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University of Notre Dame |
Highlighted Lectures
Slides from my talk at the 2023 Luminy Workshop on Set Theory. 2022 International Congress of Mathematicians Invited Lecture on Ramsey theory of homogeneous structures, with related paper Ramsey theory of homogeneous structures: current trends and open problems in the Proceedings of the 2022 ICM. National Academy of Sciences, Engineering, and Medicine webinar series with Julia Knight called Mathematical Frontiers on Logic and Foundations. |
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.
Coordinating Editor for Annals of Pure and Applied Logic and handling papers in set theory.
Editor for Order, handling papers in set theory, infinite combinatorics, and Boolean algebras.
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 infinite structures which have analogues of the infinite Ramsey theorem. This is the area of "big Ramsey degrees" and "infinite-dimensional Ramsey theory." My article Ramsey theory of homogeneous structures: current trends and open problems gives an introduction to and overview of this 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. In 2022, I finished typing the proof extending this to structures with SDAP^+ and relations of arity at most two. In 2023, Andy Zucker and I have developed the infinite-dimensional Ramsey theory of binary free amalgamation classes, culminating much work in this area while showing that a weaker version of Todorcevic's Axiom A.3(2) suffices, along with his other axioms, to guarantee Ramsey spaces.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. With Cholak and McCoy, last summer we showed that the statement, "All finite triangle-free graphs have finite big Ramsey degrees in the Henson graph" has at least the strength of ACA_0 over RCA_0. This result and related ones are being written up.
Highlights of Recent Work (still being updated).
"The Ramsey theory of the universal homogeneous triangle-free graph," Journal of Mathematical Logic. 20 (2020), no. 2, 2050012, 75 pp.
"Ramsey theory of the Henson graphs," Journal of Mathematical Logic, 23 (2023), no. 1, Paper No. 2250018, 88 pp.
"Ramsey theory of homogeneous structures: current trends and open problems." Proc. Int. Cong. Math. 2022, Vol. 3, pp. 1462-1486.
(with Coulson and Patel) "Fraisse structures with SDAP^+", Part I (56 pp) and Part II (59 pp) Submitted.
(with Balko, Chodounsky, Hubicka, Konecny, Vena, and Zucker) Exact big Ramsey degrees for binary relational free amalgamation classes, Journal of the European Mathematical Society, accepted 4/18/2024, 53 pp.
(with Balko, Chodousnky, Hubička, Konecny, Vena, and Zucker). Characterisation of the big Ramsey degrees of the generic partial order, 24 pp, Transactions of the American Mathematical Society, accepted
(with Zucker) Infinite-dimensional Ramsey theory for binary free amalgamation classes, 49 pp, Submitted.
(with Balko, Chodounsky, Hubicka, Konecny, Vena, and Zucker) Ramsey theorem for trees with successor operation, 39 pp.
(with Wang) Inverse limit structures, Arch. Math. Logic 62 (2023), no. 3-4, 471–503.
"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.
Recent work on the Tukey theory of ultrafilters:
(with Benhamou) Cofinal types of ultrafilters on measurable cardinals, 34 pp, to appear in Journal of Symbolic Logic. This is the first paper on Tukey structure of ultrafilters on measurable and other uncountable cardinals, opening up the field to further study.
(with Benhamou) On the Tukey types of Fubini products, 15 pp, Submitted.
Invited Talks, Workshops, and Panels. For slides from 2015-2021, see Invited Talks Page. (more slides to be added soon)
Arctic Set Theory Workshop 7, Kilpisjärvi, Finland, February 16-22, 2025.
Oberwolfach Workshop on Set Theory, January 12-17, 2025.
KGRC Logic Colloquium, January 9, 2025
KGRC Research Seminar in Set Theory, January 9, 2025
Chinese Annual Conference on Mathematical Logic (CACML 2024) at Peking University, Beijing, China, October 25-27, 2024 (plenary). "Infinite structural Ramsey theory and logic"
Tsinghua University Logic Salon, October 24, 2024. "Big Ramsey degrees and logic"
Nankai University Logic Colloquium, October 23, 2024. "Forcing and Ramsey theory on trees"
Auburn University Mathematics Colloquium, October 18, 2024. "Infinite structural Ramsey theory: logic and combinatorics"
AMS-UMI Meeting, Palermo, Italy, July 23-26, 2024. Special session on set theory. "Forcing applied to Ramsey theory of Fraisse structures"
"120 Years of Choice", University of Leeds, UK, July 8-12, 2024 (plenary). "Well orderings and infinite structural Ramsey theory"
XX Simposio Latinoamericano de Lógica Matemática (SLALM2024), Montevideo, Uruguay, July 1-5, 2024. Tutorial on "Infinite structural Ramsey theory and logic".
"RaTLoCC24 - Ramsey Theory in Logic, Combinatorics and Complexity", University of Pisa, Italy, June 11-14, 2024 (plenary).
Carnegie Mellon Logic Seminar, April 2, 2024.
57th Spring Topology and Dynamical Systems Conference, University of North Carolina, Charlotte, March 6-9, 2024 (semiplenary).
SEALS, University of Florida, Gainesville, March 2-3, 2024 (conference talk and departmental colloquium).
Logic Seminar, University of Waterloo, October 2023.
Mathematics Colloquium, University of Waterloo, October 2023.
