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Talks by Tom Ilmanen

Evolving Networks

ETH Zürich

We construct planar networks evolving by their curvature. A network consists of arcs meeting at 120 degrees; any other kind of junction is called singular. A notable feature of the flow is that at each singularity, a bounded region ("bubble") must disappear, and is not allowed to re-appear; the only exception is the "cross" singularity, where two junctions merge, then re-expand in an orthogonal direction.

Main results: (1) twenty or thirty computer pictures of self-similarly shrinking networks, and a universal bound on their gaussian density and topology (2) the flow has at most a finite number of singularities at each time (3) the flow can be continued for a short time after a singularity forms (4) treelike flows (i.e. flows with no bubbles) exist for all time and have only isolated cross singularities (5) general flows exist for all time and have at most a finite number of bubble singularties plus a countable discrete set of cross singularities.

Conjecture: there are only a finite number of singularities in spacetime; but hypothetical buzzing honeycomb figures (many 6-sided regions that occasionally switch to 5 or 7 sides) present an obstruction to the proof.

(With J. Haettenschweiler, A. Neves and F. Schulze)

Tom Ilmanen

ETH Zürich

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