University of Denver
I am excited to be speaking in the logic section at the 2022 International Congress of Mathematicians which will be held online.
In September of 2019, I had the honor of being on a National Academy of Sciences, Engineering, and Medicine webinar series with Julia Knight called Mathematical Frontiers on Logic and Foundations.
An article on being a mathematician
DU Achieves R1 Status.Proud to be a part of helping the University of Denver achieve Research 1 status.
Annals of Pure and Applied Logic.
In January 2020, I became an editor for the Annals of Pure and Applied Logic, handling papers in set theory and related areas.
As of January 1, 2022, I am serving as Vice President of the Association for Symbolic Logic. Previous service includes 6-years as a member of the Committee on Logic in North America, with three of those years as Chair, as well as a two-year term on the ASL Web Advisory Committee.
Advising and Postdoctoral Mentoring.
Timothy Trujillo PhD in Spring 2014 with thesis: "Topological Ramsey spaces, associated ultrafilters, and their applications to the Tukey theory of ultrafilters and Dedekind cuts of nonstandard arithmetic".
Sonia Navarro Flores (co-advised with Michael Hrusak) PhD in January 2021 with thesis: "Topological Ramsey spaces and Borel ideals".Postdoctoral Mentoring.
Jose Mijares DU Postdoctoral Visiting Assistant Professor: Fall 2013 - Spring 2015.
Daniel Hathaway DU Postdoctoral Visiting Assistant Professor: Fall 2015 - Spring 2018.
Dr. Kaiyun Wang, College of Mathematics and Information Science, Shaanxi Normal University, was a DU visiting research scholar from November 2018 - November 2019.Masters Thesis Advising.
Sonia Navarro Flores (co-advised with David Meza-Alcantara) in 2015 on her Masters Thesis: "Ramsey numbers for ultrafilters associated to topological Ramsey spaces."
Research. I work in Set Theory, Ramsey Theory, Topology, and related areas. See my Research Page for more information.
Ramsey Theory on Infinite Structures. One of my current research interests is mapping out those infinite structures which have analogues of the infinite Ramsey theorem. This is the area of "big Ramsey degrees" and "infinite dimensional Ramsey theory." This area has its beginnings in Sierpinski's example of a coloring of (unordered) pairs of rationals into two colors so that any subset forming a dense linear order without endpoints (i.e. an isomorphic subcopy the rationals) still has pairs with both colors. An article on Ramsey theory of homogeneous structures: current trends and open problems will appear in the 2022 ICM Proceedings. This article gives a gentle introduction to the subject and an overview of current research in the area.
An expository article on the method of strong coding trees, which I developed to prove that the triangle-free Henson graph has finite big Ramsey degrees and extended to prove that all Henson graphs have finite big Ramsey degrees, appeared in "Contemporary Logic and Computing," in 2020. This method has proved to be a key in unlocking big Ramsey degrees of various binary relational structures as well as proving indivisbility for higher arities with certain strong amalgamation properties. With Gasarch, we wrote an open problems article geared towards undergraduates, appearing in ACM SIGACT News 51 (2020), no. 4, 30--46.
In the summer of 2020, I finished a proof characterizing the exact big Ramsey degrees of the triangle-free Henson graph. This was proved via a minor modification in the triangle-free coding trees developed here. The same result was around the same time independently obtained by Balko, Chodounsky, Hubicka, Konecny, Vena, and Zucker. Our joint paper characterizing exact big Ramsey degrees of all free amalgamation classes of structures with finitely many binary (and unary) relations with finitely many irreducible forbidden substructures is available here. This work builds on Zucker's upper bounds as well as new combinatorial methods for upper bounds developed by Hubicka using parameter spaces.With Coulson and Patel, we formulated a property (SDAP^+) strengthening the Disjoint Amalgamation Property and proved that any Fraisse class with finitely many relations of arity at most two satisfying SDAP^+ admits a big Ramsey structure, which moreover is simply characterized by two additional relations. The second main result in that paper is that if a Fraisse structure with finitely many relations of any arity satisfies SDAP^+, then it is indivisible. (See Zucker's Big Ramsey degrees and topological dynamics for the definition of "big Ramsey structure".) This work characterizes exact big Ramsey degrees for many new Fraisse classes, while streamlining several previous results. Key features of the work include development of coding trees of 1-types to prove Ramsey theorems directly on the structures, and the first "envelope-free" proofs. Extending the infinite dimensional Ramsey theory of the natural numbers (Galvin-Prikry and Ellentuck Theorems) to infinite structures is in its nascent stages. A paper to appear in the proceedings of the 2016 Czech DocCourse on Ramsey Theory proved an analogue of the Galvin-Prikry theorem for the Rado graph in Borel sets of Rado graphs and Ramsey's Theorem. I have recently finished typing the proof extending this to structures with SDAP^+ and relations of arity at most two. Working with Kaiuyn Wang, we have extended results of Huber-Geschke-Kojman and of Zhang on big Ramsey degrees of the profinite graph to more general binary relational structures. We characterize the exact big Ramsey degrees for ordered graphs, ordered k-clique-free graphs, ordered oriented graphs, and ordered tournaments inside their inverse limits. Inverse limit structures
Computability and reverse mathmatical aspects of the Halpern-Lauchli and Milliken theorems and big Ramsey structures of the rationals and the Rado graph have been analysed by Angles d'Auriac, Cholak, Dzhafarov, Monin, and Patey in this treatise.
