## Research

Research interests: Set theory, mathematical logic, large cardinals, forcing, inner model theory, applications of set theory to other branches 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 two topics, which often overlap:

• 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.

Papers and preprints (most recent listed first):

1. On the universality of the nonstationary ideal, to appear in Mathematical Logic Quarterly.
2. (with Monroe Eskew) Strongly proper forcing and some problems of Foreman, submitted December 2016.
3. (with John Krueger) Namba forcing, weak approximation, and guessing, submitted October 2016.
4. Chang’s Conjecture and semiproperness of nonreasonable posets, submitted May 2016.
5. (with Dominik Adolf and Philip Welch) Lower consistency bounds for mutual stationarity with divergent uncountable cofinalities, to appear in Israel Journal of Mathematics.
6. (with John Krueger)  Indestructible guessing models and the continuum, Fund. Math. 239 (2017), no. 3, 221–258.
7. (with Philipp Lücke) Characterizing large cardinals in terms of layered posets, Annals of Pure and Applied Logic, Volume 168, Issue 5, pages 1112-1131.
8. Layered posets and Kunen’s universal collapse, to appear in Notre Dame Journal of Formal Logic.
9. (with John Krueger) Quotients of strongly proper forcings and guessing models, J. Symb. Log. 81 (2016), no. 1, 264-283.
10. (with Brent Cody) Indestructibility of generically strong cardinals, Fundamenta Mathematicae 232 (2016), no. 2, 131–149.
11. Prevalence of Generic Laver Diamond, Proceedings of the AMS, Volume 143, Number 9, September 2015, Pages 4045–4058.
12. (with Martin Zeman) Ideal projections and forcing projections, Journal of Symbolic Logic, Volume 79, Issue 04, December 2014, pp 1247-1285.
13. (with Matteo Viale) Martin’s Maximum and Tower ForcingIsrael Journal of Mathematics Volume 197, Issue 1 , pp 347-376.
14. 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).
15. PFA and ideals on $\omega_2$ whose associated forcings are proper Notre Dame J. Formal Logic Volume 53, Number 3 (2012), 397-412.
16. Consistency Strength of Higher Changs Conjecture, without CH, Archive for Mathematical Logic 50 (2011), no. 7, pp. 759-775.
17. Nonregular ultrafilters on $\omega_2$, Journal of Symbolic Logic 76 (2011), no. 3, pp. 827-845.
18. Covering Theorems for the core model, with an application to stationary set reflectionAnnals of Pure and Applied Logic 161 (2009), pp. 66-93.

Reviews (most recent listed first)