Set Theory and Its Applications

Cornell University, Ithaca NY

Investigators

Abstract

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, compactness, and large cardinals. This project includes the training of graduate students. The first part of the research project involves using set-theoretic tools to study groups of piecewise linear and piecewise projective homemorphisms. This includes 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 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. This part of the research project also concerns use of set-theoretic tools such as compactness and the algebra of elementary embeddings to study the amenability problem for F. The second part of the research project concerns further developing techniques in pure and applied set theory: methods for studying the vanishing of higher derived limits in homological algebra; the role that Jensen's diamond principle plays in the theory of the sets of hereditary cardinality at most aleph1. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.