Work Published or accepted in 2020 - present.
"Ramsey theory of the Henson graphs," Journal of Mathematical Logic. To appear.
(With Sonia Navarro Flores) "Ramsey degrees of ultrafilters, pseudointersection numbers, and the tools of topological Ramsey spaces," Archive for Mathematical Logic. Online link.
"Ramsey theory of homogeneous structures: current trends and open problems." To appear in the 2022 ICM Proceedings.
"Continuous and other finitely generated canonical cofinal maps on ultrafilters," Fundamenta Mathematicae. 249 (2020), no. 2, 111--147.
(With Dan Hathaway) "Forcing and the Halpern-Lauchli theorem," Journal of Symbolic Logic. 85 (2020), no. 1, 87--102.
(With Dan Hathaway and Karel Prikry) "Perfect tree forcings for singular cardinals," Annals of Pure and Logic. 171 (2020), no. 9, 102827, 25 pp.
"The Ramsey theory of the universal homogeneous triangle-free graph," Journal of Mathematical Logic. 20 (2020), no. 2, 2050012, 75 pp.
"Topological Ramsey Spaces Dense in Forcings," in the book, "Structure and Randomness in Computability and Set Theory," pp. 3-58. World Scientific, Editors: Douglas Cenzer and Jindrich Zapletal. 2020. Available online.
(With Dan Hathaway) "Barren extensions," Journal of Symbolic Logic. 86 (2021), no. 1, 178--209. online.
"Ramsey theory on infinite structures and the method of strong coding trees," in "Contemporary Logic and Computing," pp 444-467. College Publications, Editor: Adrian Rezus. 2020. Table of Contents
(With William Gasarch) "When Ramsey Theory fails settle for more colors (big Ramsey degrees!), ACM SIGACT News. 51 (2020), no. 4, 30--46.
(With Balko, Chodounsky, Hubicka, Konecny, Vena, and Zucker) "Big Ramsey degrees of the generic partial order," in "Extended Abstracts EuroComb 2021," pp 637--643. Springer, 2021.
Invited Talks, Workshops, Tutorials, and Panels. For slides from 2015-2018, see Invited Talks Page.
Young Set Theory Workshop, May 29 - June 3, 2023 (tutorial).
European Set Theory Conference, Torino, August 29 - September 2, 2022 (plenary).
International Congress of Mathematicians, St Petersburg, July 6 - 14, 2022 (invited lecture).
ESI Set Theory Workshop, Vienna, July 4 - 8, 2022.
AIM Workshop on Descriptive Graph Theory, Palo Alto, June 27 - July 1, 2022.
UltraMath, Pisa, June 6 - 11, 2022 (plenary).
Spring Topology and Dynamical Systems Conference, Baylor University, March 9 - 13, 2022 (set-theoretic topology special session).
Combinatorics Seminar, University of Cambridge, February 17, 2022. Chalk talk on, "Ramsey theory of homogeneous structures". Host: Imre Leader
Oberwolfach Workshop on Set Theory, January 9 - 15, 2022. "Infinite dimensional Ramsey theory of homogeneous structures: a progress report"
16th International Luminy Workshop in Set Theory, September 13 - 17, 2021. "Exact big Ramsey degrees via coding trees"
Models and Sets Seminar, University of Leeds, May 5, 2021.
