Abstract: A classical problem in differential geometry asks whether the smallest surface bounded by a circle which does not introduce any strictly shorter paths is the hemisphere. It is only known for when the surface in question is homeomorphic to a disk (and some specializations I do not understand), suggesting that the main difficulty is topological in nature. 

I will discuss some ideas towards a first general lower bound for arbitrary smooth manifolds by translating the problem into an unfaithful (but independently interesting) combinatorial setting.

This talk will assume no background beyond graph theory I, although some maturity from convex geometry or topology II may help.

Based on joint work with Chris Wells.