what gets a lot of geometers and topologists to play well together, besides bringing toroidal pastries to a gathering, is to talk about local-to-global results. a central result of this kind for surfaces is the gauss-bonnet theorem, which relates the gaussian curvature of a surface (a local thing) to its euler characteristic (a global thing). instead of going into a thoroughly modern definition of euler characteristic, it is possible to instead relate the total curvature to another closely related concept for which we can draw lots of pretty pictures: singular points of vector fields and their indices. in this talk, i'll introduce vector fields on surfaces, and the concept of index, via pretty pictures and decorated oranges. then i'll talk about gaussian curvature, as intuitively as i can (which will involve some, ahem... hand-waving, and i'll try keep another kind of index juggling to a minimum!). next up is the theorem and its proof via juggling indices. finally, i'll give a little peek under the hood by mentioning triangulations of surfaces and how it relates to all this (with more hand-waving).