Natasha Dobrinen's Research


Research Interests

My research interests are Set Theory, Ramsey Theory, Topology, and Boolean Algebras, with applications to Discrete Mathematics and Analysis. Earlier works investigated when the ground model can be forced to be co-stationary in P_kappa(lambda), equiconsistencies of large cardinals with combinatorial statements such as the tree property, 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. Current projects include developing canonical Ramsey theory, especially in regard to topological Ramsey spaces, and applications to forcing, Analysis, Discrete Mathematics, and Tukey and Rudin-Keisler structures in topological spaces.


Grants

National Science Foundation Grant DMS-142470. "Conference on Infinitary Ramsey Theory, May 24-28, 2014". $10,200.

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.

National Science Foundation grant to start the BLAST Conference series (Co-PI) (University of Denver). $27,390.


Papers

PhD Thesis. Natasha Dobrinen. "Generalized weak distributive laws in Boolean algebras and issues related to a problem of von Neumann". University of Minnesota, 2001, Adviser: Karel Prikry. Dobrinen PhD Thesis

[1] Natasha Dobrinen. "Games and general distributive laws in Boolean algebras," Proc. Amer. Math. Soc. 131 (2003) 309-318. Distributive Laws

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

[2] 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 Algebra Embeddings

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

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

[5] 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 Distributivity

[6] 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. Uncountable Length Games

[7] 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. kappa-stationarity

[8] 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

[9] Natasha Dobrinen. "Global co-stationarity of the ground model from a new countable length sequence," Proceedings of the American Mathematical Society 136 (2008), no. 5, 1815--1821. Global_Costationarity

[10] 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

[11] 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. Tree Property

[12] 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.) Tukey types of ultrafilters

Natasha Dobrinen. "Continuous cofinal maps on ultrafilters," (2010) 23 pp. (This paper was never published. The results are incorporated into the new and extended paper "Continuous and other finitely generated canonical cofinal maps on ultrafilters", see [23] below.)

[13] 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," Electronic Notes in Discrete Mathematics, 43 (2013) 107--112. (Extended abstract from poster at the Erdos Centennary Conference, Budapest, July 2013).

[14] 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, 366 (2014), no. 3, 1659--1684. Ramsey Classification Theorems Part 1

[15] 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, 367 (2015), no. 7, 4627--4659. Ramsey-Classification Theorems Part 2

[16] Natasha Dobrinen. "Survey on the Tukey theory of ultrafilters," Selected Topics in Combinatorial Analysis, Zbornik Radova, Mathematical Institutes of the Serbian Academy of Sciences, 17(25) (2015) 53--80. Tukey Survey (Invited submission)

[17] Andreas Blass, Natasha Dobrinen, and Dilip Raghavan. "The next best thing to a p-point," Journal of Symbolic Logic. 80 (2015), no.3, 866--900. Next best thing

[18] Natasha Dobrinen. "High dimensional Ellentuck spaces and initial chains in the Tukey structure of non-p-points," Journal of Symbolic Logic, 28 pp, to appear. High dimensional Ellentuck spaces

[19] Jennifer Brown and Natasha Dobrinen. "Spectra of Tukey types of ultrafilters on Boolean algebras," Algebra Universalis, 20 pp, to appear. Tukey spectra in Boolean algebras

[20] Natasha Dobrinen and Jose Mijares. "Topological Ramsey spaces and metrically Baire sets," Journal of Combinatorial Theory, Series A, 135 (2015) 161--180. Metrically Baire Sets.

[21] Natasha Dobrinen, Jose G. Mijares, and Timothy Trujillo. "Topological Ramsey spaces from Fraisse classes, Ramsey-classification theorems, and initial structures in the Tukey types of p-points," 35 pp. To appear in Archive for Mathematical Logic, special issue in honor of James E. Baumgartner. Ramsey spaces from Fraisse classes (Invited submission)

[22] Natasha Dobrinen, Claude Laflamme, and Norbert Sauer. "Rainbow Ramsey simple structures," (2014) 12 pp. Accepted to Discrete Mathematics, subject to revisions. Rainbow Ramsey Simple Structures

[23] Natasha Dobrinen. "Continuous and other finitely generated canonical cofinal maps on ultrafilters," (2015) 41 pp, submitted. Canonical Cofinal Maps (updated 1/13/2016 with the version from 11/2015)

[24] Natasha Dobrinen. "Infinite dimensional Ellentuck spaces and Ramsey-classification theorems." (2015) 37 pp, submitted. Infinite dimensional Ellentuck spaces

[25] Natasha Dobrinen. "Creature forcing and topological Ramsey spaces." (2015) 13 pp, submitted. Creature forcing and topological Ramsey spaces (Invited submission)

Please note: If there are any problems with the links to papers, the post-2007 articles may be found at the Mathematics Department Preprint Series http://www.du.edu/nsm/departments/mathematics/research/preprintseries.html


Book Reviews

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

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

[3] "Appalachian Set Theory, 2006-2012," James Cummings and Ernest Schimmerling, Editors. Bull. Symbolic Logic. 20(1) (2014) 94--97.


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