Set theory helps lay the foundation for all of mathematics.
At its inception, set theory dealt with axiomatics, clarifying and studying the axioms on which mathematics is based, and discovering their consequences as well as their limitations. Modern set theory continues this line of investigation as well as others. In particular, modern set theory provides precise methods for studying real analysis and measures theory and topology. The main tools of modern set theory are cardinal invariants, combinatorics, forcing, forcing axioms, inner models and large cardinal axioms.
At DU, we work on set theory involving all of the above. Of particular interest are ultrafilters and their applications in logic, set theory and topology, including the Stone-Cech compactification of the natural numbers. The classification of ultrafilters up to Tukey (cofinal) type is one current focus of research. This study is connecting Ramsey theory to ultrafilters in an interesting manner.