The popular YouTube video "Outside In" hosted by user ssgelm is a short film created by the University of Minnesota's Geometry Center to explain the intuition behind sphere eversion and related concepts.
Could anyone provide more insight as to the name and formal definition of the formula $D + B - S = 1$ mentioned between minutes 8 and 9 of the video where $D$ is the number of 'domes' on the shown surface, $B$ is the number of 'bowls' on the surface, and $S$ is the number of 'saddles' on the surface?
A partial answer: This formula is some version of the Euler characteristic of the surface using a version of Morse theory. There are two wrinkles from the usual Morse theory: (1) this is for spheres immersed in 3D space where the height function is from the z-axis and (2) we seem to only count those singularities (domes, bowls, and saddles) whose surface normal is pointing in some fixed direction (using the video's colors, those singularities where the yellow side is on top). If you remove this latter restriction, the formula is $D + B - S = 2$. If you've seen Euler characteristic for polyhedra, there is a formal correspondence where vertices <-> bowls, edges <-> saddles, and domes <-> faces.