Constructive mathematics, higher computability (a.k.a. recursion) theory, set theory
Constructive mathematics can be understood as mathematics without the use of the Principle of the Excluded Middle; that is, without assuming that every assertion is either true or false. Without Excluded Middle, theorems of classical mathematics (i.e. with Excluded Middle) are often no longer valid.
I show non-validity results, by building models in which these classical theorems do not hold. A side benefit of so doing is that the technology for model building is thereby extended.
Higher computability studies generalized computation, which involves procedures that cannot actually be carried out on physical machines, such as transfinitely many steps, but still retain much of the flavor and spirit of real-life computation. The goal is often to find the ordinal closure point of a natural class of such machines.
My interests in set theory are with forcing and large cardinals.
Separating fragments of WLEM, LPO, and MP, joint with Matt Hendtlass, Journal of Symbolic Logic, 81, No. 4 (2016), pp. 1315-1343, doi: 10.1017/jsl.2016.38
On Extensions of Supercompactness, joint with Norman Lewis Perlmutter, Mathematical Logic Quarterly, 61, No. 3 (2015), pp. 217-223, doi: 10.1002/malq.201400030
Separating the Fan Theorem and Its Weakenings, joint with Hannes Diener, in Proceedings of LFCS '13, Lecture Notes in Computer Science #7734 (Sergei N.Artemov and Anil Nerode, eds.), Springer, 2013, pp. 280-295; also Journal of Symbolic Logic, 79, No. 3 (2014), pp. 792-813, doi: 10.1017/jsl.2014.9
Principles Weaker than BD-N, joint with Hannes Diener, Journal of Symbolic Logic, 78, No. 3 (2013), pp. 873-885
Walker's Cancellation Theorem, joint with Fred Richman, Communications in Algebra, 42, No. 4 (2014), pp. 1644-1649, doi: 10.1080/00927872.2012.747598