His primary research area is
Ramsey theory of infinite sets. He is known for solutions to the basis problem for uncountable linear orders and to the L space problem from general topology and for his work in determining the consequences of relating the continuum to certain values of the aleph function. Moore, together with his PhD student Yash Lodha, produced the first torsion-free and finitely presented counterexample to the
von Neumann-Day problem, originally described by mathematician
John von Neumann in 1929. Lodha presented this solution at the London Mathematical Society's Geometric and Cohomological Group Theory symposium in August 2013. ==Awards, distinctions, and recognitions==