Title: S^1 and S^3 and S^2, oh fy! A digital Hopf fibration
Abstract: Digital images surround us. They are found in our computers, iPhones, televisions, and more. Because they are so integrated into our lives, there is a constant need to manipulate and investigate these images. Anything that one might want to do with a digital image will inevitably involve some kind of mathematics, whether it be linear algebra, geometry, or topology. To that end, we will introduce topology in the digital setting, noting some places where it is similar and different than in the smooth setting. In particular, we will work with digital homotopy between digital images by viewing a digital image as a graph. Although there is a notion of digital fibration in this context, there seem to be very few non-trivial examples of digital fibrations. We will construct a digital analogue of the Hopf fibration, the most important single example in the history of algebraic topology. Because the 3-sphere in this setting consists of only 24 pixels, this example is robust yet small enough to be written down and investigated explicitly. This talk will be accessible to undergraduates.