Set Theory and Its Applications

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A century ago, there was a movement to put mathematics on a rigorous, unified foundation. Because the notion of a set is among the most primitive in mathematics, it was used as the basic fabric with which to build the more complicated objects of mathematics. Since that time, it has been realized that the properties of infinite sets are themselves quite subtle and defy a complete axiomatization. Moreover, these set-theoretic complexities sometimes manifest themselves in more complex mathematical structures, such as those studied in algebra, analysis, and geometry. The aim of this project is to further develop both our understanding of set-theoretic methods and also how they can be applied to problems arising in fields of mathematics such as algebra, analysis, and topology. While the project involves several lines of investigation, a central theme will be to develop a deeper understanding of the structure of the algebra of all piece-wise linear functions from the unit interval to itself using the lens of transfinite ordinal numbers.The first part of the research project involves attempting to prove the following conjecture of Matthew Brin and Mark Sapir: if G is a group of piece-wise linear homeomorphims of the unit interval, then either G is elementary amenable or else G contains an isomorphic copy of Richard Thompson's group F. It is the PI's thesis that not only is this conjecture true, but that it will be a consequence of a much finer analysis of subgroup structure of PLoI, the group of piece-wise linear homeomophisms of the unit interval. This analysis is expected to have a number of other consequences: that the finitely generated subgroups of F are well quasi-ordered by embeddability; that any finitely presented subgroup of PLoI is either abelian or contains a copy of F; that Peano Arithmetic does not prove that F is amenable. Central to the analysis will be the countable transfinite ordinals. The second part of the research project concerns further developing techniques in pure set theory: methods for studying the combinatorics of the second uncountable cardinal via forcing axioms; the role that Jensen's diamond principle plays in the theory of the sets of hereditary cardinality at most aleph1. The project also supports graduate students working in other fields of logic, including computability theory and model theory.