One of the magical parts of being a mathematician is seeing how different fields of mathematics are intertwined. Sometimes a surprising fact in one field comes from an insight in another. I will give some examples of the philosophy that when geometry comes from algebra or arithmetic, the topology can be unexpectedly constrained, which lets you prove results by "cutting and pasting". I will conclude by trying to indicate how Bott periodicity (which I won't assume you understand) is a fact about algebra, not analysis. I will discuss work of a number of mathematicians, including Melanie Wood, Aaron Landesman, Hannah Larson, Jim Bryan, and more. (This talk is intended as a survey for a broad audience.)
