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## Algebraic structures related to closed curves on
surfaces.

### Moira Chas

SUNY Stony Brook

###
Thursday, April 19, 2007

007 Kemeny Hall, 4 pm

Tea 3:30 pm, 300 Kemeny Hall

**Abstract: ** It is well known that one can attach a
non-negative integer to any pair of free homotopy classes of curves on
a surface: the minimal intersection number between any pair of
representatives of these two classes.

It is also well known that
given two curves with the same basepoint one can multiply them and so
the set of based homotopy classes becomes a group.

In the
eighties, Goldman discovered a way to combine these two structures
which yields a Lie algebra structure on the space of linear
combinations of free homotopy classes.

I will discuss several
aspects of this Lie algebra of curves on surfaces: definition,
presentation, and some of the consequences of the algebraic properties
of this structure on the topology. In particular, I will adress the
problem whether it is possible to characterize simple closed curves in
terms of this Lie algebra and also, under which conditions one can
recover from this Lie algebra the minimal intersection number.

This talk will be accessible to graduate students.