Research Monographs
Research Announcements
Research Papers

Bad oracles in higher computability and randomness, with L. Bienvenu and B. Monin, submitted.

Cupping and jump classes in the computably enumerable degrees, with K.M. Ng and G. Wu, submitted.

Cardinal invariants, nonlowness classes, and Weihrauch reducibility, with R. Kuyper and D. Turetsky, submitted.

Effectively closed subgroups of the infinite symmetric group, with A. Melnikov, A. Nies and D. Turetsky.
Proceedings of the American Mathematical Society 146 (2018) pp. 54215435.

Maximality and collapse in the hierarchy of αc.a. degrees, with K. Arthur and R. Downey, submitted.

MartinLöf reducibility and cost functions, with J. Miller, A. Nies and D. Turetsky, submitted.
 Dimension 1 sequences are close to randoms, with J. Miller, A. Shen and L.B. Westrick.
Theoretical Computer Science 705 (2018) pp. 99112.

Uniform procedures in uncountable structures, with J. Knight, A. Melnikov and D. Turetsky.
Journal of Symbolic Logic 83 (2018) pp. 529550.

Computing from projections of random points: A dense hierarchy of subideals of the Ktrivial degrees, with J. Miller and A. Nies, submitted.

Higher randomness and genericity, with B. Monin.
Forum of Mathematics: Sigma 5 (2017), 41 pages.

Finding bases of uncountable free abelian groups is usually difficult, with D. Turetsky and L.B. Westrick.
Transactions of the American Mathematical Society 370 (2018) pp. 44834508.

Proper divisibility in computable rings, with A. Melnikov.
Journal of Algebra 474 (2017) pp. 180212.

Generic Muchink reducibility and presentations of fields, with R. Downey and J. Miller.
Israel Journal of Mathematics 216 (2016) pp. 371387.

Two more characterizations of Ktriviality, with J. Miller, B. Monin and D. Turetsky.
Notre Dame Journal of Formal Logic 59 (2018) pp. 189195.

Continuous higher randomness, with L. Bienvenu and B. Monin.
Journal of Mathematical Logic 17 (2017), 53 pages.

DNR and incomparable Turing degrees, with M. Cai and M. McInerney.
Forum of Mathematics: Sigma 4 (2016), 44 pages.

The finite intersection principle and genericity, with D. Diamondstone, R. Downey and D. Turetsky.
Mathematical Proceedings of the Cambridge Philosophical Society 160 (2016) pp. 279297.

Models of Cohen measurability, with S. Shelah.
Annals of Pure and Applied Logic 165 (2014) pp. 15571576.

Coherent randomness tests and computing the Ktrivial sets, with L. Bienvenu, A. Kučera, A. Nies and D. Turetsky.
Journal of the European Mathematical Society 18 (2016) pp. 773812.

Computability and uncountable linear orderings II: degree spectra, with A. Kach, S. Lempp, and D. Turetsky.
Journal of Symbolic Logic 80 (2015) pp. 145178.

Joining nonlow c.e. sets with diagonally noncomputable functions, with L. Bienvenu, A. Kučera, J. Miller, A. Nies and D. Turetsky.
Journal of Logic and Computation 23 (2013) pp. 11831194.

A van Lambalgen theorem for Demuth randomness, with D. Diamondstone and D. Turetsky.
In Downey et al. (eds.), Proceedings of the 12th Asian Logic Colloquium, Wellington, World Scientific (2013), pp. 115124.

Natural large degree spectra, with D. Diamondstone and D. Turetsky.
Computability 1 (2012), pp. 99106.

Characterising lowness for Demuth randomness, with L. Bienvenu, R. Downey, A. Nies, and D. Turetsky.
Journal of Symbolic Logic 79 (2014), pp. 526560.

Galvin's "racing pawns" game, internal hyperarithmetic comprehension, and the law of excluded middle, with C. Conidis and D. Turetsky.
Notre Dame Journal of Formal Logic 54 (2013), pp. 233252.

Relative to any nonhyperarithmetic set, with A. Montalbán and T.A. Slaman.
Journal of Mathematical Logic 13 (2013), 26 pages.

Inherent enumerability of strong jumptraceability, with D. Diamondstone and D. Turetsky.
Transactions of the American Mathematical Society 367 (2015), pp. 17711796.

Computability and uncountable linear orderings I: computable categoricity, with A. Kach, S. Lempp, and D. Turetsky.
Journal of Symbolic Logic 80 (2015) pp. 116144.

Pseudojump inversion, upper cone avoidance, and strong jumptraceability, with R. Downey.
Advances in Mathematics 237 (2013), pp. 252285.

Strong jumptraceability and Demuth randomness, with D. Turetsky.
Proceedings of the London Mathematical Society 108 (2014), pp. 738779.

Anticomplexity, lowness and highness notions, and reducibilities with tiny use, with J. Franklin, F. Stephan and G. Wu.
Journal of Symbolic Logic 78 (2013), pp. 13071327.

