# The set theory of classification problems

Bronx Community College Mathematics Seminar, New York, December 2008

*Abstract*: Classification problems arise everywhere in mathematics: one might wish to classify some groups or rings up to isomorphism, some group actions up to conjugacy, homeomorphisms of some space up to homotopy, etc. Descriptive set theory gives us a very simple (and totally appropriate) definition of the “complexity of a classification problem,” which makes sense in a wide variety of cases. Studying things from this point of view, we can learn how close two related problems are, or when one problem is outright hopeless. I’ll give all the definitions, and some important examples of recent repute.