Hamkins research work is cited, and he gives talks, including events for the general public. Hamkins was interviewed on his research by Richard Marshall in 2013 for , as part of an ongoing interview series for that magazine of prominent philosophers and public intellectuals, and he is occasionally interviewed by the popular science media about issues in the philosophy of mathematics.
Set theory
In set theory, Hamkins has investigated the indestructibility phenomenon of large cardinals, proving that small forcing necessarily ruins the indestructibility of supercompact and other large cardinals and introducing the lottery preparation as a general method of forcing indestructibility. Hamkins introduced the modal logic of forcing and proved with Benedikt Löwe that if ZFC is consistent, then the ZFC-provably valid principles of forcing are exactly those in the modal theory known as S4.2. Hamkins, Linetsky and Reitz proved that every countable model of Gödel-Bernays set theory has a class forcing extension to a pointwise definable model, in which every set and class is definable without parameters. Hamkins and Reitz introduced the ground axiom, which asserts that the set-theoretic universe is not a forcing extension of any inner model by set forcing. Hamkins proved that any two countable models of set theory are comparable by embeddability, and in particular that every countable model of set theory embeds into its own constructible universe.
Philosophy of set theory
In his philosophical work, Hamkins has defended a multiverse perspective of mathematical truth, arguing that diverse concepts of set give rise to different set-theoretic universes with different theories of mathematical truth. He argues that the Continuum Hypothesis question, for example, "is settled on the multiverse view by our extensive knowledge about how it behaves in the multiverse, and as a result it can no longer be settled in the manner formerly hoped for." Elliott Mendelson writes of Hamkins's work on the set-theoretic multiverse that, "the resulting study is an array of new fantastic, and sometimes bewildering, concepts and results that already have yielded a flowering of what amounts to a new branch of set theory. This ground-breaking paper gives us a glimpse of the amazingly fecund developments spearheaded by the author and...others..."
Infinitary computability
Hamkins introduced with Jeff Kidder and Andy Lewis the theory of infinite-time Turing machines, a part of the subject of hypercomputation, with connections to descriptive set theory. In other computability work, Hamkins and Miasnikov proved that the classical halting problem for Turing machines, although undecidable, is nevertheless decidable on a set of asymptotic probability one, one of several results in generic-case complexity showing that a difficult or unsolvable problem can be easy on average.
Group theory
In group theory, Hamkins proved that every group has a terminating transfinite automorphism tower. With Simon Thomas, he proved that the height of the automorphism tower of a group can be modified by forcing.
Infinite chess
On the topic of infinite chess, Hamkins, Brumleve and Schlicht proved that the mate-in-n problem of infinite chess is decidable. Hamkins and Evans investigated transfinite game values in infinite chess, proving that every countable ordinal arises as the game value of a position in infinite three-dimensional chess.
Hamkins is the top-rated user by reputation score on MathOverflow. Gil Kalai describes him as "one of those distinguished mathematicians whose arrays of MO answers in their areas of interest draw coherent deep pictures for these areas that you probably cannot find anywhere else."
