My interest in topology comes from two directions: first, it was the reason that I got started in math (and subsequently the topic that broke me), and second, I found out over the last year that topology plays a prominent role in several research areas of computer science. These applications to CS are what really got me interested in the topic again. Algebraic (or combinatorial) topology is useful in studying the structure of data like meshes and graphs. Differential geometry is applicable when your data lives in another space, or on something called a manifold, and you want to do the same kinds of things that you’re used to doing in regular, old-fashioned space. Ultimately, both have to do with geometry, albeit in a very general way. That’s why these topics interest me; I want to know how to better get at the essence of data, and much of the time you can do that geometrically.
He actually enjoys thinking about Cartan-Hadamard manifolds and representation theory, but I don't hold it against him :).