Georgia Institute of Technology College of Sciences

The problem of classifying collections of objects (graphs, manifolds, operators, etc.) up to some notion of equivalence (isomorphism, diffeomorphism, conjugacy, etc.) is central in every domain of mathematical activity. Invariant descriptive set-theory provides a formal framework for measuring the intrinsic complexity of such classification problems and for deciding, in each case, which types of invariants are “too simple” to be used for a complete classification. It also provides a very interesting link between topological dynamics and the meta-mathematics of classification. In this talk I will discuss various forms of classification which naturally occur in mathematical practice (concrete classification, classification by countable structures, classification by cohomological invariants, etc.) and I will provide criteria for showing when some classification problem cannot be solved using these forms of classification.