Circle
Packings and the Weierstrass
P-function
Dr. Edward Crane
Abstract
In the
1930s Koebe proved that any triangulation of the
sphere can be represented by a finite collection of geometric discs in the
standard 2-sphere, one for each vertex, such that adjacent vertices correspond
to externally tangent discs. Moreover the resulting circle packing is unique up
to Mobius transformations. The theorem was
rediscovered in the context of polyhedra in
hyperbolic 3-space by Andreev, and then reinterpreted by Thurston, who viewed
it as a discretization of the conformal structure on
the Riemann sphere. He also gave an algorithm for computing the packing.
Thurston's idea has been developed into an interesting theory of "discrete
analytic functions", which captures the geometry but not the arithmetic of
holomorphic functions. There are interesting
connections with 3-dimensional hyperbolic geometry, integrable
systems, and discrete minimal surfaces.
I will describe the main results of this
area and also some open problems, then demonstrate a new construction of
discrete versions of Weierstrass P-functions.
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Date: |
June 10, 2008 (Tuesday) |
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Time: |
3:00 – 4:00pm |
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Place: |
Room 517, Meng Wah Complex, HKU |
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All are
welcome |
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