### Computing with curves on surfaces

#### Dylan Thurston, Barnard College, Columbia University

Abstract: In order to better understand the geometric intersection
number of curves, we introduce a new tool, the \emph{smoothing lemma}.
This lets us write i(X, A), where X is a simple curve and A is
any curve, as a maximum of various i(X, A'), where the A' are also
simple and independent of X.

We can use this for several purposes, including a new derivation for
the behaviour of Dehn-Thurston coordinates under an elementary change
in the pair of pants decomposition (first described by Penner). This
gives, in particular, an efficient algorithm for the word problem in
the mapping class group of a closed surface.