Research Monographs

  1. A transfinite hierarchy of lowness notions in the computably enumerable degrees, unifying classes, and natural definability, with R. Downey.
    Annals of Mathematics Studies 206 (2020), Princeton. 240 pages.
  2. The role of true finiteness in the admissible recursively enumerable degrees.
    Memoirs of the American Mathematical Society 181 854 (2006), 99 pages.

Research Announcements and Surveys

  1. Iterated priority arguments in descriptive set theory, with A. Day, M. Harrison-Trainor and D. Turetsky.
    Bulletin of Symbolic Logic, to appear.
  2. Two applications of admissible computability.
    In A. Rezuş (ed.), Contemporary Logic and Computing, College publications, London 2020, Landscapes in Logic, vol. 1, pp. 604-637.
  3. A hierarchy of computably enumerable degrees, with R. Downey.
    Bulletin of Symbolic Logic 24 (2018) pp. 53-89.
  4. Strong jump-traceability, with D. Turetsky.
    Bulletin of Symbolic Logic 24 (2018) pp. 147-164.
  5. Computing K-trivial sets by incomplete random sets, with L. Bienvenu, A. Day, A. Kučera, J. Miller, A. Nies and D. Turetsky.
    Bulletin of Symbolic Logic 20 (2014), pp. 80-90.
  6. The role of true finiteness in the admissible recursively enumerable degrees.
    Bulletin of Symbolic Logic 11 (2005), pp. 398-410.

