What Is... Seminar
"Algebra is the offer made by the devil to the mathematician. The devil says: I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvelous machine." -- Sir Michael Atiyah
Two of the first topological invariants one typically learns are de Rham cohomology (for manifolds) and singular homology (for general topological spaces). Both are very geometric theories; it's easy to visualize a singular chain or get a feeling for a de Rham form. Putting them together gives singular cohomology, which is very useful but hard to visualize elements of.
I'll define "K-theory" (in both homology and cohomology versions) of algebraic varieties, which has an undeserved reputation of being harder than cohomology; in many situations it is easier. In particular, both K-homology and K-cohomology elements have geometric meaning, and it is much easier to bring group actions into the picture in K-theory.