Sean Cox
Research page
Research interests: Set theory, mathematical logic, large cardinals, forcing, inner model theory, applications of set theory to other branches of mathematics.
I am especially interested in combinatorial properties of small cardinals which mimic the behavior of large cardinals. For example, strong forcing axioms make the 2nd uncountable cardinal exhibit properties typically associated with much larger ("supercompact") cardinals. What is a forcing axiom? Menachem Magidor gave a nice informal definition: it asserts that if a mathematical object can be imagined in a reasonable way, then it exists! The strength of the axiom depends on how you define "imagined in a reasonable way". For more details, including a non-technical overview of set theory, please see my Research Statement.
Publications and Preprints (most recent are listed first):
"Ideal projections as forcing projections" (with Martin Zeman), in preparation. Abstract
"Martin's Maximum and tower forcing II" (with Matteo Viale), in preparation.
"Martin's Maximum and tower forcing," (with Matteo Viale), submitted.
"The Diagonal Reflection Principle,'' Proceedings of the AMS (to appear). This paper and some related results are summarized in this poster (from the 3rd European Set Theory Conference).
"PFA and ideals on $\omega_2$ whose associated forcings are proper,'' Notre Dame Journal of Formal Logic (to appear).
"Consistency strength of higher Chang's Conjectures, without CH,'' Archive for Mathematical Logic 50 (2011), no. 7, pp. 759-775.
"Nonregular ultrafilters on \omega_2 ,'' Journal of Symbolic Logic 76 (2011), no. 3, pp. 827-845.
"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 and/or abstracts to some recent presentations:
Ideal projections as forcing projections, University of Torino logic seminar (November 2011).
The Tree Property and elementary embeddings, University of Denver Graduate Colloquium (September 2011). NOTE: some web browsers seem to have trouble displaying this file due to its size (about 2MB); in that case you may need to download it first.
Martin's Maximum and limitations on presaturated towers, University of Colorado (Boulder) logic seminar (September 2011).
Martin's Maximum and Tower Forcing, Muenster Oberseminar in Logic (June 2011).
Diagonal stationary reflection and generic ultrapowers, invited talk at the 2011 Mid Atlantic Mathematical Logic Seminar at Harvard University (February 2011).
Forcing axioms and correct models, Muenster Oberseminar in Logic (December 2010, January 2011).
The Diagonal Reflection Principle, invited talk in Torino, Italy (November 2010).
(A topic distantly related to) Natural ideals under PFA, invited talk at Sectional meeting of the AMS, UCLA (October 2010).
Catching slender functions, invited talks (2) at the UC-Irvine Logic Seminar (October 2010).
Nonregularity, Amsterdam Set Theory Workshop (June 2010).
Moore's Map Reflection Principle, Muenster Logic Oberseminar (May 2010).
The Chang Ideal, invited talk at the University of Bonn Oberseminar in Logic (April 2010).
Reviews authored:
MathSciNet review of Andrew Brooke-Taylor's "Indestructibility of Vopenka's Principle".
MathSciNet review of Thilo Weinert's "The Bounded Axiom A Forcing Axiom".
MathSciNet review of Sy-David Friedman's "Forcings which preserve large cardinals".
MathSciNet review of Arthur Apter's "Indestructibility, instances of strong compactness, and level by level inequivalence".