University of Birmingham > Talks@bham > Theoretical computer science seminar > From curves to train tracks to compressed words

From curves to train tracks to compressed words

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NB 217 not Sloman Lounge

Abstract – A curve is a smooth embedding of a circle into some other space.  A famous theorem of Jordan and Sch\”onflies states that all curves in the plane bound disks.  Thus, to the eyes of a topologist, all of these curves are really the same—- they are equivalent up to isotopy.  Curves in the once- or twice-punctured plane are not much more interesting; each of these is isotopic into a small neighborhood of one of the punctures.

In the three-times punctured plane, and in surfaces in general, there is a much more interesting story due to Dehn, Nielsen, Thurston, and others.  Thurston’s theory of train tracks allows us to describe curves purely combinatorially. This will lead us to the idea of straight-line compression of curves.  Combining this with a theorem of Plandowski, and some work, we will arrive at a new polynomial-time solution to the word problem in the mapping class group.

This talk is part of the Theoretical computer science seminar series.

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