I received my Ph.D. in 2013 from the University of Minnesota, under the supervision of Dennis Stanton; the title of my dissertation was "Cataland". I then had the opportunity to spend 2+1 wonderful years at LaCIM as a postdoc—my original offer was for two years, but I was able to stay for one additional year because Hugh Thomas came to LaCIM with a Canada Research Chair and generously extended my postdoc. (Thanks to this extra year, Hugh and I were able to invert the sweep map!)
After one year at the University of California, Santa Barbara as Jon McCammond's postdoc, I started as an assistant professor at the University of Texas at Dallas in 2017. I became a tenured associate professor in 2024.
I am roughly Γ(0) years old. I am neither the most distinguished Nathan, nor the most distinguished Williams in algebraic combinatorics. (Nor, as it turns out, am I the most well-known Nathan Williams at UTD.)
I have been the department's mathematics graduate advisor for 0 days. Visit my graduate advising page for more information.
After one year at the University of California, Santa Barbara as Jon McCammond's postdoc, I started as an assistant professor at the University of Texas at Dallas in 2017. I became a tenured associate professor in 2024.
I am roughly Γ(0) years old. I am neither the most distinguished Nathan, nor the most distinguished Williams in algebraic combinatorics. (Nor, as it turns out, am I the most well-known Nathan Williams at UTD.)
I have been the department's mathematics graduate advisor for 0 days. Visit my graduate advising page for more information.
Selected Papers
My research is in algebraic combinatorics, centered around reflection, braid, and Lie groups. I have been very fortunate to have had many brilliant collaborators, with whom I have written some very nice papers. Here are a few of my favorites:
| Promotion and Rowmotion (with J. Striker) unifies various results in what J. Propp calls "Dynamical Algebraic Combinatorics". Later related work includes Rowmotion in Slow Motion and Independence Posets (with H. Thomas), and Semidistrim Lattices (with C. Defant). | |
| Doppelgängers (with Z. Hamaker, R. Patrias, and O. Pechenik) uses K-theoretic jeu-de-taquin to give an explicit bijection between plane partitions in a rectangle and plane partitions in a trapezoid. This solves an old problem considered by both R. Proctor and J. Stembridge--as R. Proctor wrote "the question of a combinatorial correspondence...seems to be a complete mystery." | |
| Sweeping up Zeta (with H. Thomas) inverts D. Armstrong, N. Loehr, and G. Warrington's sweep map, with applications to (q,t)-combinatorics. Inverting sweep is unexpectedly equivalent to a different problem that Hugh and I solved in 2012. Later related work includes Fixed Points of Parking Functions (with J. McCammond and H. Thomas). | |
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Cataland: Why the Fuss? (with C. Stump and H. Thomas) was a 132-page monograph that extends N. Reading's Coxeter-sortable elements and the cluster complex to the positive braid monoid---extending D. Armstrong's Fussification of noncrossing partitions (after P. Edelman) to the other Coxeter-Catalan families. |
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Rational Noncrossing Catalan Combinatorics (with P. Galashin, T. Lam, and M.-T. Trinh) is a 42-page preprint that uses braid varieties to give the first uniform construction of rational noncrossing Catalan objects and the first uniform enumeration of noncrossing objects. These were both longstanding open problems, considered in both Why the Fuss? and Rational associahedra and noncrossing partitions (with D. Armstrong and B. Rhoades). |
| Coxeter Pop-Tsack Torsing (with C. Defant) studies the Bessis dual version of C. Defant's Coxeter Pop-Stack Sorting map--that is, we modify the definitions by replacing all instances of S (the simple reflections) with T (the set of all reflections). Later related work with C. Defant includes Crystal pop-stack sorting and type A crystal lattices and The Ungar games (with N. Kravitz). | |
| Charmed Roots and the Kroweras Complement (with B. DequĂȘne, G. Frieden, A. Iraci, F. Schreier-Aigner, and Hugh Thomas) continues the saga of trying to find a bijection between nonnesting partitions and noncrossing partitions. The title is an unfortunate portmanteau of Germain Kreweras's last name and rowmotion. As I said at FPSAC: "I was hired for this project as the marketing consultant. They told me to write the introduction to the paper. At some point, about halfway through writing this introduction, I decided I should check out what the paper was all about. And that's what I'm here to tell you about today." |
Papers and Preprints
These profiles likely have a fairly up-to-date list of my work:
Selected Talks
Selected Posters
| Poster | Event | Year |
|---|---|---|
Pop, Crackle, Snap (and Pow): Some Facets of Shards ![]() |
FPSAC 2023 ![]() |
2021 |
Strange Checkspectations ![]() |
FPSAC 2021 ![]() |
2021 |
Independence Posets ![]() |
FPSAC ![]() |
2020 |
Summary
Since Fall 2017, I have taught 33 courses over 15 semesters at UTD— In 2020, I won UTD's School of Natural Science and Mathematics tenure-track teaching award. I completed the ACUE course in "Effective College Instruction" in Spring 2021.
