The research area of this project is set theory, in particular, large cardinals and forcing. Set theory contains classical mathematics in the sense that every theorem of classical mathematics can 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 some problems on which mathematicians have worked are independent, i.e. neither provable nor disprovable from ZFC. Some of these independent statements are nevertheless known to be provable when ZFC is augmented by axioms asserting the existence of large infinite cardinal numbers. The instant project has three parts: first, to continue the study of a class of very large cardinals; second, to continue the study of the relation of these cardinals to problems in groups and finite combinatorics; and third, to study some unrelated open questions in forcing and infinite combinatorics. The main subject of this project is connected both to set theory and to algebra and topology. On the algebra/topology side, one of the topics to be studied is the classical braid groups, which arise in many contexts, including statistical mechanics (see for instance Kauffman, New invariants in the theory of knots, AMS Monthly, March 1988, 195-241). The connection of this area with the algebras studied in this project has led to new results about the braid groups. Moreover, results on the set theory side of the project have opened the question whether some properties of knots and braids will require the large cardinal tools of set theory for their solution.
