Most of my research is in computable (recursive)
algebra and computable model theory (see my Chapter 1
in Handbook of Recursive Mathematics), and in computability
(recursion) theory, which are
subfields of mathematical logic (see
Crossley's tutorial).
Computability theory is the mathematical theory of
algorithms. (See computability
resource 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 on structures (see
Hirschfeldt's paper
in the Bulletin of Symbolic Logic), complexity of
structures (see my paper
in the Bulletin of Symbolic Logic), and complexity of
isomorphisms (see our paper
with Fokina and Turetsky in the Annals of Pure and
Applied Logic). I also study orders on magmas (see our
co-authored paper
in JKTR). I am also interested in theoretical computer
science, in particular, complexity theory, approximate
computations, quantum
computing, and inductive
inference and algorithmic learning theory. My
other interests include philosophy of mathematics and
linguistic analysis (see our chapter
with Kaufmann and Condoravdi in The
Expression of Modality book).
North
American Annual Meeting,
Co-organizer (with J. Freitag) of the Special
Session Aspects
of Logic and Machine Learning, Cornell
University, April 7-10,
2022.
Mal'cev
Meeting, Sobolev Institute of Mathematics,
Novosibirsk, Russia, November 16-20, 2020. Member of
the Program Committee.
Logic
Colloquium, European Summer Meeting of the
Association for Symbolic Logic, Evora, Portugal,
July 22-27, 2013. Member of the Program Committee.
Co-organizer (with
Jose Felix Costa) of the Special Session on
Computability.