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Poincaré series are natural functions that arise in Riemannian geometry

when one wants to count the number of geodesic arcs of length less than

T between two given points on a compact manifold. I will begin with an

introduction on this topic. Then I will discuss some recent results with

N.V. Dang (Univ. Paris Sorbonne) showing that, in the case of negatively

curved manifolds, these series have a meromorphic continuation to the

whole complex plane. This can be shown by relating Poincaré series with

the resolvent of the geodesic vector field and by exploiting recent

results on this resolvent obtained through microlocal methods. If time

permits, I will also explain how the genus of a surface can be recovered

from the analysis of these series.