I have advised four independent study courses, two senior capstone projects, five honors senior theses (two of which resulted in publications), and one Ph.D. thesis.
I have advised four independent study courses, two senior capstone projects, five honors senior theses (two of which resulted in publications), and one Ph.D. thesis.
Selected Student Supervision
| Name | Degree | Project | Year |
|---|---|---|---|
| Weston Miller | Honors Bachelor's Thesis | Rational Catalan Numbers for Complex Reflection Groups Project code available here |
2024 |
| Priyojit Palit | Ph.D. | Dual Braid Presentations and Cluster Algebras Won ``Best Graduate Blitz'' for his video presentation at the DFW Area 2021 Virtual Poster Competition in Mathematics |
2022 |
| Yusuf Ahsanullah | Honors Bachelor's Thesis | Quantum Groups and Cluster Fans | 2022 |
| Deep Desai, Ramesh Kanakala, Ryan Lofton, Savitha Venkatesh, Ariba Tahsin | Bachelor in Data Science - Capstone Project | Software implementation of independence posets Capstone Project Presentation Award: Honorable Mention - Creativity |
2022 |
| Michelle Sferrazza | Honors Capstone Project | EL-shellability of noncrossing partition lattices | 2022 |
| Joshua Marsh | Honors Bachelor's Thesis | Nesting Nonpartitions Published in Journal of Integer Sequences |
2020 |
| Benjamin Cotton | Honors Bachelor's Thesis | A core model for G2 Published in Involve |
2020 |
Algebraic Combinatorics
| Organization | Title |
|---|---|
|
Annals of Combinatorics |
Associate Editor |
Outreach
UTD Department of Mathematical Sciences
|
Chair of the website and computer committee
Member of the outreach committee:
Faculty organizer for the Putnam Club and William Lowell Putnam Mathematical Exam |
Community
| Link | Subject | News Stations |
|---|---|---|
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2022 Mega Millions lottery jackpot | CBS 11 (also NBC 5) |
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2018 NCAA men's basketball bracket | WFAA and on over 15 television stations nationwide |
Grants under Review
| Title (click for proposal) | Funding agency | Reviews | Year |
|---|---|---|---|
| No current submissions listed | - | - | - |
Grants Funded
| Title (click for proposal) | Funding agency | Reviews | Year |
|---|---|---|---|
| New Techniques in Algebraic Combinatorics | Simons | (Not provided) | 2018 |
| Combinatorics and Braid Varieties | NSF | Reviews Panel Summary Officer Summary | 2023+ |
Grants Declined
| Title (click for proposal) | Funding agency | Reviews | Year |
|---|---|---|---|
| Independence Posets and Dynamical Algebraic Combinatorics | NSF | Reviews Panel Summary Officer Summary | F21 |
| CAREER: Dynamical Algebraic Combinatorics | NSF | Reviews Panel Summary Officer Summary | S21 |
| Independence Posets | NSF | Reviews Panel Summary Officer Summary | F20 |
| Combinatorial Tools in Representation Theory | NSF | Reviews Panel Summary Officer Summary | F19 |
| Geometry of Braid Groups in Combinatorics | NSF | Reviews Panel Summary Officer Summary | F18 |
| New bijective techniques in algebraic combinatorics | NSF | Reviews Panel Summary Officer Summary | F17 |






























