I am interested in the interactions between set theory and mathematical analysis.

Set theory, the study of infinite sets, was initiated by Georg Cantor in the 1870's when he discovered that there are different "sizes" of infinity, such as the infinity of the set of natural numbers, and the "larger" infinity of the set of real numbers. Concerns regarding the sizes of infinite sets form part of combinatorial set theory. Cantor was also interested in the "complexity" of sets of real numbers, a subject vastly expanded by Nikolai Luzin and others in the 1920s, into what is now called descriptive set theory.

Cantor's work, and the paradoxes it introduced, required studying the foundations of mathematics, leading to the development of axiomatic set theory, and placing set theory within the larger field of mathematical logic. The logical nature of set theory is perhaps most dramatically seen in the discovery, by Kurt Gödel, Paul Cohen, and many others since, of statements whose truth is independent of the axioms of set theory.

Mathematical analysis is the study of functions on the real numbers and related spaces, and encompasses calculus, differential equations, and their generalizations. Of particular interest to me is functional analysis, the study of spaces of functions, and the functions, or "operators", between them. Many of the objects of study, including Banach spaces, Hilbert spaces, and C*-algebras, are often infinite dimensional, and connections between set theory and analysis can be made by exploiting the analogies between infinite sets and infinite dimensional spaces.

Much of my work focuses on finding set theoretic structures within objects from functional analysis. For example, in set theory, ultrafilters are a way of assigning a notion of "largeness" to subsets of a given set. They naturally arise when considering notions of convergence, points in compactifications of topological spaces, and maximal ideals in algebras of functions. What notions correspond to different kinds of ultrafilters when we move from subsets of a set to closed subspaces of a Banach space? I'm particularly interested in analogues of so-called selective ultrafilters, and their relationship to the Ramsey theory of Banach spaces developed by Timothy Gowers in the 1990's.

Another fruitful area of study has been that of analytic or Borel equivalence relations on spaces. The tools of descriptive set theory can be used to analyze the complexity of such relations, providing a notion of "difficulty of classification". In functional analysis, many equivalence relations arise naturally from ideals of operators. I'm interested in their complexity, and their relationship to other "benchmark" equivalence relations studied in descriptive set theory.