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 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 classifications of 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). $27,390. Papers [1] Natasha Dobrinen. "Generalized weak distributive laws in Boolean algebras and issues related to a problem of von Neumann" thesis.pdf [2] Natasha Dobrinen. "Games and general distributive laws in Boolean algebras," Proc. Amer. Math. Soc. 131 (2003) 309-318. Distributive_laws.pdf Natasha Dobrinen. "Errata to `Games and general distributive laws in Boolean algebras'," Proc. Amer. Math. Soc. 131 (2003) 2967-2968. Distributive_laws_errata [3] Natasha Dobrinen. "Complete embeddings of the Cohen algebra into three families of c.c.c., non-measurable Boolean algebras," Pacific Jour. Math. 214(2) (2004) 201-222. Cohen_algebras.pdf [4] Natasha Dobrinen and Stephen G. Simpson. "Almost everywhere domination," Jour. of Symbolic Logic 69(3) (2004) 914-922. Almost everywhere domination.pdf [5] Natasha Dobrinen and Sy-David Friedman. "Co-stationarity of the ground model," Jour. Symbolic Logic 71(3) (2006) 1029-1043. co-stationarity.pdf [6] James Cummings and Natasha Dobrinen. "The hyper-weak distributive law and a related game in Boolean algebras," Annals of Pure and Applied Logic 149 (2007), no. 1-3, 14--24 hyper-weak.pdf. [7] Natasha Dobrinen. "More ubiquitous undetermined games and other results on uncountable length games in Boolean algebras," Note di Matematica 27 (2007), suppl. 1, 65--83. unctbl_games.pdf. [8] Natasha Dobrinen. "Co-stationarity of the ground model and new$\omega$-seqences," Proc. Amer. Math. Soc. 136 (2008), no.5, 1815--1821. new_omega_seq.pdf. [9] Natasha Dobrinen. "$\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.

[10] 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.

[11] 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

[12] 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.

[13] 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.pdf.

Natasha Dobrinen. "Continuous cofinal maps on ultrafilters," (2010) 23 pp. continuous_maps.pdf (Due to sundry roadblocks, 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 below.)

[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 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.

[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. "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.

[19] 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

[20] Natasha Dobrinen, Jose Mijares, and Timothy Trujillo. "Topological Ramsey spaces from Fraisse classes, Ramsey-classification theorems, and initial structures in the Tukey types of p-points," (2014) 34 pp, To appear. Invited submission to Archive for Mathematical Logic special volume in honor of James E. Baumgartner. Ramsey Spaces from Fraisse classes

[21] 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

[22] Natasha Dobrinen and Jose Mijares. "Topological Ramsey spaces and metrically Baire sets," Journal of Combinatorial Theory, Series A, 135 (2015) 161--180. For the next few weeks you can obtain a free copy of the actual article here. Or see Metrically Baire Sets for the preprint.

[23] Natasha Dobrinen, Claude Laflamme, and Norbert Sauer. "Rainbow Ramsey simple structures," (2014) 12 pp, submitted. Rainbow Ramsey Simple Structures

[24] Natasha Dobrinen. "Continuous and other finitely generated canonical cofinal maps on ultrafilters," (2015) 37 pp, submitted. Canonical Cofinal Maps

[25] Natasha Dobrinen. "Infinite dimensional Ellentuck spaces and Ramsey-classification theorems." (2015) 36 pp, submitted. Infinite dimensional Ellentuck spaces

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

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

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. 20(1) (2014) 94--97.

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