square-summable diamond shears is very close but slightly smaller than the Weil-Petersson class. This is joint work with Dragomir Saric and Yilin Wang.

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