Research

Research interests: Set theory, mathematical logic, and their connections to other areas of mathematics. I am interested in figuring out which mathematical problems are, at least in principle, solvable by mathematical methods.  The most famous results in this area are Gödel’s Incompleteness Theorems, the independence of the Continuum Hypothesis, and the independence of the Axiom of Choice (from the standard axioms of mathematics). My recent work focuses mostly on the following areas, which often overlap:

  • Connections of logic and set theory with other areas of math, particularly algebra, analysis, and category theory.
  • The properties and forcing effects of certain natural partial orders. I’m especially interested in Boolean algebras which arise as quotients of natural ideals (such as the nonstationary ideal and Shelah’s approachability ideal).
  • Large-cardinal-like behavior of small, accessible cardinals. This includes topics such as Forcing Axioms and generic elementary embeddings between models of set theory.

Preprints

  1. Kaplansky classes and stability
  2. (with Kamsma and Rosický): Cellular generation revisited
  3. (with David Aspero, Asaf Karagila, and Christoph Weiss)  The \kappa-Strongly Proper Forcing Axiom

Refereed Publications:

  1. Approximation Theory and Elementary Submodels. Bulletin of Symbolic Logic (to appear).
  2. (with Feigert, Kamsma, Mazari-Armida, and Rosický) Cofibrant generation of pure monomorphisms in preasheaf categories.  Advances in Mathematics 497 (2026), Article 111010, DOI 10.1016/j.aim.2026.111010.
  3. The Flat Cover Conjecture for Monoid Acts.  Journal of the London Mathematical Society (2) 112 (2025), no. 6, Paper No. e70404.
  4. Vopenka’s Principle, Maximum Deconstructibility, and singly-generated torsion classes.  Applied Categorical Structures  33 (2025), no. 3, Paper No. 19.
  5. (with Marco Aldi, Jeff Buffkin, and Cody Cline)  Sparse systems of functions and quasi-analytic classes.  Fundamenta Mathematicae 270 (2025), no. 2, 99–111.
  6. (with Alejandro Poveda and Jan Trlifaj) Approximation Properties of Torsion Classes. Bull. London Math. Soc. 56 (2024), no. 12, 3819–3828.
  7. (with Brent Cody and Kayla Lee) Sparse analytic systems.  Forum of Mathematics, Sigma 11 (2023).
  8. (with Gunter Fuchs) The diagonal strong reflection principle and its fragments.  The Journal of Symbolic Logic, 88(3), 1281-1309.
  9. Filtration Games and Potentially Projective Modules.  Fundamenta Mathematicae 260(2023), no.3, 199–232.
  10. (with Matthew Elpers) How robustly can you predict the future?  Canadian Journal of Mathematics 75(2023), no. 5, 1493–1515. 
  11. (with Monroe Eskew) Compactness versus hugeness at successor cardinals.   Journal of Mathematical Logic Vol. 23, No. 01, 2250016 (2023)
  12. (with Jiří Rosický) Fibrantly generated weak factorization systems.  Rend. Semin. Mat. Univ. Padova 152 (2024), 133–144.
  13. (with Philipp Lücke) Forcing axioms and the complexity of non-stationary ideals.  Monatsh. Math. 199 (2022), no. 1, 45–84.
  14. Salce’s problem on cotorsion pairs is undecidable.  Bull. London Math. Soc., 54 (2022): 1363-1374.
  15. Maximum Deconstructibility in module categories.  Journal of Pure and Applied Algebra 226 (2022), no. 5.
  16. Adjoining only the things you want: a survey of Strong Chang’s Conjecture and related topics.  To appear in Research Trends in Contemporary Logic.
  17. The Pi^1_1 downward Lowenheim-Skolem-Tarski property of Stationary Logic.  RIMS Kôkyûroku No. 2164 (2020).
  18. (with Hiroshi Sakai) Martin’s Maximum and the Diagonal Reflection Principle, RIMS Kôkyûroku No. 2141 (2019).
  19. Forcing axioms, approachability, and stationary set reflection, J. Symbolic Logic 86 (2021), no. 2, 499–530.
  20. (with Hiroshi Sakai)  A variant of Shelah’s characterization of Strong Chang’s Conjecture, Mathematical Logic Quarterly 65 (2019): 251-257.
  21. On the universality of the nonstationary ideal, Mathematical Logic Quarterly 64 (2018), no. 1-2, 103-117.
  22. (with Monroe Eskew) Strongly proper forcing and some problems of Foreman, Transactions of the American Mathematical Society 371 (2019), no. 7, 5039–5068.
  23. (with John Krueger) Namba forcing, weak approximation, and guessing, Journal of Symbolic Logic, Volume 83, Issue 4 December 2018 , pp. 1539-1565.
  24. Chang’s Conjecture and semiproperness of nonreasonable posets, Monatshefte für Mathematik 187 (2018), no. 4, 617-633.
  25. (with Dominik Adolf and Philip Welch) Lower consistency bounds for mutual stationarity with divergent uncountable cofinalities, Israel Journal of Mathematics October 2018, Volume 228, Issue 1, pp 1–19.
  26. (with John Krueger)  Indestructible guessing models and the continuum, Fundamenta Mathematicae 239 (2017), no. 3, 221–258.
  27. (with Philipp Lücke) Characterizing large cardinals in terms of layered posets, Annals of Pure and Applied Logic, Volume 168, Issue 5, pages 1112-1131.
  28. Layered posets and Kunen’s universal collapse, Notre Dame J. Form. Log. 60 (2019), no. 1, 27–60.
  29. (with John Krueger) Quotients of strongly proper forcings and guessing models, J. Symb. Log. 81 (2016), no. 1, 264-283.
  30. (with Brent Cody) Indestructibility of generically strong cardinals, Fundamenta Mathematicae 232 (2016), no. 2, 131–149.
  31. Prevalence of Generic Laver Diamond, Proceedings of the AMS, Volume 143, Number 9, September 2015, Pages 4045–4058.
  32. (with Martin Zeman) Ideal projections and forcing projections, Journal of Symbolic Logic, Volume 79, Issue 04, December 2014, pp 1247-1285.
  33. (with Matteo Viale) Martin’s Maximum and Tower Forcing, Israel Journal of Mathematics Volume 197, Issue 1 , pp 347-376.
  34. The Diagonal Reflection Principle, Proceedings of the AMS 140 (2012), no. 8, pp. 2893–2902.  This paper and some related results are summarized in this poster (from the 3rd European Set Theory Conference in Edinburgh, Scotland).
  35. PFA and ideals on omega_2 whose associated forcings are proper Notre Dame J. Formal Logic Volume 53, Number 3 (2012), 397-412.
  36. Consistency Strength of Higher Changs Conjecture, without CH, Archive for Mathematical Logic 50 (2011), no. 7, pp. 759-775.
  37. Nonregular ultrafilters on omega_2, Journal of Symbolic Logic 76 (2011), no. 3, pp. 827-845.
  38. Covering Theorems for the core model, with an application to stationary set reflection, Annals of Pure and Applied Logic 161 (2009), pp. 66-93.

Slides from selected talks

Reviews