The systole of a compact Riemannian manifold (M,g) is the length of the shortest non contractible loop of M, it is attained by a periodic geodesic. A systolic inequality is a lower on the volume of any Riemannian metric depending only on the systole. If one consider the systole as a kind of ‘belt size’ of (M,g), a systolic inequality just says ‘the bigger the belt, the bigger the guy’. In this talk, we will discuss systolic inequalities for surfaces. We will prove optimal results for the torus and the projective plane (which go back to Loewner and Pu in the 50’s) and non optimal results for higher genus surfaces (due to Hebda and Gromov in the 80’s). This topic involve a nice mixture of metric geometry and elementary topology. If time permits, we will say a few words about higher dimensions.

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Last updated: 22 May 2019