Mathematical Sciences: Problems in Large Cardinals, Forcing and Combinatorics
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The area of research is set theory, in particular, large                cardinals and forcing.  Set theory may be viewed as containing                  mathematics, that is, every theorem of classical mathematics may                be formulated as a statement about sets, and then formally                      derived from the standard collection ZFC of axioms for set                      theory.  Beginning with the work of Godel (1936) and Cohen                      (1963), set theorists have shown that certain problems, on which                mathematicians have worked, are independent, i.e. neither                       provable nor disprovable from ZFC.  The first and most well known               example of this is the Godel-Cohen result that Cantor's continuum               hypothesis is independent.  Some statements, independent of ZFC,                are nevertheless known to be provable when ZFC is augmented by                  axioms asserting the existence of large infinite cardinal                       numbers.  This project has two parts:  firstly, to continue the                 study of a class of very large cardinals, and secondly, to study                the consistency and independence of von Neumann's conjecture and                a collection of related problems.  Such studies deepen our                      understanding of the limits of mathematical proof.
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