Caltech Logic Seminar, March 29, 2021 (online). "Fraisse classes with simply characterized big Ramsey degrees"
Midwest Model Theory Seminar, March 9, 2021 (online). "Fraisse classes with simply characterized big Ramsey degrees"
SEALS 2021, University of Florida, Gainesville, February 27 - 28, 2021 (online). "Ramsey theory on infinite structures" (plenary).
UltraMath 2020, Pisa, Italy, May 31 - June 6, 2020. Cancelled due to Covid-19.
Online Logic Seminar. April 9, 2020. "Ramsey properties on infinite structures".
Oberwolfach Workshop on Set Theory, Oberwolfach, Germany, April 5-11, 2020. Cancelled due to Covid-19.
ASL North American Meeting, UC Irvine, March 25-28, 2020, Special Session on Forcing and Ramsey Theory. Cancelled due to Covid-19, but they held a Virtual Meeting instead. Ramsey Theory on Fraisse structures
Mathematics Department Colloquium, Boise State University, March 10, 2020. "Ramsey Theory for Infinite Structures and Set Theoretic Methods."
UCLA Logic Colloquium. February 7, 2020. Talk on Strong coding trees and Ramsey theory on infinite structures.
I gave invited talks at the JMM in the two AMS-ASL Special Sessions: "Choiceless Set Theory and Related Areas", and "Logic Facing Outward". Denver, January 15-16, 2020. Slides: Barren extensions slides and Logic and Combinatorics
Set Theory Seminar at The Hebrew University. Chalk talks on December 11 and 18, 2019 on my work on big Ramsey degrees of Henson graphs.
Set Theory Seminar, Bar-Ilan University, December 9, 2019. Chalk talk on "Strong coding trees and applications to Ramsey theory on infinite graphs."
Carlos di Prisco Birthday Meeting, Bogota, Colombia, November 28-29, 2019. Barren extensions
Fall Southeast AMS Sectional Meeting, Special Session on Cech-Stone compactification of semigroups: Algebra, Topology, Dynamics, and Combinatorics, University of Florida at Gainesville, November 2-3, 2019. I gave a chalk talk on "Ramsey ultrafilter and friends", comparing results about Ramsey ultrafilters in regard to Rudin-Keisler, Tukey, barren extensions, etc. to similar results for ultrafilters with finite Ramsey degrees.
National Academies of Sciences, Engineering, and Medicine. "Mathematical Frontiers" Webinar Series. With Julia Knight, we were panelists on the topic of Logic and Foundations, September 10, 2019.
15th International Luminy Workshop in Set Theory, Luminy, France, September 23-27, 2019. Borel sets of Rado graphs are Ramsey
Casa Matematica Oaxaca, Workshop on Reverse Mathematics of Combinatorial Principles, September 15-20, 2019. Chalk talk on "Open problems in the reverse mathematics of Ramsey theory on trees and graphs."
AMS Fall Central Sectional Meeting, Special Session on "Topology and Descriptive Set Theory," Madison, WI, September 14-15, 2019. Borel partitions of a space of Rado graphs are Ramsey
Universidad de los Andes, research visit to Carlos DiPrisco. I gave a chalk talk in the set theory seminar titled, "Tukey theory of ultrafilters."
Casa Matematica Oaxaca, Set theory of the reals, August 4-9, 2019. (unfortunately unable to attend, due to a prior commitment)
Midsummer Combinatorial Workshop XXIV, July 29 - August 2, 2019, Prague, Czech Republic. Ramsey theory on infinite graphs
Logic Fest in the Windy City, May 30 - June 2, 2019. Tutorial: Ramsey theory on trees and applications to infinite graphs
50 Years of Set Theory in Toronto, May 13-17, 2019. Strong coding trees and applications to Ramsey theory on infinite graphs
Spring Topology and Dynamical Systems Conference, March 14-16, 2019. Special Session on Set Theoretic Topology, "Borel colorings of the Rado graph". (unable to attend because of flights cancelled due to a blizzard in Denver)
Notre Dame Logic Seminar, March 5, 2019.
Miami University, February 25 - March 1, 2019. Colloquium talk.
