Mathematics Colloquia and Seminars

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Geodesic currents are measures that realize a suitable closure of the space of weighted curves on a surface.
They were introduced by Bonahon in 1986, when he proved that the Teichmueller space embeds inside the space of currents and, furthermore, hyperbolic length of curves
can be extended to currents.
Since then, many other geometric structures on surfaces have been realized as currents and other functions on curves have been extended to currents,
such as flat lengths, some word lengths and, more recently, certain notions of length coming from representations of surface groups into higher rank Lie groups (so called ``Anosov representations'').
One of the key properties of these functions on curves is that they decrease under surgery of an essential crossing of a curve, a phenomenon we refer to as the ``smoothing property''.
In this talk we introduce the concept of geodesic current and show that functions on weighted curves that satisfy the smoothing property, together with some other mild conditions, can be extended to geodesic currents continuously. We briefly discuss an application to curve counting problems. This is joint work with Dylan Thurston.