Research

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Research Interests

Algebraic combinatorics, particularly lattice theory in connection with Coxeter groups and representation theory.

Research Narrative

I am an algebraic combinatorialist, which means I organize discrete data in ways that highlight useful structures or patterns. Most often the “data” I consider consists of a family of objects that appear in vastly disconnected fields of mathematics or physics. My research begins with two superficially simple questions:

  • In each family, how many objects are there?
  • In each family, are some objects bigger than others?

The type of answer we expect from the first question is straightforward. No matter how complicated the objects we are counting, we expect that the number of objects is a non-negative whole number. Since we typically consider an infinite family of objects, the solution to our counting problem is an infinite sequence of such numbers. The Fibonacci numbers or the binomial coefficients are famous examples of such sequences of numbers.

The type of answer we expect from the second question is a little more complicated, but roughly we would like a rule which tells us when and how to compare different objects. It is possible that some of our objects may not be comparable at all. For this reason, we say that a set of objects together with a rule $(\le)$ that compares some of the objects is a partially ordered set, or a poset for short.

Figure 1 shows a picture of the poset of subsets of the set ${1,2,3}$. A set $A$ is “bigger” than a set $B$ if all of the elements of $A$ are also in $B$.

I am particularly interested in posets that come “from nature.” For me, a poset is “from nature” if it grows from the study of some mathematical object, be it a polytope or a category of modules. A poset “from nature” could be a poset whose Hasse diagram (or underlying graph) is the one-skeleton of a polytope; or a poset whose elements are collections of certain subcategories. My research program leverages the geometric and algebraic structure of such posets to answer combinatorial questions.

Publications and Peer Reviewed Conference Proceedings

You can also find my publications on the arXiv and Google Scholar.

  1. “Cambrian combinatorics on quiver representations (type A).” (with E. Gunawan, E. Meehan, and R. Schiffler) Adv. in Appl. Math. 143 (2023), no. 102428. arXiv:1912.02840.
  2. “Pairwise Compatibility for 2-Simple Minded Collections II: Preprojective Algebras and Semibrick Pairs of Full Rank.” (with E.J. Hanson) Ann. Comb. (2022). arxiv: 2010.08645.
  3. “Dynamical combinatorics and torsion classes.” (with G. Todorov and S. Zhu) J. Pure and Applied Algebra, 25 (2021), no. 9. arxiv: 1911.10712.
  4. “Lattices from graph associahedra and subalgebras of the Malvenuto–Reutenauer algebra.” (with T. McConville) Algebra universalis 82 (2021), no. 2. arxiv: 1808.05670.
  5. “The canonical join complex of the Tamari lattice.” J. Combinatorial Theory. Series A (2020), 174.
  6. “Lattices from graph associahedra.” (with T. McConville) Sem. Lothar. Combin. proc. FPSAC 2019
  7. “The canonical join complex.” Electron. J. Combin. (2019). arxiv: 1610.05137.
  8. “Minimal inclusions of torsion classes.” (with A. Carroll and S. Zhu.) Algebraic Combin. 2 (2019), no. 5, 879-901. arxiv: 1710.08837.
  9. “The canonical join complex of the Tamari lattice.” Sem. Lothar. Combin. proc. FPSAC 2018
  10. “Coxeter-biCatalan Combinatorics.” (with N. Reading) J. Algebraic Combin. 47 (2018), no. 2, 241–300. arxiv: 1605.03524.
  11. “Universal geometric coefficients for the four-punctured sphere.” (with E. Meehan, N. Reading, and S. Viel.) Ann. Comb. 22 (2018), no. 1, 1–44. arxiv: 1602.03448.
  12. “Coxeter-biCatalan Combinatorics.” (with N. Reading) DMTCS Proceedings FPSAC 2015
  13. “Universal geometric coefficients for the four-punctured sphere.” (with E. Meehan, N. Reading, and S. Viel.) DMTCS Proceedings FPSAC 2015

Preprints

  1. “Pop-Stack Operators for Torsion Classes and Cambrian Lattices.” (with C. Defant and E.J. Hanson). Preprint arXiv:2312.03959.
  2. “Exceptional sequences in semidistributive lattices and the poset topology of wide subcategories.” (with E.J. Hanson). Preprint arXiv:2209.11734.