University of Florida, February 18-23, 2019. A talk in the Colloquium, and a talk in the Logic Seminar.
Arctic Set Theory Workshop 4, Kilpisjaervi, Finland, January 21-26, 2019. Ramsey theory of the Henson graphs.
ASL Winter Meeting, at the Joint Mathematics Meetings, Baltimore, January 18-19, 2019, (plenary). Ramsey theory of the Henson graphs.
University of Oxford, Analytic Topology in Mathematics and Computer Science Seminar, January 16, 2019. Ramsey theory and infinite graphs.
University of Vienna, at the invitation of Vera Fischer, January 7-12, 2019. I gave a Mini-course on Infinitary Ramsey Theory (slides and video available here), and a lecture on the Ramsey theory of Henson graphs.
Grants and Awards.
I am grateful to have received a third National Science Foundation Grant DMS-1901753, this one on "Logic, Ramsey Theory, and Relational Structures," which started September 1, 2019. $158,476.
In 2016, I was awarded the Excellence in Research Award, University of Denver, Division of Natural Sciences and Mathematics, for the academic year 2015-2016.See Grants and Awards for the listing of these and previous grants and awards.
In Fall 2021 I am teaching Real Analysis I and a graduate course in Infinitary Ramsey Theory, drawing from Todorcevic's book, "Introduction to Ramsey Spaces". In Spring 2022 I am teaching Calculus 3 and Real Analysis.
Women in Mathematics.
The Mathematics Department at DU now has an AWM Student Chapter. The Faculty Sponsor is Professor Mei Yin and I am the Faculty Co-Sponsor. I also run quarterly mentoring meetings for the women graduate students in math.
Dana Bartosova and I are organizing a Special Session on "Ramsey theory of infinite structures" for the Spring AMS Western Sectional to be held virtually at the University of Denver, May 14-15, 2022.
Lynne Yengulalp and I were the Program Committe for the Special Session in Set-theoretic Topology at the 54th Spring Topology and Dynamical Systems Conference, Murray State University, March 18-21, 2020. This meeting was postponed due to Covid-19 and will take place online during May 12-16, 2021.
I chaired the Program Committee for the ASL Winter Meeting at the Joint Mathematics Meeting, Denver, January 15-18, 2020.
I am on the Program Committee for the BEST conference series, 2018-2023.
Nick Galatos and I (Chair) organized BLAST 2018. This 10-year anniversary conference was held at the University of Denver, August 6-10, 2018.
Dan Hathaway and I organized a special session on Set Theory of the Reals (including Ramsey Theory) at the AMS Western Sectional Meeting at the Unviersity of Denver in October 2016.
The DU Sesquicentennial Ramsey Theory Conference 2014 was held at the University of Denver, May 24-28, 2014. The meeting was organized by Jose Mijares and myself. We thank the National Science Foundation and The DU Provost Sesquicentennial Fund for funding to support many participants.
DU is the birthplace of the BLAST Conference Series, which started in 2008 when Professor Nikolaos Galatos and myself co-founded and co-chaired BLAST 08. BLAST is an interdisciplinary conference focused on Boolean algebras, Logic, Lattices, algebraic Logic, universal Algebra, Set theory, and Topology (Set-theoretic, point free, and general). These are logic-related areas of interest to researchers at Chapman University; New Mexico State University, Las Cruces; University of Colorado, Boulder; University of Denver; University of Kansas, Lawrence; University of Missouri, Kansas City; University of Nebraska, Omaha; and University of North Texas. BLAST conferences are good for graduate students, early career mathematicians, and anyone interested in expanding their research to include neighboring areas. We are grateful for National Science Foundation funding for the BLAST Conference Series.
Front Range Logic Saturdays (formerly known as the Maximally Informal Gathering or MIG) is a consortium of logicians from the Boulder, Denver, Golden and Fort Collins areas. We gather on selected Saturdays to share research talks, ideas, and lunch. MIG was started in 2009 by Bart Kastermans and myself. FRLS was run by Dan Hathaway and Timothy Trujillo from 2015-2018.
I have also organized various special sessions and served on program committees for conferences of the ASL and the AMS. These are listed in my CV.
From 2008 until May 2014, I was the coordinator and supervisor for the Graduate Colloquium which meets on Fridays at 2pm. See Graduate Colloquium for a list of past and present speakers.