# 2018, Malavika Mukundan’s internship

Since Gromov’s works on systolic geometry, there has been a flurry of papers devoted to bounding the length of a shortest non-contractible closed curve in Riemannian surfaces (and higher-dimensional manifolds) in terms of other quantities, such as the area. This has expanded in many directions, in particular an investigation of the length of shortest decompositions of surfaces, together with algorithms to efficiently compute them.

Malavika Mukundan first familiarized herself with basic low-dimensional topology (Stillwell 1993), systolic geometry (in particular via the relation with the girth of a graph: Bollobàs and Szemerédi, 2002), and algorithmic questions related to graphs on surfaces.

Then she was intrigued by a preprint by Kowalick, Lafont and Minemyer (2018), which considers the following question: Let T be a triangulation of an orientable surface with m triangles. Find an upper bound, depending on m, in the number of tetrahedra of a 3-manifold with boundary T. This paper achieves two upper bounds n = f(g) • m (for some function f, where g is the genus of the surface) and n = O(m log^2 m). A part of the strategy for the first bound is to iteratively cut the surface into shortest non contractible closed curves and fill the resulting boundary with a disk until the sphere is obtained.

Malavika read and understood the proof of this result. A natural idea to improve over it is to use decompositions of surfaces, instead of iteratively simplifying the topology of the surface, and trying this idea led her to learn a construction by Buser and Seppälä (2002) to construct short homology bases of surfaces. She also considered using pants decompositions of surfaces, in the spirit of Buser (1992); she read and understood the paper by Guth, Parlier and Young (2011) that provides bounds on the length of such pants decompositions.