Résumé : We will first introduce translation surfaces, which are Riemann surfaces built from gluing polygons in the plane via translations. The torus is the most basic example of a translation surface. The closed geodesics we count are called saddle connections, and are found by following geodesics which start and end at a marked point. In the case of the torus, the saddle connections correspond to pairs of integers (a,b) which are coprime to each other. We will present probabilistic results counting saddle connections with length conditions, as well as counting pairs of saddle connections with various pairing conditions. We will finish with highlighting the open questions and difficulties of counting triples of closed geodesics.
[Slides.pdf] [arXiv] [vidéo]
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