Research Papers

  1. Some open questions and recent results on computable Banach spaces, with R. Downey and L. Qian, to appear in proceedings of CiE 2024.
  2. Effective separation and reduction properties of Borel Wadge classes, with D. Turetsky, submitted.
  3. An effective classification of Borel Wadge classes, with A. Day, M. Harrison-Trainor and D. Turetsky, submitted.
  4. Algorithmically random seris, with R. Downey and A. Tanggara. .
    Computability, to appear.
  5. Many forcing axioms for all regular uncountable cardinals, with S. Shelah.
    Israel Journal of Mathematics, to appear.
  6. A hierarchy of computably enumerable degrees, II: some recent developments and new directions, with R. Downey and E. Hammatt.
    New Zealand Journal of Mathematics 52 (2022) pp.175-231.
  7. Cousin's lemma in second-order arithmetic, with J.M. Barrett and R. Downey.
    Proceedings of the American Mathematical Society B 9 (2022) pp.125-134.
  8. A note on computable distingushing colorings, with N. Bazhenov, A. Melnikov, R. Miller and and K.M. Ng.
    Lobachevskii Journal of Mathematics 42 (2021) pp.693-700.
  9. Realizing computably enumerable degrees in separating classes, with P. Cholak, R. Downey, and D. Turetsky, submitted.
  10. Computing sets from all infinite subsets, with M. Harrison-Trainor, L. Patey, and D. Turetsky.
    Transactions of the American Mathematical Society 374 (2021) pp. 8131-8160.
  11. Enumerations of families closed under finite differences, with M. Harrison-Trainor, J. Miller, and D. Turetsky.
    Computability 13 pp. 89-104.
  12. Scott complexity of countable structures, with R. Alvir, M. Harrison-Trainor, and D. Turetsky.
    Journal of Symbolic Logic 86 (2021) pp. 1706-1720.
  13. Non-density in punctual computability, with M. Harrison-Trainor, A. Melnikov, and D. Turetsky.
    Annals of Pure and Applied Logic 172 (2021) #102985, 17 pages.
  14. More on bases of uncountable free abelian groups, with L. Richter, S. Shelah, and D. Turetsky, submitted.
  15. Highness properties close to PA-completeness, with J. Miller and A. Nies.
    Israel Journal of Mathematics 244 (2021) pp.419-465.
  16. Punctual categoricity and universality, with R. Downey, A. Melnikov, K.M. Ng, and D. Turetsky.
    Journal of Symbolic Logic 85 (2020) pp.1427-1466.
  17. Cuts in the ML degrees, with K. Arthur and J. Miller.
    In Aspects of Computation and Automata Theory with Applications Lecture Notes Series of the IMS, Singapore 42 (2023), pp.67-78.
  18. Completeness of the hyperaritmetic isomorphism equivalence relation, with D. Turetsky.
    Advances in Mathematics 403 (2022), 56 pages.
  19. Relationships between computability-theoretic properties of problems, with R. Downey, M. Harrison-Trainor, L. Patey and D. Turetsky.
    Journal of Symbolic Logic 87 (2022) pp.47-71.
  20. Bad oracles in higher computability and randomness, with L. Bienvenu and B. Monin.
    Israel Journal of Mathematics 8 (2021) pp.229-276.
  21. Cupping and jump classes in the computably enumerable degrees, with K.M. Ng and G. Wu.
    Journal of Symbolic Logic 85 (2020) pp.1499-1545.
  22. Cardinal invariants, non-lowness classes, and Weihrauch reducibility, with R. Kuyper and D. Turetsky.
    Computability 8 (2019) pp.305-346.
  23. 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. 5421-5435.
  24. Maximality and collapse in the hierarchy of α-c.a. degrees, with K. Arthur and R. Downey.
    Computability 10 (2021) pp. 181-214.
  25. Martin-Löf reducibility and cost functions, with J. Miller, A. Nies and D. Turetsky.
    Israel Journal of Mathematics 260 (2024), pp. 261-301.
  26. Dimension 1 sequences are close to randoms, with J. Miller, A. Shen and L.B. Westrick.
    Theoretical Computer Science 705 (2018) pp. 99-112.
  27. Uniform procedures in uncountable structures, with J. Knight, A. Melnikov and D. Turetsky.
    Journal of Symbolic Logic 83 (2018) pp. 529-550.
  28. Computing from projections of random points, with J. Miller and A. Nies.
    Journal of Mathematical Logic 20 (2020), 41 pages.
  29. Higher randomness and genericity, with B. Monin.
    Forum of Mathematics: Sigma 5 (2017), 41 pages.
  30. 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. 4483-4508.
  31. Proper divisibility in computable rings, with A. Melnikov.
    Journal of Algebra 474 (2017) pp. 180-212.
  32. Generic Muchink reducibility and presentations of fields, with R. Downey and J. Miller.
    Israel Journal of Mathematics 216 (2016) pp. 371-387.
  33. Two more characterizations of K-triviality, with J. Miller, B. Monin and D. Turetsky.
    Notre Dame Journal of Formal Logic 59 (2018) pp. 189-195.
  34. Continuous higher randomness, with L. Bienvenu and B. Monin.
    Journal of Mathematical Logic 17 (2017), 53 pages.
  35. DNR and incomparable Turing degrees, with M. Cai and M. McInerney.
    Forum of Mathematics: Sigma 4 (2016), 44 pages.
  36. The finite intersection principle and genericity, with D. Diamondstone, R. Downey and D. Turetsky.
    Mathematical Proceedings of the Cambridge Philosophical Society 160 (2016) pp. 279-297.
  37. Models of Cohen measurability, with S. Shelah.
    Annals of Pure and Applied Logic 165 (2014) pp. 1557-1576.
  38. Coherent randomness tests and computing the K-trivial sets, with L. Bienvenu, A. Kučera, A. Nies and D. Turetsky.
    Journal of the European Mathematical Society 18 (2016) pp. 773-812.
