Most of my research is in computable (recursive) algebra and computable model theory (see Harizanov's Handbook of Recursive Mathematics chapter), and in computability (recursion) theory, which are subfields of mathematical logic (see Crossley's 2005 tutorial).
Computability theory is the mathematical theory of algorithms. (See computability theory home page.) Problems which can be solved algorithmically are called decidable. Undecidable problems can be more precisely classified by considering generalized algorithms, which require external knowledge. Turing degrees provide an important measure of the level of such knowledge needed. Computable model theory explores algorithmic properties of objects and constructions arising within mathematics.
I am especially interested in computability theoretic complexity of relations (see Hirschfeldt's Bulletin of Symbolic Logic paper) and structures (see Harizanov's Bulletin of Symbolic Logic paper), including their Turing degrees. I am also interested in quantum computing and in theoretical computer science, in particular, complexity theory, frequency computations, and inductive inference and algorithmic learning theory. My other interests include natural language semantics and philosophy of mathematics.
My research has been supported by the NSF research grants DMS-1202328 (2012–2015), DMS-0904101 (2009–2012), DMS-0704256 (2007–2009), and DMS-0502499 (2005–2007), as well as by the NSF binational research grants for collaboration with Russia/Kazakhstan (2000–2014).