### Abstract or Additional Information

Moduli spaces of constant curvature surfaces, i.e. of Riemann surfaces, i.e. of smooth curves are central objects in several subfields of topology, analysis, and algebra. In this talk I will introduce "constant curvature" point of view on these spaces by considering a simple but non-trivial example: Euclidean structures on an oriented two-dimensional torus. I will define and topologize the "Teichmueller space" of marked Euclidean structures, show it is naturally homeomorphic to the upper half-plane, and describe the action of the mapping class group. This leads to the result of the title (after we learn what the "modular orbifold" is).

The talk is aimed at a general audience, so it should require few prerequisites and be mostly self-contained.