  39. Computability and uncountable linear orderings II: degree spectra, with A. Kach, S. Lempp, and D. Turetsky.
    Journal of Symbolic Logic 80 (2015) pp. 145-178.
  40. Joining non-low c.e. sets with diagonally non-computable functions, with L. Bienvenu, A. Kučera, J. Miller, A. Nies and D. Turetsky.
    Journal of Logic and Computation 23 (2013) pp. 1183-1194.
  41. 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. 115-124.
  42. Natural large degree spectra, with D. Diamondstone and D. Turetsky.
    Computability 1 (2012), pp. 99-106.
  43. Characterising lowness for Demuth randomness, with L. Bienvenu, R. Downey, A. Nies, and D. Turetsky.
    Journal of Symbolic Logic 79 (2014), pp. 526-560.
  44. 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. 233-252.
  45. Relative to any non-hyperarithmetic set, with A. Montalbán and T.A. Slaman.
    Journal of Mathematical Logic 13 (2013), 26 pages.
  46. Inherent enumerability of strong jump-traceability, with D. Diamondstone and D. Turetsky.
    Transactions of the American Mathematical Society 367 (2015), pp. 1771-1796.
  47. Computability and uncountable linear orderings I: computable categoricity, with A. Kach, S. Lempp, and D. Turetsky.
    Journal of Symbolic Logic 80 (2015) pp. 116-144.
  48. Pseudo-jump inversion, upper cone avoidance, and strong jump-traceability, with R. Downey.
    Advances in Mathematics 237 (2013), pp. 252-285.
  49. Strong jump-traceability and Demuth randomness, with D. Turetsky.
    Proceedings of the London Mathematical Society 108 (2014), pp. 738-779.
  50. Anti-complexity, lowness and highness notions, and reducibilities with tiny use, with J. Franklin, F. Stephan and G. Wu.
    Journal of Symbolic Logic 78 (2013), pp. 1307-1327.
  51. Martin-Löf 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. 3623-3628.
  52. Computable structure theory on ω1 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. 50-80.
  53. Strong jump-traceability II: K-triviality, with R. Downey.
    Israel Journal of Mathematics 191 (2012), pp. 647-666.
  54. Characterising the strongly jump-traceable sets via randomness, with D. Hirschfeldt and A. Nies.
    Advances in Mathematics 231 (2012), pp. 2252-2293.
    (This is a slightly expanded version.)
  55. The Slaman-Wehner theorem in higher recursion theory, with A. Montalbán and T.A. Slaman.
    Proceedings of the American Mathematical Society 139 (2011), pp. 1865-1869.
  56. K-trivial 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. 51-58.
  57. A random set which only computes strongly jump-traceable c.e. sets.
    Journal of Symbolic Logic 76 (2011), pp. 700-718.
  58. 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. 2977-3018.
  59. Binary subtrees with few labelled paths, with R. Downey, C. Jockusch, and K. Milans.
    Combinatorica 31 (2011), pp. 285-303.
  60. Diagonally non-recursive functions and effective Hausdorff dimension, with J. Miller.
    Bulletin of the London Mathematical Society 43 (2011), pp. 636-654.
  61. Benign cost functions and lowness notions, with A. Nies.
    Journal of Symbolic Logic 76 (2011), pp. 289-312.
  62. Lowness for Kurtz randomness, with J. Miller.
    Journal of Symbolic Logic 74 (2009), pp. 665-678.
  63. K-trivial degrees and the jump-traceability hierarchy, with G. Barmpalias and R. Downey.
    Proceedings of the American Mathematical Society 137 (2009), pp. 2099-2109.
  64. Working with strong reducibilities above totally ω-c.e. and array computable degrees, with G. Barmpalias and R. Downey.
    Transactions of the American Mathematical Society 362 (2010), pp. 777-813.
  65. Turing degrees of reals of positive packing dimension, with R. Downey.
    Information Processing Letters 108 (2008), pp. 198-203.
  66. The upward closure of a perfect thin class, with R. Downey and J. Miller.
    Annals of Pure and Applied Logic 156 (2008), pp. 51-58.
  67. Strong jump-traceability I: the computably enumerable case, with R. Downey and P. Cholak.
    Advances in Mathematics 217 (2008), pp. 2045-2074.
  68. Totally <ωω-computably enumerable degrees and m-topped 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. 46-60.
  69. Totally ω-computably enumerable degrees and bounding critical triples, with R. Downey and R. Weber.
    Journal of Mathematical Logic 7 (2007), pp. 145-171.
  70. Models of real-valued measurability, with S. Fuchino and S. Shelah.
    Annals of Pure and Applied Logic 142 (2006), pp. 380-397.
  71. 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. 79-86.
  72. Every 1-generic computes a properly 1-generic, with B. Csima, R. Downey, D. Hirschfeldt, and J. Miller.
    Journal of Symbolic Logic 71 (2006), pp. 1385-1393.
  73. 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. 59-78.
  74. Uniform almost everywhere domination, with P. Cholak and J. Miller.
    Journal of Symbolic Logic 71 (2006), pp. 1057-1072.
  75. Ranked structures and arithmetic transfinite recursion, with A. Montalbán.
    Transactions of the American Mathematical Society 360 (2008), 1265-1307.
  76. Embedding and coding below a 1-generic degree, with A. Montalbán.
    Notre Dame Journal of Formal Logic 44 (2003), pp. 200-216.
  77. Generalized high degrees have the complementation property, with A. Montalbán and R.A. Shore.
    Journal of Symbolic Logic 69 (2004), pp. 1200-1220.
  78. The elementary theory of the metarecursively enumerable degrees, with T.A. Slaman and R.A. Shore.
    Journal of Mathematical Logic 6 (2006), pp. 380-397.