Natasha Dobrinen's Research


Research Interests

My research interests mainly fall under the broad category of Logic and Foundations of Mathematics. I do research in Set Theory, Ramsey Theory, Boolean Algebras, and Measure Theory. I have investigated relationships between random reals, eventually dominating functions, measure, generalized weak distributive laws, infinitary two-player games, and complete embeddings of the Cohen algebra into complete Boolean algebras. Currently, I am working on problems in Ramsey theory, problems regarding the structure of the Tukey types of ultrafilters, and problems involving both.


Grants

National Science Foundation Grant DMS-1301665. "Ramsey Theory, Set Theory, and Tukey Order." September 1,2013 - August 31, 2016. $114,368.

Support for Travel and Living Expenses to participate in the Fields Institute Thematic Program in Forcing and Its Applications, National Science Foundation, July 2012 - December 2012. $5000.

Simons Foundation Collaboration Grant #245286. "Classification of Tukey types of ultrafilters." September 1, 2012 - August 31, 2017. $35,000. (Terminated September 1, 2013 due to NSF grant.)

Association for Women in Mathematics/National Science Foundation Mentoring Travel Grant. November/December 2010 (Paris). $2800.

University of Denver Faculty Research Fund grant. "Subtle Measurements on Large Spaces." July/August 2010 (Paris). $3000.


Papers

Natasha Dobrinen. "Generalized weak distributive laws in Boolean algebras and issues related to a problem of von Neumann" thesis.pdf

Natasha Dobrinen. "Games and general distributive laws in Boolean algebras," Proc. Amer. Math. Soc. 131 (2003) 309-318. Distributive_laws.pdf

Natasha Dobrinen. "Errata to `Games and general distributive laws in Boolean algebras'," Proc. Amer. Math. Soc. 131 (2003) 2967-2968. Distributive_laws_errata

Natasha Dobrinen. "Complete embeddings of the Cohen algebra into three families of c.c.c., non-measurable Boolean algebras," Pacific Jour. Math. 214(2) (2004) 201-222. Cohen_algebras.pdf

Natasha Dobrinen and Stephen G. Simpson. "Almost everywhere domination," Jour. of Symbolic Logic 69(3) (2004) 914-922. Almost everywhere domination.pdf

Natasha Dobrinen and Sy-David Friedman. "Co-stationarity of the ground model," Jour. Symbolic Logic 71(3) (2006) 1029-1043. co-stationarity.pdf

James Cummings and Natasha Dobrinen. "The hyper-weak distributive law and a related game in Boolean algebras," Annals of Pure and Applied Logic 149 (2007), no. 1-3, 14--24 hyper-weak.pdf.

Natasha Dobrinen. "More ubiquitous undetermined games and other results on uncountable length games in Boolean algebras," Note di Matematica 27 (2007), suppl. 1, 65--83. unctbl_games.pdf.

Natasha Dobrinen. "Co-stationarity of the ground model and new $\omega$-seqences," Proc. Amer. Math. Soc. 136 (2008), no.5, 1815--1821. new_omega_seq.pdf.

Natasha Dobrinen. "$\kappa$-stationary subsets of $\mathcal{P}_{\kappa^+}\lambda$, infinitary games, and distributive laws in Boolean algebras," Journal of Symbolic Logic 73 (2008), no. 1, 238--260. games_kstat.pdf.

Natasha Dobrinen and Sy-David Friedman. "Internal consistency and co-stationarity of the ground model," Journal of Symbolic Logic 73 (2008), no. 2, 512--521. Internal Consistency Costationarity.pdf.

Natasha Dobrinen and Sy-DavidFriedman. "Homogeneous iteration and measure one covering relative to HOD," Archive for Mathematical Logic 47 (2008), no. 7-8, 711--718. Homogeneous Iterations

Natasha Dobrinen and Sy-David Friedman. "The consistency strength of the tree property at the double successor of a measurable," Fundamenta Mathematicae 208 (2010), 123--153. treeproperty.pdf.

Natasha Dobrinen and Stevo Todorcevic. "Tukey types of ultrafilters," Illinois Journal of Mathematics 55(3) (2011), 907--951. (This paper now has an official publication date of 2011, even though it is appearing in 2013 due to the journal's backlog.) Here is a revised and improved version: tukey.pdf.

Natasha Dobrinen. "Continuous cofinal maps on ultrafilters," (2010) 23 pp, submitted. continuous_maps.pdf

Natasha Dobrinen and Stevo Todorcevic. "A new class of Ramsey-Classification Theorems and their applications in the Tukey theory of ultrafilters, Part 1," Transactions of the American Mathematical Society, 26 pp, to appear. A Ramsey Classification Theorem

Natasha Dobrinen and Stevo Todorcevic. "A new class of Ramsey-classification Theorems and their applications in the Tukey theory of ultrafilters, Part 2," Transactions of the American Mathematical Society, 34 pp, to appear. Ramsey-Classification Theorems

"Natasha Dobrinen and Stevo Todorcevic. "A new class of Ramsey-classification Theorems and their applications in the Tukey theory of ultrafilters, Parts 1 and 2," (Extended Abstract from poster at the Erdos Centennary conference, Budapest, July 2013), Electronic Notes in Discrete Mathematics, 43 (2013) 107--112.

Andreas Blass, Natasha Dobrinen, and Dilip Raghavan. "The next best thing to a p-point," (2013) 35 pp, submitted.

Natasha Dobrinen. "Survey on the Tukey theory of ultrafilters," Mathematical Institutes of the Serbian Academy of Sciences," (2013) 29 pp, submitted.

Natasha Dobrinen. "The Abstract Nash-Williams Theorem and applications to initial structures in the Tukey types of non-p-points satisfying weak partition properties," preprint.

Natasha Dobrinen, Jose Mijares, and Timothy Trujillo. "A general framework for topological Ramsey spaces, Ramsey-classification theorems, and initial structures in the Tukey types of p-points, Part 1," in preparation.

Please note: If there are any problems with the links to papers, the latter ones may be found at the Mathematics Department Preprint Series http://www.du.edu/nsm/departments/mathematics/research/preprintseries.html


Book Reviews

Herbert B. Enderton. "A Mathematical Introduction to Logic," Bull. Symbolic Logic. 9(3) (2003) 406--407.

Steven Givant and Paul Halmos. "Introduction to Boolean Algebras," Bull. Symbolic Logic 16(2) (2010) 281--282.

"Appalachian Set Theory, 2006-2012," James Cummings and Ernest Schimmerling, Editors. Bull. Symbolic Logic. 4pp, To appear.


"For the Lord God is my strength and song, and He has become my salvation." Isaiah 12:2