MartinLöf random points satisfy Birkhoff's ergodic theorem with respect to effectively closed sets, with J. Franklin, J. Miller, and K.M. Ng.
Proceedings of the American Mathematical Society 140 (2012), pp. 36233628.

Computable structure theory on ω_{1} using admissibility, with J. Knight.
In Greenberg, Hamkins, Hirschfeldt and Miller (eds.), Effective Mathematics of the Uncountable, Lecture Notes in Logic, ASL and Cambridge University Press (2013), pp. 5080.
 Strong jumptraceability II: Ktriviality, with R. Downey.
Israel Journal of Mathematics 191 (2012), pp. 647666.

Characterising the strongly jumptraceable sets via randomness, with D. Hirschfeldt and A. Nies.
Advances in Mathematics 231 (2012), pp. 22522293.
(This is a slightly expanded version.)

The SlamanWehner theorem in higher recursion theory, with A. Montalbán and T.A. Slaman.
Proceedings of the American Mathematical Society 139 (2011), pp. 18651869.

Ktrivial sets are never continuously random, with G. Barmpalias, A. Montalbán, and T.A. Slaman.
In Arai et al. (eds.), Proceedings of the 11th Asian Logic Conference, Singapore, World Scientific (2011), pp. 5158.

A random set which only computes strongly jumptraceable c.e. sets.
Journal of Symbolic Logic 76 (2011), pp. 700718.

Extensions of embeddings below computably enumerable degrees, with R. Downey, A.E.M. Lewis, and A. Montalbán.
Transactions of the American Mathematical Society 365 (2013), pp. 29773018.

Binary subtrees with few labelled paths,
with R. Downey, C. Jockusch, and K. Milans.
Combinatorica 31 (2011), pp. 285303.

Diagonally nonrecursive functions and effective Hausdorff dimension,
with J. Miller.
Bulletin of the London Mathematical Society 43 (2011), pp. 636654.

Benign cost functions and lowness notions,
with A. Nies.
Journal of Symbolic Logic 76 (2011), pp. 289312.

Lowness for Kurtz randomness,
with J. Miller.
Journal of Symbolic Logic 74 (2009), pp. 665678.

Ktrivial degrees and the jumptraceability hierarchy,
with G. Barmpalias and R. Downey.
Proceedings of the American Mathematical Society 137 (2009), pp. 20992109.

Working with strong
reducibilities above totally ωc.e. and array computable degrees,
with G. Barmpalias and R. Downey.
Transactions of the American Mathematical Society 362 (2010), pp. 777813.

Turing degrees of reals of positive packing dimension,
with R. Downey.
Information Processing Letters 108 (2008), pp. 198203.

The upward closure of a perfect thin class,
with R. Downey and J. Miller.
Annals of Pure and Applied Logic 156 (2008), pp. 5158.

Strong jumptraceability I: the computably enumerable case,
with R. Downey and P. Cholak.
Advances in Mathematics 217 (2008), pp. 20452074.

Totally
<ω^{ω}computably enumerable degrees and mtopped degrees,
with R. Downey.
In Cai, Cooper, Li (eds.),
Theory and Applications of Models of Computation (proceedings of TAMC 2006, Beijing),
Lecture Notes in Computer Science 3959, Springer (2006), pp. 4660.

Totally ωcomputably enumerable degrees and
bounding critical triples,
with R. Downey and R. Weber.
Journal of Mathematical Logic 7 (2007), pp. 145171.

Models of realvalued measurability,
with S. Fuchino and S. Shelah.
Annals of Pure and Applied Logic 142 (2006), pp. 380397.

Lowness for computable machines,
with R. Downey, A. Nies and N. Mihailović.
In Chong et al. (eds.), Computational Prospects of Infinity, Lecture Notes Series of the Institute for Mathematical Sciences, NUS, vol. 15, World Scientific (2008), pp. 7986.

Every 1generic computes a properly 1generic,
with B. Csima, R. Downey, D. Hirschfeldt, and J. Miller.
Journal of Symbolic Logic 71 (2006), pp. 13851393.

Prompt simplicity, array computability, and cupping,
with R. Downey, J. Miller and R. Weber.
In Chong et al. (eds.), Computational Prospects of Infinity, Lecture Notes Series of the Institute for Mathematical Sciences, NUS, vol. 15, World Scientific (2008), pp. 5978.

Uniform almost everywhere domination, with P. Cholak and J. Miller.
Journal of Symbolic Logic 71 (2006), pp. 10571072.

Ranked structures and arithmetic transfinite recursion,
with A. Montalbán.
Transactions of the American Mathematical Society 360 (2008), 12651307.

Embedding and coding below a 1generic degree, with A. Montalbán.
Notre Dame Journal of Formal Logic 44 (2003), pp. 200216.

Generalized high degrees have the complementation property, with A. Montalbán and R.A. Shore.
Journal of Symbolic Logic 69 (2004), pp. 12001220.

The elementary theory of the
metarecursively enumerable degrees, with T.A. Slaman and R.A. Shore.
Journal of Mathematical Logic 6 (2006), pp. 380397.