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An Informal Introduction to Computing with Chern Classes

Chern classes are objects associated to vector bundles. They describe relationships between the vector bundle and the topology of the base space. They straddle the worlds of topology and algebra or geometry; as such, they can be subtle, and difficult to compute. On the other hand, they satisfy some nice properties, and they are useful tools. One goal of these notes is to give examples of several different applications of Chern classes. In particular, Chern classes of certain bundles on Grassmanians can be used to solve enumerative problems, and this will be the main goal of these notes. I hope these notes will be accessible to a student who has some basic knowledge of algebraic geometry, say at the level of Shafarevich [14], and who is familiar with vector bundles.