The proposed research aims to deepen our understanding of the combinatorics of infinite sets, to develop new techniques, and to explore new applications of infinite combinatorics in other fields of mathematics. One component of the research is foundational: it aims to better understand the relationship between classification problems (such as the basis problem for the uncountable linear orders and the metrization problem for compact convex sets) and the value of the cardinality of the continuum. Currently we have an acceptable understanding of how to produce models of set theory in which the continuum is the second uncountable cardinal; the methods for producing models of set theory in which the continuum has some other value is considerably more limited. Part of the present proposal aims to develop better techniques for producing models of set theory in which the continuum is larger than the second uncountable cardinal, while maintaining the ability to control other combinatorial properties of sets in the model. Additionally, the proposal aims to further develop techniques in Ramsey theory needed to solve problems arising in other fields and in particular arising in the study of the amenability of groups. Specifically the amenability of Thompson's group F - the subject of a longstanding problem in the field - is equivalent to a new type of Ramsey-theoretic statement relating to generalizations of Hindman's theorem to nonassociative operations.
Frequently in mathematics it is necessary to find a guiding heuristic in order to gain intuition in a setting where our own human experience falls short. Two examples of this need are provided by the study of the combinatorial properties of infinite sets and the geometry of high dimensional spaces. In both cases, guiding intuition is provided by a classical piece of mathematics known as Ramsey's theorem, concerning colorings of edges in a graph, a mathematical abstraction relating to networks. The proposal aims to develop new techniques in Ramsey theory in order to improve our understanding of foundational issues relating to infinite sets and high dimensional geometric objects.