We know from systolic geometry that on a compact Riemannian manifold a lot of geometric information is contained in the length of the shortest non-homologically trivial closed curve. Naturally, even more geometric information can be extracted from the data of the shortest curve with prescribed homology class, for each homology class. This data is called the stable norm: it is a norm on the first homology group of the manifold that extends the notion of length to homology classes. Unfortunately, the stable norm is very difficult to compute in practice, and there are extremely few known explicit examples.
In this talk I will explain how we can explicitly compute the stable norm of a family of flat surfaces. The core argument is the computation of the stable norm of flat slit tori. Surprisingly, the stable norm of slit tori is linked to the Farey sequence, a number sequence famous for its remarkable arithmetical properties. In a second part, I will show how we can glue together several slit tori and flat cylinders to construct half-translation surfaces on which the stable norm is known. Finally, I will discuss a curve-counting problem related to the stable norm. More precisely I will show that, contrary to all the previously known examples, on the previous surfaces the growth of the number of homology classes whose stable norm is realized by the length of a simple curve (aka simple homology classes) is sub-quadratic.