The 17th International Workshop in Set Theory, CIRM Luminy, France, October 2023
Casa Matematica Oaxaca, Workshop on Set-theoretic Topology, August 2023 (plenary).
Young Set Theory Workshop, May 29 - June 3, 2023 (tutorial).
MAMLS Spring Fling, Rutgers, May 2023 (plenary).
Computability and Combinatorics, University of Connecticut, May 2023 (plenary).
American Institute of Mathematics, San Jose, CA, May 2023, "From aleph_2 to infinity".
Mathematics Colloquium, The Ohio State University, April 2023.
Logic Seminar, University of Illinois, Chicago, April 2023.
Mathematics Colloquium, University of Southern Illinois, April 2023.
ASL North American Annual Meeting, UC Irvine, March 2023 (tutorial).
Spring Topology and Dynamical Systems Conference, Rhodes College, March 2023, Special Session on Set-theoretic Topology.
Logic Seminar, University of Waterloo, March 2023.
Set Theory Seminar, Fields Institute, March 2023.
UCLA Distinguished Women in Math Lecture Series, February 2023.
VIG Meeting in honor of Tony Martin, UCLA, February 2023 (plenary).
Conference on Geometry and Dynamics of Groups of Transformations, Fields Institute, January 2023 (plenary).
Fields Institute, Toronto, Canada, "Thematic Program on Set Theoretic Methods in Algebra, Dynamics and Geometry," January--June 2023
Logic Seminar, University of Wisconsin, Madison, November 2022.
The Mathematics of Julia Knight, Notre Dame, October 2022 (plenary).
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"
Tutorials.
XX Simposio Latinoamericano de Lógica Matemática (SLALM2024), Montevideo, Uruguay, July 1-5, 2024. Slides to be uploaded soon.
Young Set Theory Workshop, May 29 - June 3, 2023. Slides 1, Slides 2, Slides 3.
ASL North American Annual Meeting, UC Irvine, March 2023. Slides 1, Slides 2.
Logic Fest in the Windy City, May 30 - June 2, 2019. Tutorial on Ramsey theory on trees and applications to 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).
RIMS Workshop on Infinite Combinatorics and Forcing Theory, Research Institute for Mathematical Sciences, Kyoto University, Japan, November 28 - December 1, 2016. I gave a tutorial on Ramsey theory in forcing. Tutorial 1 Tutorial 2 Tutorial 3 Expository paper corresponding to Tutorial 3, appeared in the RIMS Symposium Proceedings.
Teaching.
In Fall 2023 I taught Basic Logic I (graduate level, MATH 60510) and Calculus 2. In Spring 2024 I taught a graduate Topics in Logic course on forcing and Ramsey theory. In Fall 2024 I am teaching Beginning Logic (MATH 10130). In Spring 2025 I will teach Basic Logic I (MATH 60510) again.
Grants and Awards.
I am grateful to have received a fourth National Science Foundation Grant DMS-2300896; this is a continuation of my previous grant on "Logic, Ramsey Theory, and Relational Structures." See Grants and Awards for the listing of these and previous grants and awards.
Advising and Postdoctoral Mentoring.
PhD Students.
Tan Özalp, University of Notre Dame (current).
Jenny Xu, University of Notre Dame (current).
Sonia Navarro Flores (co-advised with Michael Hrusak) PhD in January 2021 with thesis: "Topological Ramsey spaces and Borel ideals".
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". 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."
Undergraduate Thesis Advising.
Gavin Dooley, University of Notre Dame, "Easton's Theorem". Fall 2022 - Spring 2024. This thesis won a Taliaferro Prize, given by the Notre Dame Math Department.
Women in Mathematics.
At the University of Notre Dame, we have an ACMS-MATH AWM Student Chapter. The faculty sponsor is Professor Martina Bukac, and I am faculty co-sponsor.
At the University of Denver, I was faculty co-sponsor of the AWM Student Chapter. The faculty sponsor is Professor Mei Yin. I also ran quarterly mentoring meetings for the women graduate students in math.
Conference Organizing.
Co-organizer with Jan Hubicka, Stevo Todorcevic, and Andy Zucker, "Infinite Structural Ramsey Theory," BIRS Workshop, Banff 2025.
Program Committee Member, ASL Winter Meeting at the JMM, January 2025.
Program Committee Member, Logic Colloquium, Gothenburg, Sweden, June 2024.
Program Committee Member, Young Set Theory, Budapest, Hungary, June 17-21, 2024.
I chaired the Program Committee for the ASL Winter Meeting at the Joint Mathematics Meeting, Denver, January 15-18, 2020.
Professor Nikolaos Galatos and myself co-founded the BLAST Conference Series when we co-chaired BLAST 08 at the University of Denver. We also co-organized BLAST 2018 at the University of Denver. 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; University of North Texas, and others. 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.
More conference, seminar, and special session organizing is listed in my CV.
An article on being a mathematician.
I am a member of the Notre Dame Ceili Band. The band consists of undergraduate and graduate students, staff and faculty, and community members. We play Fall, Halloween, Christmas, Valentine's, and Spring Ceilis free for all Notre Dame and broader community members. We also play on Sunday's at Fiddler's Hearth for the dancers, as well as other gigs. Here is an article about the Ceili Band from our university newspaper, The Irish Rover.
Go Irish!