**Research Interests**

My research interests are Set Theory, Ramsey Theory, Topology, and Boolean Algebras, Analysis and their inter-relations. Earlier works investigated algebraic characterizations of Boolean algebras with probability measures and realted eventually dominating functions; infinitary two-player games characterizing distributive laws; and consistency strengths of combinatorial statements such as the tree property and co-stationarity of the ground model.

Current projects include the Ramsey theory of homogeneous structures, applying set-theoretic methods to develop the area of Ramsey theory for homogeneous universal structures with forbidden substructures; interactions between Ramsey theory and forcing; developing canonical Ramsey theory for equivalence relations on relational structures; constructions of new Banach spaces using barriers on high and infinite dimensional Ellentuck spaces; and the general theory of topological Ramsey spaces, including applications to finding exact Tukey and Rudin-Keisler structures in the Stone-Cech compactification of the natural numbers.

**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, 81 (2016), no. 1, 237--263. High dimensional Ellentuck spaces

[19] Jennifer Brown and Natasha Dobrinen. "Spectra of Tukey types of ultrafilters on Boolean algebras," Algebra Universalis. 75 (2016), no. 4, 419--438. 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," Archive for Mathematical Logic, special issue in honor of James E. Baumgartner, 56 (2017), no. 7-8, 733--782. Ramsey spaces from Fraisse classes (Invited submission)

[22] Natasha Dobrinen, Claude Laflamme, and Norbert Sauer. "Rainbow Ramsey simple structures," Discrete Mathematics. 339 (2016), no. 11, 2848--2855. Rainbow Ramsey Simple Structures

[23] Natasha Dobrinen. "Continuous and other finitely generated canonical cofinal maps on ultrafilters," Fundamenta Mathematicae, 249 (2020), no. 2, 111--147. Canonical Cofinal Maps

[24] Natasha Dobrinen. "Infinite dimensional Ellentuck spaces and Ramsey-classification theorems," Journal of Mathematical Logic, 16 (2016), no. 1, 37 pp. Infinite dimensional Ellentuck spaces

[25] Natasha Dobrinen. "Creature forcing and topological Ramsey spaces," Topology and Its Applications, special issue in honor of Alan Dow's 60th birthday. 213 (2016), 110--126. Creature forcing and topological Ramsey spaces (Invited submission)

[26] Natasha Dobrinen and Dan Hathaway. "The Halpern-Lauchli Theorem at a measurable cardinal," Journal of Symbolic Logic, 82 (2017), no.4, 1560--1575. Halpern-Lauchli at measurable

[27] Natasha Dobrinen. "Topological Ramsey spaces dense in forcings," in "Structure and Randomness in Computability and Set Theory", World Scientific (2020), 3--58. Editors: Douglas Cenzer, Christopher Porter, and Jindrich Zapletal. SEALS

[28] Natasha Dobrinen. "The Ramsey theory of the universal homogeneous triangle-free graph," Journal of Mathematical Logic. 20 (2020), no. 2, 75pp. Big Ramsey degrees H_3

[29] Natasha Dobrinen. "Forcing in Ramsey theory," RIMS Kokyuroku 2042, (2017), 17--33. This is an expository paper corresponding to my third tutorial at the RIMS Symposium on Infinite Combinatorics and Forcing Theory, Kyoto 2016. RIMS Tutorial III

[30] Natasha Dobrinen, Daniel Hathaway, and Karel Prikry. "Perfect tree forcings for singular cardinals," Annals of Pure and Applied Logic, 171 (2020), no. 9, 102827, 25pp. Perfect tree forcings

[31] Alvaro Arias, Natasha Dobrinen, Gabriel Giron-Garnica, and Jose G. Mijares. "Banach spaces from barriers in high dimensional Ellentuck spaces," Journal of Logic and Analysis, 10 (2018), no. 5, 42 pp. Banach spaces from h.d. Ellentuck spaces

[32] Natasha Dobrinen and Dan Hathaway. "Forcing and the Halpern-Lauchli Theorem," Journal of Symbolic Logic, 85 (2020), no. 1, 87--102. Forcing and HL

[33] Natasha Dobrinen. "The Ramsey Theory of Henson graphs," (2019) 75 pp. Submitted. Ramsey Theory of Henson graphs

[34] Natasha Dobrinen and Sonia Navarro-Flores. "Ramsey degrees of ultrafilters, pseudointersection numbers, and the tools of topological Ramsey spaces," (2019) 25 pp. Submitted. Ramsey degrees

[35] Natasha Dobrinen and Dan Hathaway. "Classes of barren extensions," Journal of Symbolic Logic. 32 pp. (to appear) Barren extensions

[36] Natasha Dobrinen. "Borel sets of Rado graphs and Ramsey's Theorem," To appear in European Journal of Combinatorics, special issue for the 2016 Prague DocCourse on Ramsey Theory, 29 pp. Borel sets of Rado graphs

[37] Natasha Dobrinen. "Ramsey theory on infinite structures and the method of strong coding trees," 24 pp. Contemporary Logic and Computing, College Publications, London (2020). Coding Trees

[38] Natasha Dobrinen and Kaiyun Wang. "Big Ramsey degrees in universal inverse limit structures," (2020) 20 pp. Submitted. Universal Limit Structures

[39] Rebecca Coulson, Natasha Dobrinen, and Rehana Patel. "Fraisse classes with simply characterized big Ramsey structures," (2021) 70 pp. Submitted. SDAP

[40] Natasha Dobrinen. The Ramsey theory of the universal homogeneous triangle-free graph, Part II: Exact big Ramsey degrees," 22 pp. Preprint. Triangle-free Exact big Ramsey degrees The work in this paper will appear as part of the larger paper [42].

[41] Natasha Dobrinen. "Ramsey theory of homogeneous structures: current trends and open problems," 22 pp. To appear in the Proceedings of the 2022 International Congress of Mathematicians. ICM

[42] Balko, Chodounsky, Dobrinen, Hubicka, Konecny, Vena, and Zucker. "Exact big Ramsey degrees for binary relational free amalgamation classes," 97 pp. Submitted. Exact big Ramsey degrees

[43] Natasha Dobrinen. "Baire spaces of Fraisse structures in which all Borel sets are Ramsey," 30 pp. Preprint.

[44] Natasha Dobrinen and Saharon Shelah. "Halpern-Lauchli Theorem at singular cardinals". In preparation.

[45] Natasha Dobrinen, Michael Hrusak, and Sonia Navarro Flores. "Ideals related to topological Ramsey spaces." In preparation.

Please note: Many of these articles may also 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