Presentations

  1. Combinatorics of $\tau$-exceptional sequences, University of Waterloo Combinatorics Algebraic and Enumerative Seminar — Jan. 2023
  2. Triangulations and maximal almost rigid representations, Special Session on Representation Theory of Algebras, CMS Winter Meeting, Toronto, Canada — Dec. 2022
  3. New-biCambrian Lattices, TACO Seminar, Loyola University — Nov. 2022
  4. New-biCambrian Lattices, Algebraic Combinatorics Seminar, Universit`e Gustave Eiffel, Paris — Sept. 2022
  5. Dynamics and Topology of Lattice Posets Related to the Weak Order, Discrete Math Seminar, IIT — Oct. 2022
  6. Kappa-Rowmotion for Semidistributive Lattices, Dynamical Algebraic Combinatorics Workshop, UBC Okanagan, Canada — Nov. 2021
  7. Fundamentally Semidistributive Lattices, UMass Lowell Math Department Colloquium (Online) — Oct. 2021
  8. Invited Course Teacher - Tamari Lattices and Posets, Summer School in Algebraic Combinatorics, Universit'e du Qu'ebec `a Montr'eal — June 2021
  9. Pairwise completability for 2-Simple minded collections, Minisymposium on Flow Polytopes of Graphs, CanaDAM 2021 — May 2021
  10. Pairwise completability for 2-Simple minded collections, Special Session on Algebraic and Combinatorial Aspects of Polytopes, AMS Spring Sectional Meeting, San Francisco State University — April 2021
  11. The Kreweras Complement on the Lattice of Torsion Classes, Dynamical Algebraic Combinatorics Virtual Workshop, BIRS, Alberta, Canada — Oct. 2020
  12. Pairwise completability for 2-Simple minded collections, Discrete CATS Seminar, University of Kentucky (Online) — March 2021
  13. Cover Relations in the Lattice of Torsion Classes: Dynamics and Completability, Online seminar on representation theory of finite-dimensional algebra — Feb. 2021
  14. The Kreweras Complement on the Lattice of Torsion Classes, 30th Meeting on Representation Theory of Algebras, Universit'e de Sherbrooke, Sherbrooke, Qu'ebec, Canada — Sept. 2020
  15. Graph Associahedra and the Poset of Maximal Tubings, Combinatorics Seminar, University of Michigan, Ann Arbor, MI — Oct. 2019
  16. Graph Associahedra and the Poset of Maximal Tubings, Algebra and Combinatorics Seminar, Loyola University, Chicago, IL — Sept. 2019
  17. Graph Associahedra and the Poset of Maximal Tubings, FPSAC, University of Ljubljana, Slovenia — July 2019
  18. Graph Associahedra and the Poset of Maximal Tubings, CanaDAM, Vancouver, British Columbia, Canada — May 2019
  19. Graph Associahedra and the Poset of Maximal Tubings, Discrete Math Seminar, University of Massachusetts, Amherst, MA — April 2019
  20. Graph Associahedra and the Poset of Maximal Tubings, Special Session on Cluster Algebras and Related Topics, AMS Spring Sectional Meeting, University of Connecticut, Hartford, CT — Feb. 2019
  21. Graph Associahedra and the Poset of Maximal Tubings, Combinatorics Seminar, Universit'e du Qu'ebec `a Montr'eal, Montr'eal, QC — Feb. 2019
  22. The combinatorics of torsion classes, Special Session on Representation Theory of Algebras, CMS Summer Meeting, Fredericton, New Brunswick, Canada — June 2018
  23. Graph Associahedra and the Poset of Maximal Tubings, Algebra and Combinatorics Seminar, NC State University, Raleigh, NC — May 2018
  24. The combinatorics of torsion classes, Special Session on Noncommutative Algebra and Representation Theory, AMS Spring Sectional Meeting, Northeastern University, Boston, MA — May 2018
  25. Graph Associahedra and the Poset of Maximal Tubings, Special Session on Algebraic, Geometric, and Topological Methods in Combinatorics, AMS Spring Sectional Meeting, Northeastern University, Boston, MA — April 2018
  26. The biCatalan Kreweras Complement, Special Session on Dynamical Algebraic Combinatorics, Joint Mathematics Meetings, San Diego, CA — Jan. 2018
  27. The combinatorics of torsion classes, Cluster Algebra Seminar, University of Connecticut, Storrs, CT — Dec. 2017
  28. The combinatorics of torsion classes, Geometry-Algebra-Singularities-Combinatorics Seminar, Northeastern University, Boston, MA — 2017
  29. Counting and the canonical join representation, Pick My Brain Seminar, Northeastern University, Boston, MA — 2017
  30. The canonical join complex of the Tamari lattice, Algebraic Combinatorixx 2 Workshop, BIRS, Alberta, Canada — 2017
  31. Quick Fire Talk: The canonical join complex, Connections for Women: Geometric and Topological Combinatorics, MSRI — 2017
  32. The canonical join complex, Combinatorics Seminar, Massachusetts Institute of Technology, Cambridge, MA — 2017
  33. The canonical join complex, Representation Theory Seminar, Northeastern University, Boston, MA — 2017
  34. The canonical join complex, Special Session Algebraic and Enumerative Combinatorics, AMS Fall Sectional Meeting, Bowdoin College, Brunswick, ME — 2016
  35. The canonical join complex, Maurice Auslander Distinguished Lectures and International Conference, Woods Hole Oceanographic Institution, Quissett Campus, Woods Hole, MA — 2016
  36. Coxeter-biCatalan Combinatorics, Combinatorics Seminar, University of Michigan, Ann Arbor, MI — 2015
  37. Coxeter-biCatalan Combinatorics, Combinatorics Seminar, University of Minnesota, Minneapolis, MN — 2015
  38. Coxeter-biCatalan Combinatorics, Algebra and Combinatorics Seminar, NC State University, Raleigh, NC — 2015
  39. Coxeter-biCatalan Combinatorics, Algebra and Combinatorics Seminar, DePaul University, Chicago, IL — 2015
  40. Coxeter-biCatalan Combinatorics, Combinatorics Seminar, Universit'e du Qu'ebec `a Montr'eal, Montr'eal, QC — 2015
  41. Coxeter-biCatalan Combinatorics, FPSAC, Korea Advanced Institute of Science and Technology, Daejeon, South Korea — 2015
  42. Coxeter-biCatalan Combinatorics, Special Session on Cluster Algebras, Joint Mathematics Meetings, San Antonio, TX — 2015